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The Search For Periodic Gravitational WavesGregory Mendell, LIGO Hanford Observatory

on behalf of the LIGO Scientific Collaboration

The Laser Interferometer Gravitational-Wave Observatory

http://www.ligo.caltech.edu

Supported by the United States National Science Foundation

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Gravitational Waves• Gravitation = metric tensor:• Weak Field Limit:• Gauge choice:• Quadrupole field:• At the detector:

dxdxgds 2

01

2

2

22

htc

hg

)/(2

0000

00

00

0000

cztifTT ehh

hhh

]2)/(2cos[1

)2/(ˆˆ

crtfr

hTT

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suspended test masses

free masses

Laser Interferometer Gravitational-wave Detection

Gravitational-wave Strain: LLh /•LIGO’s arms: L = 4 km.

•Sensitivity: L nominally around 10-18 m. (Depending on frequency and duration of the signal this can be much larger or much smaller!)

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Inside LIGO

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Sources of Continuous Gravitational Waves

Mountain on neutron starPrecessing neutron star

Accreting neutron star

Oscillating neutron star

A B

C

D

Credits:

A. image by Jolien Creighton; LIGO Lab Document G030163-03-Z.

B. image by M. Kramer; Press Release PR0003, University of Manchester - Jodrell Bank Observatory, 2 August 2000.

C. image by Dana Berry/NASA; NASA News Release posted July 2, 2003 on Spaceflight Now.

D. image from a simulation by Chad Hanna and Benjamin Owen; B. J. Owen's research page, Penn State University.

Search methods can detected any type of periodic source.

Upper limits are set on gravitational-wave amplitude, h0, of rotating triaxial ellipsoid.

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Nature of gravitational wave signal• The GW signal from a triaxial pulsar can be modelled as

• The unknown parameters are

• h0 - amplitude of the gravitational wave signal

• - polarization angle of signal; embedded in Fx,+

• - inclination angle of the pulsar

• 0 - initial phase of pulsar (0)

• In the targeted searches we currently only look for signals at twice the rotation frequency of the pulsars

• For blind searches the location in the sky and the source’s frequency evolution are unknown.

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Figure: D. Sigg LIGO-P980007-00-D

);();( tFtF

Amplitude Modulation

Beam Pattern Response Functions:

Polarization Angle

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Periodic Signals

nc

trtT ˆ

)(

T = arrival time at SSBt = arrival time at detector

Solar SystemBarycenter

GW approaching from direction n

Relativistic corrections are included in the actual code.

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Phase and Frequency Modulation

0

100

0

100

)ˆ)(

()!1(

)()!1(

s

ss

s

ss

Tnc

trt

s

f

TTs

f

Phase at SSB:

Phase at detector:

...)ˆ)(

(!

ˆ)(

)(

)ˆ)(

(!

ˆ)(

1)(

010

100

s

ss

s

ss

Tnc

trt

s

ffn

c

tatf

Tnc

trt

s

ffn

c

tvtf

Frequency at detector:

df/dt :

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Analysis of Hardware Injections

RA (hours) RA (hours)

DE

C (

degr

ees)

DE

C (

degr

ees)

PRELIMINARY APS April 2006

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Fake gravitational-wave signals corresponding to rotating neutron stars with varying degrees of asymmetry were injected for parts of the S4 run by actuating on one end mirror. Sky maps for the search for an injected signal with h0 ~ 7.5e-24 are below. Black stars show the fake signal’s sky position.

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All Sky Loudest Events 1000 SFTs, Fake Inst. Line & Noise:

DE

C (

rad

ian

s)

RA (radians)

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Maximum Likelihood

0,0,0cos

,0

),,cos,(

2

12

)(

...2

1

2

1

2

1)|(

2

0

22

0

2

00

222

22

2

)(

3

2

)(

2

2

)(

1

23

233

22

222

21

211

h

hfh

hhhxhx

eeehxP

j

j j

jj

j j

jj

j j

jj

hxhxhx

Likelihood of getting data x for model h for Gaussian Noise:

Chi-squared

Minimize Chi-squared = Maximize the Likelihood

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Coherent Matched Filtering

1

0

1

0

)0(M N

j

ijb

jbeFxX

1

0

/21 N

k

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Nx

1

0 )(2

)2cos1(2sin)0( 0

M

k b

bb

k

SFTki

b k

i

S

XeFX

Jaranowski, Krolak, & Schutz gr-qc/9804014; Schutz & Papa gr-qc/9905018; Williams and Schutz gr-qc/9912029; Berukoff and Papa

LAL Documentation

222

*2222 24

FFFF

XXeFFXFXF

MF

F-statistic

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Time Domain Coherent Search Using Bayesian Analysis

Ch

hxhx

dxxhPC

hxPxP

hPxhP

eehxP

baPbPabPaP

0

2

)(

2

2

)(

1

)|(

)|()(

)()|(

...2

1

2

1)|(

)|()()|()(

22

222

21

211

Bayes’ Theorem:

Likelihood:

Confidence Interval:

Posterior Probability:

Prior

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Semi-coherent Methods

Time

Fre

quen

cy

Time

Track Doppler shift and df/dt

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• Break up data into segments; FFT each segment.

• Track the frequency.

• StackSlide: add the power weighted by the noise inverse.

• Hough: add 1 or 0 if power is above/below a cutoff (advanced version includes weighted average of 1’s & 0’s).

• PowerFlux: add power using weights that maximize SNR

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Short Fourier Transforms (SFTs) & Periodograms:

Hough Number Count & Weighted Number Count:

StackSlide Power:

Power Flux:

kk S

FF22

1

0'

'24

'

221

0

1

0

1

0

1

0

1

0

21

0

2

'/)(

/)()()(

1

/1

//1

/0

~2~

M

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k

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M

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tifjk

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T

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P

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nNnNn w

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Frequentist Confidence Region

• Estimate parameters by maximizing likelihood or minimizing chi-squared

• Injected signal with estimated parameters into many synthetic sets of noise.

• Re-estimate the parameters from each injection

ftBftAts 2sin2cos)( Matlab simulation for signal:

Figure courtesy Anah Mourant, Caltech SURF student, 2003.

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Bayesian Confidence Region

x

Matlab simulation for signal: ftBftAts 2sin2cos)( Uniform Prior: )2/exp();,( 2xBAP

Figures courtesy Anah Mourant, Caltech SURF student, 2003.

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Frequentist Population-based Confidence Region From The Loudest Event.

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GWs from triaxial pulsar

zz

yyxx

I

II

r

fIh zz

20

4

2

0 c

G16

• For upper limits have to select a model. (This is not needed for detection!)• Ellipticity, , measures asymmetry in triaxially shaped pulsar

equatorial ellipticity

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The back of the envelope please…For a triaxial ellipsoid (or think of I as a way of scaling a the time varying quadrupole Q):

• Maximum expected ellipticities:

•Solid quark stars: 10-4

•Hybrid hyperon/heutron stars or buried B fields: 10-5

•Neutron stars with fluid core and solid crust: 10-6

• Estimated strain:

h = 42(G/c4) If 2/r

~ 22(vesc2/c2)(vR

2/c2)(R/r)

~10–25( /10-6)(I/1045 g.cm2) (f/300 Hz)2 (1kpc/r)

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Results1. S1 coherent time-domain and F-stat search targeting pulsar J1939+2134:

The LIGO Scientific Collaboration, Phys.Rev. D69 (2004) 082004; gr-qc/0308050

Best UL: h_0 < 1.4 10-22; < 2.9 10-4 (1045 g.cm2/I)

2. S2 coherent time-domain search targeting 28 pulsars: The LIGO Scientific Collaboration: B. Abbott, et al, M. Kramer, A.G. Lyne, Phys.Rev.Lett. 94 (2005) 181103;gr-qc/0410007

Best UL: h_0 < 1.7 10-24; Best UL < 4.5e 10-6

3. S2 semi-coherent all-sky, 200-400 Hz, Hough search: LIGO Scientific Collaboration, Phys.Rev. D72 (2005) 102004; gr-qc/0508065

Best UL: h_0 < 4.4 10-23

4. S3 coherent all-sky, 160-728.8 Hz and Sco X-1, 464-484 Hz & 604-624 Hz, F-stat search: LIGO Scientific Collaboration, submitted to Phys. Rev. D (2006); gr-qc/0605028

Best all-sky UL: h_0 < 6.6e-23 ; Best Sco X-1 UL: h_0 < 1.7e-22

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Preliminary Results and Plans5. S3 Einstein@Home search: preliminary S3 results posted at

http://einstein.phys.uwm.edu/. Final S3 and S4 results in preparation. Initial start of S5 search is running.

6. S3/S4 Targested search including pulsars: 32 isolated and 44 in binaries; results in preparation.

April 2005 APS Best UL: h_0 < few 10-25, and < 10-6 for one pulsar

7. S4 all-sky, 50-1000 Hz, PowerFlux, StackSlide, Hough search: results in preparation; S5 preliminary all-sky PowerFlux search

April 2005 APS Best UL: h_0 < few 10-24.

8. Start of S5 preliminary coherent time-domain targeted pulsar search:

Best UL h0 < few 10-25; Best UL: < few x 10-7

9. S4/S5 F-stat targeted isolated x-ray sources: RX J1856.5-3754 (nearest known NS) and Cas A (youngest known NS) search started.

10. S5 Multi-IFO PowerFlux & Multi-IFO Hierarchical Einstein@Home Search under development.

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S5 targeted h0 Results

• Spin-down upper limit calculated with intrinsic spin-down value if available i.e. corrected for Shklovskii transverse velocity effect

• Closest to spin-down upper limit

Crab pulsar ~ 2.1 times greater than spin-down (fgw = 59.6 Hz, dist = 2.0 kpc)

h0 = 3.0x10-24, = 1.6x10-3

Assumes I = 1038 kgm2

• Sensitivity curves use:

Crab pulsar

Frequency (Hz)

h 0

PRELIMINARY APS April 2006

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Einstein@Home S3 ResultsPRELIMINARY

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Many searches are underway. You can join via Einstein@home:

http://einstein.phys.uwm.edu/• Like SETI@home, but for LIGO/GEO matched filter search for GWs from rotating compact stars.

• Support for Windows, Mac OSX, and Linux clients

• Our own clusters have thousands of CPUs.

• Einstein@home has many times more computing power at low cost.

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End

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S5 And Future Prospects

• PowerFlux will give quick views of S5 data, and provide candidates for follow-up if any unexplained signals turn up.

• Hierarchical searches, with coincidence, other vetoes, and follow-up searches, will improve sensitivity.

Divide data into ~30 hr segments and generate coherent F-statistic for each.

Apply StackSlide/Hough of the F-statistic segments. Automatic follow-up of candidates will be done. The code will run under Einstein@Home.

• Advanced LIGO will have 10x sensitivity, better a lower frequencies. Using signal recycling can further improve sensitivity in a narrow frequency band.

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Einstein@Home S3 ResultsPRELIMINARY

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Line Avoiding: Doppler Skybands (e.g., for no-spindown)

Skyband 0 (good)

Skyband 10 (worst)

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All Sky Loudest Events 1000 SFTs, Fake Pulsar & Noise:

DE

C (

rad

ian

s)

RA (radians)

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Dealing With Instrument Lines: Cleaning

Frequency (Hz)

Spe

ctra

l Den

sity

[1/

Sqr

t(H

z)]

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Astrophysical Sources

Black Holes Dense Stars

Supernovae Stochastic Background

Photos: http://antwrp.gsfc.nasa.gov; http://imagine.gsfc.nasa.gov

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Neutron and Strange-quark Stars

http://chandra.harvard.edu/resources/illustrations/neutronstars_4.html

NASA/CXC/SAO

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Calibration• Measure open loop gain G,

input unity gain to model• Extract gain of sensing

function C=G/AD from model

• Produce response function at time of the calibration, R=(1+G)/C

• Now, to extrapolate for future times, monitor single calibration line in AS_Q error signal, plus any changes in gain beta, and form alphas

• Can then produce R at any later time t, given alpha and beta at t

• A photon calibrator, using radiation pressure, gives results consistent with the standard calibration.

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S4 StackSlide “Loudest Events” 50-225 Hz

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•Searched 450 freq. per .25 Hz band, 51 values of df/dt, between 0 & -1e-8 Hz/s, up to 82,120 sky positions (up to 2e9 templates). The expected loudest StackSlide Power was ~ 1.22 (SNR ~ 7)

•Veto bands affected by harmonics of 60 Hz.

•Simple cut: if SNR > 7 in only one IFO veto; if in both IFOs, veto if abs(fH1-fL1) > 1.1e-4*f0

Sta

ckS

lide

Pow

erS

tack

Slid

e P

ower

Frequency (Hz)

APS April 2006

PRELIMINARY

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S4 StackSlide “Loudest Events” 50-225 Hz

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Sta

ckS

lide

Pow

erS

tack

Slid

e P

ower

Frequency (Hz)

APS April 2006

PRELIMINARY

FakePulsar3=>

FakePulsar3=>

FakePulsar8=>

FakePulsar8=>

<=Follow-up Indicates Instrument Line

<=Follow-up Indicates Instrument Line

•Searched 450 freq. per .25 Hz band, 51 values of df/dt, between 0 & -1e-8 Hz/s, up to 82,120 sky positions (up to 2e9 templates). The expected loudest StackSlide Power was ~ 1.22 (SNR ~ 7)

•Veto bands affected by harmonics of 60 Hz.

•Simple cut: if SNR > 7 in only one IFO veto; if in both IFOs, veto if abs(fH1-fL1) > 1.1e-4*f0

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Best: 4.4e-24 For 139.5-139.75 Hz Band

PRELIMINARY

Frequency (Hz)

S4 StackSlide h0 95% Confidence All Sky Upper Limits 50-225 Hz

APS April 2006

Note that h0 is the gravitational-wave amplitude for a rotating triaxial ellipsoid. Monte Carlo injections of triaxial sources over the search parameter space & source orientations set the ULs.

Best: 5.4e-24 For 140.75-141.0 Hz Band

h0 U

pper

Lim

it h

0 Upp

er L

imit

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Linear amplitude (=0.5*h0

worst-pulsar-orient.)

Doppler Skyband 0

Color coding

BLUE – Non-Gaussian

Diamond – Wand. Line

Green – Upper Limit

Red point – Loose Candidate (SNR > 7)

PRELIMINARYPowerFlux 95% CL limits – S4 H1 (50-1000 Hz)

APS April 2006

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PowerFlux S5 H1 40-800 Hz (no spindown)

PRELIMINARY APS April 2006

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Linear amplitude (=0.5*h0

worst-pulsar-orient.)

Doppler Skyband 0

Color coding

BLUE – Non-Gaussian

Diamond – Wand. Line

Green – Upper Limit

Red point – Loose Candidate (SNR > 7)

Detector artifacts worse at low frequencies for L1 39.87 Hz comb

(RF oscillator)

*For APS

PRELIMINARY95% CL limits – L1 (50-1000 Hz)

APS April 2006

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Sensitivity of PeriodicSearch Techniques• False alarm & false

dismissal rates determine SNR of detectable signal.

• Coherent matched filtering tracks phase.

• Incoherent power averaging tracks frequency only.

• Optimal search needs 1023 templates per 1 Hz for 1 yr of data; Hierarchical approach needed.

4/17

2/12325

7

2/12326

101800

10102

10

10103

/

obscoh

n

incohc

obs

n

cohc

nobsc

T

s

T

s

Hz

Sh

T

s

Hz

Sh

SThSNR

These are for 1% false alarm & 10% false dismissal rates, an average sky position and source orientation. A hierarchical search employs a much large false alarm rate, then use coincidence and other vetoes to maximize sensitivity and narrow follow-up searches to the most promising candidates.

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S2 Time Domain Search Results

Best ho UL = 1.7 x10-24. Best ellipticity UL = 4.5 x 10-6 (I = 1045 gcm2)

LIGO Scientific Collaboration & The Pulsar Group, Jodrell Bank Observatory, gr-qc/0410007, Phys. Rev. Lett. 94 (2005) 181103.

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S2 All-sky Hough Search Results

Best ho Upper Limit = 4.43 x10-23

LIGO Scientific Collaboration, gr-qc/0508065, submitted to Phys. Rev. D (2005).

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Estimate for population of objects spinning down due to gravitational waves

birthratebirthrategalaxy

galaxyage

birthrate

age

yrs

c

GI

Rh

Rr

fc

fGI

rh

ffKffQIE

10010

8

51

~

8

||201

||4/||

243

min

3

52

Blanford (1984) as cited by Thorne in 300 Years of Gravitation; see also LIGO Scientific Collaboration, gr-qc/0508065, submitted to Phys. Rev. D (2005); and LIGO Scientific Collaboration, S2 Maximum Likelihood Search, in prepartion.

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Simulation:What might detection sound or look like?

Play Me

(AM & FM modulation greatly exaggerated!)