Binary inspiral event rates - Istituto Nazionale di Fisica Nuclearedragon.roma2.infn.it/lectures/rome_lorimer.pdf · 2004. 9. 22. · White Dwarf (WD) Neutron Star (NS) Black Hole

Binary inspiral event rates

Dunc Lorimer, Jodrell Bank Observatory, UK

Villa Mondragone International School of Gravitation and Cosmology September 8, 2004

our motivation as pulsar astronomers...

Discovery of J0737–3039A (Burgay et al. 2003) −→ “A”Discovery of J0737–3039B (Lyne et al. 2004) −→ “B”


Mass–mass diagram for the double pulsar


Detection of Shapiro delay−60−40

−20

0

20

40

60

Res

idua

ls (µ

s)

0 60 120 180 240 300 360Longitude from ascending node (deg)

−40

−20

0

20

40

60

80

100

Res

idua

ls (µ

s)

Determination of masses allows test of GR...

sobssGR

= 1.0001± 0.00220

(Kramer et al. 2004 — astro-ph/0405179)

...this edge-on orbit has other consequences also...


...the A pulsar shines through its companion (B)

..magnetosheath

radio beam of B

wind of A

A B

absorbing plasma

to Earth


High time resolution observations of the eclipse

-0.50

0.51

1.5

-20 0 20

-0.50

0.51

1.5

-0.50

0.51

1.5

89 90 91

0

0.5

1

Orbital phase (degrees)

Time (s)

Puls

ed fl

ux d

ensi

ty (a

rbitr

ary

units

)M

cLaughlin et al. (2004) A

pJ submitted (astro−ph/0408297)


...but back to the subject of this talk...

• Compact object census– Observed sample– Merging plane– Simple evolutionary scenarios

• Radio pulsar statistics– The pulsar phenomenon– Surveys and selection effects– Correction techniques

• Coalescence rates– Empirical estimates NS-NS, NS-WD, WD-NS– Population synthesis estimates– WD-WD binaries– NS-BH and BH-BH binaries

• Future directions


A quick census of Galactic compact objectsSystem We observe Nobs Merger rate?

yr−1

WD–WD brighter of the two WDs 13 10−2

WD–NS PSR and sometimes the WD 2 > 7× 10−6NS–WD PSR and sometimes the WD 40 > 4× 10−6NS–NS one or both neutron stars as PSRs 8 8× 10−5NS–MS PSR and main-sequence star 2 —NS–BH nothing so far ? < 7× 10−6NS–GS low-mass X-ray binary 120 —NS–SG high-mass X-ray binary 70 —BH–GS low-mass X-ray binary 3 —BH–SG high-mass X-ray binary 7 —

Key quantity is GW coalescence timescale:

τGW ∼ 107 yr(

Pbhr

)8/3 (µ

M�

)−1 (M

M�

)−2/3(1− e2)7/2,

where M = mp + mc and µ = m1m2/M .


The compact object “merging plane”


Simplified binary evolution scenarios

Main Sequence (MS)

Red Giant

White Dwarf (WD)

Neutron Star (NS)

Black Hole (BH)

WD−WDe.g. 0957−666

WD−NSe.g. J1141−6545

WD NS

WD NS

NS−NS BH−WD

NS−BH

BH−BHe.g. J1012+5307 e.g. B1913+16

NS−WD

WD BH

BHWD NS

NS

−more massive−shorter lifetime

−less massive−longer lifetime

primary secondary

primary forms compact object

secondary forms compact object

Mass transfer via Roche lobe overflow


The pulsar phenomenon

Original pen chart recording (Hewish et al. 1968)

P

ΩB

Pulsars are rapidly rotating highly magnetised neutron stars...

Individual pulses

1.5 ms < P < 8.5 s


The pulsar distance scalePulse dispersion caused by free electrons in the interstellar medium...

Observed projection onto Galactic plane(heavily observationally biased sample!)


Pulsar searching (in a nutshell)

DM

Filterbank Data

17.9 Hz (55.69 ms) S/N ~ 9

Standard pulsar search procedure...

Time Series Folded Spectrum

FFT


Major radio pulsar surveys carried out so far

search signal-to-noise ratio �binary, the degradation factorF ¼ �binary=�control. Significant degradation occurs, there-fore, when F5 1. Since accumulated Doppler shift, andtherefore F, is a strong function of the orbital phase atthe start of a given observation, for both binary systems,we calculate the mean value of F for a variety of startingorbital phases appropriately weighted by the time spentin that particular part of the orbit.

A similar analysis was made by Camilo et al. (2000)for the millisecond pulsars in 47 Tucanae. In this paper,where we are interested in the degradation as a functionof integration time, we generate time series with a varietyof lengths between 1 minute and 1 hr using samplingintervals similar to those of the actual surveys listed inTable 1. The results are summarized in Figure 1, wherewe plot average F versus integration time for both sets oforbital parameters. As expected, surveys with the longestintegration times are most affected by Doppler smearing.For the Parkes Multibeam Survey (Lyne et al. 2000;Manchester et al. 2001), which has an integration time of35 minutes, mean values of F are 0.7 and 0.3 for PSRB1913+16 and PSR B1534+12, respectively.4 The greaterdegradation for PSR B1534+12 is due to its mildly

eccentric orbit (e � 0:3 vs. 0.6 for PSR B1913+16), whichresults in a much more persistent change in apparentpulse period when averaged over the entire orbit. For theJodrell Bank and Swinburne surveys (Nicastro et al.1995; Edwards et al. 2001), which both have integrationtimes of the order of 5 minutes, we find F � 0:9 for bothsystems. For all other surveys, which have significantlyshorter integration times, no significant degradation isseen, and we take F ¼ 1.

5. STATISTICAL ANALYSIS

In this section we describe in detail the derivation of theprobability distribution of the Galactic coalescence rate R.The analysis method makes use of Bayesian statistics andtakes into account the rate contributions of both observedNS-NS binaries. At the end of the section we derive theassociated detection rates for LIGO.

5.1. The Rate Probability Distribution forEach Observed NS-NS Binary

As already mentioned in x 2, for each of the twoobserved NS-NS binaries (PSR B1913+16 or PSRB1534+12), we generate pulsar populations in physicaland radio luminosity space with pulse periods and widthsfixed to the observed ones and with different absolutenormalizations, i.e., total number Ntot of pulsars in thegalaxy. We generate large numbers of ‘‘ observed ’’ pulsar

4 In order to improve on the sensitivity to binary pulsars, the ParkesMultibeam Survey data are now being reprocessed using various algo-rithms designed to account for binary motion during the integration time(Faulkner et al. 2003).

TABLE 1

Simulated Pulsar Surveys

Year Telescope

�a

(MHz)

D�b

(MHz)

tobsc

(s)

tsampd

(ms)

Smine

(mJy) Detectedf References

1972 .............. Lovell 76 m 408 4 660 40 10 51/31 1, 2

1974 .............. Arecibo 305m 430 8 137 17 1 50/40 3, 4

1977 .............. Molonglo 408 4 45 20 10 224/155 5

1977 .............. Green Bank 300 inch 400 16 138 17 10 50/23 6, 7

1982 .............. Green Bank 300 inch 390 16 138 17 2 83/34 8

1983 .............. Green Bank 300 inch 390 8 132 2 5 87/20 9

Lovell 76 m 1400 40 524 2 1 61/40 10

1984 .............. Arecibo 305m 430 1 40 0.3 3 24/5 9

1985 .............. Molonglo 843 3 132 0.5 8 10/1 11

1987 .............. Arecibo 305m 430 10 68 0.5 1 61/24 12

1988 .............. Parkes 64m 1520 320 150 0.3 1 100/46 13

1990 .............. Arecibo 305m 430 10 40 0.5 2 2/2 14

1992 .............. Parkes 64m 430 32 168 0.3 3 298/101 15, 16

1993 .............. Arecibo 305m 430 10 40 0.5 1 56/90 17–20

1994 .............. Lovell 76 m 411 8 315 0.3 5 5/1 21

1995 .............. Green Bank 140 inch 370 40 134 0.3 8 84/8 22

1998 .............. Parkes 64m 1374 288 265 0.1 0.5 69/170 23

Parkes 64m 1374 288 2100 0.3 0.2 �900/600 24, 25

a Center frequency.b Bandwidth.c Integration time.d Sampling time.e Sensitivity limit at the survey frequency for long-period pulsars (calculated for each trial in the simulations).f Total number of detections and new pulsars.References.—(1) Davies, Lyne, & Seiradakis 1972. (2) Davies, Lyne, & Seiradakis 1973. (3) Hulse & Taylor 1974. (4) Hulse

& Taylor 1975. (5) Manchester et al. 1978. (6) Damashek, Taylor, & Hulse 1978. (7) Damashek et al. 1982. (8) Dewey et al.1985. (9) Stokes et al. 1986. (10) Clifton et al. 1992. (11) D’Amico et al. 1988. (12) Nice, Fruchter, & Taylor 1995. (13)Johnston et al. 1992. (14)Wolszczan 1991. (15)Manchester et al. 1996. (16) Lyne et al. 1998. (17) Ray et al. 1996. (18) Camilo etal. 1996. (19) Foster et al. 1995. (20) Lundgren, Zepka, & Cordes 1995. (21) Nicastro et al. 1995. (22) Sayer et al. 1997. (23)Edwards et al. 2001. (24) Lyne et al. 2000. (25)Manchester et al. 2001.

988 KIM, KALOGERA, & LORIMER Vol. 584

Recent Parkes multibeam surveys dominating flood of discoveries

(1/2 of all known PSRs incl. 3 NS-NS; one WD-NS; many NS-WD)


as an aside... some of the surveying telescopes...

...pulsar searching is a lot of fun! ...many still to be found


Radio pulsar survey selection effects

• Inverse square law: S ∝ L/d2

We are here!

Galactic centre




• Interstellar: dispersion, scattering

Emitted Pulse Detected Pulse

Pulsar Telescope





• The sky is hot!





• The sky is hot!

• Nulling and scintillation





• The sky is hot!

• Nulling and scintillation

• Doppler smearing of binary systems

Signal Frequency

Am

plitu

de

Signal Frequency

Am

plitu

de


Acceleration searching‘‘raw’’ data for PSR B1913+16

constant acceleration removed

~inertial frame

raw time series

resample time series

T = t ( 1 + v(t)/c ) ; v(t)=at

t

T


Accounting for S/N losses due to acceleration

S/N i S/N b

S/N b S/N iDefine sensitivity reduction factor as /

...results need to be averaged over all orbital phases...


Accounting for S/N losses due to acceleration

...this effect particularly crucial for deep searches...


If that wasn’t enough - pulsars are beamed!

...and there is debate about the shape of the beam!


Mapping beam shape through geodetic precession

...note geodetic precession does not effect number estimates


Accounting for selection biases: zeroth-order!

• Guess at where incompleteness sets in (say 1 kpc!)

• Count pulsars that are within this distance (100)

• Multiply by ∼ 5 to “correct” for beaming

• Multiply by 202 to scale over whole Galaxy

Simulated sample(no selection effects)

Observed sample

N ~ 100 x 5 x 400

This is about the

than anything :)more by accident ‘right’ answer, but

= 200,000

N.B. Divide by mean pulsar lifetime to get BPSRs ∼ 0.01 yr−1.


Accounting for selection biases: Monte Carlo

Interstellarmediummodel

AccelerationCurve

sampleTrueModel

sample

Pulsar surveys

Smin

RotationCurve

The Galaxy

Neutron starproperties...

V0,P(t),B(t),L(t)


Using Monte Carlo simulations to invert a population

R

(2) Model this region

(1) Set up a well−defined sample

(within this region selection effects well understood)

(e.g. uniform surface density and exponential scale height)

(3) Detect each pulsar over the modelled region

(use pulsar parameters and accurate survey models)

Create ‘‘N’’ cells

‘‘n’’ detections

. ... .. ..

....... . .

..

...

. .. . .

.

..


Constructing luminosity functions

Normal pulsars Millisecond pulsars

N.B. it’s very useful to perform consistency check:

• Take luminosity function and distribution

• Run sample through Monte Carlo detection code

• Does the model observed sample look reasonable?

N.B. method only provides luminosities down to observed minimum


The observed sample of DNS binaries

PSR P Pb a sin i e ω̇ M τGWms days lt-s deg yr−1 M� Gyr

B1913+16 59.0 0.323 2.34 0.617 4.227 2.83 0.31B1534+12 37.9 0.421 3.73 0.274 1.756 2.75 2.69B2127+11C 30.5 0.335 2.52 0.681 4.457 2.71 0.22J1518+4904 40.9 8.634 20.04 0.249 0.011 2.62 9600J1811−1736 104.2 18.779 34.78 0.828 0.009 2.6 1700J0737−3039A 22.7 0.102 1.42 0.088 16.88 2.58 0.087J0737−3039B 2773.5 0.102 1.51 0.088 16.88 2.58 0.087J1829+2456 41.0 1.17 7.24 0.14 0.28 2.53 60J1756−2251 28.5 0.319 2.75 0.18 2.59 2.57 1.7

Let’s now use these techniques to calculate the GalacticNS-NS coalescence rate...

(use most plausible pulsar distribution parameters in calculations)


Determining the NS-NS coalescence rate R

Take some Monte Carlo (physical) model of the Galaxy, then for eachobserved object...

• Calculate scale factor → Si = Ni/ni

• Correct for finite fraction of sky covered by pulsar beam → fi ∼ 5

• Estimate lifetime until GW coalescence → Ti = τc + τGW

R =nDNS∑i=1

SifiTi

Narayan et al. (1991) ApJ 379 L17; Phinney (1991) ApJ 380 L17

however, there are problems/uncertainties...


The small-number (cf. Malmquist) bias

As pointed out by Kalogera et al. (2001) ApJ, 556, 340...

Flux-limited sample

Model Detected PopulationSun

Model Galactic Population

Sun

...need to investigate the distribution of detections


Making the most of the available information...

The number of detections follows a Poisson distribution:


Making the most of the available information...

So for a given model, we have

P (Nobs;λ) =λNobs exp(−λ)

Nobs!,

where λ = 〈Nobs〉 = αNtot, and can show that

P (R) =(

ατlifef

)2R exp(−Rατlife/f).

For the first time, have probability distribution of R...

(for details see Kim et al. 2003, ApJ, 584, 985).


Most recent such analysis including J0737–3039

Galactic rate RNS−NS = 83+210−66 Myr−1.Predict a further 4 NS-NS systems in Parkes survey.


Scaling these results to LIGO...

Model R IRF Rdet of LIGOinitial advanced

Myr−1 kyr−1 yr−1

1 23.2+59.4−18.5 6.4 9.7+24.9−7.7 52.2

+133.6−41.6

6 83.0+209.1−66.1 6.4 34.8+87.6−27.7 186.8

+470.5−148.7

9 7.9+20.2−6.3 6.6 3.3+8.4−2.6 17.7

+45.4−14.1

10 23.3+57.0−18.4 5.8 9.8+23.9−7.7 52.4

+128.2−41.3

12 9.0+21.9−7.1 6.0 3.8+9.2−3.0 20.2

+49.4−15.9

14 3.8+9.4−2.8 5.4 1.6+3.9−1.2 8.5

+21.1−6.2

15 223.7+593.8−180.6 7.1 93.7+248.6−75.6 503.2

+1336.0−406.3

17 51.6+135.3−41.5 6.9 21.6+56.7−17.4 116.1

+304.4−93.4

19 14.6+38.2−11.7 7.0 6.1+16.0−4.9 32.8

+86.0−26.3

20 89.0+217.9−70.8 6.2 37.3+91.2−29.6 200.3

+490.3−159.3

Bad news: initial LIGO � 1 event in 3 yr :(Good news: advanced LIGO � 1 event per yr!!!

Bottom line: advanced LIGO should resolve problem (by inversion)


Apply same method to NS-WD and WD-NS samples

PSR P Pb mwd e τc τGW NPSRJ0751+1807 3.479 6.315 0.18 < 10−4 6.8 7.6 2900J1757−5322 8.870 10.88 0.67 10−6 5.1 7.8 1200

J1141−6545 393.9 4.744 0.986 0.172 1.5×10−3 0.6 400

• Select same set of physical models

• For each PSR, calculate scale factor (no beaming correction)

• Sum up contribution to RNS−WD and generate rate PDF

Of particular interest is contribution to LISA curve...

hrms(f) ' 1.7× 10−26(MM�

)5/6(f

mHz

)−7/6(No

Mpc−3

)1/2 (Tobsyr

)−1/2For further details, see Kim et al. (2004; astroph/0402162)


Apply same method to NS-WD and WD-NS samples

99%

95%

68%

68%

95%

99%

99%

95%

68%

68%

95%

99%

WD-WD

Bottom line: these sources fall below LISA noise curve (see also Cooray 2004)


Population synthesis estimates

P P

M

P

P

PP

i

i

iM , m , P , e , ti i

i

i

i

e

t ( )

log t

M ( )

Initial distributions

Population Synthesis

Until N = 500 000

Stellar evolution and binary evolution

Current binary

Choose realization i:

m/M

P ( )

m

M

P

e

log

m/M ( ) e ( )

(Figure taken from Nelemans’ PhD thesis)


Population synthesis estimates

Pros:

• Extremely powerful diagnostic/educational tool

• Potentially explain observed correlations etc.

• Can predict new evolutionary outcomes (e.g. NS-BH)

• Best applied to isolated problems if possible

Cons:

• Extremely sensitive to input physics

• At the mercy of normalisation scheme

• Do not always include selection effects


WD-WD binaries and LISA

from Nelemans et al. (2001) A&A 375, 890...

• Originally shown by Evans et al. (1987) ApJ 323, 129

• Note excess contribution is from AM CVn-like systems

• Confusion sets in below about 2 mHz


A note on NS-BH and BH-BH systems

Both are prime sources for LIGO/VIRGO/GEO due to greater distance

but have not been observed electromagnetically. Simple upper limitfor NS-BH from radio pulsar observations:

BNS−BH <BPSRsNPSRs

=0.011500

∼ 7× 10−6 yr−1

is exactly at the level of recent population synthesis predectionsSipior et al. (2004; astro-ph/0407268) predict 1 every 1500 pulsars!

Only candidate so far is PSR J1740–3052(Stairs et al. 2001, MNRAS 325, 797)→ still awaiting confirmation of K-giant companion

BH-BH systems could be extremely common in globular clusters(advanced LIGO could see up to one event per day; Portegies Zwart& McMillan 2000)


Future directions in general

A better understanding of the following

• WD sample biases

• radio pulsar luminosity function

• radio pulsar Galactic distribution

• radio pulsar velocity distribution

• extrapolation out to initial/advanced LIGO distances


Future directions for radio observers

Uncovering the rest of the Galactic NS population...

• NOW: Re-analysis of Parkes data (3–4 more NS-NS)

• NOW: Globular cluster surveys (NS-BH binary????)

• NOW: LIGO/VIRGO/GEO600 running

• NOW: All-sky Arecibo surveys (500 pulsars expected)

• 2008?: LOFAR in operation (several thousand pulsars)

• 2008?: Advanced LIGO

• 2012?: LISA

• 2015?: SKA in operation (20,000+ pulsars expected)

Can we find the elusive NS–BH binary???


Perhaps by 2030, we’ll see this!

−15 −10 −5 0 5 10 15X (kpc)

−15

−10

−5

0

5

10

15

Y (k

pc)

...and perhaps that’s a good place to stop!


Documents

Binary inspiral event rates - Istituto Nazionale di Fisica Nuclearedragon.roma2.infn.it/lectures/rome_lorimer.pdf · 2004. 9. 22. · White Dwarf (WD) Neutron Star (NS) Black Hole