Observation of Gravitational Waves from a Binary Black Hole Merger

B. P. Abbott et al.*

(LIGO Scientific Collaboration and Virgo Collaboration)(Received 21 January 2016; published 11 February 2016)

On September 14, 2015 at 09:50:45 UTC the two detectors of the Laser Interferometer Gravitational-WaveObservatory simultaneously observed a transient gravitational-wave signal. The signal sweeps upwards infrequency from 35 to 250 Hz with a peak gravitational-wave strain of 1.0 × 10−21. It matches the waveformpredicted by general relativity for the inspiral and merger of a pair of black holes and the ringdown of theresulting single black hole. The signal was observed with a matched-filter signal-to-noise ratio of 24 and afalse alarm rate estimated to be less than 1 event per 203 000 years, equivalent to a significance greaterthan 5.1σ. The source lies at a luminosity distance of 410þ160−180 Mpc corresponding to a redshift z ¼ 0.09þ0.03−0.04 .

In the source frame, the initial black hole masses are 36þ5−4M⊙ and 29þ4−4M⊙, and the final black hole mass is

62þ4−4M⊙, with 3.0þ0.5−0.5M⊙c2 radiated in gravitational waves. All uncertainties define 90% credible intervals.These observations demonstrate the existence of binary stellar-mass black hole systems. This is the first directdetection of gravitational waves and the first observation of a binary black hole merger.

DOI: 10.1103/PhysRevLett.116.061102

I. INTRODUCTION

In 1916, the year after the final formulation of the fieldequations of general relativity, Albert Einstein predictedthe existence of gravitational waves. He found thatthe linearized weak-field equations had wave solutions:transverse waves of spatial strain that travel at the speed oflight, generated by time variations of the mass quadrupolemoment of the source [1,2]. Einstein understood thatgravitational-wave amplitudes would be remarkablysmall; moreover, until the Chapel Hill conference in1957 there was significant debate about the physicalreality of gravitational waves [3].Also in 1916, Schwarzschild published a solution for the

field equations [4] that was later understood to describe ablack hole [5,6], and in 1963 Kerr generalized the solutionto rotating black holes [7]. Starting in the 1970s theoreticalwork led to the understanding of black hole quasinormalmodes [8–10], and in the 1990s higher-order post-Newtonian calculations [11] preceded extensive analyticalstudies of relativistic two-body dynamics [12,13]. Theseadvances, together with numerical relativity breakthroughsin the past decade [14–16], have enabled modeling ofbinary black hole mergers and accurate predictions oftheir gravitational waveforms. While numerous black holecandidates have now been identified through electromag-netic observations [17–19], black hole mergers have notpreviously been observed.

The discovery of the binary pulsar systemPSR B1913þ16by Hulse and Taylor [20] and subsequent observations ofits energy loss by Taylor and Weisberg [21] demonstratedthe existence of gravitational waves. This discovery,along with emerging astrophysical understanding [22],led to the recognition that direct observations of theamplitude and phase of gravitational waves would enablestudies of additional relativistic systems and provide newtests of general relativity, especially in the dynamicstrong-field regime.Experiments to detect gravitational waves began with

Weber and his resonant mass detectors in the 1960s [23],followed by an international network of cryogenic reso-nant detectors [24]. Interferometric detectors were firstsuggested in the early 1960s [25] and the 1970s [26]. Astudy of the noise and performance of such detectors [27],and further concepts to improve them [28], led toproposals for long-baseline broadband laser interferome-ters with the potential for significantly increased sensi-tivity [29–32]. By the early 2000s, a set of initial detectorswas completed, including TAMA 300 in Japan, GEO 600in Germany, the Laser Interferometer Gravitational-WaveObservatory (LIGO) in the United States, and Virgo inItaly. Combinations of these detectors made joint obser-vations from 2002 through 2011, setting upper limits on avariety of gravitational-wave sources while evolving intoa global network. In 2015, Advanced LIGO became thefirst of a significantly more sensitive network of advanceddetectors to begin observations [33–36].A century after the fundamental predictions of Einstein

and Schwarzschild, we report the first direct detection ofgravitational waves and the first direct observation of abinary black hole system merging to form a single blackhole. Our observations provide unique access to the

*Full author list given at the end of the article.

Published by the American Physical Society under the terms ofthe Creative Commons Attribution 3.0 License. Further distri-bution of this work must maintain attribution to the author(s) andthe published article’s title, journal citation, and DOI.

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properties of space-time in the strong-field, high-velocityregime and confirm predictions of general relativity for thenonlinear dynamics of highly disturbed black holes.

II. OBSERVATION

On September 14, 2015 at 09:50:45 UTC, the LIGOHanford, WA, and Livingston, LA, observatories detected

the coincident signal GW150914 shown in Fig. 1. The initialdetection was made by low-latency searches for genericgravitational-wave transients [41] and was reported withinthree minutes of data acquisition [43]. Subsequently,matched-filter analyses that use relativistic models of com-pact binary waveforms [44] recovered GW150914 as themost significant event from each detector for the observa-tions reported here. Occurring within the 10-ms intersite

FIG. 1. The gravitational-wave event GW150914 observed by the LIGO Hanford (H1, left column panels) and Livingston (L1, rightcolumn panels) detectors. Times are shown relative to September 14, 2015 at 09:50:45 UTC. For visualization, all time series are filteredwith a 35–350 Hz bandpass filter to suppress large fluctuations outside the detectors’ most sensitive frequency band, and band-rejectfilters to remove the strong instrumental spectral lines seen in the Fig. 3 spectra. Top row, left: H1 strain. Top row, right: L1 strain.GW150914 arrived first at L1 and 6.9þ0.5−0.4 ms later at H1; for a visual comparison, the H1 data are also shown, shifted in time by thisamount and inverted (to account for the detectors’ relative orientations). Second row: Gravitational-wave strain projected onto eachdetector in the 35–350 Hz band. Solid lines show a numerical relativity waveform for a system with parameters consistent with thoserecovered from GW150914 [37,38] confirmed to 99.9% by an independent calculation based on [15]. Shaded areas show 90% credibleregions for two independent waveform reconstructions. One (dark gray) models the signal using binary black hole template waveforms[39]. The other (light gray) does not use an astrophysical model, but instead calculates the strain signal as a linear combination ofsine-Gaussian wavelets [40,41]. These reconstructions have a 94% overlap, as shown in [39]. Third row: Residuals after subtracting thefiltered numerical relativity waveform from the filtered detector time series. Bottom row:A time-frequency representation [42] of thestrain data, showing the signal frequency increasing over time.

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propagation time, the events have a combined signal-to-noise ratio (SNR) of 24 [45].Only the LIGO detectors were observing at the time of

GW150914. The Virgo detector was being upgraded,and GEO 600, though not sufficiently sensitive to detectthis event, was operating but not in observationalmode. With only two detectors the source position isprimarily determined by the relative arrival time andlocalized to an area of approximately 600 deg2 (90%credible region) [39,46].The basic features of GW150914 point to it being

produced by the coalescence of two black holes—i.e.,their orbital inspiral and merger, and subsequent final blackhole ringdown. Over 0.2 s, the signal increases in frequencyand amplitude in about 8 cycles from 35 to 150 Hz, wherethe amplitude reaches a maximum. The most plausibleexplanation for this evolution is the inspiral of two orbitingmasses, m1 and m2, due to gravitational-wave emission. Atthe lower frequencies, such evolution is characterized bythe chirp mass [11]

M ¼ ðm1m2Þ3=5ðm1 þm2Þ1=5

¼ c3

G

5

96π−8=3f−11=3 _f

3=5

;

where f and _f are the observed frequency and its timederivative and G and c are the gravitational constant andspeed of light. Estimating f and _f from the data in Fig. 1,we obtain a chirp mass of M≃ 30M⊙, implying that thetotal mass M ¼ m1 þm2 is ≳70M⊙ in the detector frame.This bounds the sum of the Schwarzschild radii of thebinary components to 2GM=c2 ≳ 210 km. To reach anorbital frequency of 75 Hz (half the gravitational-wavefrequency) the objects must have been very close and verycompact; equal Newtonian point masses orbiting at thisfrequency would be only ≃350 km apart. A pair ofneutron stars, while compact, would not have the requiredmass, while a black hole neutron star binary with thededuced chirp mass would have a very large total mass,and would thus merge at much lower frequency. Thisleaves black holes as the only known objects compactenough to reach an orbital frequency of 75 Hz withoutcontact. Furthermore, the decay of the waveform after itpeaks is consistent with the damped oscillations of a blackhole relaxing to a final stationary Kerr configuration.Below, we present a general-relativistic analysis ofGW150914; Fig. 2 shows the calculated waveform usingthe resulting source parameters.

III. DETECTORS

Gravitational-wave astronomy exploits multiple, widelyseparated detectors to distinguish gravitational waves fromlocal instrumental and environmental noise, to providesource sky localization, and to measure wave polarizations.The LIGO sites each operate a single Advanced LIGO

detector [33], a modified Michelson interferometer (seeFig. 3) that measures gravitational-wave strain as a differ-ence in length of its orthogonal arms. Each arm is formedby two mirrors, acting as test masses, separated byLx ¼ Ly ¼ L ¼ 4 km. A passing gravitational wave effec-tively alters the arm lengths such that the measureddifference is ΔLðtÞ ¼ δLx − δLy ¼ hðtÞL, where h is thegravitational-wave strain amplitude projected onto thedetector. This differential length variation alters the phasedifference between the two light fields returning to thebeam splitter, transmitting an optical signal proportional tothe gravitational-wave strain to the output photodetector.To achieve sufficient sensitivity to measure gravitational

waves, the detectors include several enhancements to thebasic Michelson interferometer. First, each arm contains aresonant optical cavity, formed by its two test mass mirrors,that multiplies the effect of a gravitational wave on the lightphase by a factor of 300 [48]. Second, a partially trans-missive power-recycling mirror at the input provides addi-tional resonant buildup of the laser light in the interferometeras a whole [49,50]: 20Wof laser input is increased to 700Wincident on the beam splitter, which is further increased to100 kW circulating in each arm cavity. Third, a partiallytransmissive signal-recycling mirror at the output optimizes

FIG. 2. Top: Estimated gravitational-wave strain amplitudefrom GW150914 projected onto H1. This shows the fullbandwidth of the waveforms, without the filtering used for Fig. 1.The inset images show numerical relativity models of the blackhole horizons as the black holes coalesce. Bottom: The Keplerianeffective black hole separation in units of Schwarzschild radii(RS ¼ 2GM=c2) and the effective relative velocity given by thepost-Newtonian parameter v=c ¼ ðGMπf=c3Þ1=3, where f is thegravitational-wave frequency calculated with numerical relativityand M is the total mass (value from Table I).

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the gravitational-wave signal extraction by broadening thebandwidth of the arm cavities [51,52]. The interferometeris illuminated with a 1064-nm wavelength Nd:YAG laser,stabilized in amplitude, frequency, and beam geometry[53,54]. The gravitational-wave signal is extracted at theoutput port using a homodyne readout [55].These interferometry techniques are designed to maxi-

mize the conversion of strain to optical signal, therebyminimizing the impact of photon shot noise (the principalnoise at high frequencies). High strain sensitivity alsorequires that the test masses have low displacement noise,which is achieved by isolating them from seismic noise (lowfrequencies) and designing them to have low thermal noise(intermediate frequencies). Each test mass is suspended asthe final stage of a quadruple-pendulum system [56],supported by an active seismic isolation platform [57].These systems collectively provide more than 10 ordersof magnitude of isolation from ground motion for frequen-cies above 10 Hz. Thermal noise is minimized by usinglow-mechanical-loss materials in the test masses and their

suspensions: the test masses are 40-kg fused silica substrateswith low-loss dielectric optical coatings [58,59], and aresuspended with fused silica fibers from the stage above [60].To minimize additional noise sources, all components

other than the laser source are mounted on vibrationisolation stages in ultrahigh vacuum. To reduce opticalphase fluctuations caused by Rayleigh scattering, thepressure in the 1.2-m diameter tubes containing the arm-cavity beams is maintained below 1 μPa.Servo controls are used to hold the arm cavities on

resonance [61] and maintain proper alignment of the opticalcomponents [62]. The detector output is calibrated in strainby measuring its response to test mass motion induced byphoton pressure from a modulated calibration laser beam[63]. The calibration is established to an uncertainty (1σ) ofless than 10% in amplitude and 10 degrees in phase, and iscontinuously monitored with calibration laser excitations atselected frequencies. Two alternative methods are used tovalidate the absolute calibration, one referenced to the mainlaser wavelength and the other to a radio-frequency oscillator

(a)

(b)

FIG. 3. Simplified diagram of an Advanced LIGO detector (not to scale). A gravitational wave propagating orthogonally to thedetector plane and linearly polarized parallel to the 4-km optical cavities will have the effect of lengthening one 4-km arm and shorteningthe other during one half-cycle of the wave; these length changes are reversed during the other half-cycle. The output photodetectorrecords these differential cavity length variations. While a detector’s directional response is maximal for this case, it is still significant formost other angles of incidence or polarizations (gravitational waves propagate freely through the Earth). Inset (a): Location andorientation of the LIGO detectors at Hanford, WA (H1) and Livingston, LA (L1). Inset (b): The instrument noise for each detector nearthe time of the signal detection; this is an amplitude spectral density, expressed in terms of equivalent gravitational-wave strainamplitude. The sensitivity is limited by photon shot noise at frequencies above 150 Hz, and by a superposition of other noise sources atlower frequencies [47]. Narrow-band features include calibration lines (33–38, 330, and 1080 Hz), vibrational modes of suspensionfibers (500 Hz and harmonics), and 60 Hz electric power grid harmonics.

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[64]. Additionally, the detector response to gravitationalwaves is tested by injecting simulated waveforms with thecalibration laser.To monitor environmental disturbances and their influ-

ence on the detectors, each observatory site is equippedwith an array of sensors: seismometers, accelerometers,microphones, magnetometers, radio receivers, weathersensors, ac-power line monitors, and a cosmic-ray detector[65]. Another ∼105 channels record the interferometer’soperating point and the state of the control systems. Datacollection is synchronized to Global Positioning System(GPS) time to better than 10 μs [66]. Timing accuracy isverified with an atomic clock and a secondary GPS receiverat each observatory site.In their most sensitive band, 100–300 Hz, the current

LIGO detectors are 3 to 5 times more sensitive to strain thaninitial LIGO [67]; at lower frequencies, the improvement iseven greater, with more than ten times better sensitivitybelow 60 Hz. Because the detectors respond proportionallyto gravitational-wave amplitude, at low redshift the volumeof space to which they are sensitive increases as the cubeof strain sensitivity. For binary black holes with massessimilar to GW150914, the space-time volume surveyed bythe observations reported here surpasses previous obser-vations by an order of magnitude [68].

IV. DETECTOR VALIDATION

Both detectors were in steady state operation for severalhours around GW150914. All performance measures, inparticular their average sensitivity and transient noisebehavior, were typical of the full analysis period [69,70].Exhaustive investigations of instrumental and environ-

mental disturbances were performed, giving no evidence tosuggest that GW150914 could be an instrumental artifact[69]. The detectors’ susceptibility to environmental disturb-ances was quantified by measuring their response to spe-cially generated magnetic, radio-frequency, acoustic, andvibration excitations. These tests indicated that any externaldisturbance large enough to have caused the observed signalwould have been clearly recorded by the array of environ-mental sensors. None of the environmental sensors recordedany disturbances that evolved in time and frequency likeGW150914, and all environmental fluctuations during thesecond that contained GW150914 were too small to accountfor more than 6% of its strain amplitude. Special care wastaken to search for long-range correlated disturbances thatmight produce nearly simultaneous signals at the two sites.No significant disturbances were found.The detector strain data exhibit non-Gaussian noise

transients that arise from a variety of instrumental mecha-nisms. Many have distinct signatures, visible in auxiliarydata channels that are not sensitive to gravitational waves;such instrumental transients are removed from our analyses[69]. Any instrumental transients that remain in the dataare accounted for in the estimated detector backgrounds

described below. There is no evidence for instrumentaltransients that are temporally correlated between the twodetectors.

V. SEARCHES

We present the analysis of 16 days of coincidentobservations between the two LIGO detectors fromSeptember 12 to October 20, 2015. This is a subset ofthe data from Advanced LIGO’s first observational periodthat ended on January 12, 2016.GW150914 is confidently detected by two different

types of searches. One aims to recover signals from thecoalescence of compact objects, using optimal matchedfiltering with waveforms predicted by general relativity.The other search targets a broad range of generic transientsignals, with minimal assumptions about waveforms. Thesesearches use independent methods, and their response todetector noise consists of different, uncorrelated, events.However, strong signals from binary black hole mergers areexpected to be detected by both searches.Each search identifies candidate events that are detected

at both observatories consistent with the intersite propa-gation time. Events are assigned a detection-statistic valuethat ranks their likelihood of being a gravitational-wavesignal. The significance of a candidate event is determinedby the search background—the rate at which detector noiseproduces events with a detection-statistic value equal to orhigher than the candidate event. Estimating this back-ground is challenging for two reasons: the detector noiseis nonstationary and non-Gaussian, so its properties mustbe empirically determined; and it is not possible to shieldthe detector from gravitational waves to directly measure asignal-free background. The specific procedure used toestimate the background is slightly different for the twosearches, but both use a time-shift technique: the timestamps of one detector’s data are artificially shifted by anoffset that is large compared to the intersite propagationtime, and a new set of events is produced based on thistime-shifted data set. For instrumental noise that is uncor-related between detectors this is an effective way toestimate the background. In this process a gravitational-wave signal in one detector may coincide with time-shiftednoise transients in the other detector, thereby contributingto the background estimate. This leads to an overestimate ofthe noise background and therefore to a more conservativeassessment of the significance of candidate events.The characteristics of non-Gaussian noise vary between

different time-frequency regions. This means that the searchbackgrounds are not uniform across the space of signalsbeing searched. To maximize sensitivity and provide a betterestimate of event significance, the searches sort both theirbackground estimates and their event candidates into differ-ent classes according to their time-frequency morphology.The significance of a candidate event is measured against thebackground of its class. To account for having searched

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multiple classes, this significance is decreased by a trialsfactor equal to the number of classes [71].

A. Generic transient search

Designed to operate without a specific waveform model,this search identifies coincident excess power in time-frequency representations of the detector strain data[43,72], for signal frequencies up to 1 kHz and durationsup to a few seconds.The search reconstructs signal waveforms consistent

with a common gravitational-wave signal in both detectorsusing a multidetector maximum likelihood method. Eachevent is ranked according to the detection statisticηc ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Ec=ð1þ En=EcÞ

p, where Ec is the dimensionless

coherent signal energy obtained by cross-correlating thetwo reconstructed waveforms, and En is the dimensionlessresidual noise energy after the reconstructed signal issubtracted from the data. The statistic ηc thus quantifiesthe SNR of the event and the consistency of the databetween the two detectors.Based on their time-frequency morphology, the events

are divided into three mutually exclusive search classes, asdescribed in [41]: events with time-frequency morphologyof known populations of noise transients (class C1), eventswith frequency that increases with time (class C3), and allremaining events (class C2).

Detected with ηc ¼ 20.0, GW150914 is the strongestevent of the entire search. Consistent with its coalescencesignal signature, it is found in the search class C3 of eventswith increasing time-frequency evolution. Measured on abackground equivalent to over 67 400 years of data andincluding a trials factor of 3 to account for the searchclasses, its false alarm rate is lower than 1 in 22 500 years.This corresponds to a probability < 2 × 10−6 of observingone or more noise events as strong as GW150914 duringthe analysis time, equivalent to 4.6σ. The left panel ofFig. 4 shows the C3 class results and background.The selection criteria that define the search class C3

reduce the background by introducing a constraint on thesignal morphology. In order to illustrate the significance ofGW150914 against a background of events with arbitraryshapes, we also show the results of a search that uses thesame set of events as the one described above but withoutthis constraint. Specifically, we use only two search classes:the C1 class and the union of C2 and C3 classes (C2þ C3).In this two-class search the GW150914 event is found inthe C2þ C3 class. The left panel of Fig. 4 shows theC2þ C3 class results and background. In the backgroundof this class there are four events with ηc ≥ 32.1, yielding afalse alarm rate for GW150914 of 1 in 8 400 years. Thiscorresponds to a false alarm probability of 5 × 10−6

equivalent to 4.4σ.

FIG. 4. Search results from the generic transient search (left) and the binary coalescence search (right). These histograms show thenumber of candidate events (orange markers) and the mean number of background events (black lines) in the search class whereGW150914 was found as a function of the search detection statistic and with a bin width of 0.2. The scales on the top give thesignificance of an event in Gaussian standard deviations based on the corresponding noise background. The significance of GW150914is greater than 5.1σ and 4.6σ for the binary coalescence and the generic transient searches, respectively. Left: Along with the primarysearch (C3) we also show the results (blue markers) and background (green curve) for an alternative search that treats eventsindependently of their frequency evolution (C2þ C3). The classes C2 and C3 are defined in the text. Right: The tail in the black-linebackground of the binary coalescence search is due to random coincidences of GW150914 in one detector with noise in the otherdetector. (This type of event is practically absent in the generic transient search background because they do not pass the time-frequencyconsistency requirements used in that search.) The purple curve is the background excluding those coincidences, which is used to assessthe significance of the second strongest event.

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For robustness and validation, we also use other generictransient search algorithms [41]. A different search [73] anda parameter estimation follow-up [74] detected GW150914with consistent significance and signal parameters.

B. Binary coalescence search

This search targets gravitational-wave emission frombinary systems with individual masses from 1 to 99M⊙,total mass less than 100M⊙, and dimensionless spins up to0.99 [44]. To model systems with total mass larger than4M⊙, we use the effective-one-body formalism [75], whichcombines results from the post-Newtonian approach[11,76] with results from black hole perturbation theoryand numerical relativity. The waveform model [77,78]assumes that the spins of the merging objects are alignedwith the orbital angular momentum, but the resultingtemplates can, nonetheless, effectively recover systemswith misaligned spins in the parameter region ofGW150914 [44]. Approximately 250 000 template wave-forms are used to cover this parameter space.The search calculates the matched-filter signal-to-noise

ratio ρðtÞ for each template in each detector and identifiesmaxima of ρðtÞwith respect to the time of arrival of the signal[79–81]. For each maximum we calculate a chi-squaredstatistic χ2r to test whether the data in several differentfrequency bands are consistent with the matching template[82]. Values of χ2r near unity indicate that the signal isconsistent with a coalescence. If χ2r is greater than unity, ρðtÞis reweighted as ρ ¼ ρ=f½1þ ðχ2rÞ3=2g1=6 [83,84]. The finalstep enforces coincidence between detectors by selectingevent pairs that occur within a 15-ms window and come fromthe same template. The 15-ms window is determined by the10-ms intersite propagation time plus 5 ms for uncertainty inarrival time of weak signals. We rank coincident events basedon the quadrature sum ρc of the ρ from both detectors [45].To produce background data for this search the SNR

maxima of one detector are time shifted and a new set ofcoincident events is computed. Repeating this procedure∼107 times produces a noise background analysis timeequivalent to 608 000 years.To account for the search background noise varying across

the target signal space, candidate and background events aredivided into three search classes based on template length.The right panel of Fig. 4 shows the background for thesearch class of GW150914. The GW150914 detection-statistic value of ρc ¼ 23.6 is larger than any backgroundevent, so only an upper bound can be placed on its falsealarm rate. Across the three search classes this bound is 1 in203 000 years. This translates to a false alarm probability< 2 × 10−7, corresponding to 5.1σ.A second, independent matched-filter analysis that uses a

different method for estimating the significance of itsevents [85,86], also detected GW150914 with identicalsignal parameters and consistent significance.

When an event is confidently identified as a realgravitational-wave signal, as for GW150914, the back-ground used to determine the significance of other events isreestimated without the contribution of this event. This isthe background distribution shown as a purple line in theright panel of Fig. 4. Based on this, the second mostsignificant event has a false alarm rate of 1 per 2.3 years andcorresponding Poissonian false alarm probability of 0.02.Waveform analysis of this event indicates that if it isastrophysical in origin it is also a binary black holemerger [44].

VI. SOURCE DISCUSSION

The matched-filter search is optimized for detectingsignals, but it provides only approximate estimates ofthe source parameters. To refine them we use generalrelativity-based models [77,78,87,88], some of whichinclude spin precession, and for each model perform acoherent Bayesian analysis to derive posterior distributionsof the source parameters [89]. The initial and final masses,final spin, distance, and redshift of the source are shown inTable I. The spin of the primary black hole is constrainedto be < 0.7 (90% credible interval) indicating it is notmaximally spinning, while the spin of the secondary is onlyweakly constrained. These source parameters are discussedin detail in [39]. The parameter uncertainties includestatistical errors and systematic errors from averaging theresults of different waveform models.Using the fits to numerical simulations of binary black

hole mergers in [92,93], we provide estimates of the massand spin of the final black hole, the total energy radiatedin gravitational waves, and the peak gravitational-waveluminosity [39]. The estimated total energy radiated ingravitational waves is 3.0þ0.5−0.5M⊙c2. The system reached apeak gravitational-wave luminosity of 3.6þ0.5−0.4 × 1056 erg=s,equivalent to 200þ30−20M⊙c2=s.Several analyses have been performed to determine

whether or not GW150914 is consistent with a binary

TABLE I. Source parameters for GW150914. We reportmedian values with 90% credible intervals that include statisticalerrors, and systematic errors from averaging the results ofdifferent waveform models. Masses are given in the sourceframe; to convert to the detector frame multiply by (1þ z)[90]. The source redshift assumes standard cosmology [91].

Primary black hole mass 36þ5−4M⊙Secondary black hole mass 29þ4−4M⊙Final black hole mass 62þ4−4M⊙Final black hole spin 0.67þ0.05−0.07

Luminosity distance 410þ160−180 Mpc

Source redshift z 0.09þ0.03−0.04

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black hole system in general relativity [94]. A firstconsistency check involves the mass and spin of the finalblack hole. In general relativity, the end product of a blackhole binary coalescence is a Kerr black hole, which is fullydescribed by its mass and spin. For quasicircular inspirals,these are predicted uniquely by Einstein’s equations as afunction of the masses and spins of the two progenitorblack holes. Using fitting formulas calibrated to numericalrelativity simulations [92], we verified that the remnantmass and spin deduced from the early stage of thecoalescence and those inferred independently from the latestage are consistent with each other, with no evidence fordisagreement from general relativity.Within the post-Newtonian formalism, the phase of the

gravitational waveform during the inspiral can be expressedas a power series in f1=3. The coefficients of this expansioncan be computed in general relativity. Thus, we can test forconsistency with general relativity [95,96] by allowing thecoefficients to deviate from the nominal values, and seeingif the resulting waveform is consistent with the data. In thissecond check [94] we place constraints on these deviations,finding no evidence for violations of general relativity.Finally, assuming a modified dispersion relation for

gravitational waves [97], our observations constrain theCompton wavelength of the graviton to be λg > 1013 km,which could be interpreted as a bound on the graviton massmg < 1.2 × 10−22 eV=c2. This improves on Solar Systemand binary pulsar bounds [98,99] by factors of a few and athousand, respectively, but does not improve on the model-dependent bounds derived from the dynamics of Galaxyclusters [100] and weak lensing observations [101]. Insummary, all three tests are consistent with the predictionsof general relativity in the strong-field regime of gravity.GW150914 demonstrates the existence of stellar-mass

black holes more massive than≃25M⊙, and establishes thatbinary black holes can form in nature and merge within aHubble time. Binary black holes have been predicted to formboth in isolated binaries [102–104] and in dense environ-ments by dynamical interactions [105–107]. The formationof such massive black holes from stellar evolution requiresweak massive-star winds, which are possible in stellarenvironments with metallicity lower than ≃1=2 the solarvalue [108,109]. Further astrophysical implications of thisbinary black hole discovery are discussed in [110].These observational results constrain the rate of stellar-

mass binary black hole mergers in the local universe. Usingseveral different models of the underlying binary black holemass distribution, we obtain rate estimates ranging from2–400 Gpc−3 yr−1 in the comoving frame [111–113]. Thisis consistent with a broad range of rate predictions asreviewed in [114], with only the lowest event rates beingexcluded.Binary black hole systems at larger distances contribute

to a stochastic background of gravitational waves from thesuperposition of unresolved systems. Predictions for such a

background are presented in [115]. If the signal from such apopulation were detected, it would provide informationabout the evolution of such binary systems over the historyof the universe.

VII. OUTLOOK

Further details about these results and associated datareleases are available at [116]. Analysis results for theentire first observational period will be reported in futurepublications. Efforts are under way to enhance significantlythe global gravitational-wave detector network [117].These include further commissioning of the AdvancedLIGO detectors to reach design sensitivity, which willallow detection of binaries like GW150914 with 3 timeshigher SNR. Additionally, Advanced Virgo, KAGRA, anda possible third LIGO detector in India [118] will extendthe network and significantly improve the positionreconstruction and parameter estimation of sources.

VIII. CONCLUSION

The LIGO detectors have observed gravitational wavesfrom the merger of two stellar-mass black holes. Thedetected waveform matches the predictions of generalrelativity for the inspiral and merger of a pair of blackholes and the ringdown of the resulting single black hole.These observations demonstrate the existence of binarystellar-mass black hole systems. This is the first directdetection of gravitational waves and the first observation ofa binary black hole merger.

ACKNOWLEDGMENTS

The authors gratefully acknowledge the support ofthe United States National Science Foundation (NSF) forthe construction and operation of the LIGO Laboratoryand Advanced LIGO as well as the Science andTechnology Facilities Council (STFC) of the UnitedKingdom, the Max-Planck Society (MPS), and the Stateof Niedersachsen, Germany, for support of the constructionof Advanced LIGO and construction and operation of theGEO 600 detector. Additional support for Advanced LIGOwas provided by the Australian Research Council. Theauthors gratefully acknowledge the Italian IstitutoNazionale di Fisica Nucleare (INFN), the French CentreNational de la Recherche Scientifique (CNRS), and theFoundation for Fundamental Research on Matter supportedby the Netherlands Organisation for Scientific Research,for the construction and operation of the Virgo detector, andfor the creation and support of the EGO consortium. Theauthors also gratefully acknowledge research support fromthese agencies as well as by the Council of Scientific andIndustrial Research of India, Department of Science and

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Technology, India, Science & Engineering Research Board(SERB), India, Ministry of Human Resource Development,India, the Spanish Ministerio de Economía yCompetitividad, the Conselleria d’Economia iCompetitivitat and Conselleria d’Educació, Cultura iUniversitats of the Govern de les Illes Balears, theNational Science Centre of Poland, the EuropeanCommission, the Royal Society, the Scottish FundingCouncil, the Scottish Universities Physics Alliance, theHungarian Scientific Research Fund (OTKA), the LyonInstitute of Origins (LIO), the National ResearchFoundation of Korea, Industry Canada and the Provinceof Ontario through the Ministry of Economic Developmentand Innovation, the Natural Sciences and EngineeringResearch Council of Canada, Canadian Institute forAdvanced Research, the Brazilian Ministry of Science,Technology, and Innovation, Russian Foundation for BasicResearch, the Leverhulme Trust, the Research Corporation,Ministry of Science and Technology (MOST), Taiwan, andthe Kavli Foundation. The authors gratefully acknowledgethe support of the NSF, STFC, MPS, INFN, CNRS and theState of Niedersachsen, Germany, for provision of compu-tational resources. This article has been assigned thedocument numbers LIGO-P150914 and VIR-0015A-16.

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Y. M. Hu,36 S. Huang,73 E. A. Huerta,105,82 D. Huet,23 B. Hughey,97 S. Husa,66 S. H. Huttner,36 T. Huynh-Dinh,6 A. Idrisy,72

N. Indik,8 D. R. Ingram,37 R. Inta,71 H. N. Isa,36 J.-M. Isac,60 M. Isi,1 G. Islas,22 T. Isogai,10 B. R. Iyer,15 K. Izumi,37

M. B. Jacobson,1 T. Jacqmin,60 H. Jang,77 K. Jani,63 P. Jaranowski,106 S. Jawahar,107 F. Jiménez-Forteza,66 W.W. Johnson,2

N. K. Johnson-McDaniel,15 D. I. Jones,26 R. Jones,36 R. J. G. Jonker,9 L. Ju,51 K. Haris,108 C. V. Kalaghatgi,24,91

V. Kalogera,82 S. Kandhasamy,21 G. Kang,77 J. B. Kanner,1 S. Karki,59 M. Kasprzack,2,23,34 E. Katsavounidis,10

W. Katzman,6 S. Kaufer,17 T. Kaur,51 K. Kawabe,37 F. Kawazoe,8,17 F. Kéfélian,53 M. S. Kehl,69 D. Keitel,8,66 D. B. Kelley,35

W. Kells,1 R. Kennedy,86 D. G. Keppel,8 J. S. Key,83 A. Khalaidovski,8 F. Y. Khalili,49 I. Khan,12 S. Khan,91 Z. Khan,95

E. A. Khazanov,109 N. Kijbunchoo,37 C. Kim,77 J. Kim,110 K. Kim,111 Nam-Gyu Kim,77 Namjun Kim,40 Y.-M. Kim,110

E. J. King,104 P. J. King,37 D. L. Kinzel,6 J. S. Kissel,37 L. Kleybolte,27 S. Klimenko,5 S. M. Koehlenbeck,8 K. Kokeyama,2

S. Koley,9 V. Kondrashov,1 A. Kontos,10 S. Koranda,16 M. Korobko,27 W. Z. Korth,1 I. Kowalska,44 D. B. Kozak,1

V. Kringel,8 B. Krishnan,8 A. Królak,112,113 C. Krueger,17 G. Kuehn,8 P. Kumar,69 R. Kumar,36 L. Kuo,73 A. Kutynia,112

P. Kwee,8 B. D. Lackey,35 M. Landry,37 J. Lange,102 B. Lantz,40 P. D. Lasky,114 A. Lazzarini,1 C. Lazzaro,63,42 P. Leaci,29,79,28

S. Leavey,36 E. O. Lebigot,30,70 C. H. Lee,110 H. K. Lee,111 H. M. Lee,115 K. Lee,36 A. Lenon,35 M. Leonardi,89,90

J. R. Leong,8 N. Leroy,23 N. Letendre,7 Y. Levin,114 B. M. Levine,37 T. G. F. Li,1 A. Libson,10 T. B. Littenberg,116

N. A. Lockerbie,107 J. Logue,36 A. L. Lombardi,103 L. T. London,91 J. E. Lord,35 M. Lorenzini,12,13 V. Loriette,117

M. Lormand,6 G. Losurdo,58 J. D. Lough,8,17 C. O. Lousto,102 G. Lovelace,22 H. Lück,17,8 A. P. Lundgren,8 J. Luo,78

R. Lynch,10 Y. Ma,51 T. MacDonald,40 B. Machenschalk,8 M. MacInnis,10 D. M. Macleod,2 F. Magaña-Sandoval,35

R. M. Magee,56 M. Mageswaran,1 E. Majorana,28 I. Maksimovic,117 V. Malvezzi,25,13 N. Man,53 I. Mandel,45 V. Mandic,84

V. Mangano,36 G. L. Mansell,20 M. Manske,16 M. Mantovani,34 F. Marchesoni,118,33 F. Marion,7 S. Márka,39 Z. Márka,39

A. S. Markosyan,40 E. Maros,1 F. Martelli,57,58 L. Martellini,53 I. W. Martin,36 R. M. Martin,5 D. V. Martynov,1 J. N. Marx,1

K. Mason,10 A. Masserot,7 T. J. Massinger,35 M. Masso-Reid,36 F. Matichard,10 L. Matone,39 N. Mavalvala,10

N. Mazumder,56 G. Mazzolo,8 R. McCarthy,37 D. E. McClelland,20 S. McCormick,6 S. C. McGuire,119 G. McIntyre,1

J. McIver,1 D. J. McManus,20 S. T. McWilliams,105 D. Meacher,72 G. D. Meadors,29,8 J. Meidam,9 A. Melatos,85

G. Mendell,37 D. Mendoza-Gandara,8 R. A. Mercer,16 E. Merilh,37 M. Merzougui,53 S. Meshkov,1 C. Messenger,36

C. Messick,72 P. M. Meyers,84 F. Mezzani,28,79 H. Miao,45 C. Michel,65 H. Middleton,45 E. E. Mikhailov,120 L. Milano,67,4

J. Miller,10 M. Millhouse,31 Y. Minenkov,13 J. Ming,29,8 S. Mirshekari,121 C. Mishra,15 S. Mitra,14 V. P. Mitrofanov,49

G. Mitselmakher,5 R. Mittleman,10 A. Moggi,19 M. Mohan,34 S. R. P. Mohapatra,10 M. Montani,57,58 B. C. Moore,88

C. J. Moore,122 D. Moraru,37 G. Moreno,37 S. R. Morriss,83 K. Mossavi,8 B. Mours,7 C. M. Mow-Lowry,45 C. L. Mueller,5

G. Mueller,5 A.W. Muir,91 Arunava Mukherjee,15 D. Mukherjee,16 S. Mukherjee,83 N. Mukund,14 A. Mullavey,6

J. Munch,104 D. J. Murphy,39 P. G. Murray,36 A. Mytidis,5 I. Nardecchia,25,13 L. Naticchioni,79,28 R. K. Nayak,123 V. Necula,5

K. Nedkova,103 G. Nelemans,52,9 M. Neri,46,47 A. Neunzert,98 G. Newton,36 T. T. Nguyen,20 A. B. Nielsen,8 S. Nissanke,52,9

A. Nitz,8 F. Nocera,34 D. Nolting,6 M. E. N. Normandin,83 L. K. Nuttall,35 J. Oberling,37 E. Ochsner,16 J. O’Dell,100

E. Oelker,10 G. H. Ogin,124 J. J. Oh,125 S. H. Oh,125 F. Ohme,91 M. Oliver,66 P. Oppermann,8 Richard J. Oram,6 B. O’Reilly,6

R. O’Shaughnessy,102 C. D. Ott,76 D. J. Ottaway,104 R. S. Ottens,5 H. Overmier,6 B. J. Owen,71 A. Pai,108 S. A. Pai,48

J. R. Palamos,59 O. Palashov,109 C. Palomba,28 A. Pal-Singh,27 H. Pan,73 Y. Pan,62 C. Pankow,82 F. Pannarale,91 B. C. Pant,48

F. Paoletti,34,19 A. Paoli,34 M. A. Papa,29,16,8 H. R. Paris,40 W. Parker,6 D. Pascucci,36 A. Pasqualetti,34 R. Passaquieti,18,19

D. Passuello,19 B. Patricelli,18,19 Z. Patrick,40 B. L. Pearlstone,36 M. Pedraza,1 R. Pedurand,65 L. Pekowsky,35 A. Pele,6

S. Penn,126 A. Perreca,1 H. P. Pfeiffer,69,29 M. Phelps,36 O. Piccinni,79,28 M. Pichot,53 M. Pickenpack,8 F. Piergiovanni,57,58

V. Pierro,87 G. Pillant,34 L. Pinard,65 I. M. Pinto,87 M. Pitkin,36 J. H. Poeld,8 R. Poggiani,18,19 P. Popolizio,34 A. Post,8

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J. Powell,36 J. Prasad,14 V. Predoi,91 S. S. Premachandra,114 T. Prestegard,84 L. R. Price,1 M. Prijatelj,34 M. Principe,87

S. Privitera,29 R. Prix,8 G. A. Prodi,89,90 L. Prokhorov,49 O. Puncken,8 M. Punturo,33 P. Puppo,28 M. Pürrer,29 H. Qi,16

J. Qin,51 V. Quetschke,83 E. A. Quintero,1 R. Quitzow-James,59 F. J. Raab,37 D. S. Rabeling,20 H. Radkins,37 P. Raffai,54

S. Raja,48 M. Rakhmanov,83 C. R. Ramet,6 P. Rapagnani,79,28 V. Raymond,29 M. Razzano,18,19 V. Re,25 J. Read,22

C. M. Reed,37 T. Regimbau,53 L. Rei,47 S. Reid,50 D. H. Reitze,1,5 H. Rew,120 S. D. Reyes,35 F. Ricci,79,28 K. Riles,98

N. A. Robertson,1,36 R. Robie,36 F. Robinet,23 A. Rocchi,13 L. Rolland,7 J. G. Rollins,1 V. J. Roma,59 J. D. Romano,83

R. Romano,3,4 G. Romanov,120 J. H. Romie,6 D. Rosińska,127,43 S. Rowan,36 A. Rüdiger,8 P. Ruggi,34 K. Ryan,37

S. Sachdev,1 T. Sadecki,37 L. Sadeghian,16 L. Salconi,34 M. Saleem,108 F. Salemi,8 A. Samajdar,123 L. Sammut,85,114

L. M. Sampson,82 E. J. Sanchez,1 V. Sandberg,37 B. Sandeen,82 G. H. Sanders,1 J. R. Sanders,98,35 B. Sassolas,65

B. S. Sathyaprakash,91 P. R. Saulson,35 O. Sauter,98 R. L. Savage,37 A. Sawadsky,17 P. Schale,59 R. Schilling,8,b J. Schmidt,8

P. Schmidt,1,76 R. Schnabel,27 R. M. S. Schofield,59 A. Schönbeck,27 E. Schreiber,8 D. Schuette,8,17 B. F. Schutz,91,29

J. Scott,36 S. M. Scott,20 D. Sellers,6 A. S. Sengupta,94 D. Sentenac,34 V. Sequino,25,13 A. Sergeev,109 G. Serna,22

Y. Setyawati,52,9 A. Sevigny,37 D. A. Shaddock,20 T. Shaffer,37 S. Shah,52,9 M. S. Shahriar,82 M. Shaltev,8 Z. Shao,1

B. Shapiro,40 P. Shawhan,62 A. Sheperd,16 D. H. Shoemaker,10 D. M. Shoemaker,63 K. Siellez,53,63 X. Siemens,16 D. Sigg,37

A. D. Silva,11 D. Simakov,8 A. Singer,1 L. P. Singer,68 A. Singh,29,8 R. Singh,2 A. Singhal,12 A. M. Sintes,66

B. J. J. Slagmolen,20 J. R. Smith,22 M. R. Smith,1 N. D. Smith,1 R. J. E. Smith,1 E. J. Son,125 B. Sorazu,36 F. Sorrentino,47

T. Souradeep,14 A. K. Srivastava,95 A. Staley,39 M. Steinke,8 J. Steinlechner,36 S. Steinlechner,36 D. Steinmeyer,8,17

B. C. Stephens,16 S. P. Stevenson,45 R. Stone,83 K. A. Strain,36 N. Straniero,65 G. Stratta,57,58 N. A. Strauss,78 S. Strigin,49

R. Sturani,121 A. L. Stuver,6 T. Z. Summerscales,128 L. Sun,85 P. J. Sutton,91 B. L. Swinkels,34 M. J. Szczepańczyk,97

M. Tacca,30 D. Talukder,59 D. B. Tanner,5 M. Tápai,96 S. P. Tarabrin,8 A. Taracchini,29 R. Taylor,1 T. Theeg,8

M. P. Thirugnanasambandam,1 E. G. Thomas,45 M. Thomas,6 P. Thomas,37 K. A. Thorne,6 K. S. Thorne,76 E. Thrane,114

S. Tiwari,12 V. Tiwari,91 K. V. Tokmakov,107 C. Tomlinson,86 M. Tonelli,18,19 C. V. Torres,83,c C. I. Torrie,1 D. Töyrä,45

F. Travasso,32,33 G. Traylor,6 D. Trifirò,21 M. C. Tringali,89,90 L. Trozzo,129,19 M. Tse,10 M. Turconi,53 D. Tuyenbayev,83

D. Ugolini,130 C. S. Unnikrishnan,99 A. L. Urban,16 S. A. Usman,35 H. Vahlbruch,17 G. Vajente,1 G. Valdes,83

M. Vallisneri,76 N. van Bakel,9 M. van Beuzekom,9 J. F. J. van den Brand,61,9 C. Van Den Broeck,9 D. C. Vander-Hyde,35,22

L. van der Schaaf,9 J. V. van Heijningen,9 A. A. van Veggel,36 M. Vardaro,41,42 S. Vass,1 M. Vasúth,38 R. Vaulin,10

A. Vecchio,45 G. Vedovato,42 J. Veitch,45 P. J. Veitch,104 K. Venkateswara,131 D. Verkindt,7 F. Vetrano,57,58 A. Viceré,57,58

S. Vinciguerra,45 D. J. Vine,50 J.-Y. Vinet,53 S. Vitale,10 T. Vo,35 H. Vocca,32,33 C. Vorvick,37 D. Voss,5 W. D. Vousden,45

S. P. Vyatchanin,49 A. R. Wade,20 L. E. Wade,132 M. Wade,132 S. J. Waldman,10 M. Walker,2 L. Wallace,1 S. Walsh,16,8,29

G. Wang,12 H. Wang,45 M. Wang,45 X. Wang,70 Y. Wang,51 H. Ward,36 R. L. Ward,20 J. Warner,37 M. Was,7 B. Weaver,37

L.-W. Wei,53 M. Weinert,8 A. J. Weinstein,1 R. Weiss,10 T. Welborn,6 L. Wen,51 P. Weßels,8 T. Westphal,8 K. Wette,8

J. T. Whelan,102,8 S. E. Whitcomb,1 D. J. White,86 B. F. Whiting,5 K.Wiesner,8 C. Wilkinson,37 P. A. Willems,1 L. Williams,5

R. D. Williams,1 A. R. Williamson,91 J. L. Willis,133 B. Willke,17,8 M. H. Wimmer,8,17 L. Winkelmann,8 W. Winkler,8

C. C. Wipf,1 A. G. Wiseman,16 H. Wittel,8,17 G. Woan,36 J. Worden,37 J. L. Wright,36 G. Wu,6 J. Yablon,82 I. Yakushin,6

W. Yam,10 H. Yamamoto,1 C. C. Yancey,62 M. J. Yap,20 H. Yu,10 M. Yvert,7 A. Zadrożny,112 L. Zangrando,42 M. Zanolin,97

J.-P. Zendri,42 M. Zevin,82 F. Zhang,10 L. Zhang,1 M. Zhang,120 Y. Zhang,102 C. Zhao,51 M. Zhou,82 Z. Zhou,82 X. J. Zhu,51

M. E. Zucker,1,10 S. E. Zuraw,103 and J. Zweizig1

(LIGO Scientific Collaboration and Virgo Collaboration)

1LIGO, California Institute of Technology, Pasadena, California 91125, USA2Louisiana State University, Baton Rouge, Louisiana 70803, USA

3Università di Salerno, Fisciano, I-84084 Salerno, Italy4INFN, Sezione di Napoli, Complesso Universitario di Monte S. Angelo, I-80126 Napoli, Italy

5University of Florida, Gainesville, Florida 32611, USA6LIGO Livingston Observatory, Livingston, Louisiana 70754, USA

7Laboratoire d’Annecy-le-Vieux de Physique des Particules (LAPP), Université Savoie Mont Blanc, CNRS/IN2P3,F-74941 Annecy-le-Vieux, France

8Albert-Einstein-Institut, Max-Planck-Institut für Gravitationsphysik, D-30167 Hannover, Germany9Nikhef, Science Park, 1098 XG Amsterdam, Netherlands

10LIGO, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

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11Instituto Nacional de Pesquisas Espaciais, 12227-010 São José dos Campos, São Paulo, Brazil12INFN, Gran Sasso Science Institute, I-67100 L’Aquila, Italy13INFN, Sezione di Roma Tor Vergata, I-00133 Roma, Italy

14Inter-University Centre for Astronomy and Astrophysics, Pune 411007, India15International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bangalore 560012, India

16University of Wisconsin-Milwaukee, Milwaukee, Wisconsin 53201, USA17Leibniz Universität Hannover, D-30167 Hannover, Germany

18Università di Pisa, I-56127 Pisa, Italy19INFN, Sezione di Pisa, I-56127 Pisa, Italy

20Australian National University, Canberra, Australian Capital Territory 0200, Australia21The University of Mississippi, University, Mississippi 38677, USA

22California State University Fullerton, Fullerton, California 92831, USA23LAL, Université Paris-Sud, CNRS/IN2P3, Université Paris-Saclay, Orsay, France

24Chennai Mathematical Institute, Chennai, India 60310325Università di Roma Tor Vergata, I-00133 Roma, Italy

26University of Southampton, Southampton SO17 1BJ, United Kingdom27Universität Hamburg, D-22761 Hamburg, Germany

28INFN, Sezione di Roma, I-00185 Roma, Italy29Albert-Einstein-Institut, Max-Planck-Institut für Gravitationsphysik, D-14476 Potsdam-Golm, Germany

30APC, AstroParticule et Cosmologie, Université Paris Diderot, CNRS/IN2P3, CEA/Irfu, Observatoire de Paris,Sorbonne Paris Cité, F-75205 Paris Cedex 13, France

31Montana State University, Bozeman, Montana 59717, USA32Università di Perugia, I-06123 Perugia, Italy

33INFN, Sezione di Perugia, I-06123 Perugia, Italy34European Gravitational Observatory (EGO), I-56021 Cascina, Pisa, Italy

35Syracuse University, Syracuse, New York 13244, USA36SUPA, University of Glasgow, Glasgow G12 8QQ, United Kingdom

37LIGO Hanford Observatory, Richland, Washington 99352, USA38Wigner RCP, RMKI, H-1121 Budapest, Konkoly Thege Miklós út 29-33, Hungary

39Columbia University, New York, New York 10027, USA40Stanford University, Stanford, California 94305, USA

41Università di Padova, Dipartimento di Fisica e Astronomia, I-35131 Padova, Italy42INFN, Sezione di Padova, I-35131 Padova, Italy

43CAMK-PAN, 00-716 Warsaw, Poland44Astronomical Observatory Warsaw University, 00-478 Warsaw, Poland

45University of Birmingham, Birmingham B15 2TT, United Kingdom46Università degli Studi di Genova, I-16146 Genova, Italy

47INFN, Sezione di Genova, I-16146 Genova, Italy48RRCAT, Indore MP 452013, India

49Faculty of Physics, Lomonosov Moscow State University, Moscow 119991, Russia50SUPA, University of the West of Scotland, Paisley PA1 2BE, United Kingdom51University of Western Australia, Crawley, Western Australia 6009, Australia

52Department of Astrophysics/IMAPP, Radboud University Nijmegen, P.O. Box 9010, 6500 GL Nijmegen, Netherlands53Artemis, Université Côte d’Azur, CNRS, Observatoire Côte d’Azur, CS 34229, Nice cedex 4, France

54MTA Eötvös University, “Lendulet” Astrophysics Research Group, Budapest 1117, Hungary55Institut de Physique de Rennes, CNRS, Université de Rennes 1, F-35042 Rennes, France

56Washington State University, Pullman, Washington 99164, USA57Università degli Studi di Urbino “Carlo Bo,” I-61029 Urbino, Italy58INFN, Sezione di Firenze, I-50019 Sesto Fiorentino, Firenze, Italy

59University of Oregon, Eugene, Oregon 97403, USA60Laboratoire Kastler Brossel, UPMC-Sorbonne Universités, CNRS, ENS-PSL Research University, Collège de France,

F-75005 Paris, France61VU University Amsterdam, 1081 HV Amsterdam, Netherlands62University of Maryland, College Park, Maryland 20742, USA

63Center for Relativistic Astrophysics and School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332, USA64Institut Lumière Matière, Université de Lyon, Université Claude Bernard Lyon 1, UMR CNRS 5306, 69622 Villeurbanne, France

65Laboratoire des Matériaux Avancés (LMA), IN2P3/CNRS, Université de Lyon, F-69622 Villeurbanne, Lyon, France66Universitat de les Illes Balears, IAC3—IEEC, E-07122 Palma de Mallorca, Spain

67Università di Napoli “Federico II,” Complesso Universitario di Monte S. Angelo, I-80126 Napoli, Italy68NASA/Goddard Space Flight Center, Greenbelt, Maryland 20771, USA

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69Canadian Institute for Theoretical Astrophysics, University of Toronto, Toronto, Ontario M5S 3H8, Canada70Tsinghua University, Beijing 100084, China

71Texas Tech University, Lubbock, Texas 79409, USA72The Pennsylvania State University, University Park, Pennsylvania 16802, USA

73National Tsing Hua University, Hsinchu City, 30013 Taiwan, Republic of China74Charles Sturt University, Wagga Wagga, New South Wales 2678, Australia

75University of Chicago, Chicago, Illinois 60637, USA76Caltech CaRT, Pasadena, California 91125, USA

77Korea Institute of Science and Technology Information, Daejeon 305-806, Korea78Carleton College, Northfield, Minnesota 55057, USA

79Università di Roma “La Sapienza,” I-00185 Roma, Italy80University of Brussels, Brussels 1050, Belgium

81Sonoma State University, Rohnert Park, California 94928, USA82Northwestern University, Evanston, Illinois 60208, USA

83The University of Texas Rio Grande Valley, Brownsville, Texas 78520, USA84University of Minnesota, Minneapolis, Minnesota 55455, USA

85The University of Melbourne, Parkville, Victoria 3010, Australia86The University of Sheffield, Sheffield S10 2TN, United Kingdom

87University of Sannio at Benevento, I-82100 Benevento, Italy and INFN, Sezione di Napoli, I-80100 Napoli, Italy88Montclair State University, Montclair, New Jersey 07043, USA

89Università di Trento, Dipartimento di Fisica, I-38123 Povo, Trento, Italy90INFN, Trento Institute for Fundamental Physics and Applications, I-38123 Povo, Trento, Italy

91Cardiff University, Cardiff CF24 3AA, United Kingdom92National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan

93School of Mathematics, University of Edinburgh, Edinburgh EH9 3FD, United Kingdom94Indian Institute of Technology, Gandhinagar Ahmedabad Gujarat 382424, India

95Institute for Plasma Research, Bhat, Gandhinagar 382428, India96University of Szeged, Dóm tér 9, Szeged 6720, Hungary

97Embry-Riddle Aeronautical University, Prescott, Arizona 86301, USA98University of Michigan, Ann Arbor, Michigan 48109, USA

99Tata Institute of Fundamental Research, Mumbai 400005, India100Rutherford Appleton Laboratory, HSIC, Chilton, Didcot, Oxon OX11 0QX, United Kingdom

101American University, Washington, D.C. 20016, USA102Rochester Institute of Technology, Rochester, New York 14623, USA

103University of Massachusetts-Amherst, Amherst, Massachusetts 01003, USA104University of Adelaide, Adelaide, South Australia 5005, Australia105West Virginia University, Morgantown, West Virginia 26506, USA

106University of Biał ystok, 15-424 Biał ystok, Poland107SUPA, University of Strathclyde, Glasgow G1 1XQ, United Kingdom

108IISER-TVM, CET Campus, Trivandrum Kerala 695016, India109Institute of Applied Physics, Nizhny Novgorod, 603950, Russia

110Pusan National University, Busan 609-735, Korea111Hanyang University, Seoul 133-791, Korea

112NCBJ, 05-400 Świerk-Otwock, Poland113IM-PAN, 00-956 Warsaw, Poland

114Monash University, Victoria 3800, Australia115Seoul National University, Seoul 151-742, Korea

116University of Alabama in Huntsville, Huntsville, Alabama 35899, USA117ESPCI, CNRS, F-75005 Paris, France

118Università di Camerino, Dipartimento di Fisica, I-62032 Camerino, Italy119Southern University and A&M College, Baton Rouge, Louisiana 70813, USA

120College of William and Mary, Williamsburg, Virginia 23187, USA121Instituto de Física Teórica, University Estadual Paulista/ICTP South American Institute for Fundamental Research,

São Paulo SP 01140-070, Brazil122University of Cambridge, Cambridge CB2 1TN, United Kingdom

123IISER-Kolkata, Mohanpur, West Bengal 741252, India124Whitman College, 345 Boyer Avenue, Walla Walla, Washington 99362 USA

125National Institute for Mathematical Sciences, Daejeon 305-390, Korea126Hobart and William Smith Colleges, Geneva, New York 14456, USA

127Janusz Gil Institute of Astronomy, University of Zielona Góra, 65-265 Zielona Góra, Poland

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128Andrews University, Berrien Springs, Michigan 49104, USA129Università di Siena, I-53100 Siena, Italy

130Trinity University, San Antonio, Texas 78212, USA131University of Washington, Seattle, Washington 98195, USA

132Kenyon College, Gambier, Ohio 43022, USA133Abilene Christian University, Abilene, Texas 79699, USA

aDeceased, April 2012.bDeceased, May 2015.cDeceased, March 2015.

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View the table of contents for this issue, or go to the journal homepage for more

2000 Rep. Prog. Phys. 63 1317

(http://iopscience.iop.org/0034-4885/63/9/201)

Home Search Collections Journals About Contact us My IOPscience

Rep. Prog. Phys. 63 (2000) 1317–1427. Printed in the UK PII: S0034-4885(00)07909-4

Detection of gravitational waves

L Ju, D G Blair and C ZhaoDepartment of Physics, The University of Western Australia, Nedlands, WA 6907, Australia

Received 4 January 2000

Abstract

Gravitational wave detectors have been under development since the pioneering work of Weberin the 1960s. The long and painstaking research effort has yielded enormous improvementsin detector sensitivity. Astronomical observations of binary pulsar systems have confirmedthe existence of gravitational radiation. Direct detection is inevitable once planned detectorsreach sensitivity goals.

This review begins by introducing the concept of gravitational waves, and discusses theirsignificance. Section 2 discusses sources of gravitational waves, giving estimates of signalcharacteristics and signal strengths. Section 3 presents an overview of gravitational wavedetection and the critical issues of data processing.

In the fourth section the physics of resonant-mass gravitational wave detectors is discussedin some detail, covering all areas from antenna materials to transducers and the quantum limitsto measurement. This section reviews the major operating antennas in the existing worldwidearray but also discusses the prospects for achieving substantial increases in sensitivity in thefuture.

The fifth section presents the concepts and designs for laser interferometer gravitationalwave detectors. Large-scale devices will be in operation in the first decade of the twenty-firstcentury and should eventually be certain of detecting a known class of gravitational wavesource. At their predicted sensitivity, space interferometers will be able to detect numerousknown galactic sources of gravitational waves and also will be able to detect black hole mergersthat are thought to have occurred as primordial galaxies merged and grew in the early universe.

(Some figures in this article are in colour only in the electronic version; see www.iop.org)

0034-4885/00/091317+111$90.00 © 2000 IOP Publishing Ltd 1317

1318 L Ju et al

Contents

Page1. Introduction to gravitational waves 1320

1.1. Listening to the universe 13201.2. Gravitational waves in stiff-elastic spacetime 13211.3. Gravitational waves in general relativity 1326

2. Sources of gravitational waves 13282.1. Introduction 13282.2. Classification of sources 13292.3. Supernovae 13312.4. Rough guide to signal amplitudes 13332.5. Neutron star coalescence sources 13342.6. Low-frequency sources 13352.7. Gravitational waves from binary systems 13362.8. Stochastic background from the era of early star formation 13372.9. Cosmological gravitational waves from the big bang 1339

3. Detection of gravitational waves 13413.1. An overview of detector technology 13413.2. Space laser interferometer gravitational wave detectors 13443.3. The world array of resonant-mass detectors 13463.4. Laser interferometer detectors 13493.5. Issues of data processing and signal detection 1350

4. Resonant-mass detectors 13554.1. Introduction 13554.2. Intrinsic noise in resonant-mass antennas 13564.3. The signal-to-noise ratio for burst detection 13594.4. Transducers 13634.5. Antenna materials 13664.6. Antenna suspension and isolation systems 13684.7. Present status of resonant-mass detectors 13704.8. Performance of resonant bars 13714.9. Multiple antenna correlation 13744.10. Future prospects 13774.11. Vibration isolation and suspension developments 13814.12. Conclusion 1382

5. Interferometer detectors 13825.1. History 13825.2. Configurations 13835.3. Recycling 13885.4. Vibration isolation 13945.5. Thermal noise 14025.6. Control systems 1406

Detection of gravitational waves 1319

5.7. Laser stabilization 14095.8. Optics 14125.9. High-power lasers 1414

6. Conclusion 1416Acknowledgments 1416References 1416

1320 L Ju et al

1. Introduction to gravitational waves

1.1. Listening to the universe

Our sense of the universe is provided predominantly by electromagnetic waves. Duringthe twentieth century the opening of the electromagnetic spectrum has successively broughtdramatic revelations. For instance, optical astronomy gave us the Hubble law expansion ofthe universe. Radio astronomy gave us the cosmic background radiation, the giant radio jetsemerging from black holes in galactic nuclei and neutron stars in the form of radio pulsars.X-ray astronomy gave us interacting neutron stars and black holes. Infrared astronomy gaveus evidence for a massive black hole in the nucleus of our own Galaxy.

Gravitational waves offer us a new sense with which to understand our universe. Ifelectromagnetic astronomy gives us eyes with which we can see the universe, then gravitationalwave astronomy offers us ears with which to hear it. We are presently deaf to the myriadgravitational wave sounds of the universe. Imagine you are in a forest: you see a steep hillside,massive trees and small shrubs, bright flowers and colourful birds flitting between the trees.But there is much more to a forest: the sound of the wind in the treetops, the occasional crashof a falling branch, the thump thump of a fleeing kangaroo, the pulse of cicadas, the whistles ofparrots and honking of bell frogs. The sense of hearing dramatically enriches our experience.

The gravitational wave universe is likely to be rich with ‘sounds’ across a frequency rangefrom less than one cycle per month (below one microhertz) up to tens of kilohertz. Frequenciesin the audio frequency band will be detectable using Earth-based detectors. But lowerfrequencies will require observatories in space. Gravitational waves are produced wheneverthere is non-spherical acceleration of mass–energy distributions. The lowest frequencies willconsist of extremely red-shifted signals from the very early universe, as well as the slowinteractions of very massive black holes, and a weak background from binary star systems.Signal frequencies often scale inversely as the mass of the relevant systems. Black holesbelow 100 solar masses, and neutron stars will produce gravitational waves in the audiofrequency range: nearly monochromatic whistles from millisecond pulsars, short bursts fromtheir formation, and chirrups from the coalescence of binary pairs.

During the twentieth century, at each opening of a new window in the electromagneticspectrum, the universe surprised us with unexpected phenomena. Our imagination and abilityto predict is limited. The sources we predict today are probably just a fraction of what we willhear when our detectors reach sufficient sensitivity.

Gravitational waves are waves of tidal force. They are vibrations of spacetime whichpropagate through space at the speed of light. They are registered as tiny vibrations of carefullyisolated masses. Their detection is primarily an experimental science, consisting of thedevelopment of the necessary ultra-sensitive measurement techniques. While the gravitationalwaves can be considered as classical waves, the measurement systems must be treated quantummechanically since the expected signals generally approach the limits set by the uncertaintyprinciple.

The binary pulsar PSR 1913+16 has played a key role in the unfolding story of gravitationalwaves. This system has proved Einstein’s theory of general relativity to high precision,including the quadrupole formula which states that the total emitted gravitational wave powerfrom any system is proportional to the square of the third time derivative of the system’squadrupole moment. The pulsar loses energy exactly as predicted by this formula [1]. Hulseand Taylor, who discovered the system more than 20 years ago [2], were rewarded by a Nobelprize in 1993, by which time careful monitoring had shown gravitational wave energy lossfrom the system in agreement with theory to better than 1%.

Detection of gravitational waves 1321

1.2. Gravitational waves in stiff-elastic spacetime

In Newtonian physics spacetime is an infinitely rigid conceptual grid. Gravitational wavescannot exist in this theory. They would have infinite velocity and infinite energy densitybecause in Newtonian gravitation the metrical stiffness of space is infinite. Conversely generalrelativity introduces a finite coupling coefficient between curvature of spacetime, described bythe Einstein curvature tensor, and the stress energy tensor which describes the mass–energywhich gives rise to the curvature. This coupling is expressed by the Einstein equation

T = c4

8πGG. (1.1)

Here T is the stress energy tensor and G is the Einstein curvature tensor, c is the speed of lightand G is Newton’s gravitational constant. The coupling coefficient c4/(8πG) is an enormousnumber, of order 1043. This expresses the extremely high stiffness of space which is the reasonthat the Newtonian law of gravitation is an excellent approximation in normal circumstances,and why gravitational waves have a small amplitude, even when their energy density is veryhigh.

The existence of gravitational waves is intuitively obvious as soon as one recognizes thatspacetime is an elastic medium. The basic properties of gravity waves can be easily deducedfrom our knowledge of Newtonian gravity, combined with knowledge that spacetime curvatureis a consequence of mass distributions.

First, consider how gravitational waves might be generated. Electromagnetic waves aregenerated when charges accelerate. Because a negative charge accelerating to the left isequivalent to a positive charge accelerating to the right, it is impossible to create a time-varying electric monopole. The process of varying the charge on one electrode always createsa time-varying dipole moment. Hence it follows that electromagnetic waves are generated bytime-varying dipole moments. In contrast to electromagnetism, gravity has only one charge:there is no such thing as negative mass! Hence it is not possible to create an oscillating massdipole. Action equals reaction. That is, momentum is conserved and the acceleration of onemass to the left creates an equal and opposite reaction to the right. For two equal masses,their spacing can change but the centre of mass is never altered. This means that there is atime-varying quadrupole moment, but there is no variation in monopole or dipole moment.

To be certain of the quadrupole nature of gravitational waves, think of a system whichcollapses under its own gravity. First think of a spherically symmetrical array of masses thatcollapse gravitationally towards a point. At a distance there is no difference between thegravitational field of a point mass and that of the same mass distributed in a uniform sphericaldistribution. (This is a consequence of the inverse square law, and is also true for electric fields.)Hence the process of gravitational collapse of a spherical distribution creates no variation inthe external gravitational field, and hence no gravitational waves. Clearly gravitational wavesmust be created by non-spherical motions of masses. Consider a ring of eight test masses,such as the one illustrated in figure 1.

The simplest non-spherical motion is one in which the edge masses move inwards andthe top and bottom masses move apart as shown in figure 1(a). Such a quadrupole motiondoes vary the external field and does create gravitational waves. For a small amount of verticalstretching, and an equal horizontal shrinking, it is obvious that the diagonally placed masseshave no radial motion. There is clearly a second polarization rotated 45 from the first inwhich the diagonal masses move radially, and the top, bottom and edge masses have no radialmotion. Unlike electromagnetic waves, gravitational wave polarizations are just 45 apart.

Gravitational wave detection can be easily understood from the symmetry between sourcesand detectors—time reversal invariance. A gravitational wave will distort a ring of test masses

1322 L Ju et al

h+

h×

0 π/23π/2π 2π

0 3π/2π 2ππ/2

(b)

(a)

Figure 1. (a) The lowest order non-spherical deformation of a ring: the diagonal masses are notmoved. (b) The deformation of a ring of test particles in one cycle of a gravitational wave field.

in exactly the same way that the distortion of a ring of test masses creates gravitational waves.The non-spherical deformation pattern we just observed is exactly like the tidal deformationof the Earth created by the gravity gradient due to the Moon. A gravitational wave is indeed awave of time-varying gravity gradient. The amplitude of a gravitational wave is measured bythe relative change in spacing between masses. That is, the wave amplitude, usually denotedh, is given by L/L, where L is the equilibrium spacing and L is the change of spacingof two test masses. Whereas electromagnetic luminosity depends on the square of the secondtime derivative of the electric dipole moment, the gravitational wave luminosity is proportionalto the square of the third time derivative of the mass quadrupole moment. The extra derivativearises because gravitational wave generation is associated with the differential acceleration ofmasses.

The above deformation patterns also apply to solid or fluid bodies. The rigidity of normalmatter is so low compared with that of spacetime that the stiffness of the matter is utterlynegligible. Considering the deformations of figure 1(a) applying to a solid sphere, such asthe Earth, it also follows that the 45 points must involve circumferential motions since thedeformation shown acts to transfer matter from the ‘equator’ to the poles in the same way thatthe lunar tides act on the Earth.

Detection of gravitational waves 1323

ω

Figure 2. A rotating dumbbell or a binary star system, viewed edge-on, has a maximal variationof quadrupole moment.

x

y

z

(a)

x

y

z

(b)

©

©

Figure 3. Gravitational wave field force lines. (a) ‘+’ polarization; (b) ‘×’ polarization.

The gravitational wave has an effective force field determined by the displacement vectorsof the test masses. The force field is discussed further below, and is shown in figure 3. The forcefield indicates that detectors can be designed to couple to gravity waves in several differentways. They may detect straight linear strains, orthogonal strains, or circumferential strains.

The weak coupling of gravitational waves to matter is a consequence of the enormouselastic stiffness of spacetime. If the elastic stiffness of spacetime were infinite (Newtonianphysics) the coupling would be zero. In general relativity the generation of gravitationalwaves is given quantitatively by combining the third time derivative of the quadrupole momentdescribed above, with the appropriate coupling constant. The latter can only depend on theconstants G and c (for classical waves) and by dimensional analysis this constant must havethe form G/c5. The luminosity of a source is given by

LG ∼ G

c5

(d3D

dt3

)2

. (1.2)

Except for a numerical factor, this is the Einstein quadrupole formula [3]. There are twouseful formulae one can derive from equation (1.2). The first is the formula for a hypotheticalterrestrial source or binary star system. The second is for an interacting black hole system.The terrestrial source might be a pair of oscillating masses joined with a spring. Ideally, thespacing of the masses should change from zero to L. This is achieved in the edge on view of arotating dumbbell or binary star system in a circular orbit as shown in figure 2. Viewed edge-onthe masses appear to move in and out periodically twice per rotation cycle. The quadrupolemoment for two masses distance x apart is Mx2. If the motion is sinusoidal at an angularfrequency of ω, the square of the third time derivative is ∼M2L4ω6. Thus the gravitational

1324 L Ju et al

wave luminosity of such a system is

Lc ∼ G

c5M2L4ω6. (1.3)

This equation applied to any natural or artificial source in our solar system gives a depressinglysmall luminosity, due to the extraordinarily small value of G/c5. However, the situation isdifferent in an astrophysical context.

Suppose that the system is a similar binary system, except that it consists of a pair ofgravitationally bound masses, of size such that their escape velocity approaches c and theirradius is near to the Schwartzchild radius: that is, a pair of black holes. In this case we canexpress L in units of the Schwartzchild radius, using rs = 2GM/c2 and replace Lω withvelocity expressed in light speed units. Then it follows immediately that

LG ∼ c5

G

(vc

)6 ( rsr

)2. (1.4)

The remarkable difference between equation (1.3) and equation (1.4) expresses the differencebetween the physics of normal matter and black holes. Equation (1.3) is scaled by the tinyfactor G/c5, while equation (1.4) is scaled by its enormous reciprocal. Normal matter inour solar system creates negligible curvature of spacetime. A black hole creates an extremedistortion of spacetime. Hence normal matter sources are intrinsically extremely weak, whilevery large amplitude waves are created in events such as the coalescence of a pair of blackholes (for which we would expect v ∼ c when rs ∼ r). The factor c5/G is roughly thetotal electromagnetic luminosity of the universe. This is the upper limit to the gravitationalwave luminosity of black hole systems. In reality, equation (1.4) does not take into accountthe gravitational redshift effects and other spacetime curvature effects which act to reduce themaximum luminosity. However, to order of magnitude, equation (1.4) indicates the extremeluminosity of gravitational waves that can be expected in short bursts when gravitationallycollapsed systems with strong gravity, such as black holes (escape velocity = c) and neutronstars (escape velocity ∼0.1c), are involved.

As we saw above, a gravitational wave is a wave of gravity gradient which causes relativedisplacements, or strains between test masses. The detection of gravitational waves requiresthe detection of small strain amplitudes. We should now consider the typical size of such strainamplitudes. One can very crudely estimate this by scaling the amplitude of the gravitationalwave relative to the extreme amplitude at the point of coalescence of two masses to forma black hole. At the point of black hole formation spacetime curvature is very large. Forexample, the deflection of light for a light beam passing near to the event horizon can approacha complete orbit of a black hole. At the point of generation the dynamic curvature of spacethat will become the outgoing gravitational wave is unlikely to be able to exceed the staticcurvature represented by the maximal deflections of light past a black hole. The strain L/L

represented by such deflections can be estimated from the difference in light travel time for thedeflected path around the black hole (say half an orbit) and the direct path between the samepoints in the absence of the black hole. For a half-orbit (in Euclidean geometry) the circularpath is π/2 longer than the direct path, so roughly L ∼ L, and the maximum possible strainamplitude is ∼unity. But by the inverse square law, the amplitude of the wave reduces as 1/r .(The energy density which is proportional to the square of the amplitude reduces as 1/r2.) Sofor such a black hole source we can give the strain amplitude at distance r as simply h ∼ rs/r .For more realistic sources only a fraction of the total energy will participate in quadrupolemotion. Thus it is more reasonable to include an efficiency factor ε which characterizes thefraction of the total system rest mass which can convert to gravitational waves. In this case we

Detection of gravitational waves 1325

can write

h ∼ ε1/2 rs

r. (1.5)

Since the Schwartzchild radius of a solar mass is a few kilometres, the maximum strainamplitude that can be expected from any stellar source is numerically equal to the reciprocalof its distance in kilometres. Because rs is linearly proportional to the mass, gravitationalwave amplitudes from very high mass sources, such as colliding 109 solar mass black holes ingalactic nuclei, will be of correspondingly larger amplitude. Putting in numbers, equation (1.5)gives h ∼ 10−16 for 10 solar masses and 100% efficiency at the galactic centre, and h ∼ 10−13

for 3 billion solar masses at 3 Gpc (towards the edge of the visible universe).Clearly these maximal numbers are very small. It might seem that the supermassive

black hole sources might be much more detectable than the stellar mass source. The strainamplitude in this case corresponds to the detection of a motion equal to the size of an atomicnucleus on a one-metre baseline, or one metre between here and Neptune. In fact the detectionof such small strains on Earth is probably impossible, because the frequency of the wavesfrom supermassive black hole sources must always be very low. The peak frequency, or itsreciprocal, the burst duration, can be estimated from the time the binary black hole systemtakes to complete its final orbit before coalescence. Its value is about 10 kHz for one solarmass, reducing inversely as the mass. Thus, the peak frequency will be about 1 kHz forthe above galactic centre source, and 3 × 10−6 Hz for the distant massive black holes. Thelatter frequency will be reduced towards 10−6 Hz by cosmological redshifts. At such lowfrequencies environmental effects, and particularly gravity gradients associated with tides andweather variations in the surrounding environment, create perturbations which greatly exceedthe desired signal. The only known means around this obstacle is by using drag-free satellitetechnology to create very stable free-floating masses in space, and laser interferometry betweenthem. In this case detection does look relatively straightforward, though expensive, since itrequires several widely spaced spacecraft. For frequencies above 1 Hz, terrestrial detectionappears to be possible, limited only by fundamental quantum measurement limits.

For even lower frequencies than 10−6 Hz it is possible that radio pulsars can replace man-made spacecraft in detection systems. The pulsar ideally provides a perfect monochromatictiming signal. The radio beams from the pulsar are traversed by incoming gravitational waves.If several pulsars are observed in the same part of the sky, gravitational wave signals wouldappear as correlated arrival time variations of pulses from pulsars in different directions. Inthis case it is more convenient to consider the gravitational wave acting not on the pulsar itself,but on spacetime geometry near to the Earth through which the pulsar signal propagates.

Today detectors are in long-term operation which exceed the 10−16 sensitivity indicatedabove by more than two orders of magnitude. Advanced detectors of two types are underdevelopment which should achieve another three orders of magnitude in amplitude sensitivity.

For 50 years after Einstein predicted gravitational waves [3] physicists considered themto be of academic interest only. It was not until after the pioneering work of Joseph Weber [4],and his reported discoveries [5,6] that a growing number of physicists around the world startedto develop different types of antennas to search for gravitational waves. Since Weber’s firstreports, which were never confirmed, the improvement in detectors has been quite remarkable.Relating them to optical telescopes, the improvement achieved so far is equivalent to the stepfrom a 3 cm diameter optical telescope to a 3 m diameter instrument. In the next decade itis hoped that the improvement will be equivalent to a step up in size from 3 m to 3 km. Atthis sensitivity gravitational wave detection is practically certain, and the field of gravitationalastronomy will be able to slowly map and explore the new spectrum, and the objects that itreveals.

1326 L Ju et al

In the following sections we will discuss gravitational wave sources in more detail beforegoing on to discuss detectors, first in the form of a general overview, and then with specificemphasis on existing ground-based detectors and their future prospects.

1.3. Gravitational waves in general relativity

Here we give a brief summary of the mathematical basis for gravitational waves. For furtherdetails on the theory of gravitational waves, readers are referred to [7–12].

The geometry of spacetime can be expressed by the metric tensor gαβ which connects thespacetime coordinate dxα(α, β = 1, 2, 3, 4) to the spacetime interval ds by way of the relation

ds2 = gαβ dxα dxβ. (1.6)

In reality, gravitational waves in the vicinity of the Earth will always be very weak. Thebackground curvature can be ignored and the background metric can be approximated as thatof the Minkowski flat metric η. An approximation of the gravitational wave field can then beexpressed in the form [7]

gαβ = ηαβ + hαβ, (1.7)

where ηαβ is the metric of the flat background, and |hαβ | 1 is the perturbation on thisbackground. If there is no stress-energy source term in Einstein’s field equation, i.e. T = 0 inequation (1.1), we are left with the weak field vacuum approximation to the Einstein equations.To obtain an explicit statement of the metric perturbations h it is necessary to make a gaugechoice. The most useful gauge is the transverse traceless gauge in which the coordinates aredefined by the geodesics of freely falling test bodies. In this gauge, and in the weak fieldlimit discussed above, the equations of general relativity become a system of linear equations,specifically a system of wave equations [7](

∇2 − 1

c2

∂2

∂t2

)hαβ = 0. (1.8)

Equation (1.8) is a three-dimensional wave equation, telling us that gravitational waves travelat the speed of light c. The gravitational wave curvature tensor h can be considered as thegravitational wave field. The wave field is transverse and traceless, and for waves travelling inthe z-direction may be expressed as follows:

hαβ =

0 0 0 00 hxx hxy 00 hyx hyy 00 0 0 0

. (1.9)

There is no z-component due to the transverse nature of the waves, and to be traceless h satisfies

hxx = −hyy. (1.10)

Because the Riemann tensor is symmetric, h also satisfies

hxy = hyx. (1.11)

The symmetry of hmeans that there are just two possible independent polarization states whichare usually denoted h+ and h×. In the case of sinusoidal gravitational waves we can expressthese polarizations as

h+ = hxx = Re [A+e−iω(t−z/c)], (1.12)

h× = hxy = Re [A×e−iω(t−z/c)]. (1.13)

Here A+ and A× are the strain amplitudes of each polarization.

Detection of gravitational waves 1327

We have already seen that a gravitational wave field moves masses in the same way that anelectromagnetic wave sets charged particles in motion. Each wave field exerts tidal forces onobjects through which it passes. The corresponding lines of force have a quadrupole patternas shown in figure 3. Figure 3(a) shows the force lines of the ‘+’ polarization and (b) showsthe ‘×’ polarization which is rotated 45 with respect to the ‘+’ state. These time-varying tidalforces can deform an elastic body or change the distance between mass points in free space.A ring of particles placed perpendicular to the wave propagation direction will be distorted aswe already saw in figure 1.

Einstein’s famous quadrupole formula describes the gravitational wave amplitude from asource. Einstein derived his formula in a slow-motion weak field approximation, but Thorne [8]emphasizes that the result is accurate as long as the reduced wavelength exceeds source size.This condition applies to all but the most compact sources such as forming or coalescing blackholes. The latter are potentially the strongest and most detectable sources. It is unfortunatethat these are just the ones where the nonlinearity of general relativity, and in particular thegravitational redshift of the outgoing gravitational waves due to the gravitational energy of thespacetime curvature itself, makes the gravitational radiation amplitude extremely difficult toestimate.

The quadrupole formula states that the gravitational wave amplitude h at a distanceR froma source is proportional to the second time derivative of the transverse traceless projection ofthe quadrupole moment evaluated at the retarded time t − r/c. That is

hjk = 2

r

G

c4

∂2

∂t2[Djk(t − R/c)]T T , (j, k = 1, 2, 3) (1.14)

where [Djk(t − R/c)]T T is the transverse traceless projection of the quadrupole momentevaluated at retarded time (t − R/c). The transverse traceless requirement relates to thetransverse nature of gravitational waves, and the lack of wave generation from sphericallysymmetrical motions. For weak fields, for which gravitational self-energy is small (seeDamour [13]) D is given by the second moment of the source mass density ρ:

Djk =∫

ρ(t)[xjxk − 13x

2δjk] d3x. (1.15)

In this equation the term with the Kronecker delta ensures that D is trace-free.The total gravitational wave power is proportional to the square of the third time derivative

of the mass quadrupole moment [14]. In general, the total energy radiation rate LG is givenby the sum of the squares of all the projections of the quadrupole moment

LG = 1

5

G

c5

∑jk

∣∣∣∣d3Djk

dt3

∣∣∣∣2

. (1.16)

The very small universal constant

G

5c5= 5.49 × 10−54 s J−1 (1.17)

sets the characteristic gravitational radiation power output.The presence of the factor (G/c5) in equation (1.16) indicates that unless the

...

Djk involvesenergy of astronomical proportions, the gravitational wave power will be extremely small. It iseasy to show that it is impossible to generate detectable gravitational waves on the laboratoryscale, even at extreme limits of known technology. We can only hope to observe gravitationalwaves emitted by astrophysical sources.

It is useful, however, to consider a laboratory source simply as an application ofequations (1.15) and (1.16). Suppose the source consists of a pair of masses distance L apart,

1328 L Ju et al

and joined by a spring to allow sinusoidal oscillation of their spacing at angular frequency ω.From equation (1.15), D = ML2, and if L = L0 + a sin ωt , it follows that

D = ML20 + 2ML0a sin ωt + Ma2 sin2 ωt. (1.18)

From equation (1.18), taking the third time derivative, it follows that this source will producegravitational waves at the frequencies of ω and 2ω. If the amplitude a is small compared withL0, the 2ω term is small and the gravitational wave luminosity is given by

LG = G

5c54M2L2

0a2ω6. (1.19)

For any practical harmonic oscillator on Earth LG is infinitesimal. However, such massquadrupole oscillators have been created as sources of near-field dynamic gravity gradients(not waves) for the purpose of calibrating gravitational wave detectors. Such systems has beensuccessfully used for low-frequency detectors tuned to the Crab pulsar [15] and also by theRome group to calibrate their resonant-bar detector and measure the inverse square law ofgravitation [16].

2. Sources of gravitational waves

2.1. Introduction

Astrophysics provides us with a variety of candidate systems which should be observable inthe spectrum of gravitational waves. However, it is important to remember that our powers ofprediction of new phenomena are poor, so any list of sources is almost certain to be incomplete.

Amongst stellar mass systems we expect detectable gravitational radiation from theformation of black holes and neutron stars, and the coalescence of binary neutron stars andfinal collapse of such binaries to form a black hole. We would expect not only discrete sources,but also continuous stochastic backgrounds created from large numbers of discrete sources.In our Galaxy the very large populations of binary stars create a stochastic background in the10−2 to 10 Hz range. In the universe as a whole all of the above neutron star and black holeformation events are likely to merge to form a continuous background in the audio frequencypart of the spectrum. This particular background provides an exciting opportunity to observethe earliest epochs of Galaxy formation, and the birth and growth of the supermassive blackholes that appear to reside in the nuclei of many galaxies and quasars. We may also be ableto observe gravitational waves from the big bang, amplified during the inflationary era, andpossible signatures of cosmological phase transitions and topological defects such as cosmicstrings. These very earliest sources in the universe would constitute a probe of physics atenergy scales far beyond those accessible in particle accelerators and hence represent the bestopportunity we have to obtain experimental data from the era of inflation.

Back in our own Galaxy we would also expect to find many quasi-monochromatic sourcesof gravitational waves such as individual binary star systems, including binary neutron stars asthey evolve towards coalescence, and various rotating neutron star systems such as millisecondand x-ray pulsars.

Figure 16 shows the gravitational wave spectrum across nine decades. The spectrumconveniently divides into a terrestrial detection band, above 1 Hz (generally within the audiofrequency band), mainly associated with stellar mass compact objects, and a space detectionband, from 10−6 to 10−1 Hz, where sources include both binary star systems in our Galaxy,and cosmological sources associated with massive black hole interactions.

Detection of gravitational waves 1329

2.2. Classification of sources

All the above sources can be naturally divided into three distinct classes, according to themethods of data processing and signal extraction. The first class consists of catastrophic burstsources such as the final coalescence of compact binary star systems, or the formation ofneutron stars and black holes in supernova events. The binary coalescence events can consistof binary neutron stars, binary black holes, or neutron-star–black-hole binaries. The burstsignal consists of a very short single event, consisting of one or very few cycles, and hence ischaracterized by a broad bandwidth, roughly determined by the reciprocal of the event duration.

The second class consists of narrow-band sources. These include the rotation of singlenonaxisymmetric stars, particularly pulsars and accreting neutron stars, as well as binary starsystems far from coalescence. All such systems are quasi-periodic because gravitational waveenergy loss must cause period evolution, and in general they are also periodically Dopplershifted by binary motion and Earth’s orbital motion. Such sources are generally weaker thanthe burst sources, but in principle they are always amenable to long-term integration to extractsignals from the noise. This requires accurate knowledge of the frequency modulations tomaintain a coherent integration. Assuming a white noise background and perfect knowledgeof the frequency evolution, the signal-to-noise ratio increases asN1/2 whereN is the number ofcycles. For narrow-band sources it may be possible to integrate for 108 s, compared with lessthan 10 ms for a burst source. Thus, at 100 Hz, N can be 1010 allowing a 105-fold improvementin signal-to-noise ratio.

The third class of sources are the stochastic backgrounds produced from the integratedeffects of many weak periodic sources in our Galaxy, or from a large population of burstsources at very large distances, as well as the above-mentioned cosmological processes in theearly universe. Stochastic backgrounds are difficult to detect in a single detector because theyare practically indistinguishable from instrument noise. If the source was not isotropicallydistributed (such as a population of binary stars towards the centre of our Galaxy), it might bedetectable from the variation of observed instrument noise as the detector orientation variedon the rotating Earth. However, a much better way of detecting stochastic backgrounds isby cross-correlating two nearby detectors. In this case the correlated stochastic signal willintegrate up in relation to the uncorrelated instrument noise (assuming both detectors to betruly independent). In this case the signal-to-noise ratio increases as N1/4, where N is theeffective number of cycles, determined by the observation frequency. This technique allows a300-fold improvement in signal-to-noise ratio in 108 s of integration, (compared again with a10 ms burst source).

Binary neutron star systems can produce gravitational waves in all the three classes. First,a large population of binary neutron stars in our Galaxy, with orbital periods in the range fromdays to minutes, can produce a stochastic background of individually unresolvable sourcesin our Galaxy in the frequency band ∼10−2–10−5 Hz. Nearby individual systems which arefar from binary coalescence could produce detectable nearly monochromatic waves at anyfrequency up to 0.1 Hz. In addition to the binary orbit, the individual rotation of the starsthemselves (if they are nonaxisymmetric), will also give rise to quasi-periodic gravitationalwaves. For example, the spin-down of a millisecond pulsar can be entirely due to gravitationalwave emission if just 10−7M is located in a nonaxisymmetric configuration on the star [17].As the binary evolves and radiates away gravitational potential energy, it will gradually spiralinwards. As a result, the frequency of the gravitational wave signal will increase with time. Atthe same time, any periodic waves from the rotation of the individual stars will cause loss ofrotational kinetic energy, so that this frequency will decrease with time. Eventually the starswill coalesce, resulting in a short intense burst of gravitational waves.

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-2 10-19

-1 10-19

0

1 10-19

2 10-19

0.4 0.5 0.6 0.7 0.8 0.9 1

h+

Time (sec)

Figure 4. Predicted gravitational waveform from the inspiral of 10M black hole binaries [18]. Toshow the individual cycles near coalescence, the orbital frequency in this graph has been artificiallyreduced.

Figure 4 is a predicted gravitational waveform produced by the inspiral of a binary madeof two black holes [18]. During the final minutes of a coalescing binary the waveform is highlydistinctive.

The time evolution of the frequency of two 1.4M neutron stars in a binary system isshown in figure 5. Over a period of about 1000 s the frequency rises from about 10 Hz to1 kHz as the neutron stars spiral together. This part of the merger begins when the stars arewithin about 1000 km of each other. The orbital velocity is ∼0.1c. Signal detection canmake use of exactly the same principles used to extract narrow-band signals due to the factthat the time evolution of the signal frequency and phase is predictable. Matched filtering,based on the existence of a family of accurately predictable waveforms, can allow integrationover all of the observed signal cycles. A terrestrial detector may be able to observe morethan 1000 gravitational wave cycles from a neutron star binary. The total number of cyclesobservable increases strongly as the lower cut of frequency is reduced. This provides a strongincentive for creating detectors at the lowest possible frequency. For 1000 observable cycles,the signal-to-noise ratio is improved by the square root of this number, or about 30.

Exactly the same concepts may be applied to supermassive black hole binaries. The signalfrequency decreases inversely with the black hole mass. Thus a pair of 108 solar mass blackholes would produce a chirp of gravitational waves rising from one cycle per year to a cycleper day over a period of 1011 s! This is much too long to observe the entire event, so inreality one could only expect to observe rather few cycles. However, since there appear to bea large population of quasars and galaxies containing massive black holes, as well as a largepopulation of interacting galaxies, such events may not be uncommon, and may give rise tonumerous strongly detectable sources at very low frequencies.

Detection of gravitational waves 1331

101

102

103

10-3 10-2 10-1 100 101 102 103

GW

fre

quen

cy (

Hz)

Time to coalescence (s)

Figure 5. The frequency evolution for coalescence of a binary system consisting of a pair of 1.4Mneutron stars.

2.3. Supernovae

Supernovae have long been considered a primary source of gravitational wave bursts.Unfortunately, astrophysics gives us few clues to their efficiency in producing gravitationalwaves during core collapse. Indeed, the true nature of the various supernova classes is stilluncertain. In particular, it is uncertain whether a type I(a) supernova occurs through detonationor collapse in a solitary or binary white dwarf system. Some supernovae, such as the Crabsupernova of the year 1054AD, do create neutron stars, but others such as supernova 1987Ahave failed to yield an identifiable neutron star. It is also unknown whether supernovae candirectly create black holes.

The possibility of strong gravitational radiation emission only occurs if the event consistsof gravitational collapse to a neutron star or black hole. Even in this case the efficiency ofgravitational radiation emission is contentious. (The efficiency ε is defined as the fraction of therest mass of the system concerned converted to gravitational waves.) Estimates of gravitationalwave emission have been based on two quite independent approaches. The first follows thegravitational collapse of a system in two or three dimensions, considering as much of thephysics as possible—magnetohydrodynamics, neutrino physics, and general relativity—in anattempt to deduce the time dependence of the quadrupole moment and hence the conversionefficiency to gravitational waves.

The second approach assumes that the collapse has occurred and follows the time evolutionof a newly formed hot and rapidly rotating neutron star. Any phenomenon that creates non-axial symmetry will convert rotational kinetic energy into gravitational waves. In addition,convective motions, vibrational modes of oscillation of the star, and nonaxisymmetric emissionof neutrinos can convert into gravitational wave emission. To date, many core collapse

1332 L Ju et al

calculations have predicted rather low efficiency—say 10−6 to 10−10 [19, 20], while thepost-collapse evolution calculations have predicted far higher efficiencies, of order 10−3 to10−4. We will discuss some of these results below. However, it must be emphasized thatall the models used so far are deficient due to uncertainty regarding the equation of stateand the viscosity and the difficulty in constructing a full 3D numerical general relativistichydrodynamical code which must also include magnetic and neutrino phenomena [21]. Dueto the enormous difficulties involved it seems most unlikely that theory alone will be able toanswer the primary observational questions on the efficiency and the waveforms generated insupernovae. However, almost all models show an unsurprising common feature: the efficiencyof gravitational wave production depends on the angular momentum of the progenitor star.

Lai and Shapiro [22] have considered the time evolution of a new-born rapidly rotatingneutron star. They have shown that the new-born star is driven by gravitational radiationinto a non-asymmetric configuration due to a bar-mode instability. A unique gravitationalwave signature ensues: the wave frequency sweeps rapidly downward from a few hundredhertz towards zero, while the wave amplitude increases rapidly from zero at the onset of theinstability to a maximum at a few hundred hertz, and then reduces steadily as the frequencyfalls. Additional gravitational wave signals can also arise in rapidly rotating neutron stars. Therotating stars are modelled as nonaxisymmetric ellipsoids. A secularly unstable Maclaurinspheroid [23] will evolve away from the axisymmetric configuration due to gravitationalradiation, and proceed ultimately toward a Dedekind ellipsoid [23].

According to Lai and Shapiro the characteristic amplitude of a gravitational wave duringthe evolution from a Maclaurin spheroid to a Dedekind ellipsoid is given (within 20% accuracy)by

hc ∼ 1.8 × 1022

(10 Mpc

R

)(M

10M

)3/4 ( r0

10 km

)f 1/2 (2.1)

where M and r0 are the mass and radius of the star, respectively. Here hc refers to the effectiveamplitude which takes into account the number of cycles that the signal is within the detectorbandwidth (see below).

At high frequency, gravitational radiation can be expected from the evolution of a Jacobi-like ellipsoid [23] toward a Maclaurin spheroid. This Maclaurin spheroid can evolve furtherto a Dedekind ellipsoid. The characteristic amplitude during the Jacobi-like evolution can befitted to the form

h ∼ 2.7 × 10−20

(10 Mpc

R

)(M

1.4M

)3/4 ( r0

10 km

)f −1/5. (2.2)

Houser et al [24] have modelled the gravitational radiation from a bar-mode instabilityin rapidly rotating neutron stars. Their calculation using Newtonian gravity and withoutconsideration of further collapse to a black hole, nor other hydrodynamic instabilities, gives agravitational radiation conversion efficiency of ε ∼ 0.1%.

The above examples seem to indicate that supernovae which produce rapidly rotatingneutron stars may be reasonably efficient sources of gravitational radiation. The nature of theproduction process is likely to be through shape instabilities such as those discussed, but itis unlikely that predictions of waveform are accurate. Large amounts of angular momentummay be radiated away in gravitational waves but if the duration and frequency evolution areunknown this presents an additional complication when it comes to trying to dig a signal outof the detector noise. The fraction of supernovae for which high gravitational wave emissionoccurs is unknown. In the following discussion where we need to use a numerical value, weshall adopt an efficiency of 0.1%. However, the uncertainty of this number must be recalled.

Detection of gravitational waves 1333

To start with, it is useful to note that a supernova of 0.1% efficiency produces a characteristicstrain amplitude of ∼10−18 at 10 kpc (within our Galaxy), and 10−21 at 10 Mpc (halfway tothe Virgo Cluster of galaxies). The chance of detecting gravitational wave bursts obviouslydepends strongly on the rate of the burst events. Due to the isolation of the Milky Way Galaxy,and to the large distances required to substantially increase the size of the target population,the amplitude distribution of bursts is extremely non-uniform. Strong events from our Galaxyare almost certainly rare, and to increase the event rate substantially one needs to be able todetect events in the Virgo Cluster. Thus, to have a chance of detecting several events per year,the sensitivity must be able to detect characteristic amplitudes of less than 10−21.

2.4. Rough guide to signal amplitudes

It is useful to have some formulae with which to make rough estimates of signal amplitudes.For a continuous gravitational wave of frequency fg , the strain amplitude h is related to thepower density w through the relation [25, 26]

w ≈ πc3

4Gf 2g 〈h2〉 = 3.18 × 1035f 2

g 〈h2〉 Wm2. (2.3)

where h2 = h2+ + h2

×. Because of the large numerical constant in equation (2.3) the amplitudeh is extremely small even for a fairly large power density. For a gravitational wave with strainamplitude of h ∼ 10−21 (typical of possible signals from the Virgo Cluster) at a frequencyof 1 kHz, the flux would be 0.3 W m−2, which is about 1020 times bigger than typical radioastrophysical energy fluxes. The strain amplitude can be written as

h = 4πR2w ∼(

G

π2c3

)1/2L1/2

fgR∼ 1.7 × 1022

(1 kHz

fg

)(10 Mpc

R

)(L

1046 W

)1/2

. (2.4)

As we saw above, the modelling of gravitational wave forms in gravitational collapse isextremely uncertain. However, for a gravitational wave burst event, the characteristic timescale of the event τg , and the total gravitational energy released Eg , provide a reasonable basisfor estimating source parameters. The energy radiation rate L is related to τg and Eg by

L ∼ Eg/τg. (2.5)

Burst sources naturally have a broadband spectral distribution. The characteristic frequencyof a burst of duration τg is roughly

fg = 1

2πτg. (2.6)

This frequency roughly defines the peak frequency in the spectrum. For a roughly Gaussianburst, the width of the spectrum f is of the same order of fg . The strain amplitude can thenbe written as [8]

h ∼(

G

π2c3

)1/2(Eg/τg)

1/2

fgR∼ 5.8 × 10−20

(Eg

Mc2

)1/2 (1 kHz

fg

)1/2 (10 Mpc

R

). (2.7)

If a gravitational collapse forms a black hole, we can be more specific in estimating the eventduration. Defining a characteristic time τ to be the time for the gravitational wave to travelacross the region of strong gravitation ds , which is assumed to be about twice the gravitationalradius 2GM/c2, and assuming τg is approximately the same as the characteristic time τ , wehave

τ ∼ 1

2πfg

∼ ds

c∼ 4GM

c3. (2.8)

1334 L Ju et al

For a system of several solar masses, this corresponds to a frequency of a few kHz. Puttingequation (2.8) into equation (2.7) and using Eg = Mc2, the strain amplitude of a burst eventthen becomes [8]

h ∼ 1

2π

c

fgRε1/2 ∼ 5 × 10−21

(1 kHz

fg

)(10 Mpc

R

)( ε

10−3

)1/2. (2.9)

2.5. Neutron star coalescence sources

The modelling of gravitational wave emission from neutron star coalescence has been studiedextensively. For most of their evolution, the neutron stars can be considered as point masses,and much of their evolution is well described by the quadrupole formula equations (1.15)and (1.16). The waveform as shown in figure 4 is quite distinctive and amenable to the methodof matched filtering for signal detection. A numerical template is used, defined by the setof possible waveforms. When this is cross-correlated with the data and correctly matched inphase, it will produce a large positive correlation. The signal-to-noise ratio is substantiallyenhanced by this means. The apparent signal enhancement achievable is expressed in terms ofthe characteristic amplitude hc. The characteristic amplitude represents the effective amplitudedetected after optimal filtering of the waveform. Roughly, hc includes an enhancement of thesignal by the square root of the number of cycles within the spectral band of interest and isroughly a factor of 30 for a neutron star coalescence detected by a laser interferometer detector,although this increases strongly if the waveform is detectable at much lower frequencies wherethe frequency evolution is slow. For example, the number of observable cycles increases almost50 times if the detector is able to observe down to 10 Hz instead of 100 Hz. This means thatwe can only roughly estimate the size of the detectable signal, as it depends on the detailedfrequency response of the detector.

Thorne [18] gives the characteristic strain amplitude of the waves from inspiralling binariesas

h ∼ 0.237µ1/2M1/3

r0f1/6c

= 4.1 × 10−22

(µ

M

)1/2 (M

M

)1/3 (100 Mpc

R

)(100 Hz

fc

)1/6

.

(2.10)

Here M and µ are the total and reduced masses: M = M1 + M2, µ = M1M2/M , and fc

is roughly the frequency of maximum detector sensitivity.Lai and Shapiro [22, 27] have modelled neutron star coalescence taking into account the

dissipative hydrodynamics of the systems. They showed that a hydrodynamical instabilityarises through tidal interactions, which significantly accelerates the coalescence at smallseparations. This leads to a reduction in the coalescence time, and an increase in h comparedwith a non-viscous system, as shown in figure 6.

The rate of coalescence events is such that the chance of an event in our Galaxy is negligible.Based on the statistics of observed single and binary neutron star systems and on the supernovaerate in external galaxies [28, 29], it is estimated that the merger rate of binary neutron stars inour Galaxy is between 10−6 yr−1 and 10−4 yr−1. For galaxies at R 200 Mpc a lower limit ofthis rate is roughly 1–3 yr−1. However, Tutukov et al [30] and Yamaoka et al [31] have shownthat the above merger rate, calculated by means of statistics of observed binaries, is probablywrong because of the short lifetimes of most new-born neutron star binaries. From modelsbased on stellar evolution, they estimate a neutron star binary merger rate up to 100 yr−1

in galaxies out to 200 Mpc (assuming a Hubble period of THubble = 1.5 × 1010 yr, whichcorresponds to a Hubble constant of H0 ∼ 66 km s−1 Mpc−1).

Detection of gravitational waves 1335

Figure 6. Waveform from neutron star binarycoalescence [22]. The thick solid curvecorresponds to zero viscosity. The thin solid curveassumes a mass-averaged shear kinetic viscosityv = 0.5(M/r0)

1/2, where M and r0 are the massand the radius of the star respectively. The dottedcurve is the case for two point masses (i.e. asfigure 4).

The neutron star coalescence signals amplitude and waveform can be predicted withreasonable confidence, so that such sources are certain to be detected when sufficient sensitivityis achieved. The burst sources are also very promising for advanced gravitational wavedetectors [32–34], but suffer from uncertainty in the value of ε. From equation (2.9), thegravitational wave strain amplitude for burst events at 200 Mpc is comparable to that frombinary coalescence if ε ∼ 0.01. In this case, the event rate will be much greater.

From the estimation of the stellar population with distance, the merger rate would be∼0.1 yr−1 for a distance of R ∼ 20 Mpc, the typical distance to the Virgo Cluster. The burstrate by contrast could be 30 yr−1 [35].

2.6. Low-frequency sources

As discussed above, very intense low-frequency gravitational wave sources can be expectedfrom gravitational waves associated with the merger of massive black holes. Rees [36,37] hasargued that massive black holes are inevitable in the cores of young galaxies. There is verystrong evidence that such black holes exist in many objects, with masses ranging from 106 to109 solar masses. Galactic mergers are likely to give rise to such black hole mergers so oneestimate of the rate of powerful gravity wave events can be obtained by estimating the rate ofgalactic mergers. For large galaxies with central black holes, Haehnelt [38] has estimated thisrate at about one per century. This does not include the far more frequent mergers of smallergalaxies for which central black holes have not been confirmed. Vecchio [39] has shown thatfor the black hole merger rate to reach one per year practically all galaxies out to z = 1 wouldhave to contribute black holes to feed the merger process. The latter is not such a strongconstraint, however, since the horizon for detecting black hole mergers could be far beyondz = 1. For example, if one considers mergers to z = 3, only a few per cent of galaxies arerequired to have a central black hole to achieve one event per year.

Potentially detectable low-frequency gravitational waves can also be created by low-massobjects orbiting massive black holes. The low-mass objects could be smaller black holes orneutron stars, white dwarfs or even main sequence stars. Such sources could exist in thenucleus of our own Galaxy, and could in principle be detectable well beyond the Virgo Cluster(which multiplies up the number of potential sources by several thousand).

The basic physics behind gravity wave emission from sources of this type relates to thequestion of whether the gravity gradient from the central black hole is sufficient to tidally disruptthe incoming object. Clearly, main sequence stars will be most easily disrupted. Only if thecentral black hole is capable of swallowing whole the incoming object, will the gravitationalradiation be strong. Otherwise the tidally disrupted star will form an accretion disc and slowly

1336 L Ju et al

accrete the material with negligible gravitational wave emission. To avoid tidal disruption, theSchwartzchild radius of the black hole must be large compared with the radius of the infallingobject. Roughly, the central black hole must be 107–108 solar masses for main sequence stars,104 solar masses for white dwarfs, and 10 solar masses for neutron stars.

Sigurdsson [40] has estimated the rate of capture of stellar mass black holes by massiveblack holes in galactic nuclei. The gravitational potential of the central black hole createsa cusp-like stellar density profile. It is difficult to estimate the space density of objects inthe central cusp. It depends on star formation in the central high-density region of galaxies.However, the population in the cusp will never achieve dynamic equilibrium because starsapproaching too close to the central black hole will be lost into the hole. Sigurdsson estimatesthe rate of black hole capture by a central object as 10−8 per year, meaning that a realisticdetectable rate (one per year) requires observations to a range of the order of 3 Gpc. If the highdensities in galactic nuclei favour higher mass star formation the event rate could increase byan order of magnitude.

2.7. Gravitational waves from binary systems

Short-period binary systems can create interesting amplitudes of gravitational waves in the10−1–10−5 Hz range. Such binaries exist in several classes. One of the most definite classesconsist of the W Ursa Majoris binaries (WUMas), which are contact binary stars, with orbitalperiods of hours. They are generally low-mass systems. About one star in 150 with mass>0.6M is a contact binary [41]. Lower mass binaries are difficult to detect: there could bean equal population of such systems with even shorter orbital periods.

A second important class of short-period binary stars are the cataclysmic variables,consisting of an interacting main sequence and white dwarf binary. Cataclysmic variableshave orbital periods in the range 1000 s to one day: the shortest-period systems are probablywhite-dwarf–white-dwarf systems.

Neutron star binaries occur in various forms, from the NS–NS binaries such as the Hulse–Taylor pulsar PSR 1913 + 16, to rather more common systems in which neutron stars havewhite dwarf or main sequence star companions. The latter often occur as interacting binaries—low-mass or high-mass x-ray binaries, in which x-ray emission occurs due to mass transfer onto the neutron star.

Verbunt [42] has summarized the density and strain amplitude expected from the short-period binary star systems. Table 1 below is based on his review. The table shows the roughnumber density and mass parameters, and the distances of typical sources. Most producegravitational waves ∼10−3–10−4 Hz, at an amplitude ∼10−20–10−22. There is clearly anabundance of sources in the categories of nearby sources (<100 pc), such that the totalpopulation creates a stochastic background of gravitational wave noise.

Several x-ray binary systems have been shown to contain neutron stars spinning in therange 250–350 Hz. It is suggested that their spin rate is determined by the balance between themass accretion which provides a source of energy and angular momentum, and gravitationalwave emission which is the dominant energy sink. Emission could occur by a variety ofsymmetry breaking instabilities. A specific suggestion is the so called r-mode instability, inwhich rotational fluid flow patterns are induced in the neutron star [43].

Stellar evolution studies suggest that globular clusters are a breeding ground for close-spaced binary black holes [44]. It has been recently suggested that these binaries get ejectedfrom globular clusters by three-body interactions, creating a halo population of binaries whichwill coalesce in less than a Hubble time. Such binaries might coalesce without electromagneticsignature, and could have a sufficiently large population that they could be detected at a

Detection of gravitational waves 1337

Table 1. Binary sources of low frequency gravitational waves. The table shows the rough galacticdensity of each source, the distance within which sources are expected, the typical mass of eachbinary component, and the typical frequency and strain amplitude from the nearest sources.

Type Density/number d (pc) M/M m/M log f (s−1) logh

WUMa (0.3–0.6M) 2 × 10−4 pc−3 15 0.6 0.3 −4.0 −20.4WUMa (0.1–0.3M) 2 × 10−4 pc−3 15 0.3 0.1 −3.7 −20.7Cataclysmic variables 10−5 pc−3 45 0.3 0.6 −3.7 −20.7Double degenerates (AM CVn) 100 0.04 0.6 −2.7 −21.2Low-mass x-ray binaries (Pb < 2 × 10−4 s) 30 1 000 0.4 1.4 −3.8 −21.7Low-mass binary pulsars (PSR 2051-08) 1 300 0.03 1.4 −3.8 −21.7High-mass x-ray binaries (Cyg X-3) 1 10 000 4.0 1.4 −3.9 −21.9NS–NS binary pulsars (1913 + 16) <10−5 500 1.4 1.4 −3.7 −20.8Binaries in globular clusters (4U1820-30) 8 100 0.06 1.4 −2.5 −22.3

reasonable rate at h ∼ 10−21.Gamma ray bursts which emit energy comparable to a solar mass at cosmological distances

could be due to the formation of black holes, or to neutron star binary coalescence, or to thecoalescence of neutron-star–black-hole binaries. Except in the case of spherically symmetricalblack hole formation, these mechanisms should all include strong gravitational wave bursts.Searches have failed to find correlations between bursts and existing detectors, but this is notunexpected as sensitivity is still not high enough.

Neutron star black hole coalescences, whether or not they are associated with gammabursts, can allow neutron star structure and microphysics to be probed because the break pointin the coalescence waveform is set by the tidal disruption of the neutron star. This dependsstrongly on the neutron star radius and equation of state.

2.8. Stochastic background from the era of early star formation

We now consider the effects of supernovae and neutron star births at cosmological distances.In this case, we are extrapolating from a radius of 10 Mpc to a radius of several Gpc. Forexample, we consider supernovae from galaxies at redshift z = 2. Such galaxies are mucholder than massive stellar lifetimes, and the rate of supernovae in such systems is generallythought to be 10–100 times greater than supernovae in contemporary galaxies [45]. This issupported by observations by Cowie et al [46] which indicate a fourfold enhancement in faintblue galaxies at z 1. At greater distances, millimetre wavelength studies of the Hubble deepfield region show the presence very-high-luminosity objects consistent with dust-enshroudedgalaxies at z ∼ 2–4. Observations indicate that star formation is occurring at rates ∼50 timesthat in the present epoch [47]. The significance of these increased star formation rates, andhence increased supernova rates, is that it leads to the possibility that the gravitational wavesfrom supernovae create a nearly continuous stochastic background.

First consider a simple case. Suppose that all supernovae have a gravitational wave burstduration τ , and a mean rate of occurrence within some horizon distance (say 3 Gpc), of R

bursts per second. Then the mean duty cycle D of supernova bursts is given by D = Rτ . Ifτ ∼ 10−3 s, then D reaches unity when R reaches 3 × 1010 y−1, or 103 s−1. When D reachesunity the supernova bursts create an effectively continuous stochastic background. To make anaccurate determination of the supernova stochastic background, one needs to take into accountboth evolutionary effects and cosmological effects. If every population I (second-generation)star was the result of a single prior supernova, then there would need to have been ∼1021

supernovae to create the observed population of second-generation stars. This corresponds

1338 L Ju et al

10-5

10-4

10-3

10-2

10-1

100

10-26 10-25

duty

cyc

le

strain amplitude h

Ω = 1

h0= 10-22 (at 10 Mpc)

Figure 7. Duty cycle versus strain amplitude for supernova-generated gravitational radiation (withan initial Gaussian burst of star formation at z = 2) [49].

to an average rate Excedrin 1000 supernovae s−1. Redshifts both stretch the pulses and themean time between bursts. The luminosity distance in non-Euclidean geometry changes theobserved amplitude of each burst.

A preliminary analysis in flat spacetime with assumed star formation rates showed that theevent rate could be as high as 104 s−1 [48]. Burman et al [49] refined the predictions of [48]using various predictions for star formation rates to determine the duty cycle of short bursts ofgravitational waves from supernovae within the observable universe for various cosmologicalmodels. Ferrari et al have separately considered supernovae [50] and gravitational waves fromyoung neuron stars [51]. They obtained event rates ∼20 s−1.

Figure 7 gives a typical result [49]. This result uses a Gaussian burst of star formationabout z = 2. It shows the burst amplitude versus duty cycle for supernovae assumed to havean amplitude h0 = 10−22 at 10 Mpc. The background can only be considered a true stochasticbackground as D tends to unity: for this model this occurs at h ∼ 10−25. Most of this signalis due to events occurring during the initial burst of star formation. The amplitude of thisstochastic background will be characteristic of the more distant sources at z = 2.

Like all stochastic backgrounds, the supernova background can in principle be detected bycross-correlation of signals between nearby detectors (less than half a wavelength separation:for τ ∼ 10−3 s, they should be less than 100 km apart) [52]. As discussed above, the signal-to-noise ratio is increased as the 1

4 power of the effective measurement time defined by thecross-correlation integration time. For 107 s, integration at 1 kHz, this represents a 300-foldimprovement. Thus, the combined effects of all supernovae is to create a signal which can bedetected at a signal-to-noise ratio comparable to that of an individual supernova at 20–30 Mpc.Thus, a detector capable of detection extragalactic supernovae can, with cross-correlation,detect a stochastic background produced at 30 times greater distance.

Detection of gravitational waves 1339

There are other aspects of the supernova stochastic background which are worthmentioning. Its spectrum represents the average spectrum over all supernovae, but it will bereddened according to the contribution of high-redshift supernovae. The duty cycle is clearlyamplitude dependent. Nearer sources will create less-frequent, larger-amplitude bursts. Atlow duty cycle the background will be like popcorn noise, while for D > 1, it will approximateGaussian noise. There is an important difference in this regard. The presence of a popcornnoise component [53] means that unlike true white noise, the individual short bursts createbroadband intensity correlations which might allow more powerful digging into the noise. Forexample, the broadband correlations might allow the background to be detected as spectralintensity correlations within a single detector. This might be combined with cross-correlationbetween two detectors to dig still deeper into the noise. Further work in this area is badlyneeded.

The energy density of the supernovae background 1SN , expressed as a fraction of closuredensity, is given by

1SN = 1fsfSN ε (2.11)

where 1 is the usual fraction of closure density for the universe as a whole, fS is the fractionof this matter which forms into stars in a Hubble time, fSN is the mass fraction which takespart in supernova events in a Hubble time, and ε the mean gravity wave conversion efficiencyfor supernovae. It is possible that 1SN could be in the range 10−6–10−8. However, if ε is lowand the duty cycle is low, 1SN could be 10−10.

If the majority of the gravitational waves are generated in relatively long duration spin-downs of neutron stars the spectrum will be dominated by a continuous stochastic component,but if it is emitted in short supernova bursts, the popcorn component will dominates. If theenergy density of the early star formation stochastic background is ∼10−8, then it should beeventually detectable by pairs of advanced detectors [54].

To show the signals on the same scale, which relates to detectability, stochastic sources areassumed to have been integrated up for times ∼108 s, binary coalescences have been integratedover the coalescence frequency range, while the burst sources signal strengths are the only onesrepresenting the instantaneous signal amplitude.

Finally, to summarize our discussion of sources, we present a graph (figure 8) containingestimates of various events. The comparison is approximate, as it compares various sourcesdetected by various techniques. Supernovae signals would appear as bursts requiring no specialsignal processing. However, estimates must allow for a wide range of efficiencies and sourcedistances. Black hole formation is similarly uncertain.

2.9. Cosmological gravitational waves from the big bang

Various sources of gravitational waves from the early universe have been hypothesized. Thesemay be thought of as the gravitational wave analogue of the microwave cosmic backgroundradiation. The cosmic microwave background originated at the epoch of last scattering, at aredshift z ∼ 103 when neutral gas first formed in the universe. Thus, the microwave backgroundprobes the universe when it was ∼105 years old. A similar background of neutrinos shouldalso exist, a relic from their epoch of last scattering, about 0.1 s after the big bang, at a redshiftz ∼ 1010. Due to the weak coupling of gravitational waves with matter, their epoch of releasewould have been much earlier still, at around the Planck time ∼10−43 s, or z ∼ 1030.

Thus primordial gravitational waves offer the tantalizing possibility of probing the universevery near to the moment of creation. Unfortunately, we do not have accurate predictions abouttheir amplitude. It has been suggested that the background could have been parametrically

1340 L Ju et al

Figure 8. Spectrum of gravi-tational wave sources [18, 22].In this figure, the abbrevia-tions are: BH, collapse to blackhole; NS/NS, neutron star coa-lescence; NS evol, secular evolu-tion of a nonaxisymmetric neu-tron star.

amplified during a period of inflation, or that phase transitions in the early universe (for whichthere is no experimental evidence) could have created an enhanced background. If there wasno process to enhance the background amplitude, then we need consider simply a thermalbackground that was in equilibrium at the extremely high energies of the Planck era. Thebackground will then have been redshifted like any other radiation. Today this radiation wouldbe in the microwave regime and have an amplitude h ∼ 10−35, which is beyond the possibilityof detection.

If the universe contained an initial inhomogeneity of amplitude hg [55], then today itwould have an amplitude at frequency f given roughly by

h ∼ 10−20hg/f. (2.12)

We have little idea of the initial amplitude, except for limits set by the cosmic microwavebackground which implies that inhomogeneities traced by matter had an amplitude ∼10−5.This would imply that the cosmological background amplitude could be 10−28 at 1 kHz, (whichis beyond terrestrial experiments) and 10−21 at 10−4 Hz, (which is experimentally accessibleby space laser interferometers).

A constraint on the cosmological background is set by considering the energy density, andrelating it to cosmological models. Thus cosmological backgrounds are often parametrized interms of the closure density fraction 1g . If the spectrum contains equal energy in each decade,it has a slope of −1 on a logh–log f plot. For example, for the universe to be closed by gravi-tational waves, 1g = 1, the amplitude hg would be 10−13 at 10−5 Hz, falling to 10−21 at 1 kHz.For 1g = 10−6, hg falls by three orders of magnitude to 10−16 at 10−5 Hz and 10−24 at 1 kHz.

It would appear most unlikely that the universe be closed by gravitational waves, and itwould be surprising if gravitational waves contributed significantly to the missing mass. Thebinary pulsar, in addition to confirming the emission of gravitational waves, also allows limits

Detection of gravitational waves 1341

to be set on the stochastic background of gravitational waves, and on the time variation ofthe gravitational constant. Taylor and co-workers [56] have used pulsar timing of the binarypulsar PSR 1913 + 16 to set limits on cosmological backgrounds. The method only worksat very low frequencies where both pulsars and atomic frequency standards have sufficientfrequency precision that measurement of the arrival time variations of pulsar signals givesuseful sensitivity to gravitational waves. Taylor’s timing measurements over several years seta limit to 1g ∼ 4 × 10−2 in the frequency range 10−9 to 10−12 Hz [56].

3. Detection of gravitational waves

3.1. An overview of detector technology

The development of gravitational wave detectors was pioneered by Joseph Weber in the early1960s [4]. He used a massive aluminium bar as the antenna. Following his work, researchersall over the world have been working hard to build different types of gravitational wavedetectors. The detection of gravitational waves is based on the following idea (as discussedin section 1.2). A gravitational wave can be considered as a time-dependent strain in spacehaving two linear polarization states (h+ and h×). When the wave passes test masses in spaceit will cause displacements of the test masses, as shown in figure 1. A measurement of therelative displacements of the test masses is a measure of the wave. The gravitational wavedoes work on electromagnetic field, such as a capacitance or a laser light field. Because thedisplacements are very small, the momentum of the gravitational wave is in general limitedby the uncertainty principle. The quantum limit presents a significant but not insurmountablebarrier for future detectors.

A gravitational wave detector can be constructed from a pair of masses which can move‘freely’ with respect to each other. They can be suspended as pendulums, so that in thehorizontal direction they can be treated as nearly free masses above the pendulum resonantfrequency. A pair of masses connected by a spring (figure 9(a)) can also be used to forma resonant gravitational wave detector. Such a mechanical resonator will be driven by agravitational wave as long as it has spectral components at the resonant frequency of the mass–spring system. If the detector is a high-Q resonator, it will continue to oscillate long after thegravitational wave has passed. That is, the resonator remembers the effect of the gravitationalwave. A measure of the oscillation of the resonator will give information about the passinggravitational wave.

The resonator need not necessarily be two masses connected by a spring. A lump of metalsuch as a cylindrical bar is well suited to the purpose (figure 9(b)). The difference between thebar detector and two point masses with a spring in between is that the bar detector has a setof higher-order resonant modes. However, for the lowest resonant frequency, the bar can bemodelled just like two masses connected by a spring, with an effective mass equal to half themass of the idealized detector. A multi-spring mass resonator (figure 9(c)) can be constructedto detect not only the amplitude of the gravitational waves but also the direction of the waves.This leads to the idea of a spherical antenna [34, 57–59], as shown in figure 9(d).

All resonant-mass detectors use cryogenic techniques to reduce the thermal noise and toenable the use of high-sensitivity superconducting transducers. A high Q-factor ensures thatthe thermal noise (which even after locking has a very large amplitude compared with thesignal) is manifested as a highly sinusoidal, and hence predictable, waveform. A gravitationalwave induces a small change in the amplitude and phase of this waveform. Typically, thememory effect is used to resolve the signal component averaged over perhaps a few hundredcycles of the antenna. The time over which the signal can be resolved depends most strongly on

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(a) AAAA

(b)

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AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

AAA

(c) (d)

Figure 9. Model of resonant-mass gravitational wave detectors. (a) Two masses joined by a spring.(b) Bar detector. (c) Multi-spring mass detector. (d) Spherical detector, in this case formulated asa truncated icosahedron, with six transducers located as indicated.

the sensitivity and electromechanical coupling of the transducer which is used on the antenna.If the transducer is sufficiently strongly coupled, the signal could be resolved in perhaps asingle cycle, and then the measured bandwidth would be comparable to the antenna frequency.Practical bar antennas to date have only demonstrated bandwidth of about 10 Hz. However,improved transducers should soon allow the bandwidth to reach ∼100 Hz and eventuallynear-quantum-limited sensitivity [60–63].

As indicated in figure 1, a passing gravitational wave will make a ring of particlesstretch and shrink alternately in orthogonal directions. An interferometer configuration whichcan detect the relative position change between two orthogonal masses is a natural detectorconfiguration [64–66], as shown in figure 10. When a gravitational wave passes, the lengthsof the two arms of the interferometer change in antiphase. This results in a change inthe interference intensity at the output. This change in light intensity is a measure of thegravitational wave. Since the test masses simply follow the passing wave pattern, these devicescan be expected to accurately trace the waveform. The advantage of this type of system is thenonresonant feature and the fact that the scale of an instrument is set, not by the velocity ofsound (which limits a resonant bar to a few metres in length if the detector is to detect radiationat ∼1 kHz), but by the velocity of light. Thus, a laser interferometer can detect gravitationalwaves over a wide band of frequencies and can in principle be scaled up to 150 km arm length,(for the same 1 kHz upper frequency) for which the absolute displacement L = hL is muchlarger. Because 150 km arm length is impossible on the surface of the Earth (due to thecurvature of the Earth), practical instruments must be scaled down to several kilometres.

In reality, the problem of noise always limits the bandwidth of laser interferometers. Intheir widest bandwidth configuration laser interferometers are limited by seismic and thermalnoise at low frequency and photon shot noise at high frequency, as we discuss in detail in

Detection of gravitational waves 1343

Vibration Isolation and Suspension System

Test Mass Mirror

Beamsplitter

High Power Laser

Light Detector

Figure 10. Schematic diagram of a Michelson interferometer for use as a gravitational wavedetector.

section 5. This leaves a high-sensitivity bandwidth in the 100–500 Hz range. Optical tuningcan in principle be used to restrict the bandwidth considerably, allowing optimum sensitivity tobe achieved in narrower bands of several tens of Hz anywhere in the 10 Hz to several kHz range.

There is not a similar size constraint if a laser interferometer detector is placed in space.Laser interferometers can be conceived, of scale size millions of kilometres, in Earth or solarorbit. In this case practical considerations make such devices best suited to search for low-frequency sources in the range of 10−1 to 10−5 Hz. Free-floating spacecraft carrying testmasses shielded from the solar wind, and low-power lasers should be able to achieve very highsensitivity, limited eventually by the confusion limit of gravitational wave ‘noise’ from thelarge number of binary star sources in the Galaxy (see section 3.2 for further discussion).

At even lower frequencies, signals from pulsars have been used as gravitational wavedetectors. Pulsars, and especially millisecond pulsars, represent precise-frequency sources,close to the limit of the best man-made clocks. While we are accustomed to thinking ofgravitational waves changing the relative spacing of test masses, this picture can be confusingwhen thinking of a pulsar as one of the test masses perhaps 1000 light years away. It iseasier to consider the pulsar beam passing through the curved spacetime due to the passageof very-low-frequency gravitational waves in the vicinity of the Earth. (Both pictures areequivalent however.) The result is that the gravitational wave causes changes in the arrivaltime of the pulsar signal. Because the pulsar signal is weak, and because atomic clocks givebest precision over long periods of time, the optimum precision of this method of detectionoccurs for frequencies ∼10−7 to 10−8 Hz [56].

Interferometric detection has the advantage that it gives intrinsic immunity to laserfrequency noise. Indeed, a laser interferometer can in principle use white light. However,any single-beam detector is sensitive to frequency fluctuations in the source. The sensitivitylimit is set directly by the frequency stability of the radiation source: h ∼ f/f . Sincefrequencies can only be compared against a standard, the limit of a single-beam detector suchas a pulsar signal is either the stability of the pulsar, or the stability of the frequency reference:today f/f ∼ 10−16.

The pulsar timing technique can to some extent avoid this single-beam clock stabilityproblem by using several pulsars in different directions. A gravitational wave creates correlatedfluctuations depending on the pulsar direction so that in principle it should be possible to dig atleast an order of magnitude below the clock stability limit. Unfortunately, solar wind refractive

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effects and interstellar scintillation are very strong in the MHz–GHz frequency range of strongpulsar signals. This provides an additional noise source that could mimic a gravitational wave.

There is no need in principle to rely on pulsars to provide the timing source for single ormultibeam detection of gravitational waves. Interplanetary spacecraft generally transpond aground-station-generated precision clock frequency back to the ground station. Comparison ofthe return signal frequency and the transmitted frequency predominantly records the spacecraftvelocity, seen as a Doppler shift. Gravitational waves will appear as a perturbation in theDoppler tracking signal. Because the signal strength is stronger than for pulsars, and the pathlengths are shorter, Doppler tracking is sensitive in the 10−4 to 10−5 Hz band. Solar windrefraction is again a very serious limit, but in this case it could be overcome if tracking tookplace at various frequencies, and if the frequencies used were increased into the millimetrewavelength range. Doppler tracking experiments [67, 68] have taken place successfully aslow-cost add-ons to space missions, but the technique is unlikely to compete with space laserinterferometers. Future interplanetary spacecraft equipped with nanosecond pulsed lasersavoid all local refractive errors from solar plasma, but solar wind buffeting causes additionalnoise which must be overcome by drag-free spacecraft technology.

3.2. Space laser interferometer gravitational wave detectors

A joint NASA–ESA space mission has been proposed, to place into solar orbit a set of three lasertransponding spacecraft which would form a space laser interferometer for gravitational wavedetection. The idea is to create a nested set of interferometers in a triangular configuration, andto use active transponding rather than passive interferometry as used in terrestrial detectors.The space environment allows the path length to be increased to 5 million kilometres. Considerfirst one vertex of the triangle. A pair of stabilized CW laser beams are transmitted throughtelescopes in 60 Y-shaped arms of the spacecraft, to two identical target spacecraft 5 millionkilometres away. Each receives a very weak signal, but one sufficient to allow an on-boardlaser to be coherently phase locked to the incoming beam with a slight offset frequency. Thislaser is then directed back to the originating spacecraft. Thus the distant spacecraft acts likean active mirror, returning the incoming signal to its origin.

LISA (laser interferometer space antenna) as proposed uses three spacecraft in a specialsolar orbit. All six laser beams create three independent Michelson interferometers. The in-coming beam at each arm has its phase compared (by beating) with some of the outgoing signal.That effectively measures the changes in the length of one arm. The same measurement is donein the second arm by the second laser beam. The phase differences are compared, to create amonitor of the arm length differences which could indicate the passage of a gravitational wave.

The above concept can give excellent sensitivity, as shown in the sensitivity curve forLISA in figure 11. This is only achieved if many noise sources are greatly reduced. The firstis the buffeting by the solar wind. To overcome this noise the spacecraft are centred arounda free-floating test mass, which is shielded from the wind. The spacecraft are controlled byminiature ion drives—field emission electric propulsion thrusters—to maintain the spacecraftlocation centred on the test mass. The forces required are ∼10−6 N.

A second potential source of noise is the gravity gradient due to the thermal expansion andcontraction of the spacecraft structure. This can be overcome with careful design and thermalshielding.

To be able to use low-power thrusters the spacecraft must be placed in an orbit in which theirrelative positions have very high intrinsic stability. An orbit that achieves this requirement [69]is a heliocentric orbit of about 1 AU, 20 behind the Earth. The orbits of the spacecraft havea small eccentricity of e = d/(D

√3), and a small inclination i = d/D, where d is the

Detection of gravitational waves 1345

Figure 11. The sensitivity curve of LISA, along with its prime gravitational wave sources [322].This is the sensitivity achieved after one year of integration.

Figure 12. The positions of the proposed LISA spacecraft in heliocentric orbit [323].

triangle arm length and D = 1 AU is the semimajor axis of the orbit. Remarkably, threepairs of spacecraft in these orbits, with careful specification of their nodes, appears to maintaina nearly ‘rigid’ configuration which rotates slowly while maintaining the triangle in a planewhich is inclined at 60 to the ecliptic. The positions of the spacecraft are shown in figure 12.

An important question is the orbital stability. If the relative spacing of the satellites changestoo rapidly the fringe rate becomes high and the noise contribution from the local oscillator(which is used to apply frequency offsets and measures the fringe rate) becomes relativelylarger. Thus it is essential that the fringe rate be low. Ideally the difference in arm lengthchanges between two arms should be reduced to mm s−1, less than the nominal metres persecond arm length changes predicted for the proposed orbits. Stabilization schemes for the

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Figure 13. Artist’s impression of space interferometer (LISA).

local oscillator through modulating the light in one arm or through the use of improved localoscillators should allow these problems to be overcome. Spacecraft manoeuvres are requiredoccasionally to compensate for accumulated orbital drift.

Figure 13 shows an artist’s impression of this remarkably ambitious conception. A largeteam of scientists is pursuing this project, which should fly in the second decade of the twenty-first century. More details are available from the LISA web page (http://lisa.ipl.nasa.gov/).

3.3. The world array of resonant-mass detectors

The improvement of resonant-bar detectors since they were first reported by Weber hasbeen enormous. Their amplitude sensitivity has been increased several hundred times,corresponding to an improvement in flux sensitivity of about five orders of magnitude. Currentdetectors are sensitive to narrow frequency bands near 700 Hz and 900 Hz, but improvementswill increase the bandwidth of each to>50 Hz. Figure 14 shows one of the present resonant-bardetectors.

An array of five resonant-mass gravitational wave detectors, coordinated under theInternational Gravitational Events Collaboration is in operation [70, 71, 110, 114, 156]. Thedetectors are located at Baton Rouge (Louisiana), CERN (Geneva), Legnaro (near Padova),Frascati (near Rome) and Perth (Australia). The data are available at a web address(http://axln01.lnl.infn.it/igec/). Since 1993 between two and four antennas have been incoincident operation searching for bursts at a strain sensitivity better than 10−18 (see figure 16).This is sufficient to detect strong galactic gravitational wave bursts, but insufficient for detectionbeyond our Galaxy. Over the past two decades the limits to the strength and rate of gravitationalwave burst events impinging on the Earth has been reduced substantially, but these limits are stillbelow astrophysical predictions. Thus, so far, it can be stated that the rate of gravitational wavebursts is not two orders of magnitude greater than expected from conservative astrophysicalpredictions, or else that their strength is not at the high end of predicted signal strengths(ε > 10−2, see figure 29 for quantitative results).

Detection of gravitational waves 1347

Figure 14. Photograph of the antenna NAUTILUS at Frascati, showing the bar and its cryogenicshields.

The resonant bars have been used for rather deep pulsar searches in certain directions(h ∼ 10−23) [71] (see figure 30) and, by using cross-correlation, have been used to setimpressive limits on the stochastic background of gravitational waves (h ∼ 10−22) [84].

Some of the resonant-bar detectors are being improved with better transducers andamplifiers, allowing their bandwidth to be increased towards 100 Hz. This improves theburst sensitivity and time resolution which in the immediate future should allow an order-of-magnitude improvement in burst sensitivity. Efforts are underway to create quantum-limitedtransducer systems (e.g. [72]) which should eventually allow a further order-of-magnitudeimprovement in amplitude sensitivity.

To match the ultimate sensitivity of long-baseline laser interferometer detectors (seebelow), it is necessary to increase detector mass from a few tonnes to a few hundred tonnes.Such massive spherical detectors have been proposed and development work is underway onsmall prototypes in Frascati, Leiden, and Sao Paulo [73–75]. As already noted they havethe advantage of omnidirectional sensitivity. They use the proven cryogenic techniques ofthe existing resonant-mass detectors, but to scale up to hundred tonnes represents an excitingmajor engineering challenge (see section 4.10 for more details).

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(a)

(b)

Figure 15. The LIGO detector at Hanford. (a) An aerial view, (b) the vacuum pipe tunnel and(c) the main cornerstation.

Detection of gravitational waves 1349

(c)

Figure 15. (Continued)

3.4. Laser interferometer detectors

Three long-baseline laser interferometer gravitational wave detectors have been or are beingconstructed. The US LIGO Laboratory consists of two 4 km arm length detectors, atHanford, Washington State, and Livingstone, Louisiana (http://www.ligo.caltech.edu). TheItalian/French VIRGO project is completing a 3 km arm length instrument at Casina,near Pisa (http://www.virgo.infn.it). Smaller detectors are under construction at Hannover(the German/British GEO project, 600 m arm length, http://www.geo600.uni-hannover.de),Tokyo (TAMA 300 m arm length, http://tamago.mtk.ano.ac.jp) and Perth (80 m arm length,http://www.gravity.pd.uwa.edu.au). Figure 15 shows some photographs of the LIGO detectorin Hanford: (a) an aerial view, (b) the vacuum pipe tunnel and (c) the main cornerstation.The long-baseline laser interferometer detectors are initially expected to achieve sensitivity asshown in figure 16.

Laser interferometers are complex instruments limited by a range of noise sources: internalthermal noise in the mirror test masses, seismic noise, radiation pressure noise, laser frequencynoise, control system noise, residual gas refraction noise etc. All noise sources must be reducedas far as possible to allow the detectors to achieve high sensitivity. The first decade of thetwenty-first century will see steady improvement of the detectors. Late in that decade it is likelythat major improvements to the detectors will be possible using improved lasers, improved testmasses and improved vibration isolation. Figure 16 shows the expected improvements.

Ultimately, both laser interferometers and resonant masses can be improved by usingvarious quantum measurement techniques. Laser interferometers can in principle also beimproved by use of cryogenic methods, and by increasing their arm length.

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Figure 16. Rough comparison of various detector technologies and some sources. Note that thesensitivity depends on the type of signal searched for: the resonant-mass sensitivities quoted referto burst sources (e.g. the narrow-band sensitivity of bars has already been demonstrated at 10−23).At low frequencies interacting white dwarf binaries (IWDB) and neutron star binaries are knownsources. At high frequency the only known source is neutron star binary coalescence: all the othershave unknown signal strength.

All gravitational wave detectors provide data dominated by noise. All face difficult dataprocessing challenges. In the following we shall briefly review some of the main issues inthe context of the various classes of gravitational wave sources, before going on to examineresonant-mass and laser interferometer detectors in more detail.

3.5. Issues of data processing and signal detection

There are many important and still unresolved issues of data processing which it is necessary tosolve to enable detection of signals to the levels anticipated from a simple analysis. These issuesaffect burst detection, stochastic background detection and narrow-band detection. Since thegeneral analysis principles are independent of the type of detector used, we will discuss someof them in generality.

The raw data from a detector must first be filtered. In section 2.2 we discussed the use ofmatched filters or optimal filters to extract particular signals. An optimal filter is one whichmaximizes the signal-to-noise power ratio for a particular signal waveform. The term matchedfilter arose because in the special case where the noise is white, the optimal filter is one withan impulse response which matches the shape of the input signal. In these ideal circumstancesthe matched filter is simply a template of the waveform one desires to detect.

Think of the data from a detector over a particular period of time, due to some inputsignal. In a perfect noise-free detector it would create an output signal s(t) which is uniquely

Detection of gravitational waves 1351

determined by incoming signal and the detector impulse response. After sampling, a realdetector output consists of a two-dimensional array of output values oi and time values ti :oi, ti. But oi = si + ni is a sum of the input signal si and the noise ni . The template is anoise-free array representing s(t), but it has an arbitrary phase, so can be represented sj , tj where the difference between i and j represents a time difference. When the template isaligned with the signal i = j and the sum of the products oisi(=s2

i + sini) over all valuesof i will be maximal. At times when si is zero there is no noise contribution. The templatewhich accurately matches the ideal signal response of the detector is therefore the optimal filterfunction.

In practice one could apply a large calibration signal and measure the output signal tocreate such an optimal template.

Often it is more convenient to consider the problem in frequency space. In frequencyspace the phases of the noise frequency components are random and uncorrelated. If youmultiply the Fourier transform of the output data O(f ) with the complex conjugate of theFourier transform of the signal waveform s∗(f ), the signal component will be positive definitewhile the noise phases remain random. Summing over frequency space, the signal will add upbut the noise frequency components will tend to cancel. Thus, in frequency space the matchedfilter transfer function is simply the complex conjugate of the Fourier transform of the inputsignal s∗(f ).

In almost all real situations the noise is not white. However, coloured noise can bewhitened by passing it through a filter with a transfer function equal to S

−1/2n where Sn is the

noise power spectral density of the noise. Thus, the optimal filter transfer function is simplys∗(f )/S

1/2n . In the time domain the same correction for coloured noise is made using the noise

autocorrelation function whose Fourier transform is Sn.All optimal filters require a sum over frequency or time. If the signal is transient, then the

sum will be zero after the signal has passed and the detector has stopped responding. For aresonant bar this will be several ring-down times of the bar (after which all memory of the signalis lost). In the same time before the signal arrives the bar loses memory of its instantaneousmechanical state. Thus it is only necessary to integrate over a modest time interval before andafter the signal arrives. The Louisiana State University (LSU) group have implemented a timedomain optimal filter by applying a large ‘signal’ pulse using a calibrator, and measuring thedetector response, as well as the detector noise in the absence of a signal. Other bar groupshave used frequency domain filters and obtained similar results [76].

There are many ways to implement optimal filters and the best choice often involvesminimization of computation requirements for the particular search. For some systems optimalfilters can be implemented in quasi-real time whereas for others the need for prior data requiresthe search to be conducted off-line.

A particular issue is that of noise stationarity. In practice, stationarity of the noise is not agood approximation. This means that the noise spectrum at the output varies with time, due,for example, to environmental effects such as variable microseismic noise from changes in theocean wave conditions [77]. To overcome slow changes in the detector noise distribution it ispossible to always use recent noise data for the creation of the optimum filter. Several groupsuse noise from the previous few hours to continually adapt the filter [78].

Another problem relates to the fact that in gravitational wave detectors it is impossible toturn off the signal, and possible signal-like noise events (such as cosmic ray events). Thus, inprinciple it is impossible to measure the noise spectrum in the absence of signal. In practicetoday this is not a critical issue (signals are rare and very small), but for future detectors it maybe important to ensure that the filter does not suppress signal by confusing it with noise. Henghas shown [79] that periodic transient bursts are indeed suppressed by an optimum filter if the

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bursts are present during the measurement of the noise distribution.For CW signals, a Fourier transform of the entire data set represents an optimal filter. By

this method the signal appears as a narrow spectral line. If the Earth’s orbital and rotationalDoppler corrections are included this method can be used to search for unknown CW signals(such as those due to isolated spinning neutron stars) in a particular direction (because theDoppler correction is direction dependent, see figure 30).

Traditionally the lock-in amplifier or phase-sensitive detector (PSD) has been used tocreate a matched filter operating in the time domain for CW or near-CW signals. For modernsystems this type of filter can be implemented by software (hence not in real time) whichallows it to be used for multiple searches through the same piece of data. It also allows CWsearches with arbitrarily long integration times. (The traditional analogue lock-in amplifier islimited by the charge storage time of a capacitor.)

The lock-in amplifier form of the matched filter multiplies the data by +1 during thepositive going signal cycle, and −1 during negative parts of the cycle. All results are summed.The sum represents the phase coherent integral of the absolute value of the signal over theobservation time: hence the alternative name for this method: coherent integration. The noisecomponents average towards zero, and a small signal component previously buried in the noisewill emerge.

Assuming that the phase of the incoming signal is unknown, it is necessary to repeat thematched filter at 90 phase shift to determine the magnitude of the orthogonal componentsof the signal. (For example, if the signal was a pure sine function, the matched filter wouldgive a zero output if the +1 and −1 multipliers were defined by a cosine function.) Clearly, ifthere were slowly varying phase errors due to errors in the timing of data acquisition, errors indirection for Doppler corrections, or errors in the prediction of the waveform, the accumulatedresult from a matched filter could average to zero.

For a CW source of fixed but unknown frequency it is usually simpler to replace matchedfiltering with the fast Fourier transform. However, if the frequency is modulated in a knownpattern (such as occurs when you search for a signal from a known binary pulsar) [80] thecoherent integration is computationally simpler.

Attempts at gravitational wave burst detection have normally followed techniques firstintroduced by Weber. By this method the data is first filtered, as discussed above, and thenthresholded to obtain a list of candidate events. Some events will be due to spurious effects(see below). These can be vetoed if the appropriate monitor channels are used. Then theevent lists for widely spaced pairs of detectors are correlated to search for coincidences. Ifthe time axes for the event lists are randomly displaced one expects all correlation due topossible gravitational wave bursts to be absent. Thus it is possible to compare true-timecoincidences with time-shifted coincidences to determine whether there is an excess of ‘zerotime delay coincidences’. This method is powerful since it allows the probability of accidentalcoincidence to be experimentally determined by simply running the random time shift analysisa sufficient number of times to obtain an accurate estimation of the probability of the observedpeak. However, it is only relevant when the number of accidental coincidences is large.

Consider, for example, a six-month coincidence run which might yield 5000 ‘candidateevents’ in each detector, most of which are assumed to be noise or interference. The coincidenceanalysis might yield 30 true-time coincidences. For such analysis with existing resonant barsthe coincidence window is usually more than 0.1 s, so that propagation delays across the Earth(∼40 ms) can be neglected. Then 1000 random time delays are applied to the data of onedetector and of these 10 might show 30 or more coincidences. If this was the case, then thezero time delay peak would have a probability that it was accidental of 10/1000 or 1%. Thisprobability could be resolved with more accuracy if more random time delays were used. In

Detection of gravitational waves 1353

this type of experiment the time delays which are chosen must either be small compared withtotal duration of the experiment, or they should be modulo the experiment duration so thatcoincidences are not lost at the ends of the record.

A serious pitfall can occur in this type of analysis if the threshold for candidate eventselection, or any other vetoing scheme, is variable during the coincidence experiment. To havesuch a variable accessible to the persons analysing the data can allow conscious or unconsciousselection which can completely invalidate the statistical significance analysis. It can be verydifficult to estimate the true significance after such selection has taken place. Since the daysof Weber, researchers have been in general very careful to pre-set all thresholds to try to avoidsuch pitfalls.

For the existing array of five resonant-mass detectors it has been suggested that the entirebody of data from all detectors should be used rather than candidate events. As discussedin the next section, the accidental coincidence rate is extremely low for five detectors. Thethresholding method means that all phase information is lost. However, it need not be lost ifthe data from separate detectors are recombined along the lines used by VLBI radioastronomy.If this was done the detectors could represent a single telescope with angular resolution forincoming bursts set by the ratio of the time resolution (in principle ∼100 µs) and the near-Earth-diameter baseline (40 ms). However, the noise increases exponentially as one digs tolower and lower energy so the overall amplitude sensitivity of the array would be limited to∼3 times.

Most data from terrestrial gravitational wave detectors can be idealized as the sum of apair of thermal distributions. The first is the intrinsic Boltzmann distribution of the detectornoise that one would measure using a spectrum analyser. This is usually due to some wellunderstood noise sources such as thermal noise, electronics noise or shot noise. The seconddistribution is described as excess noise. Excess noise arises from rare and poorly understoodsources. In the detector NIOBE at the University of Western Australia (UWA) some of theexcess noise was correlated with electromagnetic pulses and seismic noise [81]. However, themajority was not identified, but might be due to strain relief events. Surprisingly, the excessnoise distribution is rather similar for widely differing types of resonant-mass gravity wavedetector. Figure 17 shows the idealized form of these distributions. Both may be expressed as

N = N0e−Ee/kT . (3.1)

For a typical resonant-bar detector the intrinsic distribution parameters would be: N0 ∼ 105 perday, and Ee ∼ few millikelvin (Kelvin is simply a convenient energy scale). The excessdistribution typically would have N0 ∼ 10 per day, and Ee ∼ few 100 mK: that is, the excessnoise distribution has an event rate 104 times lower than the thermal distribution, and aneffective energy 100 times larger. The presence of these two distributions allow improved dataanalysis, as discussed in section 4.8.

Periodic signals can be detected by coherent integration or Fourier transform methods.Because, as always, signals are near to the limit of detection sensitivity, long integration isneeded. However, in long integrations signals will be smeared out by the Earth Doppler motionunless the source direction is known. Equally, if the source is a member of a binary systemsuch as a binary millisecond pulsar it is necessary to know the ephemeris of the system itselfas well as the precise source direction to prevent Doppler smearing of the signal. If coherentintegration can be achieved, then the signal-to-noise ratio improves as τ 1/2, which can allowvery deep searches over a 1–3 year period.

The nearest millisecond pulsar PSR 0437-4715 is a typical such potential source. In thiscase the pulsar source is a binary system, which is extremely well defined by pulsar timing [82].The observed radio pulse timing gives very accurate information which can be used to gate the

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0.001

0.01

0.1

1

10

100

1000

104

105

0 200 400 600 800 1000

Nu

mb

er o

f S

am

ple

s p

er D

ay

Energy (m illikelv in)

Figure 17. Detector noise is often characterized by a pair of Boltzmann distributions. The steepcurve is the typical antenna thermal noise distribution for burst detection in a resonant bar. Theflatter curve is the excess noise distribution due to rare disturbances.

phase-coherent integration of the detector output [80]. Such methods were pioneered by theTokyo group [83] led by Hirakawa searching for gravitational radiation from the Crab pulsar.

It is likely that many such narrow-band sources will not be detected in advance by radioastronomy. Pulsar beaming ensures that the majority of pulsars are not visible in radio, andnormal pulsars older than 108 years appear to cease to emit radio beams. There could easily be1000 rotating neutron stars within the range of the nearest observed pulsars (∼100 pc) meaningthat a few old neutron stars could exist within 10 pc of the solar system.

Unfortunately, our ability to search for such sources is very poor because of the difficultyof searching all directions in the sky. The most obvious search strategy involves applying aseparate ephemeris correction for each of typically 1010 source directions, and re-integrating thedata 1010 times. This already is a daunting computational exercise, but it becomes 1010 timesharder if the search space has to involve all possible binary orbits of the pulsar. Manyinvestigators are searching for efficient algorithms to solve this problem, based on alternativefiltering techniques or hierarchical searches [84].

As discuss in sections 2.2 and 2.8, stochastic signals can be detected by cross-correlationof nearby gravitational wave detectors. For optimum sensitivity the detectors must be locatedwithin about one reduced wavelength (λ/2) of each other. If the spacing requirementis satisfied, the signal-to-noise ratio increases as the 1

4 -power of the number of cyclesobserved [85]:

S

N=(

S2gw

S1S2Bτ

)1/4

. (3.2)

Here Sgw is the gravitational wave background power spectral density, S1 and S2 are the spectral

Detection of gravitational waves 1355

noise density of two detectors, B and τ are the bandwidth and duration of the observationrespectively. Thus, a stochastic signal of characteristic frequency ∼100 Hz can be cross-correlated to gain a factor of 102.5 in signal-to-noise ratio (compared with the observation of asingle cycle) after 108 s integration. Schutz [85] has shown that this method can be particularlyeffectively applied to signals in detectors of quite different types: specifically resonant-massdetectors and laser interferometers, which conveniently are planned to be located in sufficientlyclose proximity to each other to satisfy the above spacing requirement.

The simplest cross-correlation experiment merely gives an output consisting of a singlenumber (and a measure of its statistical significance). This can be quite misleading since thereare many ways that cross-correlation can give a false positive result. The major difficulty incross-correlation signal processing is to ensure that no correlated technical noise componentsexist in the signals. Remembering that the signal detected is generally going to be at least100 times lower than the spectral noise floor accessible using normal fast Fourier transformdiagnostic instrumentation, correlated features can exist which could never be detected inshort-term monitoring. The correlated features could be weak spectral lines such as thosecreated by electrical power harmonics, (which are phase coherent over the electrical grid) oroccasional transients such as those due to lightning flashes. To prove that a positive cross-correlation signal was associated with gravitational waves it would be necessary to showthat the correlation was preserved across the accessible frequency band, and that it was notdue to intermittent transients due to electrical interference. The individual output spectra ofthe detectors would need to be resolved for weak spectral features to the same depth as thecorrelation detection threshold. Much effort at developing algorithms and solving some ofthese practical problems is underway [53, 86].

Giazotto [87] has shown that the stochastic background signal from the combined oldradio pulsars in our Galaxy should be detectable in a single detector, due to the fact that theyare non-isotropically distributed relative to the solar system. The central concentration ofpulsars means that there should be a strong sideband modulation of the stochastic backgroundintensity as the detector sweeps the sky during Earth rotation. This signal should appear in thetechnically very demanding 1–10 Hz frequency band.

The next two sections present resonant-mass detectors (section 4), and then laserinterferometer detectors (section 5), with emphasis on techniques and the solution to varioustechnological challenges.

4. Resonant-mass detectors

4.1. Introduction

Resonant-mass detectors are designed to measure acoustic signals induced in a large mass dueto its coupling to a gravitational wave. Resonant detectors were first developed by Weber dur-ing the 1960s [4]. They consisted of large vibration-isolated aluminium cylinders instrumentedwith piezoelectric crystals glued on the surface near to the centre. A low-noise amplifier andlock-in amplifier allowed detection of the energy of the fundamental longitudinal resonance ofthe bar. A gravitational wave applies a time-varying quadrupole deformation and does mechan-ical work on the bar. The absorption cross section of the bar to gravitational waves depends onits mass and sound velocity. The cross section is highest at the fundamental resonant frequency.The latter is linked to its length and sound velocity, since its length must be half an acousticwavelength at the fundamental longitudinal resonance. Weber chose aluminium because ofits high sound velocity and availability in large pieces, and because it has quite low acousticlosses. Following Weber, many new resonant-mass detectors using similar techniques, but with

1356 L Ju et al

variations and improvements, were developed in the early 1970s. Following null results thesewere abandoned, but advanced resonant-mass detectors using cryogenics and superconductiv-ity continued to be developed. Fairbank et al [88] and Hamilton et al [89] first proposed suchcryogenic detectors, and proposed cooling to millikelvin temperatures to minimize thermalnoise. Today two detectors in Italy are in operation at temperatures below 100 mK.

In all resonant-mass detectors the large amplitude of thermal vibration considerablyexceeds the gravitational wave amplitude expected from astrophysical sources. Withoutmethods to suppress this noise the principle of detection by resonant masses would beimpossible. Weber’s key contribution was the realization that in a high-Q antenna—onewith a low acoustic loss—the effective noise energy is reduced by a factor ∼τi/τa , whereτi is the effective measurement integration time, and τa is the antenna ring-down time. Theadvantage of using a low acoustic loss antenna follows directly from the fluctuation–dissipationtheorem [90]: the greater the dissipation the greater the fluctuations or noise level imposedby the thermal reservoir. A high-Q antenna approaches an ideal harmonic oscillator, whosemotion is exactly predictable at a time in the future from the observed amplitude, frequencyand phase at an earlier time. High levels of predictability means that very small deviations fromsinusoidal behaviour can be resolved given a sufficiently sensitive transducer for monitoringits motion.

4.2. Intrinsic noise in resonant-mass antennas

To understand the operation of a resonant-mass gravitational wave detector it is convenientto start with an old-fashioned approach first introduced by Gibbons and Hawking [91]. Thisapproach is intuitively obvious but is not consistent with the optimal filter theory discussed insection 3. The instantaneous state of the antenna can be described by the pair of symmetricalharmonic oscillator coordinates X1 and X2 given by

X1 = A cosφ

X2 = A sin φ,(4.1)

where A is the antenna amplitude and φ is the phase. Experimentally, X1 and X2 can beeasily measured using two lock-in amplifiers or PSDs in a configuration shown schematicallyin figure 18. They may be analogue or digital or software devices. The state of the antenna canbe represented by a point P1 in the (X1, X2) plane. The amplitude A = |P | = (X2

1 + X22)

1/2

and phase φ = tan−1 X2/X1. This is illustrated in figure 19. A gravitational wave causes theantenna to move from P1 to P2. The direction of this motion depends on the relative phase ofthe gravitational wave and the resonant mass. To extract a signal the measuring system shouldmonitor|P1−P2| = (X2

1 −X22)

1/2. The quantity2 is described as the energy innovationand its magnitude, properly calibrated and expressed in the units of Kelvin (1.38 × 10−23 J)describes the effective temperature of the antenna.

In a noiseless antenna the motions of the vectorP would only be due to gravitational waves,but in practice P is driven by thermal fluctuations in the bar. Thermal fluctuations cause thestate vector P to execute a random walk in the X1X2 plane. A high-Q mode is weakly coupledto the thermal reservoir which is made up of all the higher modes of the system. The antennaloses energy slowly into the reservoir, and equally it is only weakly excited by the reservoir.The relaxation time τa = 2Q/ωa thus characterizes both the rate of decay after a high-energyexcitation and the rate of amplitude change when the mode is in thermal equilibrium with thereservoir.

Clearly, if τa is large and the rate of fluctuation is low, the antenna becomes moredeterministic on time scales that are short compared with τa . The mean energy is still kT ,

Detection of gravitational waves 1357

Bar Transducer

Amplifier

PSD

PSD

Reference oscillater0ß90ß

ωa

ωa

X1

X2

Figure 18. Antenna readout systems for obtaining harmonic oscillator coordinates X1 and X2.The down-conversion with the PSD was conventionally done with analogue electronics but todaycan be achieved in software using fast digital sampling [78].

AAAAAA

AAAAAA

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

AAAA

AAAA

AAAAAA

AA

X2

X1

P1P2

Figure 19. X1–X2 representation of the state of the antenna.

but the expected change in energy in a sampling time τi is kT (τi/τa). The temperatureT (τi/τa) is the effective temperature or noise temperature of the resonator, and quite clearlycan be made less than the actual temperature. Indeed, even when transducer readout noise isincluded, a noise temperature of less than 2×10−4 of the thermodynamic temperature has beendemonstrated in the detector NIOBE. To use temperature to describe the antenna noise impliesthat the distribution of 2 has a Boltzmann distribution. This is an excellent approximation(see figures 17 and 28). The slope of the distribution as well as its mean value gives the antennanoise temperature.

The above analysis describes a predictive filter for the detection of short bursts. In thiscase our prediction is that the amplitude and phase of the detector will remain unchanged overthe integration time. Today all operating detectors use optimum filters for the detection ofshort bursts. The optimum filter improves on the simple predictor discussed above because itimplicitly takes into account the dynamics of the system for times of the order of the antennaring-down time over which the motion is correlated. They are implemented as discussed insection 3 and enable the antenna noise temperature to be improved by a small factor. The mostpopular optimum filter or matched filter is the Weiner–Kolmogoroff filter which is designed todetect short delta function bursts [92]. Today, such filters are routinely used since they wereshown by Pizzella et al [93] to give substantially improved performance.

1358 L Ju et al

Every antenna must use a sensitive transducer to read out the motion. The transducer, likeall electronic devices, has a residual broadband noise floor. This noise floor is equivalent tothe Johnson or Nyquist noise of a resistor, given by V 2

n = 4kT RB, where R is the effectiveresistance of the transducer and B is the measurement bandwidth. However, the bandwidth isroughly the reciprocal of the measurement integration time τi . Thus it follows that the noisepower is 4kT /τi . Comparing this with the Brownian motion noise, we see that one noisesource varies as τi while the other varies as 1/τi . Thus there is clearly an optimum bandwidthset by the value of τi that minimizes the total noise.

Gibbons and Hawking [91] introduced a parameter β to characterize the coupling betweenthe bar and the transducer. They defined β as the proportion of the elastic energy of thedetector that can be extracted electrically through the transducer in one cycle. A bar–transducersystem with low β (weak coupling) requires more time for the signal energy to appear in thetransducer. The longer the energy transfer takes, the more time there is for fluctuations inthe antenna to dominate the noise. This point can be clarified by two alternative viewpoints.One is a thermodynamical model. The antenna is considered as a thermal bath at temperatureTeff = Taτi/τa , coupled to a transducer with noise temperature Tt which itself is coupled to anamplifier of noise temperature Tn.

A gravitational wave causes slight ‘heating’ of the fundamental mode and energy flowsthrough the coupling β. As long as β > 0 the transducer will eventually come into equilibriumwith the bar, but for a rapid response β has to be large. The thermodynamic approachemphasizes that the coupling is not unidirectional: thermal fluctuations in the amplifier orthe transducer act back on the antenna producing back-reaction noise. Indeed, it is clear thatthe transducer is a source of thermal fluctuations comparable to those originating within the bar.Voltage noise in the transducer will convert into force noise acting on the bar. Like the forceswhich act on the bar from the thermal reservoir, the back reaction will produce an additionalnoise contribution which will diminish as τi reduces to zero, as does the Brownian noise.

The second viewpoint is that the antenna–transducer system is effectively a transmissionline designed to couple energy into the transducer. One can think in terms of phonons in thebar which may be absorbed by the transducer, with the emission of a photon into the amplifier,or they may be reflected back into the bar. Then β determines the impedance match betweenthe output impedance of the bar, Zout, and the transducer’s mechanical input impedance Zll .The ratio Zll/Zout is simply the coupling coefficient β. See below for more discussion of thispoint.

Once we begin to think in terms of quanta we are led to ask: what happens if the inducedstrain in the antenna is equivalent to less than one quantum hωa? The profound significanceof the quantum mechanical limit to macroscopic measurements was realized independently byseveral groups, particularly by Braginsky [94] and Giffard [95]. Giffard used the much earlierresult of Heffner [96] who showed that, by the uncertainty principle, a linear amplifier has afundamental limit to its sensitivity, given approximately by hωa . Similarly, Giffard showed thata transducer used with a linear amplifier (an amplifier which preserves phase and changes theamplitude by a multiplicative factor) has a maximum sensitivity corresponding to a gravitationalwave which produces an equivalent of two quanta in the bar. The term equivalent is usedbecause the actual energy absorbed by the antenna depends on its instantaneous amplitude.For linear systems the signal-to-noise ratio is independent of the amplitude, and correspondsexactly to the signal produced in an ideal stationary antenna at absolute zero. Giffard’s resultmeant that the maximum achievable sensitivity of an antenna would be limited to about thesingle phonon level corresponding to a strain sensitivity ∼10−21. This is described as thestandard quantum limit.

Detection of gravitational waves 1359

Meanwhile, at least as early as 1974, Braginsky and Vorontsov proposed that in principleit might be possible to devise quantum non-demolition devices which could read out the stateof a system without disturbing it. Braginsky et al [97], Caves et al [98], Unruh [99] and otherswent on to identify methods whereby gravitational waves of amplitude less than that requiredto induce one quantum can in principle be detected using quantum non-demolition or back-action-evading techniques. The possibility of mechanical measurements down to and belowthe quantum limit in sapphire bars has been investigated in detail by Tobar et al [100]. Themost promising technique is through the use of amplitude-modulated parametric transducers.This represents a small elaboration of parametric transducers of the type we will discuss insection 4.4. The pump signal is amplitude modulated at the signal frequency to create anintrinsically phase-sensitive measuring system. However, classical noise sources need to bebeaten down close to the quantum limit before such techniques can successfully pass thequantum limit on real antennas.

4.3. The signal-to-noise ratio for burst detection

Resonant-mass detectors may be used to detect all of the signal classes discussed in section 4.2.However, most effort has concentrated on the detection of bursts. In general, when agravitational wave in the right frequency range arrives, it excites all normal modes of thebar that have a high quadrupole moment. A transducer attached to the bar will pick up thesignal, which must be discerned in the presence of transducer noise and a large Brownianmotion background, as discussed in the previous section.

The efficiency of the detector is determined by the fraction of the wave energy absorbedand converted into acoustic energy inside the bar. Clearly, it is important that the bar absorbas much as possible of the energy of the passing gravitational wave. This can be quantified interms of the antenna cross section as discussed below. Denoting the incident spectral energydensity of gravitational waves as w(f ), the energy deposited in the bar is given by

εg = σw(f ). (4.2)

The term σ in the above equation is the so-called cross section of a bar, which is the ratio of theabsorbed energy to the incoming energy, and thus a measure of the sensitivity of the bar. Thecross section is actually a function of frequency σ(f ) because the detector will absorb energymore readily around the resonant frequency fo of the antenna. The total energy deposited inthe bar is then

E =∫

σ(f )w(f ) df. (4.3)

Since for a high-Q system, σ(f ) is sharply peaked around the resonant frequency f0, only anarrow portion of the gravitational wave signal around the resonant frequency of the bar canbe picked up by the detector. In this case we may write

σ(f )w(f ) df = w(f0)

∫σ(f ) df. (4.4)

The cross section, first elaborated by Weber, can be expressed in several forms [7]. The crosssection of the bar, integrated over the frequency band, can be expressed as [101]∫

σ(f ) df = 8GM

πc

(vsc

)2m2 Hz, (4.5)

where vs is the sound velocity in the bar, and M is the mass of the bar. Clearly, to obtainhigh sensitivity it is desirable to build detectors as massive as possible and from a materialhaving a sound velocity as high as possible. Usually, the size of the bar is chosen such that

1360 L Ju et al

the fundamental mode is at about 1 kHz—about the expected frequency for the collapse of amassive stellar core to a black hole. Aluminium has been used for most resonant bars. Niobiumis used for the bar detector at the University of Western Australia. Other materials could givesubstantial advantages, as discussed in section 4.5 below.

A gravitational wave carries an energy flux S(J m−2 s−l) given by

S = c3

16πG〈h2

+ + h2x〉, (4.6)

where h+ and h× denote the dimensionless strain amplitudes of the two possible polarizationsof the wave. Since the shape of expected gravitational wave pulses from gravitational collapseevents is not accurately known, we cannot accurately determine the expected excitation of anantenna even knowing the total pulse energy. We need to know both the spectral distributionof the pulse energy, and the relationship between h and its time derivative. The details of theexpected pulses depend not only on the dynamics of the gravitational collapse, but also on themass of the collapsing object, both of which are uncertain.

If we assume only knowledge of the pulse duration τg (expected to be ∼10−3 s), andthat it is predominantly a single cycle, it is sufficient to assume that dh/dt ∼ 2h/τg . Thenequation (4.6) can be rewritten

S = c3

16πG

4h2

τ 2g

. (4.7)

The total pulse energy E is then given by S · τg:

E = c3

16πG

4h2

t2g

. (4.8)

If we assume that the spectral distribution of the pulse energy F(ω) is uniform over abandwidth ω ∼ 1/τg , it follows that

F(ω) ∼ E/ω,= Eτg ∼ c3h/4πG J m−2 Hz−l. (4.9)

Numerically F(ω) ∼ 20 × 1034 h2.The assumption used in obtaining the result must be emphasized: the result is simply an

order-of-magnitude estimate of the expected signal spectral densities. Moreover, variationsin the pulse durations could make any chosen antenna frequency only suitable for a smallproportion of actual events.

The energy deposited in an initially stationary antenna of mass M by a signal pulse F(ω)

follows directly from equation (4.5) combined with geometrical terms:

Us ∼ F(ω) sin2 θ cos2 2φ8

π

G

c

(vsc

)2M (4.10)

where θ and φ are coordinates describing the orientation of the bar relative to the incomingwave (as given in figure 20).

For a short burst of gravitational waves the bandwidth of the pulse is roughly the inverseof the pulse duration which is roughly equal to the peak frequency. Under these circumstancesthe strain amplitude δ1/1 induced in the bar is roughly equal to the incoming wave amplitudeh; there is no resonant excitation.

The incoming gravitational wave will only be detectable if the signal Us is greater than thenoise in the antenna Un. From an engineering point of view it is useful to characterize the noiseUn; we generalize the transducer to a two-port device described by a 2 × 2 impedance matrixZij . The transducer accepts force and velocity inputs F and v, giving current and voltageoutputs I and V :(

F

V

)=(Z11 Z12

Z21 Z22

)(v

l

). (4.11)

Detection of gravitational waves 1361

AAAAAA

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

AAAAA

φ

θ

x, y ,z

Figure 20. Coordinate system for resonant antenna.

The transducer has input impedance Zll , measured in kg s−1, and output impedance Z22,measured in ohms. The forward transductance Z21, measured in V m−1 s, determines thetransducer sensitivity whereas the reverse transductance Z12, measured in kg A−l, determinesthe back-acting force on the antenna due to currents in the output circuit: see [100, 102–104]for more details. Quantum mechanics tells us that it is impossible to make Z12 = 0: it isimpossible to create a perfect one-way valve.

All the noise sources in the transducer and amplifier can be expressed as equivalent spectraldensities of current and voltage noise at the input of the amplifier, denoted Si(ω) and Se(ω),respectively, as illustrated in figure 21. The current noise Si is the source of back-action noisein the antenna, whereas Se describes the series noise contribution. In terms of these quantitiesthe total system noise is given by

Un = 2kTa

τi

τa+

|Z12|22M

Si(ω)τi +2M

|Z21|2Se(ω)

τi. (4.12)

The first term in equation (4.12) is the Brownian motion or thermal noise in the antennadiscussed above. The second term describes the energy fluctuations arising from the currentnoise acting back through the reverse transductance, and giving the back-action noise, alsoproportional to integration time. The third term is the series electronics noise, which for givenSe is reduced as Z21 increases. As we saw above this term is proportional to the bandwidthτ−1I . Only the first term in equation (4.12) can ever be reduced below the quantum limit.

The problem of detecting gravitational waves with resonant-bar antennas to a large extentconsists of minimizing equation (4.12). The technical means of achieving this requires someor all of the following:

(a) Reduce the antenna temperature Ta .(b) Use a transducer with high Z21 and low Z12.(c) Use amplifiers with Se and Si as low as possible.

1362 L Ju et al

M, TA

Ta ωa

F

v

Z11 Z12

Z21 Z22

Se

SiG

V

Iτi

Bar Transducer Amplifier Recorder

Figure 21. The various quantities used to characterize a gravitational wave antenna system.

(d) Reduce the acoustic loss in the antenna as well as the acoustic and electrical losses in thetransducer to obtain high-Q or large relaxation time.

(e) Obtain a reasonable impedance match between antenna and transducer to enable τi to besmall.

The last requirement can only be achieved by good impedance matching between themechanical input impedance of the transducer and the mechanical output impedance of theantenna, which we shall discuss further below.

It is convenient to scale the noise in the system relative to the standard quantum limit ofone equivalent quantum induced in the bar. To do this we rewrite the noise equation (4.12) interms of noise number A (a quantity first used by Weber to characterize noise in masers):

A = Un/hωa = AT + AB + AS. (4.13)

Here AT , AB and AS are the equivalent numbers of noise quanta due to thermal noise, back-reaction noise and series noise in the measurement system. The experimentalists need toachieve a total system noise number A approaching unity. For a 1 kHz resonant bar thiscorresponds to a noise temperature of ∼0.1 µK. To achieve this it is necessary not only to havea low-noise transducer, but also to use a low-acoustic-loss antenna material, and to suspend andisolate the antenna so as not to increase the acoustic loss, nor to couple in excess noise fromthe environment. Because the noise number contributions are additive there is no particularadvantage in reducing one of them far below the others.

It is useful to express the thermal and transducer noise contributions directly in terms ofthe gravitational wave strain equivalent noise. The Brownian motion noise hB is given by [105]

hB ∼(

kBT τi

Mωal2Q

)1/2

. (4.14)

For a bar detector, ωal = πvs , so the Brownian noise is given by

hB ∼(

kBT τiωa

π2Mv2s Q

)1/2

. (4.15)

Expressing the above equation numerically, we have

hB = 10−21

[(f0

1 kHz

)(1010 J

Mv2s

)(109

Q

)(T

0.1 K

)( τi

10−2 s

)]1/2

. (4.16)

This means that a resonant-mass detector with the hypothetical parameters implied in theabove equation will have a Brownian motion noise strain of about 10−21. However, some ofthe parameters given above are difficult to achieve in practice. For example, the energy term issatisfied by about 10 tonnes of sapphire or 100 tonnes of bronze or niobium. In the case of theUWA niobium bar with mass of 1.5 × 103 kg, resonant frequency of 700 Hz, sound velocityof 3.4 × 103 m s−1, temperature of 4 K, Q-factor of 3 × 107 and measurement integrationtime 1 s, the Brownian noise strain amplitude is 2.2 × 10−19. It can be seen that the bar must

Detection of gravitational waves 1363

have a very high acoustic quality factor or much shorter integration time to reduce the effectof the thermal noise. With today’s multimode impedance matching techniques, τi can indeedbe reduced to 10−2 s.

The sensor noise lower limit (series noise and back-reaction noise AB + AS) is set by thestandard quantum limit [105]

hSQL ∼(

2hωa

π2Mv2s

)1/2

∼ 1.1 × 10−21

(fm

1 kHz

)1/2 (103 kg

M

)1/2 (104 m s−1

v2s

). (4.17)

This sets a strain amplitude limit ∼10−22 for a 100 tonne resonant-mass detector such as theproposed spherical detectors.

4.4. Transducers

Transducers for resonant-mass gravitational wave antennas fall into two categories: passivetransducers and parametric transducers. Passive transducers have no external power source,and their power gain is less than unity. They must always be used with a high-gain, low-noiseamplifier at the frequency of the antenna. Parametric transducers, on the other hand, have anexternal power source (a pump oscillator at frequency ωp) which is modulated by the antennamotion. They have intrinsic power gain associated with the transfer from the antenna frequencyωa to the higher frequency ωp. A laser interferometer is a parametric transducer operating atan optical pump frequency. Parametric transducers for resonant-mass readouts may be optical,microwave or radiofrequency devices.

Most parametric transducers use a high-frequency resonator combined with a low-noisehigh-frequency amplifier. Passive transducers use an inductive or capacitive readout, coupledto a SQUID amplifier. Figure 22 illustrates their basic structure. The passive transducerillustrated uses a superconducting inductor whose inductance (if constructed in a planarfashion) is proportional to the gap spacing between the coil and the superconducting groundplane on the antenna. Relative motion modulates persistent current trapped in the inductor(since the magnetic flux LI must be conserved). The changing current is detected by a SQUIDmagnetometer. The parametric transducer illustrated uses a capacitor in a resonant circuit.The capacitance is modulated by the gap spacing between the capacitor and the antenna. Thechange in capacitance due to motion modulates the resonant frequency of the circuit, creatingmodulation sidebands on the output signal. Both types of transducer may use capacitive orinductive sensing.

Fundamentally, there is little difference between passive and active transducers. Activetransducers use a transduction process that is combined with power amplification but additionalamplification of the high-frequency signal is usually necessary. Passive transducers have acomplete separation between the transduction process and the amplification process. However,the amplifier itself (such as a SQUID) makes intrinsic use of a parametric up-conversionprocess. Thus the difference between passive and active transducers is simply in the choiceof whether the parametric up-conversion occurs during or after transduction. In the case ofan optical pump frequency, amplification is unnecessary: the entire power gain is realizedthrough the up-conversion of the signal frequency to the optical frequency.

One important difference between passive and parametric transducers is in the transducerimpedance mismatch ratio or coupling factor β. For the parametric transducer

βpara1

2

CV 2pQe

mω2ax

2. (4.18)

In the limit Qe > ωp/ωa , the electrical Q-factor of the transducer resonator Qe is replaced bythe frequency ratio ωp/ωa .

1364 L Ju et al

Bar

C

PumpOscillator

Modulated Output

(a)

(b)

Persistent Current 1

SQUID

Output ωa

superconducting surface

Figure 22. Inductance or capacitance is modulated by a gap spacing. (a) Active or parametrictransducer use a low-loss resonant circuit pumped by an external oscillator. (b) Passive transducersuse an inductive or capacitive readout, coupled to a SQUID amplifier. All the circuits are madefrom superconducting and very-low-loss components.

For the passive inductive transducer

βpass = 1

2

LI 2

mω2ax

2. (4.19)

Note that the passive transducer coupling factor is not enhanced by a Q-factor term. For acapacitive passive transducer the inductive stored energy 1

2LI2 is replaced by the capacitive

stored energy 12CV 2. The parametric transducer effectively samples the incoming signal Qe

times per cycle up to a maximum value of ωp/ωa , and therefore increases its coupling by thesame factor.

Parametric coupling is reactive as a result of the position-dependent mechanical forceswhich act across the electrical resonator. The mechanical forces vary strongly over thetransducer position bandwidth, defined as the halfwidth of the electrical resonator, measuredin terms of displacement. Typically the position bandwidth is ∼pm (10−12 m). The storedelectrical energy exerts forces across the capacitance which vary strongly over the position

Detection of gravitational waves 1365

bandwidth. Hence the effective spring constant can be very large, and due to its reactive naturecan create problems in maintaining stability.

The passive transducer does not have the same coupling advantage. However, theadvantage is to some extent illusory because by moving the coupling structure to a highfrequency one reduces its size, so that the absolute value of the L or C is significantly reduced.

The practical problems of the two types of transducer are quite different. Passivetransducers are limited by a poorly understood problem of AC losses in their superconductingcircuits, and the performance of available SQUID amplifiers. Parametric transducers, onthe other hand, are limited by phase noise in the pump oscillator, tuning difficulties, noisedegradation in amplifiers at high microwave power, and sometimes by the effects of low-frequency seismic noise. All transducers, both active and passive, are limited in noiseperformance by the noise of the amplifier with which they are used. In terms of noise numberequation (4.13), at 1 kHz a noise number of 1 corresponds to a noise temperature of about100 nK, whereas at 10GHz the same performance corresponds to Tn ∼ 1K. In principle, bothtypes of transducer can reach close to the quantum limit. In practice, none have reached thislevel to date, although SQUID amplifiers have been developed close to 1µK, and amplifiers formicrowave parametric transducers have long been available with Tn ∼ 10K. The microwaveparametric transducer on NIOBE at UWA has achieved about 1 mK noise temperature, whilethe SQUID transducer on NAUTILUS has achieved similar noise performance. It is not obviouswhich type of transducer will ultimately be the most successful.

Johnson and Bocko [106–108] and Tobar et al [100] have presented designs for quantum-limited microwave transducers, while Richard and the Legnaro group have presented designfor optical transducers [109,110]. See [94,98,111,112] for further discussion of these issues.

A sensitive transducer and a low-loss resonant mass are not sufficient to create a sensitivegravitational wave antenna. There is a major problem at the interface: mechanical impedancematching. The impedance mismatch ratio (introduced as the coupling factor β) arises becausethe mechanical output impedance of the bar is very high, characterized by the elastic stiffnessof the bar itself, whereas the mechanical impedance of the electric or magnetic field whichcouples this motion into the transducer is not large. The solution is to create an acoustictransformer at the end of the bar. Such a transformer is analogous to acoustic horns used inloudspeakers, or to the mechanisms in the human ear that couple the motion of the air into thefluid of our cochlea.

All successful impedance matching schemes have consisted of low-mass secondaryacoustic resonators tuned to the antenna frequency. This creates a two-mode resonator witha pair of normal modes. The acoustic energy beats between the high-mass resonator and thelow-mass resonator, while the transducer is coupled to the low mass. The scheme can begeneralized to multimode transformers, consisting of nested sets of resonators reducing inmass by a geometric progression.

Three secondary resonator configurations have been used successfully on antennas:diaphragms (first developed by Paik at Stanford [113]), mushrooms (developed by the Romegroup [114]) and bending flaps (developed at UWA [115]). Pang and Richard [109], Hamiltonet al [116] and Tobar [63] have proposed and tested 3–5 mode systems but these have yetto be implemented. Four of the systems are illustrated in figure 23. The antenna NIOBEuses a 400 g bending flap (figure 23(a)). The bending flap is a convenient form of secondaryresonator, which has open geometry suitable for an attachment of a microwave re-entry cavityparametric transducer readout. The microwave readout system consists of a carrier suppressioninterferometer, and a microwave amplifier followed by a demodulation stage. This achievesreasonably high coupling to the microwave transducer, as demonstrated by the fact that theelastic forces provided by the transducer are sufficient to detune the mechanical oscillator by

1366 L Ju et al

φ

φ

φ

Nb bar

primarymechanical

α

microstripantenas

re-entrantca vitytransducer

cryogeniccirculator

AM noisereduction system

low noiseSLOCSCoscillatormixed with aHP 8662Asynthesizer(9.5GHz)low pass

filter

outputsignal

roomtemperaturelow-noiseamplifier

cryogeniclow-noiseamplifier

3dBHybridTEE

attenuatorphaseshifter

phasetracking

quadraturechannel

frequenc ytrackingin-phase

channel

carriersuppressioninterferometer

bending flapsecondarymechanicaloscillator

(a)

Secondary Resonator("mushroom") andTransducer

Pickup Coil

DC SQUID(Amplifier.Its output isproportionalto the motion ofthe mushroom)

(b)

Figure 23. Bar antenna transducer readout systems. (a) Bending flap [115]; (b) mushroom [114];(c) diaphragm [113]; (d) multimode transducer [116].

several Hz. The following three configurations, (b) the mushroom, (c) the diaphragm and (d)a multimode transducer, use similar SQUID amplifier readout circuits. The moving massmodulates the inductance of a flat coil, which by flux conservation leads to a modulated currentthrough a SQUID amplifier. The Rome group has used capacitive readout for a mushroom, inwhich case currents are induced by charge conservation in capacitor. The multimode transduceruses a massive ‘diaphragm’ coupled to a small tertiary mass plate supported by small niobiumcantilever springs.

4.5. Antenna materials

An ideal resonant bar would consist of a piece of nuclear matter, with high density and a velocityof sound comparable to the velocity of light! Since this is not available except in neutron stars,we must find a form of molecular matter which, to maximize coupling to gravitational waves,combines high velocity of sound vs , and high density ρ. To reduce the thermal noise we requirea low acoustic loss Q−1.

For an antenna limited by thermal noise the best antenna material (at a practical frequency)will have the largest value ofQρv3

s . This quantity is proportional to the ratio of energy absorbed(∼ρv3

s ) and the thermal noise in the antenna (∼Q−1). Of the three controlling parameters, onlythe Q-factor can be modified significantly in a particular material, depending on its preparation

Detection of gravitational waves 1367

Aluminium antenna

Niobium Coils

Niobium Diaphragm

HeatSwitch

HeatSwitch

SQUIDAmplifierinput coil

Pair ofpick-upcoils

Currentsupply

resonantsuperconductingdiaphragm

(c)

(d)

Figure 23. (Continued)

and suspension.Table 2 lists the values of ρ, vs and ρv3

s for various materials, along with the maximumachieved Q-value to date, and the signal-to-noise ratio figure of merit, Qρv3

s . The tableshows that nearly one order of magnitude improvement is obtained (at a given frequency) inρv3

s by changing from aluminium or niobium to sapphire, and when the Q-factor is includedthe very low losses in sapphire make it about 500 times superior to Nb or Al (at a givenoperating temperature). Silicon is more than 100 times better than Nb and Al. Unfortunately,at present silicon and sapphire are not available in sufficiently large masses for these apparentadvantages to be useful. Note that a lowerQ-factor can always be compensated for by sufficient

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Table 2. Comparison between antenna materials.

ρv3s Qρv3

s

Material ρ(g cm−3) vs(km s−1) Q (1013 kg s−3) (1020 kg s−3)

Aluminium 2.7 5.1 5 × 106 36 186061Aluminium 2.7 5.1 7 × 107 36 2505056Niobium 8.57 3.4 2.3 × 108 34 800Silicon 2.33 8.5 2 × 109 140 2.8 × 104

Sapphire 3.98 9.4 3 × 109 330 105

Lead 11.36 1.1 1.5Tungsten 18.8 4.3 150Copper(94)/Al(6) 8.0 4.6 2 × 107 77.8 155

cooling, so that fundamentally only theρv3s term need be considered. Copper-based alloys have

been selected as preferred materials for very-high-mass spherical antennas, chiefly becausesuperconducting materials (for which the thermal conductivity becomes very low) are verydifficult to cool to ultralow temperatures.

For comparison, table 2 also shows lead and tungsten. Lead is very poor, because of itslow sound velocity, whereas tungsten is comparable to silicon. If massive high-Q tungstenmasses could be obtained, they would have the significant advantage that the cryogenic systemnecessary to house the antenna would be smaller (and cheaper and simpler) than that neededfor lower-density materials.

4.6. Antenna suspension and isolation systems

Typical seismic noise has the spectrum of xs = αf −2 m Hz− 12 in the frequency range 1 Hz to

a few kilohertz, where f is the frequency and α is a constant. Measurements by gravitationalwave research groups at various sites have shown that the constant varies between 10−6 and10−9 (e.g. [117–119]). A vibration isolation system is needed to isolate seismic noise to wellbelow the signal level at the antenna resonant frequency. A variety of metallic suspensions havebeen developed for resonant gravitational wave detectors (e.g. [120–122]). All are designed tohave the normal-mode resonant frequencies of the isolator well below the antenna frequency,and the internal modes of the isolator elements above the frequency range of interest. Ingeneral, the normal modes define a set of low-frequency resonances. Internal modes of themass and spring elements are generally at high audio frequencies. Such isolators thereforehave good isolation above a low-frequency corner, and also below the high-frequency internalresonances. For resonant-bar antennas, the isolation band required is from a few hundred hertzto a few kilohertz. In principle, it is fairly easy to realize a mechanical isolator which willattenuate the seismic motion at 1 kHz (∼10−12 m Hz) by 1010 (e.g. [121, 122]). However,practical problems such as nonlinear up-conversion can degrade the performance [123].

Figures 24 and 25 illustrate two of the resonant-bar antennas constructed to date. Forantennas operated at 4 K the bar is supported by a low-loss multistage suspension in anexperimental chamber with which the antenna has no direct contact. A room temperaturevibration isolation stage suspends the cryogenic suspension stage. In the case of NIOBE,vibration can only act on the antenna by traversing the entire 18 stages of vibration isolation,or through transmission through the residual gas in the experimental vacuum. There areno wires connecting to the antenna (which can transmit vibration) because the transducer is

Detection of gravitational waves 1369

Intermediate Mass

Liquid Helium

Niobium Bar

MicrowaveElectronics

Transducer

Conning TowerTi alloy suspesion rodLead/Rubber vibration isolationNon-contacting radiative heat shuntBellows to decouple the dewar from the antenna suspensionAntenna suspension supportsExperimental tank suspension tube

Experimental tank

Liquid nitrogen shield30 K shield

Cryogenic cantilever suspension

Interface for electrical leads, vacuum lines and cryogenic liquids

Figure 24. Cross section of the NIOBE antenna. The cryostat is 5 m long and 3 m high.

Figure 25. View of the ultracryogenic antenna AURIGA showing cryogenic shields and the Albar.

interrogated entirely by radiative contact. In the case of antennas using passive transducerssuperconducting cables can act as transmission lines to conduct vibration, so that great caremust be taken to isolate cables using Taber isolators [124], consisting of additional mass springstages (using thin wires as springs) to which the cables are bonded.

1370 L Ju et al

The cryogenic suspension and isolation stage of an antenna is one of the most criticalcomponents. It must isolate against the noise which bypasses room temperature isolation, aswell as the thermal noise of the room temperature stage. In addition, it must suspend the antennausing low-loss elements so as to maintain a high antenna quality factor. It is essential that thecryogenic stage, at least, does not have any resonant modes near the antenna frequency. Severaldifferent systems have been used with reasonable success. Traditionally, cable suspensionshave been used: a cable slung around the belly of the resonant bar. This method has therisk of introducing vibration by the violin string mechanism: slip–stick frictional transitionsassociated with the motion of the belly cable at the point of tangential contact with the bar.This process arises because of the seismically driven low-frequency pendulum motion of thebar. Such boundary condition effects can be avoided if the antenna is somehow bonded orclamped to the bar at the tangent point. The now defunct Stanford group and the LSU groupused welded rods instead of wires to avoid this problem. The third method consists of acantilever suspension. High Q-factor curved cantilever springs such as the Catherine wheelused on NIOBE support the bar from below. This can have well defined contact points to theantenna to minimize nonlinear processes, and has given the highest Q-factor ever observed ina metal [125]. A fourth method, first suggested by Coccia [126], is the use of a nodal pointsuspension. In the case of a bar or sphere, the nodal point for the fundamental mode is locatedat the centre of mass. This means boring a hole to the centre and attaching a rod or cable.While this has many advantages in reducing sensitivity to external noise, it has not been usedin a full-scale antenna.

Antennas cooled to ultracryogenic temperatures (below 100 mK) have a particular problemto contend with. Helium exchange gas can no longer be used at such temperatures (the vapourpressure is too low) so the antenna must be cooled by conduction. This means that there mustbe direct cryogenic conduction paths to the antenna from the dilution refrigerator. Pure andnonsuperconducting metals must be used (such as OFHC copper). Yet the thermal conductionmust exist without significant vibration conduction, especially since the dilution refrigeratoris likely to be a substantial source of vibration.

Ultracryogenic detectors used to date have exhibited excess sensitivity to local vibration,due presumably to the inadequate performance of the thermal conduction/vibration isolationsystem. However, recently both NAUTILUS and AURIGA have yielded improvedperformance, down to a noise temperature of 1 mK.

4.7. Present status of resonant-mass detectors

At the time of writing five resonant-mass gravitational wave detectors are in operation. Theseconsist of three liquid helium temperature detectors, ALLEGRO, EXPLORER and NIOBEat Louisiana State University, CERN (operated by the University of Rome Group) and theUniversity of Western Australia, and two ultracryogenic detectors, NAUTILUS and AURIGAat INFN Frascati and INFN Legnaro, Italy. The latter detectors have been successfully cooledto below 100 mK. Table 3 below summarizes the basic parameters of these antennas. The noiseperformances quoted are typical/best rms noise levels for the detection of broadband bursts.

As well as undertaking long periods of operation, the antennas are all in the processof upgrade, either through installation of improved transducers, or through installation ofimproved vibration isolation. Due to the large size of the cryogenic systems, resonant antennashave a rather long cycle time (several months) of cooling and warm-up. Combined with theprobability of malfunction in experimental apparatus (a particular problem during the 1980s)the rate of progress has been slower than predicted.

Detection of gravitational waves 1371

Table 3. The resonant mass detectors which belong to the international Gravitational EventsCollaboration.

Noise Frequency StrainAntenna Location Material Temperature Temperature (mK) (Hz) sensitivity

ALLEGRO Baton Rouge Al 4 K 6 900 7 × 10−19

EXPLORER CERN Al 2 K 6 900 7 × 10−19

NIOBE Perth Nb 5 K 1 700 5 × 10−19

NAUTILUS Frascati Al 100 mK 4 900 6 × 10−19

AURIGA Legnaro Al 100 mK 1 900 3 × 10−19

4.8. Performance of resonant bars

The long-term operation of cryogenic resonant bars has been invaluable in characterizing theirinstantaneous performance, and evaluating various possible filtering techniques for extractingevents from a long data record in the presence of excess noise. With regard to instantaneousperformance it has been important to verify that the noise performance is consistent with thenoise parameters of the bar, transducer and readout system. In nearly all cases extremelygood agreement has been obtained, such as illustrated in figure 26. In the case of parametrictransducers their performance can be characterized not only by noise spectra, but also by theirvariable interaction with the antenna, as discussed in section 4.5.

From the experimentally observed noise spectral density, such as figure 26, one candetermine the sensitivity of the antenna to various signals such as stochastic background,CW signals and bursts. Figure 27 shows the calibrated burst sensitivity of NIOBE during a24 h period. The data are presented as mean noise temperature (bottom curve) and the largestnoise temperature observed in 100 s. This allows the antenna performance to be quicklyassessed, including the presence of excess noise. Figure 28 presents the same data in the formof Brownian motion noise histograms. From the single antenna data there are clearly fewevents above 10 mK, corresponding to h ∼ 10−15 (all of these could normally be eliminatedby coincidence analysis.

The resonant-bar detector network has recently been able to set new upper limits tothe strength and event rate of gravitational wave burst signals from coincidence analysis ofthree- and four-antenna data. Figure 29 shows this result in comparison with previous upperlimits [127].

We saw in section 4.2 that the noise energy for burst detection is reduced by the factor τi/τa ,where τi is the signal integration time and τa is the antenna ring-down time. The reduction ofthe noise with τa is a manifestation of the fluctuation dissipation theorem. In the case of CWsignal detection the noise energy reduces proportional to (τiτa)

−1. This means again that thebest detector is one with very high Q-factor, and that very long integration times improve theamplitude sensitivity as τ

1/2i . Figure 30 presents an FFT analysis of one month’s data from

the ALLEGRO detector in the search for pulsar signals form the globular cluster 47 Tucanae.In this case the deep integration, over a narrow frequency band, has set limits for CW pulsarsignals ∼10−23. The analysis has been repeated for various directions in the two low-noisebands of ALLEGRO.

For the detection of stochastic backgrounds one simply has to multiply the output of twonearby detectors and integrate the result. However, as discussed in section 3, the detectorsneed to be sufficiently close that the incoming waves are correlated. Their space should bewithin about λ/3 where λ is the gravitational wave wavelength c/f . Cross correlation betweenthe detectors NAUTILUS and EXPLORER (as mentioned in section 3) have yielded a limit to

1372 L Ju et al

Figure 26. The measured strain noise of ALLEGRO, shown as the irregular trace. The variousnoise contributions estimated from the noise model are shown as smooth curves. The noise isdominated by the SQUID’s wide-band and the transducer’s narrow-band noise [71].

Figure 27. The calibrated burst sensitivity of NIOBE during a 24 h period. The data are presentedas mean noise temperature (bottom curve) and the largest noise temperature observed in 100 s [78].

Detection of gravitational waves 1373

Figure 28. Histogram of the noise of NIOBE after optimal filtering for burst signals, showing theexpected Boltzmann distribution. When the weighted mean of both normal modes is evaluated theoverall system noise temperature is 0.89 mK.

Figure 29. The rate of burst events versus strain amplitude set by one-two-three- and four-antennaexperiments. The 4-antenna result is based on a short period of data and hence has weakersignificance.

1374 L Ju et al

Figure 30. Search for narrow-band continuous waves from possible pulsars in 47 Tucanae [71].Over about 1 Hz range no CW signal is visible above a strain amplitude ∼10−23.

stochastic gravitational waves of ∼10−22 at 900 Hz. The results achieved were limited becausethe detectors were too far apart. Vitale et al [128] have shown that NAUTILUS and AURIGAcan be expected to achieve improvement of more than two orders of magnitude over this figure.However, they should be located within about 100 km of each other to achieve best sensitivity.This would bring the sensitivity to within the range of possible stochastic signals from the eraof early star formation.

4.9. Multiple antenna correlation

In the future we can expect detailed correlation experiments to take place between a worldwidearray consisting of both resonant-mass and laser interferometer gravitational wave detectors.The analysis would take into account the relative amplitude of the signals observed by detectorswith different orientations relative to an incoming signal, and the phase delay due to thepropagation time of signals through the Earth. Such a combined analysis would allow sourcedirection and polarization to be accurately determined. Today we are still far from achievingthis goal. Here we will discuss the analysis performed to date on data from a far less optimalarray.

Besides seismic noise excitation, all cryogenic antennas have shown evidence of excessnoise of indeterminate origin [81], as mentioned in section 3. Low noise performance may beachieved for considerable periods of time; but interspersed are periods of excess noise which

Detection of gravitational waves 1375

is not identifiably correlated with known noise sources. This can degrade performance bymany orders of magnitude. While there are opportunities (as discussed above) for nonlinearup-conversion driven by low-frequency pendular modes, and also for thermal-stress-drivenexcitation as regions of the cryostat vary in temperature, no firm correlation is generallyapparent. Thus the seismic, acoustic, electromagnetic pulse and cosmic ray shower detectorsthat are generally used to discriminate possible noise signals are not sufficient to eliminateexcess noise, and much careful work still needs to be done. It has been shown that at a noisetemperature of ∼1 mK about 2 cosmic ray events will occur per day, and this increases to103 events per day at the 1 µK noise sensitivity [129]. The Rome group has convincinglydemonstrated the presence of cosmic ray excitation events in their data [130].

Multiple antenna coincidence correlation can minimize the effects of excess noise, asdemonstrated by many coincidence experiments (e.g. [131]), which sets new lower limits onthe gravitational wave flux. When four or more antennas are operated in coincidence the rateof accidental coincidences becomes extremely small, as we discuss further below.

Candidate gravitational wave events consist of either unknown environmentalperturbations, occasional rare Gaussian high-energy excursions, and possibly real gravitationalwave signals. These may be idealized as an independent set of background events, occurringat a constant rate R per unit time. There is evidence that the background events are not entirelyindependent but to some extent are clustered. This can occur if a local vibration source actsover a period of time. In spite of this, the data are generally well described by a Poissondistribution.

The probability of a background event in one antenna during the antenna resolving timeτr is given by

P1 = Rτr . (4.20)

Now if there are N independent antennas, the probability of accidental coincident excitationof all N antennas in a coincidence window τc that must always satisfy τc τr , is given by theproduct

PN = τNc

∏i=1,N

Ri. (4.21)

If we simplify by assuming that all antennas experience the same background at the rate R,equation (4.21) becomes

PN = RNτNc . (4.22)

From this it follows that the mean number of accidental coincidences during a coincidenceexperiment of duration ttot is given by

Nac = RNτN−1c · ttot. (4.23)

Table 4 summarizes the mean number of accidental coincidences for experiments with oneor more detectors, assuming various coincidence windows. The coincidence window dependson several factors. It cannot be smaller than the poorest clock precision of the detectors in theexperiment. Second, it depends on the timing resolution of the optimal filters in the antennareadout. The latter depends on the signal size, but for the typical thresholds used to extractcandidate events it is about 0.1 of the optimum integration time. At the improved antennasensitivity expected in the near future, cosmic ray events could produce one event per 100 s(103 events per day). To detect rare events such as gravitational waves from supernovae in ourGalaxy, we should be looking for Nac < 1 in any coincidence experiment. The probability thata single coincidence is accidental is given roughly by the value of Nac (for Nac less than 1).

Table 4 emphasizes that multiple antenna operation is essential to reduce the backgroundcoincidence rate. Two-way coincidence experiments to date have been meaningful because

1376 L Ju et al

Table 4. Mean number of accidental coincidences for R = 10−2 s−1 in a four-month (107 s)coincidence experiment.

Coincidence window (s)Number ofantennas 1 0.1 0.01 0.001

1 105 105 105 105

2 103 102 10 13 10 0.1 10−3 10−5

4 0.1 10−4 10−7 10−10

5 10−3 10−7 10−11 10−15

R < 3 × 10−4 has usually been achieved. At the increased rate of candidate events expecteddue to cosmic rays as sensitivity improves towards h ∼ 10−20, it will be difficult to obtainsignificant data unless at least three detectors are used, or else if separate vetos can eliminate thecosmic rays themselves. The very low probability of accidental coincidences achievable withfour- and five-antenna arrays apparent in table 4 demonstrates that with high time resolutionit is possible to eliminate a substantially larger background than the assumed value of R. Forexample, even if R was increased to 1 s−1, the probability of a five-way accidental coincidenceusing the smallest assumed window is 10−5 in a four-month experiment. Note that if theantennas have different resolving times, the τc used in equation (4.23) must be greater than thelongest resolving time.

The issue of multiple antenna coincidences is not as simple as indicated above due tothe fact that the antennas have varying orientations on the surface of the globe. The angulardependence of the signal S observed in a single resonant-bar antenna is given by

S(θ, φ, ε) = (0.5(1 − ε) + ε cos2 2φ) sin4 θ, (4.24)

where θ is the angle of the incoming plane wave relative to the cylinder axis of the antenna,and φ is the polarization angle of the wave measured relative to the plane of the antenna andthe source. The polarization fraction ε measures the fraction of linear polarization of the wave.For ε = 0 the wave is circularly polarized, whereas for ε = 1 the wave is 100% linearlypolarized, with polarization angle φ.

To assess the probability of multiple antenna coincidences we must investigate the antennapattern of a set of antennas on the globe. Since all antennas are horizontal, their orientationwith respect to the sky is largely determined by their locations on the Earth, and considered as awhole this leads to a complex antenna pattern when the responses S as given by equation (4.24)are combined into a synthetic multiple antenna pattern. Antenna patterns for one and tworesonant-bar antennas have been analysed by Frasca [132] and Nitti [133]; Blair and Frascaet al have analysed multiple antenna arrays [134], while Schutz and Tinto [135] have analysedantenna patterns for pairs of laser interferometers.

An analysis of antenna patterns for the geographical locations of four antennas [134] showsthat four antennas are sufficient to obtain near-100% sky coverage for two-way coincidences.That is, if we are content with only two antennas being suitably aligned for a random source, wecan observe practically 100% of the sky. On the other hand, if we are to demand four-antennacoincidences then we require the antenna orientations to be adjusted such that they optimallysearch the same part of the sky, and the sky coverage is reduced to about 50%. With eightoperating antennas one can achieve near-100% sky coverage and a minimum of four antennasin coincidence for any one event. Note, however, that since spherical antennas have 100% skycoverage, such an eight-antenna array could be replaced by three or four spherical antennas,thus avoiding the sky coverage problem.

Detection of gravitational waves 1377

We now return to the practical problems of the existing array of bars. We have seen thatthe presence of local sources of background noise lead to the minimum detection requirementthat a two-antenna zero time delay coincidence be observed. With unknown source directionwe can apply no constraint on the signal based on the relative size of the signal observed ineach antenna. However, it is possible to utilize the fact that the rotation of the Earth causesthe antenna sensitivity to be strongly modulated for specific source directions. For example,it may be reasonable to consider that the nearest sources (such as coalescing black holes) willbe concentrated near the galactic centre. Under this assumption one can reject data when theantenna sensitivity in this direction (defined by equation (4.24)) is below some predeterminedvalue. Due to the sin4 θ factor in equation (4.20) this provides a rather steep angular cut-offand in practice one can eliminate ∼50% of data by applying such a direction filter. Thisautomatically reduces the number of accidental coincidences (since these occur randomly insidereal time) and increases the statistical significance of any zero time delay excess). Such aprocedure is statistically dangerous, however. The reason for this is that there will always besome direction in which an apparent excess is statistically significant. The direction and thecut-off threshold must always be preset before the analysis.

A second means of improving the statistical significance of the data rests on the fact thatthe energy distribution for true signals (such as gravitational waves or calibration pulses) isdifferent from the distribution of excess noise events. This is easy to verify experimentally.Astone et al [136] have shown that the energy resolution achievable in a typical resonantbar with a threshold for candidate events about 10 times the mean noise energy is uncertainwithin a range typically 0.3–3 (for signal energy = 1). This uncertainty has been verifiedexperimentally, indicating that the use of relative amplitudes to determine source directioncan only be used effectively at large signal-to-noise ratio [137]. While this energy spread islarge, the energy spread of accidental coincidences is always much larger, typically spanningup to two orders of magnitude. This follows immediately from the high effective temperatureof the excess noise distribution (see figure 17). As a result one can apply an energy ratiofilter to coincidence data, thereby eliminating from consideration any coincidences for whichthe energy ratio exceeds a predetermined value. (This depends on the relative orientationof the detectors, and on their excess noise distribution.) Appropriate criteria can be set forsuch filters based on the measured distribution of non-true-time coincidences. In practice thisscheme allows about 50% of the total coincidences to be excluded, again allowing substantialimprovement in the statistical significance.

In an unpublished NIOBE–EXPLORER coincidence experiment a total of about 25 zerotime delay coincidences was reduced to a total of seven coincidences by the application of bothan energy ratio filter and a galactic centre direction filter. This figure was substantially abovethe background, but timing uncertainties and the fact that these filters were developed duringthe data analysis period ruled out attribution of statistical significance to the result. If it hadbeen a blind experiment and there had not been timing uncertainties the probability of the zerotime delay peak occurring by chance would have been improved from ∼0.01 to ∼3 × 10−4.Future experiments can utilize one or both of the above filters to improve statistics. It is atechnique that is likely to be applicable in the search for burst sources in laser interferometerdetectors.

4.10. Future prospects

4.10.1. Spherical gravitational wave detectors. Spherical gravitational wave detectors havebeen analysed extensively by Coccia and co-workers [138–143], Johnson and Merkowitz [34,144], Zhou and Michelson [145] and Stevenson et al [146–149]. A spherical detector consists

1378 L Ju et al

of a large approximately spherical mass instrumented with five or more transducers to read outthe orthogonal quadrupole modes and the gravitational wave insensitive monopole mode. Sucha detector has a high cross section for gravitational waves and omnidirectional sensitivity. Aspherical detector can be suspended close to its centre of mass to achieve quite good decouplingof the suspension from the antenna normal modes. This cannot be perfect however, because ofanother problem: the matching of transducers to the large normal mode masses. All operatingresonant-bar transducers use a secondary resonator to match the antenna normal mode mass(about one tonne) to the transducer (see section 4.4). Typically, the secondary resonator (suchas the UWA bending flap, figure 23) has a mass of several hundred grams. This is roughly themaximum mass that can be coupled to existing transducers with electromechanical couplingcoefficient in the range 1–10−3.

In the case of a 100 tonne sphere aiming for high sensitivity to bursts in the kHz range,the impedance matching problem is more complex. The maximum effective bandwidth of atwo-mode system is set by the beat frequency between the two normal modes. This is givenby f = f0(M1/M2)

1/2. For a 1000:1 mass ratio this represents a bandwidth of 3%. Sincebandwidth translates directly into burst sensitivity, it is necessary to reduce this ratio, whichcan best be done with a multistage impedance matching network. A reasonable choice is afactor of about 30, which gives a fractional bandwidth of ( 1

30 )1/2, or about 18%. However,

to match to a transducer using such mass ratios between a 50 tonne normal mode mass and a50 g transducer coupling mass requires the use of four stages, with the second-stage resonatorexceeding 1 tonne, a third-stage resonator of tens of kilos, coupled to the final low-masstransducer stage. The sphere becomes a rather hairy sphere, possessing not five fundamentalquadrupole modes that couple to gravitational waves, but 20. Care must be taken that there areno adverse couplings between the normal modes, since perfect orthogonality will be difficultto achieve in such a complex structure. Negligible cross-coupling can easily be achieved bydeliberately choosing coupling mass parameters such that all modes are offset from each otherby a few Hz.

A scheme for creating some of the additional resonators required on a sphere is illustrated infigure 31. It makes use of bending flaps, which have the advantage that they have low surface-to-mass ratio (to minimize surface losses), simple geometry for fabrication, and minimumwasted volume. The first intermediate mass is realized by simple machining of the sphere.The second could be welded or cast in situ, or it could also be cut from the first intermediatemass by suitable machining (the latter would save more space but has been omitted in the figurefor clarity). Just two of the necessary five resonator structures are shown here. This designconcept ensures that the antenna and most of the impedance matching system can be fabricatedwith complete mechanical integrity, thus avoiding unmodelled losses from bolts or glue joints.Here it is worth pointing out that the presence or absence of cylindrical symmetry is not anissue in antenna design, as demonstrated in the NIOBE bending flap at UWA. For large massratios the symmetry breaking creates a very small torsional reaction at the suspension point,which could only modify losses if the suspension itself had very large differences in acousticlosses for different degrees of freedom. In reality, suspension systems must be designed toisolate all linear and rotational degrees of freedom since intrinsic cross-coupling always occurs,determined by the Poisson ratio which has a value ∼0.3 in all pure homogeneous materials.

The final transducer stage on a spherical antenna can follow the noncontacting microwaveparametric transducer concept to avoid the need for cable isolation and save space in theultracryogenic volume. The transducer could be a sapphire transducer [150–152] or it couldbe a superconducting re-entrant cavity, as shown in figure 23. For ultralow-temperatureapplications it is necessary that the transducer have low power dissipation, generally below 1–10 µW. This requires the transducer losses to be lower than niobium transducers demonstrated

Detection of gravitational waves 1379

vibration isolation

(a)

Figure 31. (a) A 100 tonne spherical antenna with two of the five cuts required to create ∼1 tonnesecondary resonators to readout the orthogonal normal modes. To each secondary resonator isbonded a bending flap ∼30 kg to create the third normal mode of the impedance matching network.The fourth resonator (too small to show here) would consist of an additional bending flap of mass∼50 g. A parametric transducer could read out such an antenna using radiative coupling thusavoiding vibration coupling through wires. (b) [153] Noise performance of a similar sphericaldetector. Curves A–E show the increasing bandwidth as the number of secondary resonatorsincreased from 1 to 5 respectively.

so far. However, this can easily be achieved by using a larger gap spacing, and a frequencybelow 10 GHz. In the case of sapphire transducers, sideband pumping allows the dissipatedpower to be reduced below 1 µW. Figure 31(b) [153] shows that broadband noise performance∼250 Hz can be achieved with a four-mode transducer on a large sphere. The bandwidth issignificantly degraded as the number of modes is reduced.

At the transducer it becomes necessary to make a materials transition from the Al or Cu–

1380 L Ju et al

Al antenna material, to niobium or sapphire. No definitive solutions for such transitions havebeen defined, but shrink fitting (by thermal differential contraction), glue bonding or brazingare likely to allow suitably low-loss assembly. It will be important to develop bonding systemsin small-scale tests before construction of large-scale spheres.

Spherical detectors cooled to ultralow temperatures can offer difficulties in cooling dueto their small surface-to-mass ratio. Frossati has determined that forced convective cooling isessential if the cooling time is to be reasonable. If the material is superconducting, thermalconductivity freeze-out makes this a much more severe problem. For this reason Cu–Al andBe–Cu have been proposed for the antenna material.

4.10.2. Arrays of small high-frequency detectors. A part of the gravitational wave spectrumthat has had insufficient attention is the high-frequency band between 2 and 20 kHz. In thisband one expects gravitational waves from stellar mass black hole formation and their normalmodes. At high frequencies, laser interferometers lose sensitivity due to shot noise, whileresonant-mass detectors lose sensitivity due to the smaller size required for such high resonantfrequency. Frasca and Papa [154] has proposed a solution consisting of phase coherent arrays ofshort stumpy antennas, designed so that that all five lower quadrupolar modes (the longitudinalmode, the two discoidal modes and the two ‘pantograph’ modes) have comparable quadrupolemoment and sensitivity. Using five transducers, such stumpies can be read out similarly to thefive quadrupole modes of a sphere, and thus have omnidirectional sensitivity. In principle, thetotal energy sensitivity increases linearly with the total detector mass, so that an array can beenlarged arbitrarily by simply adding additional identical elements.

Whereas a large sphere achieves high sensitivity in a single device, but at the expense of arather low resonant frequency, the array of stumpies is claimed to achieve the same sensitivityat an arbitrarily high frequency, chosen by astrophysical considerations. It can have costadvantages too, through replication of identical elements.

Such an array obviously requires the individual elements to be sensitive to the same partof the spectrum. However, with adequate impedance matching the stumpies should havebandwidth ∼15% similar to that achievable in spheres and bars. This means that there isminimal difficulty in tuning individual detectors.

However, the practical limitations to achieving high sensitivity, such as the limits imposedby noise, have yet to be demonstrated. Further research in this area would be very valuable,including an in-principle demonstration of the concept using calibration signals. The array ofstumpies provides an opportunity to achieve great increases in high-frequency sensitivity in thefuture, but will probably not be implemented until positive results are obtained with existingdetectors, or astrophysics provides a very strong justification for improved high-frequencysensitivity.

4.10.3. Transducer developments and prospects. The implementation of improvedtransducers and impedance matching structures is essential to increasing the bandwidth andburst sensitivity of resonant-mass detectors. Successful transducers have been developedbased on capacitative or inductive modulated superconducting circuits with RF or DC SQUIDamplifiers, as well as microwave parametric transducers based on superconducting cavitiesand cryogenic GaAsFET amplifiers. Both systems have comparable sensitivity and a varietyof advantages and disadvantages.

The sensitivity of a SQUID-based transducer depends on the SQUID noise performance,and on the mechanical and electrical quality factors of the superconducting elements. Apersistent problem has been AC electrical losses in the niobium superconducting coils used for

Detection of gravitational waves 1381

sensing and coupling transformers in passive transducers. It has sometimes been difficult toobtain sufficient trapped persistent current in inductive sensing circuits, and sufficient electricfield across capacitative sense elements. AC losses introduce thermal noise, while low currentor electric field reduces the forward transductance, reducing the electromechanical couplingand increasing the antenna noise temperature.

A critical factor for SQUID transducers is the SQUID noise. Recently there has beenprogress in this area. Two groups have created SQUIDs with noise measured to be belowabout 30h, one to two orders of magnitude better than commercial devices. However, recentresults [155] indicate that noise is degraded in a transducer environment due probably to trappedflux.

Parametric transducer performance also depends on the mechanical and electrical qualityfactor of the transducer structure. This is less of a problem however, since the simplermechanical structures used have shown Q-factors ∼107, and electrical Q-factors ∼105–106,both of which are high enough to be negligible noise sources at current sensitivity. Microwaveamplifiers have for many years been shown in radio astronomy to have near-quantum-limitednoise, but in the only successful such implementation the amplifier noise contribution is inexcess of this [156]. Critical to achieving excellent amplifier noise is very low signal levels,requiring excellent carrier suppression, since otherwise pump power reflected from the cavitygreatly exceeds the signal sidebands.

Another critical problem is pump oscillator phase noise. The NIOBE transducer requiredan ultralow phase noise microwave oscillator to be especially developed.

To achieve sensitivity near to the quantum limit with a parametric transducer will require ahigher Q-factor electrical resonator and a lower noise pump oscillator. Fortunately, oscillators30 dB better than that used on NIOBE have now been developed, and a sapphire transducerwith mechanical and electricalQ-factor exceeding 108 should allow this technology to advanceto within a factor ∼30 of the quantum limit [100, 152, 157].

4.11. Vibration isolation and suspension developments

The vibration isolation and suspension system for resonant-mass detectors has always consistedof both room temperature and cryogenic isolation stages such as those discussed above. Inthe case of ultracryogenic detectors there has been greater emphasis on cryogenic isolationbut careful isolation of the cryostat structure itself has been necessary to prevent localdisturbances. It is tempting but incorrect to consider the vibration isolation problem solved,since improvements of more than 1000-fold in energy sensitivity are projected, and alreadydetectors show signs of inadequate isolation. In the case of NIOBE there are signs of variablenoise temperature by about a factor of two which is observed as degrading mode temperaturewithout change to the wideband noise. Sometimes diurnal variations associated with humanactivity are observed. This can be understood as arising from vibrational short circuits, perhapsdue to a small piece of superinsulation or whiskers of solid air crossing the narrow spaces inthe radiative cooler in the suspension tube, or residual conduction through the low pressure(10−5 torr) gas in the experimental volume.

EXPLORER is operated at 2 K, below the helium superfluid transition to eliminate thevibrational effects of boiling liquid helium, which otherwise causes degraded performance.NAUTILUS operates well only when local activity is low, again indicating inadequate vibrationisolation. Thus it appears that all currently operating detectors are operating close to a vibrationisolation limited noise floor, at which improved antennas and transducers will not provide theanticipated noise advantages.

In principle, isolation can be easily improved by further isolation stages, such as the

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cantilever spring stages used at UWA. However, short circuits can easily occur, especiallythrough unmodelled resonances in components such as masses, springs or wires, which cancause catastrophic degradation of performance. Not only great care, but excellent design anddeep understanding of complex mechanical structures is essential. One point in favour ofadvanced detectors such as spheres is the advantage of near-nodal-point suspension achievedusing centre of mass suspension [126], which can improve isolation by 40–60 dB dependingon the proximity of the suspension point to the node. As already emphasized, this can neverbe perfect due to the finite size of suspension elements and the multimode nature of detectors.

4.12. Conclusion

Resonant bars have been brought to a high level of development. There is now an excellentunderstanding of the technology and clear proposals for substantial future advances. Inparticular, it is likely that large spherical detectors will be found to be the best solution forobtaining high sensitivity in the 1–10 kHz range. Meanwhile, an extremely vigorous researcheffort in laser interferometer detectors is underway, as discussed in the next section.

5. Interferometer detectors

5.1. History

The Michelson interferometer has long been known as an extremely sensitive instrument tomeasure length changes. The idea of using a laser interferometer as a gravitational wavedetector was suggested as early as the 1960s [65, 158] and experimentally investigated inthe 1970s [66, 159, 160]. But the first experimental attempt, giving high sensitivity to thedisplacement of test masses was due to Forward [161]. Forward used a retro-reflector toreflect a beam to a beamsplitter and used active control for locking the interferometer to afringe. He obtained a spectral strain sensitivity of 2 × 10−16 Hz− 1

2 . The state of the art insensitive interferometers at the time of writing is represented by three prototype instrumentsof 10 to 40 m in arm length. These are at Garching [162], Glasgow [163] and CalTech [164].The 30 m delay line interferometer at Garching has achieved test mass differential positionsensitivity of 2.5 × 10−18 m Hz− 1

2 dominated by shot noise between 1 and 6 kHz. The 10 mFabry–Perot interferometer at Glasgow has reached ∼7×10−19 m Hz− 1

2 from 500 Hz to 3 kHzand is close to being limited by shot noise. The 40 m Fabry–Perot interferometer at CalTech hasachieved its best displacement sensitivity of 3 × 10−19 m Hz− 1

2 near 450 Hz. The broadbandnoise background (neglecting the narrow peaks which can be removed by appropriate filteringof the data) between 300 and 1000 Hz gives a rms differential displacement of less than 10−17 m,corresponding to an rms gravitational strain noise level of 2 × 10−19 which is comparable tothe sensitivity of current resonant-bar detectors.

In pursuit of increasing the interferometer sensitivity, several optical schemes have beeninvented. Power recycling, proposed by Drever [165], reuses the otherwise unused laser powerfrom the interferometer bright fringe. The light is reflected back towards the beamsplitter,thus increasing the total light power entering the interferometer. Dual recycling proposedby Meers [166] allows both laser power and signal power to be recycled, thus increasingsensitivity at the expense of signal bandwidth. Various other schemes such as synchronousrecycling [165], detuned resonant recycling [167], and resonant sideband extraction [168,169]provide various advantages. In general, recycling schemes allow high power built-up in theinterferometer arms (which may or may not contain optical cavities) to increase the sensitivitywhile controlling the signal sideband storage time to maintain detection bandwidth.

Detection of gravitational waves 1383

AAAA

AAAA

laser

to photon detector

end mirror M1

end mirror M2

beamsplitter

Figure 32. Simple Michelson interferometer.

Power recycling has been demonstrated on both tabletop [170–172] and suspendedinterferometers [173, 174] with success. Power recycling factors of 300 and 450 have beenachieved [173, 174]. The broadband and tuned signal recycling has been demonstratedon tabletop by ANU group [175]. Dual recycling has been tested experimentally by bothGlasgow [176] and Max Plank groups [177].

Long-baseline detectors are under construction at: Hanford, Washington and Livinston,Louisiana, USA (American LIGO project) [32], Pisa, Italy (3 km Italy/France VIRGOproject) [178], Hannover, Germany (600 m British/Germany GEO project) [179] and Tokyo,Japan (300 m TAMA project) [180].

During the past few years the development of diode-pumped Nd:YAG lasers and ultralow-loss optical coatings has offered greatly improved performance in laser interferometer devices.Nd:YAG lasers are intrinsically stable and efficient. Low-loss coatings (for mirrors working inthe infrared frequencies) offer very high recycling gain. Development of polishing techniquesfor sapphire means that improved material less susceptible to thermal lensing is becomingavailable. New ideas in suspension and isolation means that it appears likely that futureinterferometers need only be limited by fundamental limits, while there remains substantialroom for future advances.

5.2. Configurations

5.2.1. Simple Michelson. A simple Michelson interferometer detector is shown schematicallyin figure 32. The interferometer consists of three ‘free masses’—one beamsplitter, and twotest masses at right angles to form the end mirrors. These masses are vibration isolated andsuspended so that at frequencies well above resonance they can move freely as inertial testmasses in the direction of the optical path of the interferometer in the frequency range ofinterest. When a gravitational wave passes it creates relative displacements of the test masses.The relative motion of the end mirrors is read out as intensity variations in the interferometeroutput, giving information about the incoming gravitational wave.

For simplicity, consider the case of an incident gravitational wave perpendicular to theplane of the interferometer with a polarization direction parallel to the interferometer arms.The passing wave will make one arm of the interferometer shorter and the other longer in halfof the wave period, and reverse the contraction–elongation process in the other half-period.The relative change of optical length of the two arms L = L2 − L1 can be described as a

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phase shift,

F = 2πL/λ. (5.1)

This results in a change in the interference pattern at the output of the beamsplitter. Therelative difference in optical path L is proportional to the arm length L = hL. Generally,an interferometer is sensitive to a linear combination of the two polarization fields, and h inthe above equation is

h = F+h+ + F×h× (5.2)

where F+ and F× are coefficients depending on the direction to the source and the orientationof the interferometer.

Because the gravitational signal is extremely small, it is very difficult to monitor the smalltime-varying changes in the interference pattern due to the passing gravitational wave. Inpractice, the phase difference arising from the optical arm length variations is obtained by theso-called ‘nulling method’. The idea is to always keep the light returning from the two arms180 out of phase so that its output is a dark ‘fringe’. When the end mirrors are moved bythe passing gravitational waves, the error signals applied to end mirrors to maintain the darkfringe contain the information of the gravitational wave signals. In this way the effect of powerfluctuations in the laser beam can be minimized, and the shot noise level can be reduced.

Since the detection of gravitation waves with an interferometer is achieved by measuringthe relative optical path (phase) change between the two arms, and since this path differenceis proportional to the optical path L, it is clear that the size of the signal can be increased bylengthening the optical path of each arm. However, there is an optimum length Lopt. At sucha length the storage time of the light within the interferometer arms is equal to half the periodof the gravitational wave. For arms longer than Lopt, the gravitational wave signal will changesign during the light travelling time in the arm and the effect will partially cancel out. Forexample, for a gravitational wave signal of frequency fg ∼ 1 kHz, the optimum arm lengthis Lopt = c/(2fg) = 150 km. Such a long optical path length can be easily realized in aspace-based interferometer (see section 3.2). In an Earth-based interferometer this is madepossible by using multi-pass techniques. One such technique is the multi-pass Michelsoninterferometer in which an optical delay line [66] is inserted in each of the interferometerarms. The other is a Fabry–Perot Michelson interferometer [181] in which a second mirror isinserted in each arm to form a Fabry–Perot cavity.

5.2.2. Delay line Michelson interferometer. A delay line Michelson interferometer is shownschematically in figure 33. The two beams coming out of the beamsplitter are reflected manytimes between the beamsplitter and the end mirrors before they are recombined. For example,a 3 km long interferometer can have an optical length of 150 km by having 50 bounces. Apartfrom the restriction that the optical path length be shorter than Lopt, the useful number ofreflections is in practice limited by the reflection losses at the mirror.

A practical difficulty of the delay line Michelson interferometer configuration is thescattered light problem. The delay line uses a large number of beams zigzagging back andforth between mirrors. It is easy for light to scatter from the mirrors or from the side of vacuumpipes into the main beam at the photodetector. This scattered light could have a large phasedifferenceα = 2πf δl/c with respect to the main beam, where f is the laser light frequency andδl is the path difference from the main beam. When the scattered light interferes with the mainbeam, it results in a phase shift β of the recombined light given by β ∼ a sin α, where a is thefraction of the scattered light. This change of phase will be sensed by the output photodetectorwhen combined with the beam from the other arm. It can be seen that any fluctuation of

Detection of gravitational waves 1385

A

M1

M2

beamsplitter

laser

photo detector

Figure 33. Schematic diagram of a delay line Michelson interferometer.

laser

Fabry-Perot cavities

photo detector

beamsplitterAAAAAAAAAAAAAAAA

AAAAAAAAAAAA

AAAAAA

Figure 34. Schematic diagram of a Fabry–Perot cavity interferometer.

laser frequency or vibration of the vacuum pipes (which results in a change of δl) will causea phase fluctuation of each beam, and thus a fluctuation in the final signal. This effect can bereduced by stabilizing the laser frequency. Modulation of the laser light [182] can improvethe performance to some extent. This is done by changing the laser frequency f such thatover a certain measurement time the scattered light phase difference α changes from 0–2nπ .The average effect of the scattered light is then zero. This technique is limited by the dynamicrange of the modulation. Another disadvantage of a delay line Michelson interferometer isthat it needs large mirrors. However, excellent sensitivities of the order h ∼ 10−19/Hz1/2 havebeen obtained using a delay line Michelson interferometer of arm length 30 m [183].

5.2.3. Fabry–Perot cavity interferometer. The Fabry–Perot cavity interferometergravitational wave detector was first introduced by Drever et al [181]. The idea is to addtwo additional mirrors near the beamsplitter, as shown in figure 34.

The near mirror and the end mirror in each arm form an optical cavity. Light travelling

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laser

AAAA

AAAAAA

AAAAAAAAAAAAAAAA

AAAAAAAAAAAA

AAAAA

AAA

M3

AAAAAAAAAAAAA

AAAAAAAA

M3'M1

M2

D1

D1'

Recycling mirror

D2Phase modulator

polarixing beamsplitter

AAAAA

AAAAAAAAAA

AAAAAA Phase

modulatorFaradayrotatorAAAAA

AAAA

AAAAAAAA

AAAAAAAA

AAAA

Figure 35. Basic arrangement of a Fabry–Perot interferometer system [324].

in one arm is reflected between the same pair of spots on the two mirrors, forming a sharpresonance. This Fabry–Perot cavity is very sensitive to the changes in cavity length, and tothe change in frequency of the light. Because no laser has achieved the required stabilizationlevel to directly detect the change in cavity length induced by gravitational waves, two cavitiesare needed to detect differential changes. The light from a single source passes through abeamsplitter and illuminates the cavities in the two arms of the interferometer. By looking atthe differential phase change of the two arms, the effect of frequency fluctuations of the lightsource can be eliminated, and the signal is a measure of the passing of a gravitational wave.The size of the suppression of the laser frequency fluctuation noise depends on the balance ofthe two cavities. Typically, with a highly stabilized laser, the cavities must be balanced to onepart in 103 or better.

Another approach is to lock the laser in wavelength to one of the cavities, and then to lockthe second cavity to the laser wavelength. The locking signal of the second cavity then givesthe relative cavity length change with respect to the first cavity, due to the passing gravitationalwave. A practical arrangement of a Fabry–Perot cavity interferometer with such a readoutsystem is shown in figure 35.

The Fabry–Perot cavity interferometer has the advantage of having smaller mirror size,and thus smaller vacuum pipe size, than that in a delay line Michelson interferometer. Also, thescattered light problem can be reduced greatly in a Fabry–Perot cavity arrangement becausethe scattered light is made to travel the same path as the main beam in the cavity.

5.2.4. Sagnac interferometer. A Sagnac interferometer is schematically shown in figure 36 inwhich light beams travelling in opposite directions experience common optical paths. With itscommon-path nature, the Sagnac interferometer is insensitive to a range of noise sources thataffect other interferometers. Noise from low-frequency mirror displacements, laser frequencyand intensity fluctuations, laser beam pointing fluctuations, thermally induced birefringence,and reflectivity asymmetry in the arms are all suppressed. This means that the Sagnacconfiguration can have a simplified control system and reduced optical tolerance requirements.Low-temporal-coherence illumination can be used in a common-path interferometer to reduce

Detection of gravitational waves 1387

Laser

Photo detector

Figure 36. Schematic diagram of a Sagnac interferometer.

the noise caused by parasitic paths introduced by scattered light. In 1986, Weiss proposedan open-area Sagnac interferometer for gravitational wave detection. Recently, the zero-areaSagnac interferometer shown in figure 36 has been analysed and experimentally investigated asa topology for an advanced gravitational wave detector [184,185]. For 4-km 20-bounce (storagetime of 0.53 ms) LIGO-scale interferometers illuminated with a 1064 nm Nd:YAG laser, theSagnac interferometer has its first peak response at 690 Hz with 3 dB bandwidth from 220 to1250 Hz [184], as shown in figure 37. Proper setting of the storage time allows the peak responsefrequency to be tuned to the gravitational wave band of interest. A shot-noise-limited phasesensitivity of 9×10−10 rad Hz−1/2 has been achieved on a tabletop Sagnac interferometer [185].It has been demonstrated that precision phase measurement can be performed with alaser bearing a substantial amount of frequency and amplitude noise [185]. The Sagnacinterferometer with resonant sideband extraction has been demonstrated on the tabletop [186].

The comparison between the Sagnac and Michelson interferometers is detailed byMizuno [187] and colleagues. Their conclusion is that, once cavities are used either in thearms or for power- or signal-recycling, the advantages of the Sagnac interferometer over theMichelson interferometer disappear. Without power recycling, 1 kW laser with quite highstability is required to achieve the desired sensitivity for Sagnac interferometer.

5.2.5. All-reflective interferometer. Extremely high light power incident on the beamsplitter isneeded to reduced the photon shot noise which is a major limiting factor in the high-frequencyregime (∼1 kHz) in laser interferometers. However, high light power poses problems fortransmissive optics such as beamsplitters and the input/output mirrors of Fabry–Perot cavities,because there is always some optical absorption causing heating. The optical componentsare heated both by absorption in the substrate and in the reflective coatings. Heat from bothregions leads to thermal lensing and birefringence in the substrate, and also to distortionof the optical surfaces. The Sagnac interferometer might alleviate these problems since thecounterpropagating laser beams in principle share the same optical path.

A more certain method of reducing adverse thermal effects is to eliminate transmissiveoptical components completely. The losses in optical coatings are generally less thanabsorption losses, so by using reflective diffractive components instead of lenses andbeamsplitters, it should in principle be possible to make interferometers much more tolerantof high optical powers. The idea of an entirely diffractive reflective interferometer has

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0

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tude

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Gravitational Wave Frequency (kHz)

Figure 37. Theoretically calculated differential phase generated by a gravitational wave withstrain hg ∼ 10−23/

√Hz, in a 20-bounce delay line, 4 km arm length Sagnac interferometer

illuminated by a 1064 nm laser. The Michelson interferometer response (dotted curve) is shownfor comparison [184].

been discussed by Drever [188]. The concept would allow test masses to be created usingnontransmissive materials such as silicon or niobium, where low thermal noise did not haveto be combined with excellent optical properties. One possible interferometer configurationusing a diffractive beamsplitter is shown in figure 38. Veitch et al [189] have reported testsof a holographic beamsplitter used to remove optical aberrations and guarantee high fringevisibility and complete destructive interference at the interferometer output even in the presenceof aberrations. An experimental demonstration of various grating beamsplitter tabletopinterferometers (Michelson, Sagnac and Fabry–Perot) has been performed at Stanford [190].For a practical diffractive interferometer it will be necessary to perfect reflective diffractiveelements. If optical cavities are to be used, the mirror coatings will have to include a diffractivecoupling beam. For example, a mirror may be required that has 99.999% reflection, and 0.001%coupling into a beam that leaves the mirror at a suitable angle. This problem sets a challengefor optics in the next decade, which can lead to major improvements in gravitational wavedetection in the future.

5.3. Recycling

5.3.1. Power recycling. The sensitivity of an interferometer is ultimately limited by shotnoise due to photon quantum statistics. The standard quantum limit for an interferometercan be obtained from the balance of two competing quantum noise sources as described byCaves [191, 192], Braginsky et al [112] and others. The first is the photon-counting error dueto N1/2 fluctuations in the number of output photons from the interferometer. The second isthe radiation-pressure error. This arises from the perturbations on the end mirrors producedby fluctuating radiation-pressure forces which also scale as N1/2. As the input laser power Pincreases, the relative photon-counting error decreases asN1/2/N , while the radiation-pressure

Detection of gravitational waves 1389

Laser

Photo detector

Arm 1

Arm 2Diffractive Mirror

Figure 38. Schematic diagram of a Michelson interferometer based on the use of reflectivediffraction grating as a beamsplitter.

error increases as N1/2. Minimizing the total error with respect to P yields a minimum errorof order of the standard quantum limit and an optimal input power for a simple Michelsoninterferometer,

Popt = λmc

8πτ 2√η, (5.3)

at which the minimum error can be achieved. Here, m is the mass of an end mirror, λ is thewavelength of the light, τ is the measurement duration, and c is the light velocity.

With a reasonable set of values for interferometer parameters, m ∼ 102 kg, τ ∼ 10−3 s,λ ∼ 1 µm, the optimum laser power Popt is approximately 6×107 W—a power far higher thanthe power of present CW lasers. The low available input power means that the interferometerfor use as gravitational wave detectors will be limited not by the standard quantum limit, butrather by photon-counting statistics (shot noise) which scales inversely to the square root ofincident power at beamsplitter.

As mentioned above, the technique of power recycling [165,166] can be used to increasethe power incident at the beamsplitter and improve the sensitivity of interferometer detectors.The basic idea is that because the interferometer detector operates at a dark fringe output,almost all of the light (reduced only by losses in mirrors and beamsplitters) is reflected backtowards the laser, and can therefore be used again as long as it is phase coherent with the inputlaser beam. This technique is realized by inserting a recycling mirror M2 in between the laserand the beamsplitter as shown in figure 35. The position of the recycling mirror (or the laserfrequency) is then carefully adjusted so that the recycling mirror combined with the two maincavities and the beamsplitter form a large resonant optical cavity containing the interferometer.By doing so, an effective laser power perhaps 1000-fold larger than the original laser may bebuilt up inside this cavity, thus reducing the shot noise.

5.3.2. Resonant recycling. The resonant recycling technique [165] uses a mirror arrangementsuch that after each half gravitational wave period, the light in the two arms exchange armsinstead of recombining at the output photodetector. In this way, the light of each beam alwaysexperiences phase shift in the same direction. The phase shift builds up during the total storagetime over many gravitational wave cycles. In the end, a large phase difference between the twoarms can be detected at the output of the interferometer. This technique can be used in bothdelay line Michelson interferometers and in Fabry–Perot interferometers. Figure 39 showsresonant recycling arrangements for both delay line interferometer and Fabry–Perot cavityinterferometer.

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laser

beamsplitter

(a)

laser

beamsplitter

couplingmirror Mo

(b)

Figure 39. (a) Resonant recycling arrangement for a delay line interferometer. (b) Resonantrecycling arrangement for a Fabry–Perot cavity interferometer.

In the case of resonant recycling in a Fabry–Perot cavity interferometer, it is consideredthat the two cavities are coupled through the high-reflectivity mirror M0. This coupled systemthen has two modes of oscillation. The interferometer is tuned so that one of the modes of thecoupled optical cavity system matches the frequency of the laser light and the other matches thefrequency of the optical sideband produced by the motion of the end mirrors due to an incidentgravitational wave. Both the laser light and the sideband signal produced by gravitationalwave are enhanced at the output. In principle, resonant recycling gives better sensitivity fordetecting periodic signals of known frequency. The total storage time is limited by the lossesof the mirrors.

5.3.3. Dual recycling. A simple arrangement for dual recycling [193] is shown in figure 40.In addition to the power recycling mirror M3, a new signal recycling mirror M4 is placedat the output of the interferometer. Generally, the light at laser frequency (carrier) cannot

Detection of gravitational waves 1391

reach M4 when the interferometer is locked on a dark fringe. However, when the gravitationalwave arrives, it modulates the interferometer arm length difference and generate sidebands atfrequencies offset from the laser frequency by the gravitational wave frequency. The sidebandsleak from the interferometer output towards M4 as gravitational wave signal. So M4 has noeffect on the carrier but reflects the signal sidebands back to the interferometer. If we adjustthe position of the mirror M4 to allow the reflected signal sidebands and the signal in theinterferometer arms to be in phase, the signal power (sensitivity) will be increased. The relativeposition of the recycling mirror M4 determines the tuning frequency of the dual recycling. Thebandwidth is determined by the reflectivity of the signal recycling mirror and the loss of the twoarms. The dual recycling with tuning frequency at zero is called broadband dual recycling. Therelative signal-to-noise ratio of a 3 km, 16-reflection delay line interferometer with differentdegrees of tuned dual recycling is shown in figure 41. The central frequencies are at 200and 1000 Hz respectively. Even a relatively short arm interferometer can obtain impressivesensitivity. For example, figure 42 shows the predicted shot-noise-limited strain sensitivityof a 400 m, four-pass delay line [194]. The input power is 5 W. The tunned frequency is at200 Hz. Curves (a) and(c) are based on application of recycling factors (450) already achievedin the laboratory [174], curves (b) and (d) employ somewhat higher factors.

5.3.4. Simple dual recycling instruments. The advantage of the Fabry–Perot Michelsoninterferometer is that high laser power and low mechanical noise requirements arepredominantly restricted to optical cavities. Thus one needs only four very-high-performancecomponents (the main Fabry–Perot mirrors) while the beamsplitter and other components arefar less critical. When this scheme is extended to dual recycling as shown in figure 43. Theinterferometer consists of nested cavities, a pair of Fabry–Perot cavities within the overallpower recycling and signal recycling cavities. This requires rather complex control systems.

A much simpler arrangement would be a four-pass dual recycling Michelsoninterferometer (GEO project) as illustrated in figure 43 [195]. Such a system utilizes onlytwo cavities. Now, to regain the sensitivity of the Fabry–Perot Michelson, the interferometermust use high levels of power recycling, plus moderate signal recycling. Because of thethermal lensing problem [196], this arrangement requires excellent mechanical and opticalperformance of the beamsplitter.

It appears that very-low-loss silica beamsplitters may allow recycling factors up to104 [197]. An alternative would be to use sapphire beamsplitters, which have intrinsicadvantages associated with their high rigidity, and high thermal conductivity. However, thedisadvantage of sapphire is its optical birefringence which requires control of the orientationof the crystal relative to the input and output beams.

5.3.5. Resonant sideband extraction. Resonant sideband extraction [168, 169] is a similarconfiguration to dual recycling for laser-interferometric gravitational wave detectors withFabry–Perot cavities in the arms. This scheme reduces the thermal load on the beamsplitterand the coupling mirrors of the cavities and allows one to adapt the frequency response of thedetector to a variety of requirements.

To obtain a good sensitivity in interferometric gravitational wave detectors one requireshigh light power in the arms of the interferometer to increase the photon shot-noise-limitedsignal-to-noise ratio. This can be done by increasing the finesses of the arm cavities. Butsince high finesse cavities have narrow bandwidth, (i.e. long optical storage time) this sets alimit to the detector bandwidth. The same power build-up can equally be obtained in principleby using power recycling to compensate for the limitation to the power enhancement in the

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power recycling mirror M3

signal recyclingmirror M4

laser

Fabry-Perot cavities

beamsplitter

AAAAAAAAAAAAAAAA

AAAA

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Figure 40. Dual recycling arrangement of an interferometer.

Frequency (Hz)

100 1000 10000

Rel

ativ

e se

nsit

ivit

y

0.1

1

10

a

b

c

d e

f

Figure 41. Relative signal-to-noise ratio of a 3 km 16-reflection delay line with different degreesof tuned dual recycling. The tuning frequencies are at 200 Hz and 1 kHz respectively. The verticalaxis is in arbitrary units. The solid curve is for power recycling only; the dashed curve is for signalrecycling mirror reflectivity, Rs = 75%; the dash-dotted curve for Rs = 90%; the dotted curve forRs = 99% [325].

arm cavities. However, in practice the power recycling gain achievable is likely to be limitedby imperfect contrast as well as losses in the beamsplitter and the coupling mirrors of thearm cavities. Furthermore, as already discussed, high power may induce thermal lensing andbirefringence in the beamsplitter.

Resonant sideband extraction allows this dilemma to be avoided in an interferometer with

Detection of gravitational waves 1393

Frequency (Hz)

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Figure 42. Frequency response of a 400 m interferometer. Input power is 5 W. Curve (a) powerrecycling factor of 450, no signal recycling; curve (b) power recycling factor of 1700; curve (c) dualrecycling with power and signal recycling factors of 450; curve (d) power recycling factor of 1700,signal recycling factor of 800 [194].

600m

power recycling

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main mirror

compensator

beamsplitter

main mirror

detection

detection

external modulator

laser and injected optics

main mirror

Figure 43. Schematic diagram of a four-pass delay line interferometer with both power and signalrecycling (i.e. dual recycling). The angles and lengths are not to scale. The test masses of theinterferometer are formed by the main mirrors and the beamsplitter [195].

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arm cavities. The scheme resembles signal recycling, but uses a signal extraction mirrorbetween the beamsplitter and the photo-detector. The purpose of this mirror is to decreasethe storage time of the signal sidebands and therefore increase the detector bandwidth. Thisis achieved because the signal extraction mirror and arm cavities form a three-mirror-coupledcavity. Because the interferometer is locked to dark fringe the signal extraction mirror has noeffect on the carrier. The signal extraction cavity formed by the signal extraction mirror and thecoupling mirrors of the arm cavities forms a compound mirror which has frequency-dependenttransmittance and reflectivity. Tuning the signal extraction cavity allows the transmittancefor the signal frequencies of interest to be higher than that of arm cavities alone. For thesefrequencies the storage time in the three-mirror cavity is shorter than that in the unmodifiedarm cavity. In this case, the reduction of the storage time results in an increased detectionbandwidth and unchanged high-finesse arm cavities for the carrier. In principle, the powerenhancement in the arm cavities could be so great that no power recycling would be required.Yet the power passing through the beamsplitter and the coupling mirrors of the arm cavitiescould be low enough to have little thermal load. Figure 44 shows the frequency responseof 3 km arm length interferometer with resonant sideband extraction configuration at variousconditions [168].

5.4. Vibration isolation

At Earth-based sites for gravitational wave detectors, the ground is continuously vibratingwith a rms amplitude of xs ≈ αf −2 m Hz− 1

2 , where α ≈ 10−6–10−9. This is far greater thanthe signals we want to measure. Thus it is of great importance that terrestrial gravitationalwave detectors, both resonant-bar and laser interferometers, are isolated from the seismic noisebackground. High-performance mechanical vibration isolation systems are required for thispurpose. An ideal vibration isolator would not only cut out all significant seismic vibrationin the pass band of the gravitational wave detector, but also cut out the seismic noise at muchlower frequencies, so that the suspended test masses were effectively stationary with respectto the laser light field. If the total rms motion were much less than an optical fringe width, theservo control requirements would be minimized: components would be as stable as if rigidlyattached to an optical table, or placed in interplanetary space. Operation of an interferometerwould then be very simple. We show below that total rms motion from all frequencies above0.2 Hz can in principle be reduced to about 1 nm.

This ideal level of isolation has not yet been achieved but there are various approachesthat when combined, should approach the ideal performance discussed above. We subdividethe approaches to this problem under the headings passive isolation, active isolation and pre-isolation.

5.4.1. Passive mechanical isolation. Passive mechanical vibration isolators are mass–springlow-pass filters. For a multistage isolator, each stage of the isolator with resonant frequencyf0 will attenuate vibration by a factor (f0/f )2 at frequencies f f0. The total attenuationof the multistage isolator at frequencies above the corner frequency (the highest normal modefrequency of the isolator) is (f1f2 . . . fN/f

N)2, where N is the number of stages.Figure 45 shows the typical behaviour of such an isolator. The figure shows a typical

transfer function of a five-stage isolator with f0 = 2 Hz. Below the corner frequency thenormal modes amplify the seismic noise, while above it the isolation improves as the tenthpower of the frequency ratio.

Traditionally, vibration isolators for gravitational wave detectors were based on industrialvibration isolators, using systems such as alternating layers of lead (or steel) and rubber [5].

Detection of gravitational waves 1395

Figure 44. Typical frequency response of a 3 km arm length interferometer with resonant sidebandextraction configuration. (a) Dependence of the frequency response on the length of the sidebandextraction cavity (SEC) when the carrier is resonant in SEC. The curves, in order of increasingresponse at 400 Hz, are for SEC length of 1 m, 30 m, 100 m, 300 m. (b) Dependence on the tuningof SEC to carrier. The broadest response corresponds to the carrier resonant in SEC, and the othersare detuned from it (carrier resonant condition). The length of SEC is 100 m, and detuning is bysteps 2π/1000. In both figures, the vertical units are arbitrary [168].

Isolators of this type have been used for room temperature vibration isolation in bothresonant-mass gravitational wave detectors and prototype laser interferometer detectors. Thedisadvantage of these isolators is that they are not suitable for use in high vacuum (unlessthe rubber is outgassed or packaged in metal bellows); moreover, they are not suitable foruses at low temperatures where elastomer materials harden. In addition, this type of isolatorgenerally shows large temperature coefficients and drift, due to the properties of the rubber,as well as having a relatively high corner frequency, limited by the compressive yield of theelastic elements.

Ideally, a multistage pendulum could be used to provide sufficient horizontal isolation fora laser interferometer gravitational wave antenna. Several groups have used double or triplependulum suspensions combined with rubber and steel isolation stacks [198–201]. However,the curvature of the Earth creates intrinsic cross-coupling between the horizontal and verticaldirections, due to the fact that the laser beam in an interferometer can only be perpendicular tothe local vertical at one location. In a large-scale laser interferometer detector, the magnitude

1396 L Ju et al

-300

-250

-200

-150

-100

-50

0

50

10-1 100 101 102

tran

sfer

fun

ctio

n dB

frequency Hz

(a)

Figure 45. (a) Theoretical transfer function of a five-stage vibration isolator. Each stage has anatural frequency of 2 Hz. (b) Experimental upper limit of performance of a five-stage isolator [326],with a typical seismic curve of 10−6f−2. At high frequencies the isolation reaches the noise floorof the sensing transducer of 3 × 10−15 m Hz−1/2. Below 58 H the mechanical resonant frequencyof the transducer, the sensitivity of the transducer degraded. Practically all the data above 50 Hzare transducer noise.

of the cross-coupling in ideal circumstances is ∼10−3. In real isolators the cross-couplingis likely to be degraded by mechanical imperfections. Thus it is important that the verticalvibration isolation should also be very high. The isolators must also be strong enough tosupport a total weight ∼102–103 kg.

In interferometer antennas, the corner frequency needs to be pushed as low as possibleto create the broadest possible bandwidth for observations. There is a great advantage inoperating an interferometer antenna at the lowest possible frequencies. Not only does itextend the range of sources accessible to the detector, but for specific sources such as binaryneutron star coalescence events, it increases the number of cycles of the coalescence thatcan be observed, thus allowing the signal-to-noise ratio to be increased. Giazotto pioneeredthe development of low-frequency vibration isolation. A very-large-scale multistage low-frequency superattenuator based on a gas spring was developed at Pisa [202, 203]. It had ahigh load-bearing capacity and a corner frequency of 2–3 Hz. However, gas springs havestrong temperature coefficients, and so such isolators have problems of thermal stability andcomplexity. At the University of Western Australia, tapered metal cantilever spring vibrationisolators [118], were developed, which showed excellent performance but somewhat highercorner frequency. The Pisa group replaced gas springs with similar tapered cantilevers andreduced the mechanical frequencies by use of magnetic antisprings, created by using magneticrepulsion between like poles of permanent magnets. This allowed low-frequency behavioursimilar to the gas spring system to be achieved [204].

5.4.2. Active isolation. Active isolation techniques have been investigated extensively ingravitational wave research [205–210]. The basic idea is simple, as shown in figure 46: therelative displacement between the test mass to be isolated and the suspension point (the errorsignal) is sensed and fed back to an actuator to servo the suspension platform so that themotion of suspension point follows the motion of the test mass. In this way, the motion of the

Detection of gravitational waves 1397

AAAAAA

x' - x

x0A

x

x'

transducer

filter +amplifier

m

l

-

Figure 46. Schematic diagram of an active pendulum [205].

suspension platform is reduced and hence the motion of the test mass. If we look at the simplecase of figure 46, the equation of motion of the mass is (for simplicity we ignore the dampingterm)

x = ω20(x

′ − x0), ω20 = g

l. (5.4)

Assuming the error signal is fed back to the suspension point so that x ′ = x0 −A(x ′ −x), thenthe transfer function of the system is

x

x0= ω2

0

−ω2(1 + A) + ω20

= ω20/(1 + A)

−ω2 + ω20/(1 + A)

. (5.5)

It can be seen that this system behaves like a pendulum with an equivalent resonant frequencyof ω0/

√(1 + A). At frequency f fo, the attenuation is [fo/f/(1 +A)]2 instead of (fo/f )2.

This means that the isolation of mass m is improved by a factor of (1 + A).The basic arrangement for vertical active isolation is shown in figure 47. The final transfer

function is slightly complex. However, from the block diagram (figure 48), it can be seen that

x = G′x ′, G′ < 1, (5.6)

x ′ = G[xo − H(x ′ − x)] G < 1, (5.7)

where GG′ is the passive transfer function of the system and H is the loop gain of the feedback.The closed loop transfer is then

x

x0= GG

1 + H(G′ − G′G). (5.8)

The performance of this active system is improved by a factor of [1+H(G′ −G′G)]. However,it is impractical to assume very large gain H to obtain high-performance isolation. There areseveral limitations. One is the presence of internal resonances of the isolation structure. Ateach resonance there will be a phase shift added to the loop transfer function which can makethe servo unstable. To ensure stability, the gain of the servo should be within the limit that

1398 L Ju et al

filter +amplifier

AAAAAA

sensor

actuator

m

M

x

x'

x0

k'

k

fc

A

-

Figure 47. Schematic diagram of the active isolation model [206].

+

-

+

xx'

xoG' G

H

Figure 48. Block diagram of the active isolator [209].

all the internal resonant peaks are below unity gain. Another limitation is the noise in thesensor. This noise is treated the same by the servo system as the error signal between test massand suspension platform. This noise will be amplified and fedback to the platform to create adisplacement. At low frequency, where the attenuation of the test mass from the platform isnot high, sensor noise plays an important role since the test mass tends to follow the movementof the platform.

5.4.3. Residual motion and the need for ultralow-frequency pre-isolation. A suspended testmass in a laser interferometer is required to be very precisely located to within 10−6 of anoptical wavelength. Residual motion of the test mass in interferometers operated to date istypically ∼few microns. This has caused two problems. First, it makes it difficult to acquirelock, because very large forces must be applied to decelerate the test mass and locate it withinoperating range. Second, it makes it impossible to act directly on the test mass (by say amagnetic actuator) because the electronic noise of the actuator circuit is never less than 109

times smaller than the maximum signal which can be applied. (This is the dynamic range ofa low-noise amplifier.) If the maximum signal is able to correct 10−6 m of motion, then thenoise level will create noise motion just 109 times smaller, i.e. 10−15 m. This is unacceptablylarge.

There are two possible solutions to the problem. One is to create a more complex servosystem and, in particular, to apply control forces to the stage above the test mass. All detectorsconstructed to date use this approach. The second possibility is to greatly reduce the residualmotion.

Residual motion, and in particular residual accelerations, can be greatly reduced if a

Detection of gravitational waves 1399

-200

-150

-100

-50

0

50

0.01 0.1 1 10

Tra

nsfe

r fu

nctio

n

frequency

Figure 49. Comparison of transfer functions of a multistage isolator with and without an ultralow-frequency stage. Dotted curve: five-stage 2 Hz isolator, solid curve: four-stage 2 Hz isolator plusa 0.1 Hz ultralow-frequency pre-isolator.

conventional isolator is suspended by a pre-isolator with a much lower resonant frequency.This is illustrated in figure 49, where the addition of a single low-frequency stage reducesthe amplitude of the normal mode peaks by about 50 dB. To realize this advantage requiresthe development of ultralow-frequency (ULF) mechanical suspension stages. As mentionedbefore, the isolation performance of an isolator with resonant frequency of fo above thecorner frequency is (fo/f )2. Thus the lower the resonant frequency fo, the better isolationperformance and the lower the corner frequency. This does not guarantee that with one orseveral ULF stages one can obtain very high isolation performance at high frequency. Theproblem is that internal mechanical resonances usually occur at frequencies that are typicallyabout 102–103 times the fundamental resonant frequency of the mechanical resonant structures.Thus, for example, a stage with 10−1 Hz resonant frequency is likely to have an internalresonance from 100 Hz upwards and these will corrupt its isolation performance.

As a result, ULF stages are best used as pre-isolation stages in conjunction with low-frequency isolators. For an N -stage isolator, there are N normal mode peaks with amplitudeup to 100 times the seismic background, depending on the Q-factor of the isolation elements.It is predominantly these normal modes which make it difficult to control and lock theinterferometer. As discussed in section 5.4.1, different methods have been used to damp thenormal modes, such as magnetic eddy current damping [200,211] and vibration absorber [212].

1400 L Ju et al

AAAAAAAA

AAAA

m

lflexures

AAA

AAAAAA

l1

l2

me1

me2

Figure 50. Simple model of an inverted pendulum. Figure 51. Simplified diagram of a folded pendulum.

A ULF-stage pre-isolator efficiently and passively reduces the normal mode amplitude asshown in figure 49. All the normal modes are on the cut-off slope of the ULF stage and aregreatly reduced, leaving one very-low-frequency mode which is easy to control. This greatlysimplifies the control system of the interferometer.

There are two main approaches to achieving very low frequency. One is to use a negativespring or antispring. A simple version is the inverted pendulum [213,214] as shown in figure 50.The resonant frequency of an inverted pendulum is given by

ω0 =√

k

m− g

l, (5.9)

where k is the spring constant of the flexure elements, m is the mass and l is the lengthof the inverted pendulum. Gravity provides a negative spring constant. The device hasa low frequency when g/l ∼ k/m, and it becomes unstable for g/l > k/m. Invertedpendulums for gravitational wave detectors were first investigated in UWA [213]. Because amechanical Hooke’s law spring is being used to counter a gravitational spring, and because thetemperature coefficient of length differs from the temperature coefficient of Young’s modulus,such antispring devices generally have relatively large temperature sensitivity. Thus servocontrol is needed to maintain a stable operating position. A full-scale inverted pendulum pre-isolation stage about 6 m high has been built for VIRGO’s supperattanuator [215], and a 1 mstage has been developed at UWA [216].

For horizontal isolation, more elegant devices minimize the contribution of mechanicalsprings. One such device is the folded pendulum [217–219]. It combines a positive andnegative pendulum as shown in figure 51. The resonant frequency of the folded pendulum isgiven as

ω =√

1

Me

(me1g

l1− me2g

l2

)+ γ , (5.10)

where γ is a small additional term which takes into account elastic contributions from theflexures, Me = me1 + me2 is the equivalent mass of the pendulum and l1, l2 are the lengths ofthe positive and negative arm respectively. This device has low temperature sensitivity and hasachieved a resonant frequency of 15 mHz [220]. With a resonant frequency of 17 mHz, thisdevice gives isolation of more than 90 dB at 10 Hz. Above 15 Hz, the attenuation is degradeddue to internal resonances of the isolator structure, as discussed above.

Another type of ULF stage is the X-pendulum [221, 222]. It uses two cross-wire linkagearrangements that mimic the motion of a very long pendulum. A resonant frequency of

Detection of gravitational waves 1401

suspension point

ellipticalsuspension point locus

sliding joint

rigid beam

support link mass

Figure 52. Scott–Russel linkage.

50 mHz has been demonstrated. With a resonant frequency of 0.21 Hz, the X-pendulum hasdemonstrated 30 dB isolation at 3 Hz. Above 3 Hz, the internal resonances dominate.

Both the folded pendulum and X-pendulum are most easily implemented as one-dimensional isolators. Although they can be cascaded to form a two-dimensional horizontalisolator, there are significant construction problems. Winterflood and Blair [223] have usedthe Scott–Russel linkage, shown schematically in figure 51. This device mimics the motionof a very long conical pendulum, achieving two-dimensional horizontal isolation in one singlestage. A full-size prototype pre-isolator [224] has demonstrated a resonant frequency of 7 mHzand with a resonant frequency of 17 mHz, it achieves an isolation exceeding 75 dB at 0.5 Hz.

It is interesting to note that only one of the four horizontal stages discussed above isdependent on a carefully designed spring. This is the inverted pendulum, where a gravitationalantispring balances on angular mechanical spring. In the other cases, mechanical springsare eliminated, except in so far as being an intrinsic, but small, component of a flexuresuspension. All the linkage-based devices have the advantage that the temperature coefficientsand nonlinearity of springs are minimized.

As mentioned above, cross-coupling requires a high level of isolation for both horizontaland vertical isolation. Thus there is little point in building an isolator with excellent horizontalpre-isolation unless it also has good vertical pre-isolation. Since the vertical isolation mustalways counteract a large gravitational force, it is almost impossible to avoid the use of largemechanical springs for vertical load bearing. However, several practical means of counteractingthe spring constant have been demonstrated. The first is the magnetic antispring, demonstratedat Pisa [204, 225]. Pairs of magnets in a repulsive arrangement create a potential hill in themiddle line of the magnet pairs. The negative spring constant represented by the potential hillpartially cancels the positive spring constant, substantially reducing the total stiffness of thespring. Unfortunately, most magnets have high temperature coefficients, so that the magneticantispring must be very carefully temperature controlled.

An alternation which avoids magnets is the geometric antispring. This was firstdemonstrated by LaCoste in his seismometer design [226], in which a zero length coil springsuspends a horizontal arm by acting on it at an angle (figure 53). The torsion crank linkagedeveloped by Winterflood and Blair [224] uses a different geometrical antispring concept, asillustrated in figure 54. This design makes use of the nonlinearity produce by the torsionarm connected to a suspension link. It is arranged so that the effective spring constant

1402 L Ju et al

(a) (b)

Figure 53. (a) LaCoste linkage. (b) Vertical preisolation using LaCoste linkage.

crank-armsuspensionlinks

torsion rod(end view)

mass load

angle

Figure 54. The torsion crank linkage arrangement [224].

k = ∂F /∂y(F = force, y = vertical displacement) is almost zero and constant in a certainrange. In a simple model, the torsion crank achieved a resonant frequency of 50 mHz. Thisvertical ULF stage can be combined with a Scott–Russel stage to create a three-dimensionalpre-isolator as shown in figure 55.

De Salvo has demonstrated a geometric antispring based around a pair of cantilever bladesprings [227]. The torsion rods of figure 54 are replaced by cantilever blades, and usingappropriated angled suspension wires the same geometry effect causes nulling of the springconstant for a certain deflection angle.

Figure 56 shows the residual motion predicted for a passive isolator being developed atUWA [216]. This system utilizes passive eddy current damping, as well as seismic tilt control.The predicted performance is 10−9 m rms above 0.2 Hz. If such an isolator can be realized,interferometer operation will be greatly simplified.

5.5. Thermal noise

Once the seismic noise cut-off is lowered sufficiently through the use of high-performancevibration isolators, thermal noise will become the critical source of noise. From the fluctuation–

Detection of gravitational waves 1403

mass load

gimbal

torsion rodsuspension link

crank-arm

torsion-crankverticalisolator

stage

Scott-Russelhorizontal

isolatorstage

support frame

Figure 55. Three-dimensional pre-isolation stage, consisting of a torsion crank vertical stage anda Scott–Russel horizontal stage.

Figure 56. Successive reduction of residual motion. The bottom curve is the predicted overallisolation performance with pre-isolation stage, eddy current damping and tilt control.

dissipation theorem [90] the general power spectrum of the minimal fluctuation force is

F 2th = 4kBT R(ω), (5.11)

where kB is the Boltzmann constant and R(ω) is the real part of the impedance of the system.Using Z = F/v, the above equation can be expressed as

x2th = 4kBT σ(ω)

ω. (5.12)

σ(ω) is the real part of the admittance Y (ω) = 1/Z(ω). For a simple harmonic motionsystem with spring constant k, mass m and a damping r , the thermal noise displacement powerspectrum is [228]

x2th = 4kBT r

(k − mω2)2 + r2ω2. (5.13)

1404 L Ju et al

If a damping mechanism of structural damping is assumed, which can be described with acomplex modulus of elasticity [229], k = k0[1+iϕ], ϕ = constant (i.e. r = kϕ), equation (5.18)becomes

x2th = 4kBT kϕ

ω[(k − mω2)2 + k2ϕ2]= 4kBT ω0ϕ

ωm[(1 − ω2

ω20)2 + ϕ2]

, (5.14)

where ω0 =√

km

is the angular resonant frequency.In a laser interferometer gravitational wave detector, thermal noise is mainly divided into

two classes—the suspension thermal noise and the internal thermal noise of the test mass.

5.5.1. Suspension thermal noise and Q-factor of the pendulum. Suspension thermal noise ismainly the Nyquist noise of the test mass pendulum of the interferometer. The Q-factor of apendulum suspension can in principle be very high because the energy storage is predominantlyin the effectively loss-less gravitational field. However, some elastic energy must always bestored in the flexure which supports the pendulum. The Q-factor of a pendulum is limitedby the losses in this element. The Brownian motion noise amplitude of a simple pendulum atangular frequency ω, for any normal mode at frequency ωp ω is given by

x2p(ω) = 4kBT ωp2

Qpmω5. (5.15)

The Q-factor of the pendulum is given by Qp = γQ0, where Q0 = 1/ϕ is the intrinsic Q-factor of the flexure material, and γ is the enhancement factor, which depends on the geometryand material of the pendulum flexure [228, 230].

Applying equation (5.20) to a simple pendulum, it has been shown that the thermal noisein a simple pendulum scales as [228]

x2th ∝ m1/2. (5.16)

It can be seen that the suspension thermal noise can be reduced by increasing the mass ofthe pendulum and the quality factor of the test mass pendulum stage. To obtain numericalestimates, consider a typical interferometer with parameters of L = 3 km, T = 300 K,m = 30 kg, fp = ωp/2π = 1 Hz, and Qp = 109. We then have

h = x

L∼ 10−21

(10 Hz

f

)5/2/√

Hz. (5.17)

This shows that extremely-low-loss pendulums are essential. For example, to obtain thesensitivity goals of 10−23/

√Hz at 10 Hz, a pendulum Q-factor of 1010 is required.

At current sensitivity levels, it is barely possible to measure directly the thermal noisefloor for high Q-factor test masses over the frequency range of interest for gravitational wavedetectors. The thermal noise has to be inferred from Q-factor measurement of some resonanceat other frequencies. However, the Q-factor and frequency relation is model dependent, andthe calculated thermal noise floor differs with different damping mechanism assumptions.Frequency-independent Q-factors over a large frequency range have been approximatelyconfirmed in some materials [214, 231–234]. The mechanism of structural damping is nowwidely accepted. However, it still needs more investigation, particularly to confirm thephenomenon in low-loss single-crystal materials.

Various types of suspension have been studied to obtain a high pendulum Q-factor. Mostresearchers have assumed that wires are necessary. Suspensions with a thin wire double-loopsimple pendulum configuration are widely used [199,200,235–237]. A pendulum Q-factor of

Detection of gravitational waves 1405

Q ∼ 108 with wire suspension is expected [238], assuming reasonable Q-factors for the wires.Pendulum Q-factors exceed 107 had been observed with fused silica fibre suspension [239].The highest pendulum Q-factor reported so far is >3 × 108 by Braginsky et al [240] with amonolithic silica fibre and a 30 g mass.

The simple pendulum suspension system inevitably will have a set of middle-frequencyviolin string modes from the suspension wire. Those modes will contaminate the windowof gravitational wave detection. The problem of the thermal noise of pendulums suspendedby wires has been intensively studied [232–234, 241–243]. Studies have shown that highpendulum Q-factor is related to a high violin string mode Q-factor. So each violin stringmode is confined within a very narrow frequency band.

Theoretical analysis has shown that significant improvements can be achieved if thependulum is replaced by a compound pendulum supported by a thin membrane which acts asa hinge [230]. A pendulum with hinge suspension has been shown to achieve Q ∼ 107 [244].Using known high-Q materials such as niobium which has a Q-factor ∼105 in thin membrane,and assuming that Q is independent of frequency, it can be shown that Q-factors exceeding107 [245] can be expected. Also, since the test mass pendulum is suspended by a very shortmembrane, the violin string mode will be high enough to be neglected. The membrane flexurecan be made much thinner than the wire flexure and thus has lower thermoelastic effect.

5.5.2. Test mass internal resonance thermal noise. The test mass will have many internalresonances. The total thermal noise thermal noise spectral density can be expressed as [228]

x2th = 4kBT

ω

∑i

ω2i ϕi

mi[(ω2i − ω2)2 + ω4

i ϕ2i ]. (5.18)

Here i is the index of the ith mode, mi is the effective of ith mode and ϕi = 1/Qi , where Qi

is the quality factor of the ith mode. Typical internal resonances of a test mass are at aboutseveral kilohertz. The thermal noise spectral density far below the internal resonant frequencyω ωi (assuming loss factor ϕi = ϕ = constant) is given by

x2th

4kBT ϕ

ω

∑i

1

miω2i

. (5.19)

Although it can be seen from the above equation that the lowest internal resonance contributesmost to the thermal noise, detailed theoretical studies [246, 247] has shown that higher-ordermodes cannot be neglected, especially when the laser beam is not perfectly aligned.

From equation (5.19) it can be seen that to reduce internal thermal noise it is requiredthat the test masses have high internal resonances and very high Q-factors (1/ϕ). Since ωi

is proportional to the velocity of sound, high sound velocity materials are required. It isworthwhile pointing out that the dimensions of the test mass should not be too large so asto keep the frequency of internal resonances as high as possible. This is contradictory tothe requirement of using a big test mass to reduce the pendulum thermal noise. There is acompromise in choosing the size and mass of the test mass. High Q-factor materials (suchas quartz and silicon) have been investigated in prototype laser interferometer detectors [248].At present, fused silica test masses are widely used in prototype interferometer detectors. TheQ-factor of silica test masses were observed to be of the order of 106 [249,250] with a highervalue of >107 in some resonant modes recently observed by Beilby and Startin [251]. Thehighest Q-factor of silica was reported by Braginsky of 6 × 107 [252].

Using another highQ-factor material, sapphire, as test mass has been proposed [230]. Theexcellent thermal and mechanical properties of sapphire makes it a promising material for usein test masses. Compared with silica material, the high thermal conductivity means the thermal

1406 L Ju et al

lensing problem can be minimized, while the very high Young’s modulus means the internalresonance will be high. The highest reported sapphire Q-factor is 4 × 108 by Braginsky [112]and recent measurements reported a reproducible result of 2–3 × 108 Q-factors [239, 253].Theoretical analysis [254] suggests that the thermal noise can improve by a factor of >10 byusing sapphire test mass instead of silica test masses.

In practice, the suspension joint to the test mass plays an important role in obtaining bothhigh-Q pendulum motion and high internalQ-resonance. Monolithic suspension systems havebeen suggested and investigated. These include bonding the silica fibre to the silica test massusing silicate bonding method [255], and bonding niobium flexure to sapphire test mass usingactive alloy bonding [256] and possibly silicate bonding [257].

5.6. Control systems

As mentioned above, the test masses in the interferometer must be suspended to isolate againstseismic and environmental vibration. To achieve operation of an interferometer a feedbacksystem must be used control the test mass position to high precision. Firstly, a local controlsystem is necessary to suppress large-scale motions and align the mirrors to the point wherethe best interference contrast may be achieved. Secondly, to allow maximum sensitivity, aglobal control system is required to control the interferometer arm lengths to a relative motionof ∼10−12 m rms [183]. This is achieved by locking the interferometer to a dark fringe.

5.6.1. Local controls. Various damping methods to suppress low-frequency normal modein vibration isolators have been investigated. Passive damping such as magnetic eddy currentdamping has been investigated [200, 211]. The problem with this type of damping is that thedamping is usually relative to a support structure and can introduce noise into the isolatorsboth through seismic noise and resonant peaks of the support structure. This degrades thehigh-frequency performance of the isolator.

An alternative method of magnetic damping uses the narrow-band resonant absorber. Inthis case, individual modes of an isolator can be damped using a tuned resonator which is itselfmagnetically damped.

The low-frequency normal modes can also be attenuated by active damping [162, 199,258–261]. A correction force corresponding to the motion of the suspended masses (measuredby an inertial or non-inertial sensor) is applied to the appropriate part of the suspensionsystem. A non-inertial sensor such as shadow sensor [162] is widely used on prototypeinterferometers [162, 163]. It consists of a small vane mounted on the sensed surface, a light-emitting diode (LED) and an opposing photodiode (PD) mounted on the reference surface.The vane is free to move between the LED and PD and develops a signal proportional to thedisplacement of the vane by partially interrupting the light. The correction forces are applied bya small permanent magnet mounted integrally with the vane, and a coil mounted on the referencesurface. The reference surface is usually the support frame, which is attached to the ground.Then the problem arises that the seismic noise can be injected into the servo loop both in sensingand in force actuation. If motion sensing is done with respect to the frame, it is impossibleto avoid seismic noise in the sensing. This sensing noise has components in the signal bandwhich must be prevented from driving the test mass and appearing in the signal output. Thisis readily achieved by electronic filtering of the signal [162,260]. The noise injection throughthe coil can be overcome by carefully positioning the coil so that the magnetic field gradientis maximized at the magnet. This results in the best decoupling of the forces applied on thecontrolled masses [162]. Another way to overcome frame vibration is to use two coils tolinearize the field and create a magnetic force which is independent of the position and motion

Detection of gravitational waves 1407

of the frame [260]. If the reference surface is a reaction mass [262] which is well isolated fromthe seismic noise, the noise injection will not be such a problem. However, such a sensor willnot damp the common-mode motion of the test and reference masses. An inertial sensor suchas a mass-loaded piezoelectric accelerometer [263] or a wideband accelerometer [264, 265]can in principle avoid the problem of sensing frame vibrations. Inertial sensing avoids anycoupling with seismic noise. Its ultimate sensitivity is set by electronics noise, thermal noiseand thermal drifts of the accelerometer and the feedback actuators.

Another kind of non-inertial sensor is the capacitance transducer [266]. A convenientimplementation for use on dielectric test mass surfaces consists of two interleaving combs ofparallel conductors etched onto a circuit board. The dielectric constant of the sensed surfacecontributes to the capacitance between the combs. A voltage applied to the capacitor will exerta force on the test mass, while motion of the test mass modulates the capacitance. Thus thissystem can provide both sensing and feedback forces. There is no need for lossy magnets orvanes to be attached to the test mass, so the test mass acoustic losses need not be degraded [249].

5.6.2. Global controls. It is necessary to extract error signals to control the interferometerlengths or mirror positions to lock the output at the dark fringe and the recycling cavityor arm cavities on resonant with the laser frequency. In order to avoid the noise due tothe laser power fluctuations and 1/f electronics noise at low frequencies, the measurementhas to be shifted to the quieter MHz domain using modulation–demodulation techniques.Various modulation configurations have been proposed and extensively studied. Schemesknown as external [176, 267–269] and frontal [270–274] modulation have been particularlyinvestigated because their modulators are outside the interferometer. Thus, as opposed tointernal modulation, these schemes avoid introducing losses or wavefront distortion withinthe interferometer [275]. Although quantum-noise-limited sensitivity has been achievedwith internal modulation at low laser power level [276–278], all the proposed large-scalelaser interferometer detectors will not use this configuration because it introduces losses andwavefront distortion.

In the external modulation configuration (figure 36), a reference beam from theantireflective coated face of the beamsplitter is extracted, phase modulated and mixed with themain interference as in a Mach–Zehnder interferometer. This configuration needs additionaloptical components that, in order to avoid noise, must be suspended. The length of thereference Mach–Zehnder arm must be controlled to maximize the phase sensitivity. Themodulation index m in the external arm is an independent variable and can be set to maximizeJ1(m) when m reaches its optimum value of 1.84 radians. A larger modulation index leadsto more power being transferred to higher-order sidebands, which are normally not utilizedby the demodulation process. Furthermore, in power-recycling and arm-cavity configurations,another modulation is needed to extract error signals to control the recycling mirror and armcavities. Because the arm lengths of the interferometer can be equal and because the referencebeam extracted from the back face of the beamsplitter has travelled almost the same path lengthas the main beam, this configuration is insensitive to the laser frequency noise.

In the frontal modulation configuration (figure 57) the interferometer has two armswith a small difference in length. The laser beam is phase modulated before entering theinterferometer. The laser field at the interferometer input may be expressed as a superpositionof three monochromatic plane waves if the modulation index is not too high: the carrier with theoriginal laser frequency and two sidebands with the frequency shift of the modulation frequencyfm. In the simple Michelson, when the dark fringe condition is fulfilled for the carrier, thesidebands transmitted to the interferometer are maximum when arm length difference is aquarter of the modulation wavelength, c/4fm.

1408 L Ju et al

To laser Frequency Feedback

From Laser

Summing Node

I Q

I

Q

Phase Modulator

Inverting Amplifier

PD1 PD2PD3

I In-Phase Q Quadrature

Mixer

Figure 57. Global control configuration with frontal modulation [272].

If a recycling mirror is added, the modulation frequency and arm length difference mustfulfil two conditions. First, the sidebands must resonant in the recycling cavity, otherwise theeffective modulation index will be very small. Second, the signal must have maximum phasesensitivity. If Rr and Rm are respectively the intensity reflectivity of the recycling mirror andthe interferometer, the modulation frequency fm and the arm length difference l have tosatisfy the following condition [273]:

cos

(2πfm

l

c

)=√RrRm. (5.20)

This condition also ensures the optimum enhancement of the modulation index inside therecycling cavity.

Even with cavities in two arms, one modulator is enough to get all the error signals tocontrol each mirror. This configuration is simple and rather easy to realize. But because of thenonsymmetric configuration, the interferometer is sensitive to laser frequency and beam jitternoise. Frontal modulation was first demonstrated by successfully locking a tabletop prototypeof a power-recycled Michelson interferometer with Fabry–Perot cavities in the arms [272].

5.6.3. Analogue and digital controls. Implementing the control discussed above can beachieved by either analogue or digital control systems. Analogue control is a well developedtechnology. Conventional PID servo control techniques can be used. Most prototypeinterferometers use analogue control for local damping, alignment and fringe locking [279].The results have been successful, with sensitivity of 3 × 10−19 m Hz−1/2 achieved [164] andgenerally not limited by servo system noise. However, for large-scale interferometers likeLIGO and VIRGO, there are a large number of degrees of freedom (more than 200) [280]which need to be controlled and many of them are coupled each other. It is also planned thatmany error signals will be monitored and archived, to allow cross-correlation with the signalfrom the interferometer output PD. Many automated features are also required. All of the abovepoints incline towards the use of digital control systems. Since Barone et al [281] introducedthe idea of digital control into the automatic alignment of a Michelson interferometer, theyhave demonstrated theoretically and experimentally that all the specification on the noiserequirements, the dynamic range and the control bandwidth can be satisfied using all-

Detection of gravitational waves 1409

digital control systems [282]. A fully digitally controlled interferometer prototype has beensuccessfully operated in Naples, Italy for development and test of some VIRGO subsystems.

Heflin and Kawashima [283, 284] implemented both analogue and digital alignmentcontrol for the TENKO-100 DL interferometer. The results indicate that digital feedbackcontrol for application onto interferometer systems with target search frequency ∼1 kHz orless is a feasible alternative to analogue feedback systems. It has been shown to reproducemany of the same desirable features as the analogue systems.

Improved vibration isolation, especially that with very low residual motion, is likely toallow the undoubted complexity of the control problems discussed above to be substantiallysimplified.

5.7. Laser stabilization

Laser interferometers are designed to be sensitive to the optical phase difference of two arms,and should not be sensitive to common-mode fluctuations of the input light. But in practice,because of asymmetry between the two arms, fluctuations in the input light will couple into theoutput signal. In the frontal modulation scheme some asymmetry is unavoidable. In addition,differences in the optical components will cause, for example, intensity fluctuations to giverise to a differential radiation pressure force between both arms. Thus, laser intensity andfrequency fluctuations must be strongly suppressed.

5.7.1. Laser pre-stabilization. The laser frequency noise f can couple into interferometerphase fluctuation ϕ via arm length difference L [162]:

ϕ ∼= 2πfL/c. (5.21)

The interference of the scattered light with the main beam can also couple the frequency noiseinto the output signal [182]. To ensure that frequency noise is sufficiently low that it does notcompromise the sensitivity in the signal frequency regime, an active stabilization system isnecessary to reduce the laser frequency noise.

An effective method of laser frequency stabilization was proposed by Drever and Hallet al [285], modelled on a technique used in microwave systems which was first proposedby Pound [286]. In optics the technique is called Pound–Drever–Hall (PDH) modulation.Light incident on a cavity is frequency modulated. The cavity creates intensity modulationwhose phase depends on the relative frequency between the laser and the cavity resonance. Thereflected light can be thought of as containing two beams: light simply reflected from the cavityinput mirror and the light which has entered the cavity, resonated in the cavity and leaked backvia the input mirror. The cavity leakage has a strong phase shift with respect to the directlyreflected light from the input mirror, depending on the detuning from the cavity resonance. Theinterference between these two beams allows the detection of the phase difference, and hencethe frequency difference, of the laser frequency compared with the cavity resonant frequency.The light from the laser is usually modulated at a RF frequency to shift the measurement tothe quiet high-frequency domain. This helps overcome technical noise such as low-frequencyelectronics noise.

The schematic diagram in figure 58 shows a typical laser frequency stabilization scheme.A small fraction of laser light is phase modulated at a radio frequency by a Pockel cell (PC)and injected into a reference cavity. The reflected light is detected by a PD and mixed with theRF reference signal. If the laser frequency is tuned to one of the cavity resonances the reflectedlight has two balanced sidebands with opposite phase and a carrier. The mixer output is zero.If the laser frequency fluctuates the two sidebands will be unbalanced and the mixer will give

1410 L Ju et al

ReferenceCavity

Laser

Isolator PBS

PBS

λ/4

To laserinterferometer

PC

RFOscillator

Mixer

Figure 58. Frequency stabilization.

a signal proportional to the laser frequency difference with respect to the cavity resonancefrequency. This signal can be fed back to a piezomirror (for low-frequency correction) andan extra-cavity Pockel cell (for high-frequency correction) to lock the laser frequency to thecavity.

The performance achievable in PDH stabilization is theoretically set by the quantumlimit: the balance of shot noise and radiation pressure fluctuations which deform the referencecavity. However, in practice the laser shot noise limit is always dominant. This sets a limit tothe frequency noise spectral density Ssn, and if the cavity has no losses this is given by [287]

Ssn(Hz/√

Hz) = ν

J0(β)

√hν

8ηPi

. (5.22)

Here ν is the cavity linewidth, Pi is the power incident on the cavity, ν is the laser frequency,η is the quantum efficiency of the PD, β is the modulation index, and J0(β) is the zeroth Besselfunction.

Every gravitational wave detection group is involved in the development of laserstabilization. A frequency noise on the order of ∼10 mHz Hz− 1

2 at 1 kHz has been achievedwith a diode pumped Nd:YAG laser actively stabilized to a rigid reference cavity [288–291].

Another noise source in PDH locking is the mechanical and thermal noise of the referencecavity. The cavity resonant-frequency fluctuation ν is directly linked to the cavity lengthfluctuations L. That is,

ν

ν= L

L. (5.23)

If the reference cavity length is very long the noise contribution from a given cavity lengthfluctuation is much less than that of a short cavity. For this reason, the laser frequency is oftenstabilized to one arm cavity [163,289] of an interferometer for further stabilization of the lightfrequency, and a frequency noise on the order of ∼10 µ Hz Hz− 1

2 at 1 kHz has been achieved.In recent designs special-purpose mode cleaner cavities have been used [292] (see below) as areference cavity. The Max Planck group has stabilized to an interferometer’s power recyclingcavity [173].

5.7.2. Mode cleaners. It is inevitable that the laser beam has geometric fluctuations (beamjitter) because of the vibration of the laser cavity. Vibration of the injection optics may also

Detection of gravitational waves 1411

introduce the beam jitter. The beam jitter noise will couple into the interferometer output signalif the two arms are not perfectly symmetric. For example, if the beamsplitter is misalignedfrom the optimal symmetric orientation by a small angle θ the lateral movement x of incidentbeam may produce an effective differential displacement signal s given by [293]

s = 4θx. (5.24)

A simple way to clean the beam jitter noise of a low-power laser beam is to pass it througha single mode fibre [162]. In a single-mode fibre geometric fluctuations are transformedinto fluctuations of the power coupling into the fibre. The problem in that because the coreof a single-mode fibre has a small radius (of the order of a few wavelengths) the intensityinside the core is very high and this generates nonlinear effects including stimulated Brillouinscattering [294, 295]. This sets an upper limit on the maximum power that can be transportedin a single-mode fibre. In addition, fibres themselves can be subject to vibration which canintroduce additional beam jitter.

Another method to suppress beam jitter is to use a long optical cavity in transmission, calleda mode cleaner. The geometric fluctuations of the laser beam are suppressed because they arenot resonant within the cavity. The geometric beam fluctuations can be described in terms ofhigher transverse modes of the resonant cavity. In rectangular coordinates Hermite–Gaussianfunctions can be used to describe the eigenmodes of a cavity to a good approximation. Sincethe Hermite–Gaussian modes form a complete set, an incident beam with small translationalmovement x, angular fluctuation θ , beam waist size mismatch ω0, and beam waistposition mismatch b can be expanded in terms of cavity eigenmodes [292] as follows:

Ein = E00

(x

ω0+ j

kω0

2θ

)E10 +

1√2

(ω0

ω0+ j

b

kω2

)(E02 + E20). (5.25)

Here Eij are the amplitudes of the fundamental, first and second eigenmodes, while Ein is theamplitude of the incident beam and ω0 is the beam waist.

For a cavity consisting of two mirrors with radii of curvature Rc1, Rc2, intensitytransmission T1, T2, and reflectivity R1, R2, separated by a distance L, the fractionaltransmission of the incident light amplitude through this cavity is [296]

Tmn =√T1T2

1 − √R1R2

1√1 + (

2√R1R2

1−R1R2sin((m + n)L))2

, (5.26)

where L = cos−1(√(1 − L/Rc1(1 − L/Rc2).

For the fundamental mode, T00 =√T1T2

1−√R1R2

. The amplitude attenuation of the higher-ordermodes compared with the fundamental mode is given by√

1 +

(2√R1R2

1 − R1R2sin((m + n)L)

)2

≈ 2F

πsin((m + n)L), (5.27)

where F =√R1R2

1−R1R2is the finesse of the cavity. Thus a high-finesse cavity mode cleaner can

strongly suppress the higher-order modes.However, the power inside a high-finesse mode cleaner cavity will also be much higher

than the incident light power. Thus, thermal damage to the mirror coating becomes a key issuein defining the cavity configuration. If P is the power we want to transmit, and ω is the beamradius on the mirrors, then the power density on mirrors is given by I = 2FP

π2ω2 . To avoid opticaldamage to the coatings of the mirrors, the spot area should be kept above πω2

min = 2FPπImax

, whereImax is the power density limit of the coatings.

1412 L Ju et al

The power transmission of an optical cavity for incident light with frequency offset f fromthe resonant frequency is given by the well known Lorentzian response: Tf = T0

1+(2f/v)2 , whereν = c/2LF is the cavity linewidth. The cavity acts as a second-order low-pass filter. Witha large length L and high finesse F the cut-off frequency becomes low. For example, for L =100 m andF = 1000 the cut-off frequency is 1.5 kHz. Since all the laser fluctuations (amplitudeand frequency) can be understood in terms of the generation of sidebands, a mode cleaner alsoacts to suppress such fluctuations at offset frequencies higher than the cut-off frequency.

5.8. Optics

As we have already seen, a fundamental limitation of the sensitivity of interferometricgravitational wave detectors is shot noise, or photon-counting errors. In order to reduceshot noise, high light power must circulate in the interferometer. Because of the limitation oflaser power currently available from stabilized CW lasers, power recycling appears to be anindispensable technique for large-scale instruments.

The maximum power recycling factor is determined by the total losses of the interferometerwhen the transmission of the recycling mirror is properly chosen. The losses of the interferom-eter may result from energy loss due to absorption or scattering of the mirrors, or by imperfectrecombination of the two beams on the beamsplitter due to misalignments and wavefront dis-tortions. Power losses can arise from imperfect mirror surfaces, coating inhomogeneities,diffraction losses through limited apertures, beamsplitter and mirror substrate wavefront dis-tortion and depolarization. The minimization of these losses presents a formidable challenge.

5.8.1. Surface quality. The relevant specifications of the surface figure for opticalcomponents has been considered extensively by all of the groups building large-scalelaser interferometers [196, 297–299]: see, for example, the discussion by Winkler and co-workers [196], which is based on a requirement for a power build-up in a power-recycled DLinterferometer. The surface deformations can be characterized by their amplitude s and spatialwavelengthλs . Surface deformation withλs smaller than the beam diameter (micro-roughness)cause the scattering loss. The relative power loss by scattering due to the micro-roughness isgiven by

P

P=(

4πsrms

λ2

)2

, (5.28)

in which srms is the rms value of the micro-roughness amplitude. The tolerable micro-roughnessin a DL interferometer with power recycling gain of 100 and 34 reflections is srms < λ/730.

Surface deformations with λs in the order of the beam diameter contribute to the beamwavefront distortion and the deterioration of the dark fringe of a perfect interference. Thetolerable deformation in this scale for the same interferometer is srms < λ/230. For surfacedeformation with λs larger than the beam diameter (aberration) the demands are slightly less.

Using the computer-mode-based code of Vinet and Hello [297], LIGO group derivedthe requirements [299] of the optical components of the 4 km Fabry–Perot Michelsoninterferometer. The tolerable rms amplitude of the surface micro-roughness is 0.4 nm. Thesurface figure error (spatial wavelength larger than beam diameter) should be less than 0.8 nm.

Techniques of superpolishing are now well established and mirrors exceeding thetight specifications required have been developed [299–301]. LIGO optics is beingpolished by General Optics (GO) and Commonwealth Scientific and Industrial ResearchOrganization (CSIRO). Metrology indicated that the polished surfaces with rms deviation(after removeing focus and astigmatism)<1 nm over 20 cm were produced. Surface roughness

Detection of gravitational waves 1413

measurements showed a micro-roughness of ∼3 Å for CSIRO substrates and ∼0.9 Å for LIGOsubstrates [299].

The mirror coating is also critical for scattering level and wavefront preservation inthe interferometer. VIGO group has reached ∼1 ppm scattering level on 80 mm diametercoated mirror surface. The peak-to-valley coated surface deformation is ∼14 nm on 70 mmdiameter [302].

5.8.2. Absorption. Optical absorption may take place at dielectric coatings, or insidethe substrate material of transmitting optical components, such as the beamsplitter and thecoupling mirror of Fabry–Perot cavities. Apart from losses introduced by the absorptionitself, the absorbed light power will heat the local area, and consequently deform the opticalcomponents causing wavefront distortion. This arises due to the limited thermal conductivityof the substrate. The local heating forms a temperature gradient inside the substrate andconsequently introduces radius of curvature changes on surfaces. Winkler et al [196] hasanalysed the problem and shown that the change δs of the sagitta by the absorption of thereflected beam is given by

δs = α

4πκPa, (5.29)

in which Pa is the absorbed light power, α is the thermal expansion coefficient, κ is the heatconductivity of the substrate material. The crucial quantity for the magnitude of the effect isthe ratio α/κ of thermal expansion to heat conductivity. The α/κ of fused silica and sapphireare 33 and 28 respectively [196].

When the beam is transmitted through a material with a temperature gradient, because ofa temperature dependence of the refraction index, the refraction indices of the beam axisand the outer parts of the beam are different and thermal lensing may result. The pathdifference δl between beam axis and outer parts of the beam, introduced by thermal lensing,is approximately [196]

δl ≈ β

4πκPa, (5.30)

in which β = δn/δT is the temperature dependence of the refraction index and Pa is the powerabsorption there. It is clear that one wants to keep the ratio β/κ small. The β/κ of fused silicaand sapphire are 1000 and 60 [196]. As regards thermal lensing, sapphire is much better thanfused silica as a substrate material.

5.8.3. Depolarization. The temperature gradient inside the substrate will also produce a stressdistribution. The stresses generate strains in the substrate, which in turn produce refractiveindex variations (or birefringence) via the photoelastic effect. Stresses may also be introducedduring the manufacturing process. The magnitude of the birefringence may be defined by thephase difference δ introduced between orthogonal polarizations. In general, the polarization ofthe input beam will not be parallel to one of the principal axes of the birefringent component,especially since the birefringence may vary locally.

Power loss occurs due to the depolarization of the original input beam polarization. Themeasured birefringence of a very homogeneous 10 cm thick Corning 7940 grade 0A fused silicaplate is δ = 1.2 ±0.2 in the central area [303] which corresponds to a maximum loss of 10−4

per pass. The reported birefringence in sapphire is comparable, about 0.1 cm−1 [304]. Thelowest birefringences in coatings obtainable today are between 2 to 10 µrad per reflection—butso far only for mirrors and beams with a size of a few cm and mm, respectively [197].

1414 L Ju et al

Hello and Vinet [305, 306] have analysed the thermal effects in massive mirrors heatedby high-power laser beams. The complete numerical simulation of thermal effects in a GWinterferometer [307] indicate that the first-generation detectors using silica components mayrequire substrate absorption coefficients of ∼2 ppm cm−1 ± 15%, and coating absorptioncoefficient of ∼2 ppm ± 10%.These values are within present capabilities for mirror andcoating technologies. The reported lowest absorption is a level of less than 1 ppm cm−1

for light of 1 µm wavelength in a fused silica sample [308] and a level of 3.1 ppm cm−1 in asapphire sample [309]. The present coating absorption coefficient is ∼1 ppm [302]. Recyclingfactors of 300 [173] and 450 [174] have been achieved with power of ∼100 W built up insidethe recycling cavity. This corresponds to >1 kW cm−2 power on the beamsplitter and mirrors.Expected limits due to thermal effects have still not been reached.

Winkler et al [196,197] and Strain et al [308] with their co-workers at MPI have evaluatedthe effects of thermal deformation, thermal lensing and thermally induced birefringence ona recycling interferometer. Their model assumes that Gaussian-profile laser beams heat theoptical substrates through either uniform bulk absorption or uniform absorption in the coatings,and that the optics has an aperture much larger than the beam diameter. They concludedthat the power limit set by thermal lensing is a problem only for advanced interferometersoperating at higher power than those presently under development. They find that inherent andthermally induced birefringence will not be the dominant loss mechanism [197] assuming thelowest values for absorption and inherent birefringence reported. They propose that resonantsideband extraction is the best way of reducing the effects of thermal lensing to reach sensitivityappropriate to a ‘second-generation’ detector [308].

5.9. High-power lasers

The choice of the laser wavelength is an important effect on the design of a long-baselineinterferometer. At short wavelengths, the beam diffracts less and thus the diameter of theinterferometer mirrors and the vacuum tube can be reduced; thereby reducing the cost ofthe interferometer. Short wavelengths would also allow a better shot-noise-limited strainsensitivity which is ∝λ1/2 [189]. Other important factors in the choice of laser wavelength arethe losses (absorption and scattering) in the mirrors and beamsplitter, and the availability ofsuitably quiet and powerful lasers.

Contrasting with prototype interferometers which have used argon-ion lasers, all the long-baseline interferometers will use diode-laser-pumped Nd:YAG lasers (λ = 1064 nm). The useof Nd:YAG lasers is driven by their much better efficiency and generally quieter characteristicswith regard to practically all types of laser noise [293]. Diode-laser-pumped miniature ringlasers can fulfil the requirements of a interferometric gravitational wave detector concerningamplitude and frequency stability [310]. But direct use of these devices in an interferometerfor gravitational wave detector is not possible, because the output power of these system islimited to values below 2 W CW in a single axial mode [311].

One possible technique to increase the output power is injection locking which coherentlycouples a low-power master and a high-power slave oscillator resulting in a high-poweroutput with the frequency characteristics of the master [312]. As shown in figure 59, thisis accomplished by injecting the output power from the master laser into the slave laser’sresonator. The PDH reflection locking is used to lock the frequency of the slave laser to themaster’s by adjusting the position of the slave’s mirrors according to the error signal [311].

Application of the injection locking technique to Nd:YAG lasers has been extensivelyinvestigated [311,313–317]. At Laser Zentrum Hannover (LZH), a maximum single-frequencyoutput power of 20 W has been generated by injection locking to a monolithic ring laser. The

Detection of gravitational waves 1415

Slow PZT

Feedback electronic

Output

Phase modulator

Master laser

Photodetector

λ/4

λ/2

λ/2

Faraday isolator

Laser head

Mixer

Oscillator

Slave laser

Fast PZT

Figure 59. Schematic for injection locking of two lasers [311].

amplitude and frequency stability is investigated. The amplitude noise reaches the shot noiselevel beyond a few MHz. The relaxation oscillation of the miniature ring laser coupling at∼700 kHz and the disturbance in the excitation of the slave laser caused by the diode laserpower supplies at 80 kHz are two noise sources which could be further stabilized using activeamplitude stabilization [318]. Using fibre-coupled diode lasers as pump sources, 62 W CWTEM00 mode output has been achieved [319].

At Stanford, an output of 5.5 W single-frequency, ‘nearly diffraction-limited’, TEM00

power was produced by using 50.4 W of pump power. The frequency noise of the unstabilizedmaster laser of ∼20 Hz Hz− 1

2 at 1 kHz was reproduced at the output of the slave. The relativeintensity noise at the output of the slave is 1.7 × 10−6/

√Hz which is 10 times higher that that

at the output of the master laser [314]. A 40 W CW, TEM00 diode-laser-pumped, Nd:YAGminiature-slab laser has been built and demonstrated with 212 W pumping power [320]. A 10 Wlaser-diode-pumped Nd:YAG master-oscillator power amplifier is spatially and temporallyfiltered by a fixed Fabry–Perot cavity, which produced a 7.6 W TEM00 beam with 1% higher-order transverse mode content and reduced the relative power fluctuations at 10 MHz to2.8 × 10−9/

√Hz.

At VIRGO, a 10 W laser-diode-pumped Nd:YAG laser has been developed by usinginjection locking a high-power slave laser to a low power master laster [302]. The master laseris a 700 mW laser-diode-pumped miniature ring Nd:YAG laser operating at single-frequency.The slave laser is Nd:YAG laser transverse pumped on one side by 10 fibre-coupled diodes.TEM00 operation has been achieved with a slight contribution of TEM01 using a diaphragminside the X-shaped ring cavity. Using a spatial filter, the TEM00 component can be extractedto give a 9 W output power for an effective pumped power of 60 W. The 10 W laser will befrequency pre-stabilized to a reference cavity, and be actively power stabilized.

At TAMA, with 22.3 W pump input from two fibre-coupled laser diodes, combined with700 mW of power injected by a single-frequency master laser, the injection-locked slavelaser emitted 10 W of linearly polarized TEM00 beam. The measured relative intensity

1416 L Ju et al

noise and the frequency noise are 2 × 10−5/√

Hz and 50 Hz Hz− 12 respectively. When the

frequency of the injection-locked laser is stabilized to an external high-finesse reference cavity,a minimum frequency noise of 40 mHz Hz− 1

2 was measured from the locking loop error signalat 1 kHz [313].

At Adelaide, an efficient, medium power, diode-pumped Nd:YAG slab, stable resonator,ring laser based on a new diode-pumping geometry [321] was developed. Using 18 W ofabsorbed laser diode power (20 W diode output power), 5.8 W TEM00 output beam has beenproduced [189].

6. Conclusion

Gravitational wave researchers have expected to detect gravitational waves ‘within the nextdecade’ for the last three decades. Detectors have been dramatically improved and a steadilyincreasing band of physicists has been able to devote more and more resources to the problem.In the process they have uncovered new physics and new technology. Gravitational wavedetectors are the most sensitive devices ever invented.

Like the Great South Land which was rumoured for centuries before it was discovered,the spectrum of gravitational waves is a rumoured continent, first to be detected, and then tobe explored. It seems not unreasonable that the exploration will begin within the next decade,but whatever happens the search will continue to motivate physicists and drive a continuingprocess of innovation.

Acknowledgments

The authors wish to acknowledge the Australian Research Council, which has supportedgravitational wave research in Australia for many years, and our colleagues in Perth andaround the world with whom we have had rich and rewarding interactions. The field is toolarge to be able to reference all important papers, and we apologize to those whose work wehave failed to reference.

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detectors Rev. Sci. Instrum. 65 3482

Gravitational-wave astronomy: new methods of measurements

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Abstract. The current status of gravitational-wave astronomy isreviewed. Advances in ground-based antennae are discussed andnew methods of measurement that promise a considerable im-provement in sensitivity are described. The promise held out bythe improved antenna technologies is discussed.

1. Historical introduction. Searching for sourcemodels and main characteristics of antennae

The purpose of this paper is to describe to the reader the stateof the art in the part of astrophysics usually called gravita-tional-wave astronomy. This part includes the programs ofstudies and developments, as well as the discussion ofunsolved problems, which are related to ground-basedantennae using free masses. During the next 4 ± 8 years,these antennae are expected to bring qualitatively newastrophysical information: the registration of bursts ofgravitational radiation from astrophysical catastrophes thatoccurred hundreds of millions of years in the past. The mainpart of the paper is devoted to the key problem, the free-massantenna sensitivity, and to the development of qualitativelynewmethods of measurements. The first introductory sectionis devoted to the reader not acquainted with this part ofphysics, and also includes a short historical description of thedevelopment of gravitational-wave astronomy. Sections 2and 3 give the state of the art in the construction of ground-based antennae as of the moment of writing of this paper andpoint to possible ways of solving some non-trivial problems of

measurements, which determine the antenna's sensitivity. InSections 4 and 5 we give a short description of possibleconsequences resulting from successive operation of anten-nae, for gravitational-wave astronomy and for other fields ofphysics.

The emission of gravitational waves by masses movingwith alternating accelerations, predicted by A Einstein in1918 (see [1] and also [2, 3]), was first met without interest byexperimentalists. The reason for this was the smallness of theeffect for any reasonable laboratory experiment. The powerof gravitational radiation for small derivatives of theacceleration is

_Egrav G

45c5

q3Dab

qt3

2

; 1

where G is the gravitational constant, c is the speed ofelectromagnetic and gravitational wave propagation, Dab isthe quadrupole moment of masses

Dab V

r3xaxb ÿ dabr2 dv : 2

The factor Gcÿ5 10ÿ60 (in CGS units) is due to thequadrupole character of radiation from gravitational`charges' (gravitational masses). The quadrupole character,in turn, is the consequence of the experimental fact, usuallycalled the equivalence principle, which is one of the postulatesof general relativity (GR). For two equal point masses Mseparated by a distance l and rotating around the barycenterwith a frequency o, equation (1) takes the form

_Egrav 128

5

G

c5M2l 4o6 ; 3

which shows that forM 106 g, l 102 cm,o 3 102 sÿ1,the value of _Egrav is as small as 10ÿ23 erg sÿ1.

It is such estimates that underlay the pessimism ofexperimentalists in detecting this radiation in the laboratory.

Now it seems evident that the smallness of the factorGcÿ5

can be `compensated' by a high value of the factor M 2, ifM ' 1033 g (i.e. of order of the solar mass). In other words,one can expect a significant values of _Egrav from astrophysical

V B Braginski| MV Lomonosov Moscow State University,

Physics Department,

Vorob'evy gory, 119899 Moscow, Russian Federation

Tel. (7-095) 939-55 65

Received 12 April 2000

Uspekhi Fizicheskikh Nauk 170 (7) 743 ± 752 (2000)

Translated by K A Postnov; edited by M S Aksent'eva

PHYSICS OF OUR DAYS PACS numbers: 04.30. ± w, 04.80.Nn, 95.55.Ym, 95.85.Sz

Gravitational-wave astronomy: new methods of measurements

V B Braginski|

DOI: 10.1070/PU2000v043n07ABEH000775

Contents

1. Historical introduction. Searching for source models and main characteristics of antennae 6912. Thermal and non-thermal noise in free-mass antennae 6933. Antenna measurement system, the sensitivity limits of quantum origin and the problem of energy

in the system 6954. Other sources of gravitational waves. Other antennae 6975. New physical information which can be obtained with gravitational-wave antennae 698

References 699

Physics ±Uspekhi 43 (7) 691 ± 699 (2000) #2000 Uspekhi Fizicheskikh Nauk, Russian Academy of Sciences

The author is also known by the name V B Braginsky. The name used

here is a transliteration under the BSI/ANSI scheme adopted by this

journal.

sources. However, 30 years had passed after Einstein'sprediction when Fock [4] pointed out a good prospect forastrophysical sources: In 1948 he noted that for the planetJupiter orbiting around the Sun _Egrav ' 4109 erg sÿ1 ' 400 W. A simple examination of the catalogsof known binary stars [6] made at the beginning of the 1960s,immediately revealed that there are about ten binary systemswith orbital periodso ' 10ÿ3 sÿ1 (one turn per several hours)and l ' 1011 cm, which emit _Egrav ' 1030 ± 1031 erg sÿ1 (i.e.about 1% of the electromagnetic power of the Sun).

The next logical step was to answer the question:Are thereany explosive sources (i.e. astrophysical catastrophes) with amass of the order of the solar mass but with a frequency ograv

much higher than in close binary stars? At the beginning ofthe 1960s, some papers evaluated _Egrav and ograv for simplemodels of such catastrophes. At this stage, a great contribu-tion was made by Ya B Zeldovich, N S Kardashev,I D Novikov, I S Shklovski| [6, 7]. The discovery of pulsars(rotating neutron stars with a large magnetic moment whoseaxis does not coincide with the rotational one) by J Bell andAHewish in 1967 [8] was an important argument favoring theexistence of such sources. A few years later, J Taylor andRHulse discovered a pulsar in a binary systemwith a neutronstar companion [9]. Careful measurements of the orbitalperiod decrease in this binary system due to the energy lossesby gravitational radiation allowed the validity of Eqn (1) to bechecked for the first time with an accuracy of a few percent.Clearly, at the moment of coalescence of the two componentsthe power _Egrav should bemaximal. This type of cataclysm hasbecome `candidate number one' for experimentalists, who bythat time already had some experience in constructing suchantennae (see below). A detailed analysis of the coalescenceprocess of binary neutron stars has been conducted for manyyears in several groups. Although this analysis has not yetbeen completed (see the review by K S Thorne [10] andreferences therein), one can say that a certain consensusbetween theoreticians has been achieved, which states thatthe total energy emitted should be about Egrav ' 1052 ergs andthe wave packet of the radiation should last for severalseconds with an increasing mean frequency from tens toseveral hundreds Hz. The key question for experimentalists,namely, how often such events occur, has not been definitelysolved as yet, and there exist both pessimistic and optimisticprognoses.

In the `semi-optimistic' scenario (prognosis) by H BetheandGBrown [11] the coalescence of two neutron stars shouldhappen on average once per 104 years in one galaxy. Thisimplies that in a space volume with radius R 1026 cm (i.e.about 30 Mpc), which comprises near 105 galaxies, about 10coalescences (and, hence, bursts of gravitational radiation)should occur during one year. Therefore, the terrestrialobserver could register about one gravitational-wave burstwith intensity ~I ' 10 ± 0.1 erg sÿ1 cmÿ2 with changingfrequency at hundreds Hz during an integration time of onemonth. It is such intensities and event rates that are expectedto be detected by two large ground-based free-mass antennaeLIGO1 and VIRGO2 (see review [2] and references therein).These and other projects will be discussed below.

A gravitational wave is the wave of acceleration gradients(components of the Riemann curvature tensor) perpendicularto the direction of the wave vector alternating in time andspace, which propagates with the speed of light. For example,if a sine-like gravitational wave propagates along the z-axis, inone half period the acceleration gradient is positive along thex-axis and is negative along the y-axis. Over the next half aperiod, the direction of the gradients reverses. According to aconvenient expression by K S Thorne, the gravitational waveis a ripple over the static curvature. Thus, it is impossible todiscover a gravitational wave at one point. However, thisbecomes possible using two point masses separated by a finitedistance L or an extended body. In the first case one needs tomeasure the change in the distanceDLgrav between themasses,which for some optimal orientation of the wave and the twomasses is

DLgrav 1

2hL ; 4

where h is the dimensionless amplitude of the wave (metricperturbation amplitude). For further description it is impor-tant to stress that this shift in the position is due to a changingforce with an amplitude

Fgrav 1

2hLmo2

grav ; 5

where ograv is the gravitational wave frequency. The valueho2

grav is just the amplitude of the wave component of theRiemann tensor. Therefore, both gravitational-wave emittersand detectors are of quadrupole type. The term `free-massantennae' means that the frequency of real mass suspensionsis much smaller, and the frequency of their internal mechan-ical modes is much higher, than the gravitational-wavefrequency ograv. It is relevant to notice that a plain gravita-tional wave, i.e. far away from the source, has twoindependent polarizations turned with respect to each otherby 45 and usually denoted by h and h. So the totalamplitude of the wave h in equations (4) and (5) is

h h2 h2

q: 6

It is not difficult to express h through the intensity eI:h 1

ograv

16pGc3

eIr

: 7

Assuming eI 10 or 0.1 erg sÿ1 cmÿ2 and ograv 103 rad sÿ1, we obtain from Eqn (7) h ' 10ÿ21 orh ' 10ÿ22, respectively. If L 4 105 cm and m 104 g(these are the distance between the test bodies and theirmass, respectively, in the LIGO project), Eqns (4) and (5)imply that the amplitudes DLgrav ' 2 10ÿ16 cm or2 10ÿ17 cm should be measured, which are due to forcesFgrav ' 2 10ÿ6 dyne or 2 10ÿ7 dyne, respectively. Theseestimates will be used below to evaluate the requiredparameters of antenna and measurement equipment. It isimportant to emphasize that the force Fgrav reveals itself in therelative motion of one test mass with respect to another, andhence the measuring devise should be set up between themasses.

As was already noted, the antenna should not necessa-rily use two separated masses: one can take an extendedmassive body and measure variable tensions inside it causedby Fgrav. The first to suggest such an antenna was J Weber.

1 LIGO from Laser Interferometric Gravitational-wave Observatory. The

project was initiated by California and Massachusetts Institutes of

Technology.2 VIRGO is the name of the Franco-Italian project.

692 V B Braginski| Physics ±Uspekhi 43 (7)

In 1968, he announced the discovery of gravitationalradiation in the coincidence scheme with two such anten-nae [13]. The sensitivity reached in his experiments corre-sponded to h ' 10ÿ16 (i.e. five orders of magnitude (!) worsethan what is being planned at the initial stage of the LIGOand VIRGO projects). At the beginning of the 1970s,Weber's experiments were repeated by several independentgroups (and in particular, by the MSU± IKI group [14])using similar detectors. The tests gave negative results: nobursts were discovered. Despite this fact, Weber, in myopinion, deserves recognition by experimentalists as thepioneer of a new astrophysical field. One of the reasonsfor the failure of his experiments was the absence of areliable astrophysical prognosis of the event rates, theiramplitude and frequency.

The invention of lasers in 1961 by T Meiman had animportant effect on choosing the scheme for the antenna.Immediately after this discovery, in 1961, M E Gertsenshteinand V I Pustovoit [15] suggested using two widely separatedfreely-suspended massive mirrors (as the test masses), whichform an optical Fabry ± Perot resonator. The pumping of thisresonator with powerful coherent laser radiation shouldprovide a high sensitivity to slight oscillations of suchmirrors caused by a gravitational wave. This concept under-lies the LIGO and VIRGO projects. In the middle of the1970s, R Drever and R Weiss had already gained significantexperience with small laboratory models of such an antenna.Together with K S Thorne, they organized a triumvirate,which justified the necessity of developing a large-scaleantenna based on this principle (it is this antenna that hasbeen called LIGO). K S Thorne played a special role in thismatter. Being an outstanding theoretician and specialist ingeneral relativity, he simultaneously has a deep understand-ing of experimental physics. Apparently, his intuitionsuggested that the accumulation of astrophysical informa-tion should allow soon meanwhile provide experimentalistswith a reliable prediction of the values of expected h, ograv,and event rates, and that the period could be spent in gettingexperience using a large prototype of the full-scale antenna.In 1981, the National Science Foundation (NSF) in the USAstarted to finance the design and construction of a prototypeof the large antenna (which has L 4 103 cm). After 15years of work, researchers from Caltech and MIT havereached a sensitivity for this prototype of h ' 10ÿ19 (withan integration time t ' 10ÿ2 s and a signal-to-noise ratio oforder one). Thus, about two decades after Weber's experi-ments, the sensitivity has been increased by three orders ofmagnitude (see [16] and references therein). In 1996, NSFbegan financing the construction of the full-scale antennae(L 4 105 cm). The construction has been completed inthe last year, and now the gradual adjustment of two suchantennae is under way with the aim of reaching a sensitivityof h ' 10ÿ21 by the fall of 2001 (the LIGO-I project), afterwhich a long-term recording of the distance variationsbetween the mirrors in the coincidence scheme must begin.In about 2005, significant modifications are planned, amongwhich are a change of the mirrors' suspension and theirinsulation from noise. As a result, the sensitivity is expectedto increase up to h ' 10ÿ22 (the LIGO-II project). Originally,the LIGO project was purely national. However, the NSFencouraged cooperation not only between American uni-versities and institutes, but also the participation of foreigngroups (in particular, two Russian groups: one fromM V Lomonosov Moscow State University (MSU), another

from the Institute of Applied Physics of Russian Academy ofSciences). Presently, more than 200 researchers participatedirectly in the LIGO project, and there is a large exchangewith participants form the VIRGO project (which uses oneantenna with arms L 3 105 cm), as well as with the moremodest (in the size of L) projects GEO-600 and TAMA. Fora detailed description of wave solutions in general relativityand many other interesting events related to gravitational-wave astronomy, two large reviews [17, 18] are recommendedto the reader. The author has restricted himself to a veryshort historical introduction above, which, however, issufficient to get acquainted with the experimental achieve-ments over the last decade and an understanding of unsolvedproblems.

2. Thermal and non-thermal noisein free-mass antennae

Through a cross-section with area

l2grav 4p2c2

o2grav

8

during half a period of a gravitational wave a very largenumber of gravitons pass:

Ngrav eIc2

4pho4grav

' 1045eI

0:1 erg sÿ1 cmÿ2

ograv

103 rad sÿ1

ÿ4: 9

It is this ensemble of gravitons that produces the gravitationalforce Fgrav acting on the test masses of the antenna. The largevalue of Ngrav allows us to consider Fgrav as being a classicalforce. It is seen from estimate (9) that this can be done even forvalues of eImuch smaller than quoted in the previous section.The classicality of Fgrav means that there is no need toquantize the gravitational field.

According to the quantum theory of measurements, thereis no limit on detecting a small classical force acting on amass.Consequently, the gravitational antenna sensitivity isexpected to increase with improving methods of measuringsmall DLgrav (or other observables related to Fgrav). Thisimportant feature has already been used in planning the two`steps' LIGO-I and LIGO-II, which we discussed in Section 1.Apparently, after having reached the sensitivity h ' 10ÿ22,the LIGO-III stage with a smaller level of h will beconstructed. However, this does not imply that there are noconstraints at all to the antenna sensitivity, which isdependent on the Planck constant h. In reality they do existand are determined by the specific method of measurement ofthe response of the system consisting of two masses on Fgrav.These quantum limitations will be considered in Section 3,and in this section we shall describe the most importantsensitivity constraints determined by random actions ofthermal and non-thermal origin.

The mirror in the LIGO-I (a fused-quartz cylinder 25 cmin diameter with a thickness of 10 cm) has a mass of about 104

g. It is suspended on a thin steel fiber loopwith a free length ofabout 20 cm (the fiber girds about the middle of the cylindricsurface). If one considers such a suspended mirror as aGalilean pendulum with a point-like mass and takes intoaccount that the pendulum oscillation frequency is muchsmaller than ograv ' 103 sÿ1, then the obvious condition fordetection of Fgrav (provided that the only source of random

July, 2000 Gravitational-wave astronomy: new methods of measurements 693

force action is the thermostat) will be

Fgrav >

4kTH

t

r

4kTm

tmt

s; 10

where T is the thermostat temperature, H is the frictioncoefficient in the pendulum oscillation mode, and tm m=His the mode relaxation time. Formula (10) is a directconsequence of the fluctuation-dissipative theorem (FDT)for this simple model. Assuming in (10) Fgrav 2 10ÿ6 dyne(i.e. for the sensitivity level of LIGO-I) and t 10ÿ2 s, weobtain that it is necessary to have tm > 4 104 s for a signal-to-noise ratio of the order of one. In LIGO-I, such asuspension on steel fibers gives tm ' 105 s. Clearly, this isnot sufficient for LIGO-II, in which Fgrav must be smaller byan order of magnitude.

Adding the condition that the signal-to-noise ratio is 10,formula (10) implies that tm must be 4 108 s, i.e. about 12years at T 300 K, m 104 g, t 10ÿ2 s. Note thatcondition (10) holds for the so-called viscous friction model,inwhichH const. If we consider the losses in the suspensionas in the structure friction model, the requirement on t willbe an order of magnitude smaller. My colleagues from theMSU V P Mitrofanov, O A Okhramenko, and K V Tokma-kov, starting from 1991, developed the mirror suspension forthe LIGO using fibers made of a super-pure fused quartz(whose internal losses correspond to a mechanical quality ofQm ' 3 107 [19]). Several new technological tricks sug-gested and realized by them have allowed the achievementof tm ' 1:7 108 , i.e. about 5.4 years [20], which is close towhat is required in LIGO-II. Such a long relaxation time ofmechanical oscillations corresponds to a relative amplitudedecrease by 0.3% in five days. It is this value that wasmeasured in the experiment. The volume and goal of thepresent paper does not permit us to consider the technologicaldetails of these methods in more detail. However, thefollowing note is pertinent. Although FDT suggests norecommendations as to what should be done to increase tm,nevertheless, by using not too strict semi-empirical relation-ships confirmed experimentally, one can affirm that in such asuspension under real laboratory conditions tm ' 109 s andeven higher values can be reached. As will be clear from thenext section, the value of the dimensionless ratiot=tm ' 5 10ÿ11 already achieved today plays an importantrole in attaining the so-called standard quantum limit of thesensitivity.

The simplemodel considered above (a point-likemass andthe unique pendulum mode) is not full: it is also necessary totake into account the contribution to the fluctuation actionon the mirror's center of mass from thermal fluctuations overthe entire length of the fiber. In terms of the Langevinlanguage for Brownian motion, it is necessary to take intoaccount spectral components of numerous violin modes ofproper oscillations of the fiber near ograv, as well as thetransfer function of the fiber oscillations to the mirror'sbarycenter. The calculations, which we omit here, give thefollowing condition: for h ' 10ÿ22 the quality of the lowestfrequency modes of the fiber must be at the level 107. Themeasurements by Mitrofanov and Tokmakov have shownthat the quality of the low-frequency eigen modes of the samefibers made of the super-pure quartz falls within the range5 107 ± 1 108 [21].

There is onemore feature of the fluctuation force action ofthe fiber on the mirror. This feature is that the sources of

fluctuations are within the fiber itself and cause its motion.This motion (the change of the fiber coordinates relative to,say, a platform towhich the fiber is attached) can bemeasuredby a separate sensor. The calculations by S P Vyatchanin andYuM Levin [22] have shown that such a measurement allowsone to subtract about 99% of random shifts, caused bythermal fluctuations inside the fiber, from the mirror'smotion.

To conclude this part of the section, one can say that theproblem of `suppression' of random fluctuations of themirror's barycenter position, caused by the thermostat in thesuspension fibers, can also be solved in the case where theplanned sensitivity must be much better than h ' 10ÿ22.

When considering fluctuations in the mirror suspensionfibers, we have so far restricted ourselves to the case ofthermal equilibrium, to which FDT can be applied. How-ever, in addition to such fluctuations, there are others whichdo not obey FDT. It is easy to see that to attain larger tm, it isadvantageous to decrease the diameter of the fiber (of steel orfused quartz), i.e. it is profitable to approach the stress closeto the fiber disruption threshold. The free energy density inpolycrystal solids and amorphous bodies is usually of theorder of 106 erg gÿ1. As established long ago, at the tensionclose to solid body disruption, an acoustic emission isobserved, which is the consequence of a partial redistributionof the free energy (vacancy group jumps, dislocation creation,etc.) At present, no theoretical models of such processes havebeen proposed.

It is important to note that the kinetic energymo2

grav DL2grav=2 ' 2 10ÿ22 erg for LIGO-I is many orders

of magnitude smaller than the total free energy storage in thefiber. In other words, even a tiny fraction of the free energy inthe fiber transformed by tension into acoustic emission (i.e.into additional to thermal oscillations of the fiber), can mimicFgrav. Recently, A Yu Ageev and I A Bilenko (also from theMSU group), when making detailed measurements of thetemporal structure of the Brownian oscillations of the steelstring used inLIGO-I have found random short-term changesin the oscillation amplitude exceeding the ordinary values,which correspond to the Langevin model of the Brownianmotion:

dx2p

kT

mo2m

s 2ttm

s; 11

where m is the effective mass of a mode with frequency om

and relaxation time tm [23]. These rare bursts exactlyrepresent excessive (non-thermal) noise. As established fromthe measurements, such bursts, which can mimic metricvariations at the level h ' 10ÿ21, are observed in the fibers,on average, once every several hours under tensions of about50% of the breaking stress. Decreasing the tension stronglydecreases the false event rate. A coincidence scheme betweentwo antennae, of course, makes it easier to reject such `signals'imitating the action of Fgrav, however it is natural that theexperimentalists consider it as the last `line of defense'.

So far, no measurements of the excess fluctuations (ofnon-thermal origin) in the fibers of super-pure quartz havebeen performed. This is a much more complicated problem,since for such fibers the value of tm is about four orders ofmagnitude higher than for steel fibers, and, correspondingly,the sensitivity to small fiber oscillations must be at least twoorders of magnitude better. A new method of measurementsrecently suggested by I A Bilenko and M L Gorodetski| [24]will, in my opinion, allow this problem to be solved.

694 V B Braginski| Physics ±Uspekhi 43 (7)

Not only thermal fluctuations in the suspension fibersprevent accurate measurements of small Fgrav, which causesoscillations DLgrav of the mirror's center of mass. The opticalFabry ± Perot resonator is formed by two almost flat (with acurvature radius of about 10 km) surfaces of two quartzcylinders. The mirrors are constructed from a multi-layercoating of the quartz cylinders' surface, which has a highreflectivity. It is clear that the laser interferometer measuresnot the displacement of the cylinder's barycenter, but the sumof this displacement and the proper internal oscillations of thecylinder. If they are due to internal forces (for instance, due tothe inner friction inside the material which, according toFDT, is the source of the volume forces), they do not shift thebarycenter, but can give an appreciable noise contribution tothe quantity being measured.

To calculate this contribution, AGillespi and FRaab [25],using the same Langevin model for modes, estimated thefraction of the total variance of the mean coordinate of themirror (i.e. the cylinder's surface) nearograv. Note thatograv issignificantly lower than the proper frequencies of thecylinder's modes, so the contributions from low-frequencywings of several tens of modes have been taken into account.This calculations showed the possibility of attaining a levelclose to h ' 10ÿ22 only at low frequenciesograv 4 103 rad sÿ1

at a signal-to-noise ratio close to one. The same calculationsfor a sapphire cylinder (whose internal losses are an order ofmagnitude smaller than for quartz, and the density andvelocity of sound is two times as large) show that it is possibleto gain three times in sensitivity nearograv ' 103 rad sÿ1, if thelosses inside the material are of so-called structural character.

Unfortunately, fluctuations of the cylinder's surface (themean coordinate of the surface of the mirror) in the frequencydomain of interest include not only pure Brownian fluctua-tions, which can be calculated in the linear Langevin model.There is at least one more effect of a non-linear origin thatcontributes almost as much as Brownian fluctuations. Thiseffect is the consequence of anharmonism of the lattice, whichgives rise to thermal expansion and temperature fluctuationsdT. If for some reason the temperature inside the layeradjacent to the mirror's surface l changes by some value dT,the displacement of the outer surface of the mirror will besignificant even for tiny temperature changes:

dl aTldT 10ÿ17 cmaT

510ÿ7 Kÿ1l

10ÿ2 cmdT

210ÿ9 K ;

12where aT is the coefficient of linear expansion. As seen fromestimate (12), under these conditions dl is of order ofDLgrav ' 2 10ÿ17 cm for dT of about 1 nanokelvin. Suchtemperature fluctuations can result from ordinary equili-brium fluctuations, whose total variance is [26]

hdT 2i kT2

CrV; 13

where C is the specific heat capacity, r is the density, and V isthe volume. In order to find the fluctuations, correspondingto (13), of themean coordinate of themirror's surface near thefrequency ograv of interest here, it is necessary to know thefrequency dependence of the dissipation inside the mirror'smaterial. Assuming the losses inside the mirror to correspondonly to the thermoelastic model, M L Gorodetski| andS P Vyatchanin obtained analytical expressions (see [27, 28])for the spectral density of fluctuations of the mean coordinate

in the mirror's surface inside a circle of radius r0. Theintroduction of the parameter r0 is significant, because onlyfrom such a spot does the laser beam get information aboutthe mirror's motion. Calculations show that for r0 ' 1:6 cm,which was initially planned for LIGO-II, and themirrormadeof fused quartz, the sensitivity limitations due to this effect arethree times smaller than due to `purely' Brownian limitations.At present, monocrystal sapphire is being widely discussed asa possible `candidate' for the mirror material. For thismaterial, the above effect is twice as large as the Brownianone for the same r0. There are at least two possibilities toreduce the contribution of this effect to the total fluctuations,which do not relate to the mirror's barycenter motion. Thefirst possibility is to appreciably increase r0 (see [27]), thesecond suggests subtracting fluctuations of the mirror'ssurface coordinate (recall that they are due to internalforces, which do not shift the position of the mirror'sbarycenter).

It is quite possible that the analysis [27, 28] neglects otherfluctuation processes that could lead to additional randomvariations of the mirror's surface and, correspondingly, toincrease the threshold value of h (in particular, excessivenoise). Clearly, only direct measurements can give a definiteanswer.

Concluding this section, we should note that here we havedescribed the state of the art in solving two of the mostdifficult experimental problems. It is also relevant to stressthat our description had a semi-qualitative character withsimple numerical estimates. In the cited references, the readercan find a lot of details which were omitted, in particular, thespectral representation of the limiting sensitivity in thefrequency range 10 rad sÿ1 < ograv < 104 rad sÿ1. Due tolimit of space and bearing in mind the purpose of the presentpaper, we have not considered the solutions of simplerproblems, such as antiseismic insulation of the mirrors and astrategy of monitoring other possible sources of force actionon the mirrors.

Finally, we can emphasize that the problem of choice ofthemirror and its suspension proved to be, in essence, a ratherdifficult experimental task, which has required a lot of time.This can be illustrated as follows. It took in total 15 years toreach a sensitivity of h ' 10ÿ19 in the LIGO-I prototype, andabout 8 years to obtain a relaxation time of tm ' 5:4 years forLIGO-II [20].

3. Antenna measurement system,the sensitivity limits of quantum origin andthe problem of energy in the system

The scheme of the measurement system in LIGO (with somesimplifications) is shown in Fig. 1.

Its principal elements include two optical Fabry ± Perotresonators formed by two pairs of mirrors AA0 and BB0. Thedistance between the mirrors in each pair is L 4 105 cm.Optical oscillations inside them are excited by laser 1 (with thewavelength l 1064 nm) through beam-splitter 2, whichtogether with additional mirrors 3 and 4 connects theresonators. Mirrors A and B are `deaf', in other words theirreflection coefficient R is close to one. Mirrors A and B aremore transparent with a lower R. In LIGO-II, mirrors withfineness F p=1ÿ R ' 3 103 are planned. The conceptof this scheme was first suggested by Drever [29]. Later thisscheme was analyzed in detail and modified by several groups(see references cited in [12]).

July, 2000 Gravitational-wave astronomy: new methods of measurements 695

If a gravitational wave propagates along the axis x, theforce Fgrav acts between the mirrors AA 0 and their separationchanges with amplitude DLgrav ' hL=2, while the distancebetween B and B0 does not change. If the wave propagatesalong the y-axis, the roles of the mirrors reverse. When thewave propagates along the normal to the plane xy and thewave polarizations are optimal with respect to the axes x andy, both pairs of mirrors will oscillate in antiphase so that thetotal response will be twice as large.

A special control scheme (with a feed-back) allowsadjustment of the mutual positions of the mirrors and thetuning of resonance mode frequencies in these schemesaccordingly. Note that in the feed-back the upper boundaryof monitoring frequencies is much lower than ograv, so themirrors react on Fgrav as essentially free masses. The accuracyof mirror position control up to this upper boundary is smallfractions of l=F .

For a certain position adjustment of all six mirrors, theamplitude of oscillations of the phase difference of the opticalfield in two arms Dj is proportional to h.

The depth of modulation of the power flux coming intodetector 5 is, in turn, proportional toDj. Note that the tuningof the optical system is such that the detector is workingalmost in the `dark field' regime.

If a burst of gravitational radiation has a wave-packetform with duration t and frequency ograv, one can obtainsimple formulas for the two cases: when p=ograv is smallerthan the relaxation time of optical oscillations topt, andp=ograv > topt in the opposite case. The time topt can beexpressed through R and F and in one of the LIGO-IIvariants under discussion is of order

topt L

c1ÿ R FLpc' 4 10ÿ3s

F103

L

4 105 cm: 14

If p=ograv 5topt, then

Dj ' 2pLl1ÿ R

DLgrav

L' FL

lh ' 4 10ÿ10 rad

F103

L

4 105 cm

h

10ÿ22

l

10ÿ4 cm

ÿ1: 15

If p=ograv < topt, then

Dj ' 1

2hoopt

pograv

' 4 10ÿ10 radh

10ÿ22oopt

2 1015 sÿ1

p=ograv

4 10ÿ3 s

ÿ1: 16

These numerical estimates show that the values of Dj from(16) and (15) are nearly equal at p=ograv ' t ' topt.

It is important to note that measurements of smallquantities Dj over a short time interval t in such aninterferometer are essentially the matching of the opticalfield oscillations in resonators AA0 and BB0. The ratio Dj=tis the deviation of one frequencywith respect to other, and theratio DLgrav=L h=2 is the relative value of this deviationequal to Doopt=oopt. From this point of view, the sensitivityh ' 10ÿ19 already reached in the LIGO prototype [16] (inwhich L 4 103 cm) is, undoubtedly, an outstandingresult. For comparison, it is sufficient to mention that thesmallest relative deviations of two stable generators (Allan'svariance) measured so far are only about 10ÿ17.

At the end of this section, we shall use the above estimatesfor Dj ' 10ÿ10 rad (for LIGO-II).

As was noted in the Introduction, in gravitational-waveantennae at a certain sensitivity level there appear some kindof `barriers', sensitivity limitations of quantum origin. Theydepend on the choice of the observable and measurementprocedure. Such restrictions were predicted in experimentswith macroscopic masses in the coordinate measurementswith a finite averaging time in 1967 [30]. A few years later [31]it became clear that the same type of limitations can bedirectly derived from Heisenberg's uncertainty relations forcoordinate measurements, taking into account the finitenessof the measurement time t. Clearly, there exist a whole familyof such limits if measurements of an observable involve thedirect measurement of a coordinate (geometrical distance,field strength, etc.) They are called `standard quantum limits'.For example, in the case of a freemass we are interested in, thestandard quantum uncertainty of its coordinate is

DxSQL ht2m

r' 2 10ÿ17 cm

t

10ÿ2 s

1=2m

104 g

ÿ1=2;

17if two point observations of the coordinate x are made at thebeginning and the end of a time interval t.

By equating the free mass coordinate uncertainty toDxSQL, one can find an analytical expression for thestandard limit of the metric perturbation amplitude hSQL

caused by a wave packet with duration t and meanfrequency ograv in the scheme of the LIGO-II interferom-eter (with four test masses)

hSQL

8h

mo2gravL

2t

s' 2 10ÿ23

m

104 g

ÿ1=2

ograv

103 sÿ1

ÿ1L

4 105 cm

ÿ1 t10ÿ2 s

ÿ1=2: 18

Equation (18) is correct for t5 2p=ograv. As seen fromestimates (17) and (18), the planned LIGO-II sensitivityh ' 10ÿ22 is very close to hSQL. In other words, the contribu-tion of purely quantum fluctuations to the total measurementerror will be significant (see [27] for more detail).

A `recipe' to get around this obstacle has been knownfor quite a long time. As noted in Section 1, for antennae

BB0

A0

A

3

1 2

4

5

L

L

x

y

Figure 1.

696 V B Braginski| Physics ±Uspekhi 43 (7)

there are practically no sensitivity limits of quantum origin.It is only necessary to choose other observables to detectFgrav. The `recipe' usually referred to as quantum non-demolition (QND) measurements was initially suggestedfor gravitational antennae (see review [32]). However, italso proved attractive for opticians studying quantumphenomena, and QND-measurements have been success-fully realized in optical experiments, although with a ratherlarge number of photons (see review [33]). Recently, thesame measurements were performed with microwave quantaas well: the presence of a single microwave quantum in aresonator was measured without absorption of the quantum[34]. Up to the present time, this has not yet been done withmechanical test masses (a free mass or an oscillator).However, without solving this, a much more difficult,problem, it is impossible to increase the gravitationalantenna sensitivity keeping the values of L and M.

Possible schemes for QND-measurements in antennaehave been analyzed by the MSU group over last few years.The first suggestion was to use larger fineness values F inmirrors thereby drastically increasing topt [35]. At thealready reached F 2 106 [36] the value topt ' 10 s forL 4 105 cm. Such a long relaxation time opens thepossibility of using the small parameter t=topt ' 10ÿ3,which determines the attainable degree of `squeezing' ofthe quantum state of the optical field inside resonator.Naturally, the meter itself should be installed inside theresonator and should not absorb optical photons. Thevariant of such a meter, suggested in [35], was rather thedescription of a thought experiment than a realisticengineering scheme.

The second attempt to develop such a meter [37] led to thescheme shown in Fig. 2.

Three free mirrors with a maximal F (and, correspond-ingly, topt) form an optical Fabry ± Perot resonator. Anexternal laser (not shown in the figure) excites oscillations inthe resonator. Inside it, near the tilted mirror B, an additionalfourth mirror D is inserted with reduced fineness and,correspondingly, with relatively high transmittance, whichshould be specified. In such a scheme, all modes in theresonator ABC are split into doublets due to mirror D. Atsome transmittance of mirror D, when the upper componentof the doublet is excited, the ponderomotive force producedby the optical field shifts mirror D if mirror C is displaced. Ifsuch a resonator receives some amount of energy of about

what is needed to reach hSQL, the displacement isD ' hL=2 DLgrav.

The meter itself must be placed between mirror K, locatedoff the optical field (the ponderomotive force does not act onit), and mirror D. For an ideal QND device (for instance, forspeed measurements), the finite value of F , as was calculatedby M L Gorodetskii and F Ya Khalili, bounds the sensitivityof such an antenna with the limit

h hSQLogravtopt

p 10ÿ2hSQL

ograv

103 sÿ1

ÿ1=2 topt10 s

ÿ1=2:

19The analysis of the speedmeter recentlymade by theMSU

group in collaboration with K S Thorne using the parametersattainable in the modern cryogenic microwave electronics,showed that one can reach a sensitivity not better thanh ' 0:3hSQL [38]. Possibly, to find a much simpler technolo-gical solution, it would be convenient to turn the free massesD and K into a mechanical oscillator with eigen frequencynear ograv, by `adding' to them a stiffness of optical originwith a very low noise level [39], or to use symphoton quantumstates in a different antenna topology [40].

The last problem to discuss in this section is the amount ofadmissible energy inside resonators (or circulating powerW).For anordinary (coordinate)measurement scheme (seeFig. 1)in LIGO-II the projected circulating powerW ' 1013 erg sÿ1 ' 1 MW. Provided that the mirrors havemulti-layer reflection coatings of highest quality, the powerdissipated in the coating will be of order 107 erg sÿ1' 1 W.The absorption of each optical photon with energyhoopt ' 2 10ÿ12 erg will give rise to a local burst consistingof around 50 additional thermal photons, which in turn,because of a small free-path length (both in quartz andsapphire), will lead to a local shot-like heating of the mirrorsurface. Due to a non-zero thermal expansion coefficient aT,the mirror surface will fluctuate. Such a specific thermo-photon noise was analyzed in detail in paper [27]. Thecalculations showed that W ' 1013 erg sÿ1 is indeed veryclose to the limiting admissible value, if one wish to approachhSQL. Clearly, a possible solution is the use of squeezedquantum states in the main resonators, which can beprepared at t=topt 5 1.

4. Other sources of gravitational waves.Other antennae

The main purpose of the present article is to describeachievements in development of ground-based free-massantennae and the discussion of novel methods of measure-ments that can substantially increase the sensitivity. In thissection, we restrict ourselves to only a brief enumeration ofother directions of studies in gravitational-wave astronomy.

In previous sections we used a simple example to illustratethe conditions required to reach a given sensitivity: agravitational-wave burst has a duration of t ' 10ÿ2 s and amean frequency of ograv ' 103 rad sÿ1. The free-massantennae operate over a very broad frequency range10 rad sÿ1 4ograv 4 104 rad sÿ1. Bursts of radiation fromcoalescing neutron stars (the preferred source in the LIGOand VIRGO prognoses) must have t of the order of severalseconds and a changing frequency from tens to a few hundredHz. The a priori knowledge of waveforms of such burstswould significantly facilitate the detection and increase the

CK

B

A

D

x

y

DL DLL

L

Figure 2.

July, 2000 Gravitational-wave astronomy: new methods of measurements 697

signal-to-noise ratio. This is one of the problems that is beingstudied by some theoretical astrophysics groups. Ignorance ofthe exact equation of state of neutron star material makes thisproblem even more difficult. In addition, two coalescingcomponents may have strongly different values and orienta-tions of angular momentum. Experimentalists still hope thattheoreticians hold their promise to calculate around 105 burstwaveforms from such sources when the antennae startworking in a stationary observational regime.

Coalescing binary neutron stars are not the only sourcesto be sought by gravitational wave antennae. Stronger burstscan be expected from coalescing neutron stars and black holesand coalescing binary black holes. One more unsolvedtheoretical problem are the waveforms from such sourcesand their expected event rate.

Non spherically-symmetric supernova explosions canalso generate bursts of gravitational radiation. The eventrate of such events in a galaxy is about two orders ofmagnitude higher than that of coalescing neutron stars inthe model by H Bethe and G Brown. Attempts to construct areliable model of the initial stage of the supernova explosionhave failed so far.

With increasing sensitivity, ground-based antennae willbe capable of registering gravitational wave noise back-grounds. Some of them should have the same origin as thecosmic microwave background discovered in the mid-60s.The first models of the relic gravitational radiation suggestedby L P Grishchuk [41] and A A Starobinski| [42] were furtherdeveloped by other scientists. Of these new models, in myopinion, the most interesting is the model by R Brustein andG Veneziano [43] based on the superstring theory. This isessentially the first direct prediction of the superstring theory.A notable feature of this model is that it has no divergencebetween `before' and `after' the Big Bang.

At present, two other projects are also developing free-mass antennae. Both these programs invoke satellites. Thefirst of them exploits the Doppler effect as a measure of the`response' on metric perturbations, as was suggested manyyears ago [44]. These antennae have a threshold sensitivity ofhmin ' 10ÿ14 ± 10ÿ15 for gravitational wave bursts with amean frequency 10ÿ2 rad sÿ1 4ograv 4 10ÿ4 rad sÿ1. Thesensitivity is determined, first of all, by the non-stability ofmicrowave autogenerators, which are used in the Earth-satellite channel. In such a program, it would be natural tohave two pairs of free masses (the first satellite Ð Earth andthe second satelliteÐEarth) and to use a coincidence scheme.Unfortunately, there are no such dedicated programs and theexperimentalists have to use, as a `by-product', free hours ofcommunication with single satellites, whose principal goal isto study remote planets of the Solar system (see references in[45, 46] for more detail).

Another program of free-mass antennae (the LISAproject) also uses satellites and plans to search the samefrequency range of ograv as the previous one. In manyrespects this program is similar to LIGO and VIRGO: itintends to use three (drag-free) satellites at the terrestrialdistance from the Sun and separated by L ' 5 1011 cm(i.e. by six orders of magnitude larger than the mirrors inLIGO and VIRGO). Distance variations between thesatellites should be measured by a laser interferometer. Atthe present time, a significant number of ground-basedlaboratory tests of model units of such antennae and manycalculations have already been done, however the finaldecision about the practical realization of this program has

not yet been taken (see, e.g. [47] and a special issue ofjournal [48]).

5. New physical information which can beobtained with gravitational-wave antennae

In conclusion of this paper, it is relevant to enumerate whatnew can be expected by physics from using ground-basedgravitational-wave antennae.

1. The discovery of bursts of gravitational radiation andthe study of their rate of occurrence can give informationabout the space density of binary neutron stars in galaxies andthe contribution they give to the so-called dark matter. Burstwaveforms can possibly be used to study the preferentialequation of state of the neutron star matter. Short burstwaveforms will, perhaps, allow the construction of a model ofthe initial stage of supernova explosions.

2. More than twenty five years ago General Relativity(GR) turned into an engineering discipline for high-precisionspace navigation. This essentially means that GR has beenchecked to a high degree of accuracy, however high-precisionnavigation inside our Solar system implies at the same timethat GR is valid only in the case where gravitational potentialis much smaller than c2. The validity of GR in theultrarelativistic case (when gravitational potential is of orderof c2) has never been tested. The possibility of such a test willemerge if theoretical astrophysicists can predict the wave-forms of the signal from coalescing black holes and if suchbursts are really detected.

3. The detection of the relic gravitational wave back-groundwill, undoubtedly, provide an invaluable contributionto cosmology. It is however possible that detected chaoticfluctuations of the mirror's barycenter, which are not due tothermal (or non-thermal) fluctuations in the suspension orinside the mirror itself and the fluctuating action of thequantum detector, will not be correlated with the motion ofthe two pairs of mirrors in a way that the wave components ofthe Riemann tensor produce. In such a case the prediction ofSHawking [49] (see also [50] and [51]) about the interaction ofspace-time fluctuations on the Planckian scale with ordinarymatter may be realized. These fluctuations are sometimescalled the space-time foam. Hawking's prediction is that suchan interaction appears in the decoherence of the wavefunction of an ordinary body, i.e. in small random displace-ments of the body's barycenter.

2. It is natural to expect that methods of measurementsdeveloped for LIGO and VIRGO can be used in other fieldsof physics. It is worthwhile noting that when QND-detectorsof the relative velocity of motion of masses, which allow thedetermination of the linear momentum with an accuracybetter than the standard quantum limit

DPSQL hm

2t

r;

are used in antenna measurement systems, the energy,determined through the momentum, will be measured withan error DE less than h=t [52].

To conclude, it should be said that all the expenses on theLIGO project over 20 years, most of which were spent toconstruct buildings and vacuum equipment, are less than afourth part of the cost of a nuclear submarine. Yet mankindcontinues manufacturing several such submarines each year,which, in contrast to LIGO, are incapable of making it anywiser.

698 V B Braginski| Physics ±Uspekhi 43 (7)

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July, 2000 Gravitational-wave astronomy: new methods of measurements 699

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Observation of a kilogram-scale oscillator near itsquantum ground state

B Abbott1, R Abbott1, R Adhikari1, P Ajith2, B Allen2,3, G Allen4,R Amin5, S B Anderson1, W G Anderson3, M A Arain6, M Araya1,H Armandula1, P Armor3, Y Aso7, S Aston8, P Aufmuth9,C Aulbert2, S Babak10, S Ballmer1, H Bantilan11, B C Barish1,C Barker12, D Barker12, B Barr13, P Barriga14, M A Barton13,M Bastarrika13, K Bayer15, J Betzwieser1, P T Beyersdorf16,I A Bilenko17, G Billingsley1, R Biswas3, E Black1, K Blackburn1,L Blackburn15, D Blair14, B Bland12, T P Bodiya15, L Bogue18,R Bork1, V Boschi1, S Bose19, P R Brady3, V B Braginsky17,J E Brau20, M Brinkmann2, A Brooks1, D A Brown21, G Brunet15,A Bullington4, A Buonanno22, O Burmeister2, R L Byer4,L Cadonati23, G Cagnoli13, J B Camp24, J Cannizzo24,K Cannon1, J Cao15, L Cardenas1, T Casebolt4, G Castaldi25,C Cepeda1, E Chalkley13, P Charlton26, S Chatterji1,S Chelkowski8, Y Chen10,27, N Christensen11, D Clark4, J Clark13,T Cokelaer28, R Conte29, D Cook12, T Corbitt15,56, D Coyne1,J D E Creighton3, A Cumming13, L Cunningham13, R M Cutler8,J Dalrymple21, S Danilishin17, K Danzmann2,9, G Davies28,D DeBra4, J Degallaix10, M Degree4, V Dergachev30, S Desai31,R DeSalvo1, S Dhurandhar32, M Díaz33, J Dickson34, A Dietz28,F Donovan15, K L Dooley6, E E Doomes35, R W P Drever36,I Duke15, J-C Dumas14, R J Dupuis1, J G Dwyer7, C Echols1,A Effler12, P Ehrens1, E Espinoza1, T Etzel1, T Evans18,S Fairhurst28, Y Fan14, D Fazi1, H Fehrmann2, M M Fejer4,L S Finn31, K Flasch3, N Fotopoulos3, A Freise8, R Frey20,T Fricke1,37, P Fritschel15, V V Frolov18, M Fyffe18, J Garofoli12,I Gholami10, J A Giaime5,18, S Giampanis37, K D Giardina18,K Goda15, E Goetz30, L Goggin1, G González5, S Gossler2,R Gouaty5, A Grant13, S Gras14, C Gray12, M Gray34,R J S Greenhalgh38, A M Gretarsson39, F Grimaldi15, R Grosso33,H Grote2, S Grunewald10, M Guenther12, E K Gustafson1,R Gustafson30, B Hage9, J M Hallam8, D Hammer3, C Hanna5,J Hanson18, J Harms2, G Harry15, E Harstad20, K Hayama33,

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V Sannibale1, S Saraf52, P Sarin15, B S Sathyaprakash28,S Sato43, P R Saulson21, R Savage12, P Savov27, S W Schediwy14,R Schilling2, R Schnabel2, R Schofield20, B F Schutz10,28,P Schwinberg12, S M Scott34, A C Searle34, B Sears1, F Seifert2,D Sellers18, A S Sengupta1, P Shawhan22, D H Shoemaker15,A Sibley18, X Siemens3, D Sigg12, S Sinha4, A M Sintes10,51,B J J Slagmolen34, J Slutsky5, J R Smith21, M R Smith1,N D Smith15, K Somiya2,10, B Sorazu13, L C Stein15, A Stochino1,R Stone33, K A Strain13, D M Strom20, A Stuver18,T Z Summerscales53, K-X Sun4, M Sung5, P J Sutton28,H Takahashi10, D B Tanner6, R Taylor1, R Taylor13, J Thacker18,K A Thorne31, K S Thorne27, A Thüring9, K V Tokmakov13,C Torres18, C Torrie13, G Traylor18, M Trias51, W Tyler1,D Ugolini54, J Ulmen4, K Urbanek4, H Vahlbruch9, C Van DenBroeck28, M van der Sluys42, S Vass1, R Vaulin3, A Vecchio8,J Veitch8, P Veitch40, A Villar1, C Vorvick12, S P Vyatchanin17,S J Waldman1, L Wallace1, H Ward13, R Ward1, M Weinert2,A Weinstein1, R Weiss15, S Wen5, K Wette34, J T Whelan10,S E Whitcomb1, B F Whiting6, C Wilkinson12, P A Willems1,H R Williams31, L Williams6, B Willke2,9, I Wilmut38, W Winkler2,C C Wipf15, A G Wiseman3, G Woan13, R Wooley18, J Worden12,W Wu6, I Yakushin18, H Yamamoto1, Z Yan14, S Yoshida50,M Zanolin39, J Zhang30, L Zhang1, C Zhao14, N Zotov55,M Zucker15 and J Zweizig1 (LIGO Scientific Collaboration)1 LIGO—California Institute of Technology, Pasadena, CA 91125, USA2 Albert-Einstein-Institut, Max-Planck-Institut für Gravitationsphysik, D-30167Hannover, Germany3 University of Wisconsin-Milwaukee, Milwaukee, WI 53201, USA4 Stanford University, Stanford, CA 94305, USA5 Louisiana State University, Baton Rouge, LA 70803, USA6 University of Florida, Gainesville, FL 32611, USA7 Columbia University, New York, NY 10027, USA8 University of Birmingham, Birmingham B15 2TT, UK9 Leibniz Universität Hannover, D-30167 Hannover, Germany10 Albert-Einstein-Institut, Max-Planck-Institut für Gravitationsphysik,D-14476 Golm, Germany11 Carleton College, Northfield, MN 55057, USA12 LIGO Hanford Observatory, Richland, WA 99352, USA13 University of Glasgow, Glasgow G12 8QQ, UK14 University of Western Australia, Crawley, WA 6009, Australia15 LIGO—Massachusetts Institute of Technology, Cambridge, MA 02139, USA16 San Jose State University, San Jose, CA 95192, USA17 Moscow State University, Moscow, 119992, Russia

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18 LIGO Livingston Observatory, Livingston, LA 70754, USA19 Washington State University, Pullman, WA 99164, USA20 University of Oregon, Eugene, OR 97403, USA21 Syracuse University, Syracuse, NY 13244, USA22 University of Maryland, College Park, MD 20742 USA23 University of Massachusetts, Amherst, MA 01003 USA24 NASA/Goddard Space Flight Center, Greenbelt, MD 20771, USA25 University of Sannio at Benevento, I-82100 Benevento, Italy26 Charles Sturt University, Wagga Wagga, NSW 2678, Australia27 Caltech-CaRT, Pasadena, CA 91125, USA28 Cardiff University, Cardiff, CF24 3AA, UK29 University of Salerno, 84084 Fisciano (Salerno), Italy30 University of Michigan, Ann Arbor, MI 48109, USA31 The Pennsylvania State University, University Park, PA 16802, USA32 Inter-University Centre for Astronomy and Astrophysics, Pune-411007, India33 The University of Texas at Brownsville and Texas Southmost College,Brownsville, TX 78520, USA34 Australian National University, Canberra, 0200, Australia35 Southern University and A& M College, Baton Rouge, LA 70813, USA36 California Institute of Technology, Pasadena, CA 91125, USA37 University of Rochester, Rochester, NY 14627, USA38 Rutherford Appleton Laboratory, Chilton, Didcot, Oxon OX11 0QX, UK39 Embry-Riddle Aeronautical University, Prescott, AZ 86301, USA40 University of Adelaide, Adelaide, SA 5005, Australia41 University of Southampton, Southampton SO17 1BJ, UK42 Northwestern University, Evanston, IL 60208, USA43 National Astronomical Observatory of Japan, Tokyo 181-8588, Japan44 Institute of Applied Physics, Nizhny Novgorod, 603950, Russia45 University of Strathclyde, Glasgow G1 1XQ, UK46 University of Minnesota, Minneapolis, MN 55455, USA47 The University of Texas at Austin, Austin, TX 78712, USA48 Loyola University, New Orleans, LA 70118, USA49 Hobart and William Smith Colleges, Geneva, NY 14456, USA50 Southeastern Louisiana University, Hammond, LA 70402, USA51 Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain52 Sonoma State University, Rohnert Park, CA 94928, USA53 Andrews University, Berrien Springs, MI 49104 USA54 Trinity University, San Antonio, TX 78212, USA55 Louisiana Tech University, Ruston, LA 71272, USAE-mail: tcorbitt@ligo.mit.edu

New Journal of Physics 11 (2009) 073032 (13pp)Received 20 February 2009Published 16 July 2009Online at http://www.njp.org/doi:10.1088/1367-2630/11/7/073032

56 Author to whom any correspondence should be addressed.

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Abstract. We introduce a novel cooling technique capable of approachingthe quantum ground state of a kilogram-scale system—an interferometricgravitational wave detector. The detectors of the Laser InterferometerGravitational-wave Observatory (LIGO) operate within a factor of 10 of thestandard quantum limit (SQL), providing a displacement sensitivity of 10−18 min a 100 Hz band centered on 150 Hz. With a new feedback strategy, wedynamically shift the resonant frequency of a 2.7 kg pendulum mode to lie withinthis optimal band, where its effective temperature falls as low as 1.4 µK, and itsoccupation number reaches about 200 quanta. This work shows how the exquisitesensitivity necessary to detect gravitational waves can be made available to probethe validity of quantum mechanics on an enormous mass scale.

Contents

1. The LIGO interferometers 62. The cooling mechanism 63. Measurement results and discussion 94. Cooling to the quantum limit 105. Future prospects with LIGO 11Acknowledgments 11References 11

Observation of quantum effects such as ground state cooling [1]–[15], quantum jumps [16],optical squeezing [17], mechanical squeezing [18]–[20] and entanglement [21]–[26] thatinvolve macroscopic mechanical systems are the subject of intense experimental effort [27].The first step toward engineering a non-classical state of a mechanical oscillator is to coolit, minimizing the thermal occupation number of the mode. Any mechanical coupling to theenvironment admits thermal noise that randomly drives the system’s motion, as dictated by thefluctuation–dissipation theorem [28], but ‘cold’ frictionless forces, such as optical or electronicfeedback, can suppress this motion, hence cooling the oscillator.

Two types of forces have recently proven valuable for cooling. The first is africtionless damping force, originating either from an electronic servo system (‘cold damping’)[4, 29, 30] or from photothermal or radiation pressure forces in a detuned cavity (‘cavitycooling’) [1]–[3], [5, 6, 8]; this force reduces the motion of the oscillator while also diminishingits quality factor. The second is an optical restoring force, which increases the resonantfrequency of the oscillator without additional friction, effectively increasing its quality factor[7, 10]. To reach the quantum regime in experiments exploiting these techniques, a low noiseoscillator’s position must be monitored by a highly sensitive readout device. By providingboth of these features, the Laser Interferometer Gravitational-wave Observatory (LIGO)interferometers present a unique opportunity to cool kilogram-scale mirrors to enticingly lowtemperatures. Although the LIGO interferometers do not have sufficiently large optical restoringforces for the second effect to be significant, their active control systems may instead be used toreproduce the effect.

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1. The LIGO interferometers

LIGO operates three kilometer-scale interferometric detectors with the goal of directly detectinggravitational waves of astrophysical origin [31, 32]. The measurements reported here wereperformed at LIGO’s Hanford Observatory. The detector shown in figure 1 comprises aMichelson interferometer with a 4 km long Fabry–Perot cavity of finesse 220 placed in eacharm to increase the sensitivity of the detector. Each mirror of the interferometer has massM = 10.8 kg, and is suspended from a vibration-isolated platform on a fine wire to form apendulum with frequency p = 0.74 Hz, to shield it from external forces and to enable itto respond to a gravitational wave as a mechanically free mass above the natural resonantfrequency. To minimize the effects of laser shot noise, the interferometer operates with highpower levels; approximately 400 W of laser power of wavelength 1064 nm is incident on thebeam splitter, resulting in over 15 kW of laser power circulating in each arm cavity. The presentdetectors are sensitive to changes in relative mirror displacements of about 10−18 m in a 100 Hzband centered around 150 Hz (figure 2). This low noise level allows for the preparation of low-energy states for the oscillator mode considered next.

The four mirrors of the LIGO interferometer (figure 1) are each an extended object witha displacement xi (i = 1, . . . , 4) defined along the optical beam axis. The servo control systemthat keeps the interferometer mirrors at the resonant operating point is an essential componentof this study. While all longitudinal and angular degrees of freedom of the mirrors are activelycontrolled, our discussion is limited to the differential arm cavity motion, which is the degree offreedom excited by a passing gravitational wave, and hence also the most sensitive to mirrordisplacements. This mode corresponds to the differential motion of the centers of mass ofthe four mirrors, xc = (x1 − x2) − (x3 − x4), and has a reduced mass of Mr = 2.7 kg. A signalproportional to differential length changes is measured at the antisymmetric output of the beamsplitter, as shown in figure 1. This signal is filtered by a servo compensation network beforebeing applied as a force on the differential degree of freedom by voice coils that actuate magnetsaffixed to the mirrors.

2. The cooling mechanism

The degree of freedom that is of interest as a quantum particle is the differential mirror motionxc. However, optical measurements probe the location of the mirror surface (averaged over theoptical beam), which differs from center-of-mass location due to the mirror’s internal thermalnoise, and include a sensing noise due to the laser shot noise. Combining these noises into atotal displacement noise XN, the output signal is written as

xs = xc + XN. (1)

The center-of-mass motion is also subject to a noise force FN (including, for example, thethermally driven motion of the mirror suspensions and the seismic motion of the ground thatcouples through the suspensions) and a feedback force that is proportional to xs. The resultingequation of motion in the frequency domain is given by:

−Mr

[2

− i p φ() − 2p

]xc = FN − K ()xs. (2)

Here K () is the frequency-domain feedback filter kernel, and the φ() term accountsfor mechanical damping. For a viscously damped pendulum with quality factor Qp = p/0p

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Figure 1. (a) Optical layout of a LIGO interferometer. Light reflected fromthe two Fabry–Perot cavities formed by input and end mirrors, M1–M4, isrecombined at the beam splitter (BS). To control the differential degree offreedom, an optical signal proportional to mirror displacement is measured onthe photodetector (PD), and fed back as a differential force on the mirrors,after appropriate filtering to form restoring and damping forces. (b) The spectraldisplacement noise density of the differential mode of motion of the LIGO 4 kminterferometer at the Hanford Observatory is shown. Also shown is the targetsensitivity and the quantum noise contribution, which consists of shot noiseabove 30 Hz and radiation pressure noise below. The standard quantum limit(SQL) is also shown, and the closest approach to the measured sensitivity isabout a factor of 10 near 150 Hz. (c) An aerial photograph of the LIGO Hanfordsite in the state of Washington is shown. (d) A photograph of a 10.8 kg mirror isshown. Photographs courtesy of the MIT/Caltech LIGO Laboratory.

(p and 0p correspond to the real part and twice the imaginary part of the complexeigenfrequency of the pendulum), φ() = 1/Qp. If the damping is not viscous, but insteadcaused by internal friction, φ() takes on a more complex form [28]. Combining equations (1)and (2), the equation of motion for the center-of-mass is obtained:

−Mr

[2

− i p φ() − 2p − K ()/Mr

]xc = FN − K ()XN. (3)

In this experiment, the control kernel is adjusted so that

K ()/Mr ≈ 2eff + i0eff (4)

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Figure 2. Response function of mirror displacement to an applied force, forvarious levels of damping. The points are measured data, the thin lines are azero fit parameter model of the complete feedback loop, and the thick linesspanning the resonance (shown in the shaded region) are fitted Lorentzians, fromwhich the effective resonant frequency and quality factor are derived for eachconfiguration.

with eff and 0eff much larger than p and 0p, respectively, such that the modified dynamics ofxc are given by a damped oscillator driven by random forces:

−Mr[2− i0eff − 2

eff] xc = FN − K ()XN. (5)

An electro-optical potential well in which the mirrors oscillate is thus created.The output of our experiment measures xs, and in order to deduce true mirror motion xc,

the limiting sources of noise must be considered. If noise predominantly drives the center-of-mass motion, i.e. FN K ()XN, then xs ≈ xc (see equation (1)) and the measured signalcorresponds to the center-of-mass motion. However, in the case that surface or sensing noisedominates, i.e. K ()XN FN, then a correction factor must be applied to the measured signalto deduce the center-of-mass motion. Taking equations (1) and (5), in the limit that FN = 0, weobtain

xc =K ()

Mr2xs. (6)

If the levels of each noise XN and FN are not precisely known, then one can make a conservativecorrection by applying a factor max(1, |K ()/Mr

2|) to determine the worst possible center-

of mass motion, thereby accounting for the fact that the servo can inject noise back onto theoscillator. The effective temperature of the mode may then be obtained:

Teff =Mr

2effδx2

rms

kB, (7)

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9

where

δx2rms =

∫∞

0max

(1,

K ()

Mr2

)2

Sxs()d ≡

∫∞

0Sxdd. (8)

Sx s is the single-sided power spectral density of the measured motion xs and Sxd includes thecorrection factor. At large feedback gains, the measured noise Sx s may be arbitrarily suppressed,however, the mirror motion will reach a finite level as limited by the detection noise XN. This‘squashing’ effect has been explored previously [9, 33], and the calculation of Sxd avoidsunderestimates of the mirror motion. It is impossible to reliably measure the mirror motion atarbitrarily high frequencies, and the integral in equation (8) will diverge in any real system.The integration must therefore be limited in its frequency band, as is later discussed. Finally,the corresponding occupation number may be determined by

Neff =kBTeff

heff. (9)

K () of equation (4) is formed by convolving the position-dependent output signal withfilter functions corresponding to the real and imaginary parts of the feedback kernel K ().In the LIGO feedback system, there are additional filters and propagation delays that causedeviations from the ideal cold, damped spring, at high and low frequencies. Below 100 Hz,K () increases sharply to suppress seismically driven motion; at high frequencies (abovea few kilohertz), K () decreases precipitously to prevent the control system from feeding shotnoise back onto the mirrors. However, in the frequency band important for this measurement(near the electro-optical resonance), the feedback is well approximated by a spring and dampingforce, as shown in figure 2.

3. Measurement results and discussion

The servo control loops of the LIGO interferometers are optimized to minimize noise couplingto measurement of the differential mode motion of the mirrors. The modifications to the servoloops to create a nearly ideal cold spring at eff = 140 Hz do not significantly affect the noiselimits, shown in figure 1. Figure 3 shows the amplitude spectral density of mirror displacementfor varying levels of cold damping. To infer the effective temperature of the mode, its effectivefrequency eff and an estimation of the root-mean-square displacement fluctuation δxrms mustbe determined. First the differential mirror motion is driven and the response is measured, asshown in figure 2. These response functions are fit to a damped oscillator model; eff and Qeff

are products of the fit. Then δxrms is computed by integrating the spectrum in the band from 100to 170 Hz, as described in equation (8). The sensitivity in this frequency band is limited by lasershot noise that enters into XN. To correct for the finite integration band, the result is scaled bysetting our measured spectrum equal to the integral over the same frequency band of a thermallydriven oscillator spectrum,

Sxth() =4kBTeff0eff/Mr

(2eff − 2)2 + 202

eff

. (10)

In this way, a minimum effective temperature Teff = 1.4 ± 0.2 µK is measured, correspondingto thermal occupation number Neff = 234 ± 35. Systematic error of 15% in the calibration

New Journal of Physics 11 (2009) 073032 (http://www.njp.org/)

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Figure 3. Amplitude spectral density of displacement in the frequency band ofintegration. The curves (from highest to lowest) were produced by applyingincreasingly strong cold damping to the oscillator, corresponding to themeasurements of figure 2. The depression in the lowest curve is due to theshape of the background noise spectrum; the effects of the servo are correctedfor according to equations (1)–(8). The narrow line features between 100 and110 Hz are mechanical resonances of auxiliary subsystems, and a 120 Hz powerline harmonic is also visible. The predominant noise is laser shot noise.

dominates statistical error in these uncertainty estimates. The limits to integration were chosenas a compromise between having a wide limit, and choosing frequencies at which mirror motionis sensed. In the limit that the width of the integration band approaches 0, the lowest temperatureachieved approaches 0.9 µK. For larger integration limits, the temperature diverges because ofthe increased uncertainty at high frequency caused by shot noise (as occurs in all experiments).The spectra in figure 3 are predominantly limited by shot noise in the measurement band. Itmay at first appear unusual to associate a temperature with a device limited by shot noise, ratherthan thermal noise. However, the above calculations are justified, since the ultimate limit toexperiments such as this is known to arise from optical noise [34].

4. Cooling to the quantum limit

An interesting question arises as to whether this technique can lead to ground state coolingof the electromechanical oscillator. To mitigate the shot noise limit, which arises due to thefluctuating number of photons detected, the laser power could be increased. However, radiationpressure noise (a fluctuating force exerted on the mirrors due to the shot noise of the laser)increases with laser power and will ultimately limit the sensitivity. The SQL is obtained when

New Journal of Physics 11 (2009) 073032 (http://www.njp.org/)

11

shot noise and radiation pressure noise contribute equally to the total quantum noise [35].Hence, the continuous displacement measurement required for servo feedback does introduce anadditional term to the uncertainty relation for the oscillator position and momentum fluctuationsdue to measurement-induced steady state decoherence. If, however, the classical noises (such asthermal) are reduced significantly below the SQL, active feedback, with the appropriate controlkernel, is capable of cooling the electro-optic oscillator to its motional ground state [36].

5. Future prospects with LIGO

In the coming years, two upgrades of the LIGO detectors are planned. The first, EnhancedLIGO, is presently underway with an expected completion date in 2009, and seeks to improvethe sensitivity of the instruments above 40 Hz. The improvement in displacement sensitivity inthe frequency band around 150 Hz, where the cold spring measurements were performed, isexpected to be about a factor of 2. Subsequently, a major upgrade, Advanced LIGO, expectedto be completed in 2014, should give a factor of 10–15 improvement in displacement sensitivityrelative to that of the detector used for this work (with a concomitant factor of 4 increasein mass). In Advanced LIGO, the laser power circulating in the Fabry–Perot cavities shouldexceed 800 kW, permitting strong restoring forces to be generated optically. Enhanced LIGOis expected to reach ∼6 times lower occupation number, approaching 40 quanta, and withAdvanced LIGO, the detectors will be operating at the SQL, allowing the ground state to beapproached.

As they approach the SQL, these devices should enable novel experimental demonstrationsof quantum theory that involve kilogram-scale test masses [25, 37, 38]. The present work,reaching microkelvin temperatures, provides evidence that interferometric gravitational wavedetectors, designed as sensitive probes of general relativity and astrophysical phenomena, canalso become sensitive probes of macroscopic quantum mechanics.

Acknowledgments

We gratefully acknowledge the support of the United States National Science Foundationfor the construction and operation of the LIGO Laboratory and the Science and TechnologyFacilities Council of the United Kingdom, the Max-Planck-Society, and the State ofNiedersachsen/Germany for support of the construction and operation of the GEO600 detector.We also gratefully acknowledge the support of the research by these agencies and by theAustralian Research Council, the Council of Scientific and Industrial Research of India, theIstituto Nazionale di Fisica Nucleare of Italy, the Spanish Ministerio de Educacion y Ciencia,the Conselleria d’Economia Hisenda i Innovacio of the Govern de les Illes Balears, the ScottishFunding Council, the Scottish Universities Physics Alliance, The National Aeronautics andSpace Administration, the Carnegie Trust, the Leverhulme Trust, the David and Lucile PackardFoundation, the Research Corporation and the Alfred P Sloan Foundation.

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[3] Arcizet O, Cohadon P-F, Briant T, Pinard M and Heidmann A 2006 Radiation pressure cooling andoptomechanical instability of a micromirror Nature 444 71

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[10] Corbitt T, Wipf C, Bodiya T, Ottaway D, Sigg D, Smith N, Whitcomb S and Mavalvala N 2007 Opticaldilution and feedback cooling of a gram-scale oscillator to 6.9 mK Phys. Rev. Lett. 99 160801

[11] Mow-Lowry C M, Mullavey A J, Goßler S, Gray M B and McClelland D E 2008 Cooling of a gram-scalecantilever flexure to 70 mK with a servo-modified optical spring Phys. Rev. Lett. 100 010801

[12] Vinante A et al 2008 Feedback cooling of the normal modes of a massive electromechanical system tosubmillikelvin temperature Phys. Rev. Lett. 101 033601

[13] Regal C A, Teufel J D and Lehnert K W 2008 Measuring nanomechanical motion with a microwave cavityinterferometer Nat. Phys. 4 555

[14] Schliesser A, Arcizet O, Rivire R and Kippenberg T J 2009 Resolved-sideband cooling and measurement ofa micromechanical oscillator close to the quantum limit arXiv:0901.1456

[15] Groeblacher S, Hertzberg J B, Vanner M R, Gigan S, Schwab K and Aspelmeyer M 2009 Demonstration ofan ultracold micro-optomechanical oscillator in a cryogenic cavity arXiv:0901.1801

[16] Thompson J D, Zwickl B M, Jayich A M, Marquardt F, Girvin S M and Harris J G E 2008 Strong dispersivecoupling of a high-finesse cavity to a micromechanical membrane Nature 452 72

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VIEWPOINT

The First Sounds of Merging BlackHolesGravitational waves emitted by the merger of two black holes have been detected, setting thecourse for a new era of observational astrophysics.

by Emanuele Berti∗,†

F or decades, scientists have hoped they could “lis-ten in” on violent astrophysical events by detectingtheir emission of gravitational waves. The waves,which can be described as oscillating distortions in

the geometry of spacetime, were first predicted to exist byEinstein in 1916, but they have never been observed di-rectly. Now, in an extraordinary paper, scientists report thatthey have detected the waves at the Laser InterferometerGravitational-wave Observatory (LIGO) [1]. From an analy-sis of the signal, researchers from LIGO in the US, and theircollaborators from the Virgo interferometer in Italy, infer thatthe gravitational waves were produced by the inspiral andmerger of two black holes (Fig. 1), each with a mass that ismore than 25 times greater than that of our Sun. Their find-ing provides the first observational evidence that black holebinary systems can form and merge in the Universe.

Gravitational waves are produced by moving masses, andlike electromagnetic waves, they travel at the speed of light.As they travel, the waves squash and stretch spacetime in theplane perpendicular to their direction of propagation (seeinset, Video 1). Detecting them, however, is exceptionallyhard because they induce very small distortions: even thestrongest gravitational waves from astrophysical events areonly expected to produce relative length variations of order10−21.

“Advanced” LIGO, as the recently upgraded version ofthe experiment is called, consists of two detectors, one inHanford, Washington, and one in Livingston, Louisiana.Each detector is a Michelson interferometer, consisting oftwo 4-km-long optical cavities, or “arms,” that are arrangedin an L shape. The interferometer is designed so that, inthe absence of gravitational waves, laser beams traveling inthe two arms arrive at a photodetector exactly 180 out of

∗Department of Physics and Astronomy, The University of Missis-sippi, University, Mississippi 38677, USA†CENTRA, Departamento de Física, Instituto Superior Técnico,Universidade de Lisboa, Avenida Rovisco Pais 1, 1049 Lisboa, Por-tugal

Figure 1: Numerical simulations of the gravitational waves emittedby the inspiral and merger of two black holes. The coloredcontours around each black hole represent the amplitude of thegravitational radiation; the blue lines represent the orbits of theblack holes and the green arrows represent their spins. (C.Henze/NASA Ames Research Center)

phase, yielding no signal. A gravitational wave propagat-ing perpendicular to the detector plane disrupts this perfectdestructive interference. During its first half-cycle, the wavewill lengthen one arm and shorten the other; during its sec-ond half-cycle, these changes are reversed (see Video 1).These length variations alter the phase difference betweenthe laser beams, allowing optical power—a signal—to reachthe photodetector. With two such interferometers, LIGO canrule out spurious signals (from, say, a local seismic wave)that appear in one detector but not in the other.

LIGO’s sensitivity is exceptional: it can detect length dif-ferences between the arms that are smaller than the sizeof an atomic nucleus. The biggest challenge for LIGO isdetector noise, primarily from seismic waves, thermal mo-tion, and photon shot noise. These disturbances can easilymask the small signal expected from gravitational waves.

physics.aps.org c© 2016 American Physical Society 11 February 2016 Physics 9, 17

Video 1: (Animation appears online only.) A schematic depictionof LIGO’s interferometric gravitational-wave detector. Light from alaser is split in two by a beam splitter; one half travels down thevertical arm of the interferometer, the other half travels down thehorizontal arm. The detector is designed so that in the absence ofgravitational waves (top left) the light takes the same time to travelback and forth along the two arms and interferes destructively atthe photodetector, producing no signal. As the wave passes(moving clockwise from top right) the travel times for the laserschange, and a signal appears in the photodetector. (The actualdistortions are extremely small, but are exaggerated here foreasier viewing.) Inset: The elongations in a ring of particles showthe effects of a gravitational wave on spacetime. (APS/AlanStonebraker)

The upgrade, completed in 2015, improved the detector’ssensitivity by a factor of 3–5 for waves in the 100–300 Hz fre-quency band and by more than a factor of 10 below 60 Hz.These improvements have enhanced the detector’s sensitiv-ity to more distant sources and were crucial to the discoveryof gravitational waves.

On September 14, 2015, within the first two days of Ad-vanced LIGO’s operation, the researchers detected a signalso strong that it could be seen by eye (Fig. 2). The mostintense portion of the signal lasted for about 0.2 s and wasobserved in both detectors, with a combined signal-to-noiseratio of 24. Fittingly, this first gravitational-wave signal,dubbed GW150914, arrived less than two months before the100-year anniversary of the publication of Einstein’s generalrelativity theory.

Up until a few decades ago, detecting gravitational waveswas considered an impossible task. In fact, in the 1950s,physicists were still heatedly debating whether the waveswere actual physical entities and whether they could carryenergy. The turning point was a 1957 conference in ChapelHill, North Carolina [2, 3]. There, the theorist Felix Pi-rani pointed out a connection between Newton’s secondlaw and the equation of geodesic deviation, which describesthe effect of tidal forces in general relativity. This connec-tion allowed him to show that the relative accelerations ofneighboring particles in the presence of a gravitational wave

Figure 2: On September 14, 2015, similar signals were observedin both of LIGO’s interferometers. The top panels show themeasured signal in the Hanford (top left) and Livingston (top right)detectors. The bottom panels show the expected signal producedby the merger of two black holes, based on numerical simulations.(B. P. Abbott et al. [1].)

provide a physically meaningful—and measurable—way toobserve it. Sadly, Pirani, who laid the groundwork for ourmodern thinking about gravitational waves and how to de-tect them, passed away on December 31, 2015, just weeksbefore the LIGO scientists announced their discovery.

Other prominent physicists at the meeting, includingJoseph Weber, Richard Feynman, and Hermann Bondi, wereinstrumental in pushing Pirani’s ideas forward. Feynmanand Bondi, in particular, developed Pirani’s observation intowhat is now known as the “sticky bead” thought experi-ment. They argued that if beads sliding on a sticky rodaccelerated under the effect of a passing gravitational wave,then they must surely also transfer heat to the rod by friction.This heat transfer is proof that gravitational waves must in-deed carry energy, and are therefore, in principle, detectable.

Interest in carrying out such experiments wasn’t imme-diate. As Pirani noted in his 1964 lectures on gravitationalradiation [4], Weber thought that meaningful laboratoryexperiments were “impossible by several orders of magni-tude.” At about the same time, William Fowler (the futureNobel laureate) suggested that a large fraction of the en-ergy emitted by so-called massive double quasars—what wenow know as black hole binaries—might be in the form ofgravitational radiation. Pirani, however, felt that the directobservation of gravitational waves was not “necessary orsufficient” to justify a corresponding theory, arguing that un-less physicists figured out a way to quantize gravity, such atheory would not “have much to do with physics” [4].

What galvanized the field was a 1969 paper from We-ber, who claimed he had detected gravitational radiationwith a resonant bar detector (see 22 December 2005 Focusstory). The finding was controversial—physicists could notduplicate it and by the mid-1970s, most agreed that We-

physics.aps.org c© 2016 American Physical Society 11 February 2016 Physics 9, 17

ber had likely been incorrect. However, a few years later ayoung professor at the Massachusetts Institute of Technol-ogy named Rainer Weiss was preparing for his course onrelativity when he came across a proposal by Pirani for de-tecting gravitational waves. Pirani had suggested using lightsignals to see the variations in the positions of neighboringparticles when a wave passed. His idea, with one key mod-ification, led to the genesis of LIGO: rather than using thetiming of short light pulses, Weiss proposed to make phasemeasurements in a Michelson interferometer [5]. RonaldDrever, Kip Thorne, and many others made crucial contribu-tions to developing this idea into what LIGO is today. (SeeRef. [2] for a historical account.)

Now, what was once considered “impossible by sev-eral orders of magnitude” is a reality. To confirm thegravitational-wave nature of their signal, the researchersused two different data analysis methods. The first was todetermine whether the excess power in the photodetectorcould be caused by a signal, given their best estimate of thenoise, but without any assumptions about the origin of thesignal itself. From this analysis, they could say that a tran-sient, “unmodeled” signal was observed with a statisticalsignificance greater than 4.6σ. The second method involvedcomparing the instrumental output (signal plus noise) withsignals of merging black holes that were calculated us-ing general relativity. From this so-called matched-filteringsearch, the researchers concluded that the significance of theobservation was greater than 5.1σ.

The most exciting conclusions come from comparing theobserved signal’s amplitude and phase with numerical rela-tivity predictions, which allows the LIGO researchers to es-timate parameters describing the gravitational-wave source.The waveform is consistent with a black hole binary systemwhose component masses are 36 and 29 times the mass ofthe Sun. These stellar-mass black holes—so named becausethey likely formed from collapsing stars—are the largest oftheir kind to have been observed. Moreover, no binary sys-tem other than black holes can have component masses largeenough to explain the observed signal. (The most plausiblecompetitors would be two neutron stars, or a black hole anda neutron star.) The binary is approximately 1.3 billion lightyears from Earth, or equivalently, at a luminosity distance of400 megaparsecs (redshift of z ∼ 0.1). The researchers esti-mate that about 4.6% of the binary’s energy was radiated ingravitational waves, leading to a rotating black hole remnantwith mass 62 times the mass of the Sun and dimensionlessspin of 0.67.

From the signal, the researchers were also able to performtwo consistency tests of general relativity and put a boundon the mass of the graviton—the hypothetical quantum par-ticle that mediates gravity. In the first test, they used generalrelativity to estimate the mass and spin of the black hole rem-nant from an “early inspiral” segment of the signal and againfrom a “post-inspiral” segment. These two different ways of

determining the mass and spin yielded similar values. Thesecond test was to analyze the phase of the wave generatedby the black holes as they spiraled inward towards one an-other. This phase can be written as a series expansion inv/c, where v is the speed of the orbiting black holes, and theauthors verified that the coefficients of this expansion wereconsistent with the predictions of general relativity. By as-suming that a graviton with mass would modify the phaseof the waves, they determined an upper bound on the parti-cle’s mass of 1.2 × 10−22 eV/c2, improving the bounds frommeasurements in our Solar System and from observations ofbinary pulsars. These findings will be discussed in detail inlater papers.

In physics, we live and breathe for discoveries like the onereported by LIGO, but the best is yet to come. As Kip Thornerecently said in a BBC interview, recording a gravitationalwave for the first time was never LIGO’s main goal. Themotivation was always to open a new window onto the Uni-verse.

Gravitational-wave detection will allow new and moreprecise measurements of astrophysical sources. For ex-ample, the spins of two merging black holes hold cluesto their formation mechanism. Although Advanced LIGOwasn’t able to measure the magnitude of these spins veryaccurately, better measurements might be possible with im-proved models of the signal, better data analysis techniques,or more sensitive detectors. Once Advanced LIGO reachesdesign sensitivity, it should be capable of detecting binarieslike the one that produced GW150914 with 3 times its currentsignal-to-noise ratio, allowing more accurate determinationsof source parameters such as mass and spin.

The upcoming network of Earth-based detectors, compris-ing Advanced Virgo, KAGRA in Japan, and possibly a thirdLIGO detector in India, will help scientists determine thelocations of sources in the sky. This would tell us whereto aim “traditional” telescopes that collect electromagneticradiation or neutrinos. Combining observational tools inthis way would be the basis for a new research field, some-times referred to as “multimessenger astronomy” [6]. Soonwe will also collect the first results from LISA Pathfinder,a spacecraft experiment serving as a testbed for eLISA, aspace-based interferometer. eLISA will enable us to peerdeeper into the cosmos than ground-based detectors, allow-ing studies of the formation of more massive black holes andinvestigations of the strong-field behavior of gravity at cos-mological distances [7].

With Advanced LIGO’s result, we are entering the dawnof the age of gravitational-wave astronomy: with this newtool, it is as though we are able to hear, when before we couldonly see. It is very significant that the first “sound” pickedup by Advanced LIGO came from the merger of two blackholes. These are objects we can’t see with electromagnetic ra-diation. The implications of gravitational-wave astronomyfor astrophysics in the near future are dazzling. Multiple de-

physics.aps.org c© 2016 American Physical Society 11 February 2016 Physics 9, 17

tections will allow us to study how often black holes mergein the cosmos and to test astrophysical models that describethe formation of binary systems [8, 9]. In this respect, it’sencouraging to note that LIGO may have already detecteda second event; a very preliminary analysis suggests thatif this event proves to have an astrophysical origin, then itis likely to also be from a black hole binary system. Thedetection of strong signals will also allow physicists to testthe so-called no-hair theorem, which says that a black hole’sstructure and dynamics depend only on its mass and spin[10]. Observing gravitational waves from black holes mightalso tell us about the nature of gravity. Does gravity reallybehave as predicted by Einstein in the vicinity of black holes,where the fields are very strong? Can dark energy and theacceleration of the Universe be explained if we modify Ein-stein’s gravity? We are only just beginning to answer thesequestions [11, 12].

Correction (26 February 2016): An earlier version of thearticle stated that the LIGO researchers tested general rela-tivity using a measurement of the oscillations in the signalfrom the remnant black hole. However, the test was actu-ally based on parameters describing the remnant that theresearchers inferred from the “early inspiral” and “post-inspiral” segments of the signal, as the revised version ofthe article now states.

REFERENCES[1] B. P. Abbott et al. (LIGO Scientific Collaboration and the Virgo

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