NORTHWESTERN UNIVERSITY

Galactic Merger Rates of Pulsar Binaries

A DISSERTATION

SUBMITTED TO THE GRADUATE SCHOOLIN PARTIAL FULFILLMENT OF THE

REQUIREMENTS

for the degree of

DOCTOR OF PHILOSOPHY

Field of Physics and Astronomy

By

Chunglee Kim

EVANSTON, ILLINOIS

June 2006

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c© Copyright by Chunglee Kim 2006

All Rights Reserved

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ABSTRACT

Galactic Merger Rates of Pulsar Binaries

Chunglee Kim

Pulsar binaries in close orbits are strong sources of gravitational waves (GWs). With large-scale

interferometers, it will be possible to detect GW signals from these objects for the first time within the

coming decade. Astrophysical properties of sources, e.g. population sizes or event rates, are important

in accessing the detector design and performance. In this thesis, I introduce a novel statistical analysis

method to calculate Galactic merger rates of pulsar binaries based on known systems in the Galactic

field. This work involves the development of Galactic pulsar population models using Monte Carlo

methods, detailed modeling of observational selection effects for large-scale pulsar surveys, and deriving

a probability density function of the rate estimates using a Bayesian analysis. The method can be

applied for any type of binaries observed as radio pulsars. Currently, two types of pulsar binaries

have been observed: double-neutron-star (NS−NS) systems and neutron star-white dwarf (NS−WD)

binaries. Considering three merging NS−NS binaries including PSR J0737–3039, I obtain the most

likely values of Galactic NS−NS merger rate range between ∼ 4 − 220 Myr−1, depending on different

pulsar models. Extrapolating Galactic rate estimates up to the volume accessible by LIGO (Laser

Interferometer Gravitational-wave Observatory), and assuming a reference pulsar population model, I

find that the NS−NS inspiral event rates for the initial and advanced LIGO are 35 × 10−3 yr−1 and

190 yr−1, respectively. In the case of merging NS−WD binaries, the most likely rates are found in the

range ∼ 0.2 − 10 Myr−1. Based on this result, I conclude that the contribution from this population

on the confusion noise level for LISA (Laser Interferometer Space Antenna) is negligible. In addition

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to merging binaries, I estimate formation rates of eccentric NS−WD binaries, which range ∼ 0.5 − 16

Myr−1, based on different models. Due to their interesting evolutionary history, the formation rate of

eccentric NS−WD binaries provides important constraints on the theory of binary evolution. Lastly,

by means of these empirical rate estimates, I show how to constrain the parameters of theoretical

models for stellar binary evolution.

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ACKNOWLEDGEMENTS

First of all, I would like to thank my advisor, Dr. Vicky Kalogera, for her guidance and help (not

to mention her patience!). Her advice always inspired me to find the right direction, whenever I felt

lost. With her always positive and energetic manners, Vicky showed me how to deal with problems

not only in academia, but also in life.

Through my thesis projects, I was luckly to collaborate with Dr. Duncan Lorimer. He has been

crucial for this project from the beginning, and I am grateful for all his useful comments and guides

on the pulsar astronomy.

Doing astrophysics involves lots of traveling. Each trip provided me with a great chance to improve

my knowledge and, perhaps more importantly, to interact with people. I would like to thank Dr.

Andrew Lyne, Dr. Michael Kramer, and Dr. Andy Faulkner for insightful discussions during my visit

at Jodrell Bank. Also, I would like to thank Dr. Maura McLaughlin for her help and hospitality.

In the fall of 2005, Dr. Matthew Bailes kindly invited me to Swinburne University of Technology,

and introduced me to ‘real’ pulsar observations. From those lunch and tea discussions, I could learn

a lot more about pulsar observation and pulsar timing. I also would like to thank him and his family

for all the hospitality. Many thanks to Dr. Ramesh Bhat for helpful comments and detailed answers

to my random questions.

Last year was perhaps the most exiciting and challenging year during my Ph.D. period in many

ways. I appreciate Dr. Cole Miller, Dr. Melvyn Davies, and Dr. Sungsoo Kim for their encouragements

and help.

Back to Northwestern, many people helped me to keep going and to balance life in and out of

office. I am very grateful to Dr. Ron Taam for his support, particularly during my early periods at

Northwestern. Without his help, it would be impossible for me to settle down and to begin my Ph.D.

Also, I appreciate Dr. Pulak Dutta for letting me participate in his group during my first summer at

Northwestern and learn about laboratory experiments. I appreicate my thesis committee, Dr. Andre

de Gouvea and Dr. Fred Radio for their interests and their challenging questions.

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First in Tech, and later in Dearborn, I have enjoyed a friendly atmosphere in our group with

cool colleagues and nice friends. Many thanks to David Lin for his help, and warm conversations

to help me go through my first winter blizzard. I would like to thank Anne Dabrowski and Teresa

Fonseca for all the nice strolls and good wines we shared. It has been a real pleasure to share the

office with Casey Law, Hua-bai Li, Megan Krejny, and all others. Also, to the usual suspects, Atakan

Gurkan, Carol Braun, Emmanouela Rantsiou, Genya Takeda, Jeremy Sepinsky, Luis Mier y Teran,

Paul Cadden-Zimansky, Semyon Chaichenets, Sourav Chaterjee, and Tassos Fragos: thanks a lot for

many fun nights, interesting conversations, and the mafia games! Special thanks to Genya for his help

including many many rides.

I also would like to thank Bart Willems, Chris Deloye, Craig Heinke, John Fregeau, Josh Faber,

Krzysztof Belczynski, Natasha Ivanova, Marc Frietag, Philippe Grandclent, and Richard O’Shaughnessy

for interesting and helpful discussions. Esepecially, John, thanks so much for your help on retrieving

my data on sirocco!

My dear old friends, Homin, Hey-young, Eunkyung, and Hyun-Woo, I owe a lot to you. Without

you guys, it would be much difficult for me to manage those hard times.

Finally, I would like to thank to my family. Sometimes it has been difficult to be away, but they

have been always supportive and patient. Mom, Santae, and Jeong-A, I really appreciate for your love

and support!

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Table of Contents

Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.1 Pulsar Binaries as Gravitational-Wave Sources . . . . . . . . . . . . . . . . . . . . . . 15

Chapter 2 The Probability Distribution of Binary Pulsar Merger Rates. I. DoubleNeutron Star Systems in the Galactic Field∗ . . . . . . . . . . . . . . . . . . . . . 20

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2 Basic Analysis Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3 Models for the Galactic Pulsar Population . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.4 Pulsar Survey Selection Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.4.1 Survey Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.4.2 Doppler Smearing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.5 Statistical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.5.1 The Rate Probability Distribution for Each Observed NS−NS Binary . . . . . 31

2.5.2 The Total Galactic Merger Rate . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.5.3 The Detection Rate for LIGO . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

Chapter 3 Revised Merger Rates for NS−NS Systems: Implications for PSRJ0737−3039∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.1 Method for Rate Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.2 Galactic NS−NS Merger Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.3 NS−NS Inspiral Event Rates and Conclusions . . . . . . . . . . . . . . . . . . . . . . . 49

Chapter 4 A Few Remarks on the NS−NS Merger Rate Estimates∗ . . . . . . . . 52

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4.1 The Galactic NS−NS merger rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.2 Predictions for Future Discoveries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.3 Global P(R) and Supernova Constraints on NS−NS Merger Rates . . . . . . . . . . . 56

4.4 Rate Constraints from Type Ib/c Supernovae and Binary Evolution Models . . . . . . 57

4.5 Comments on PSR J1906+0746 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

Chapter 5 The Probability Distribution Of Binary Pulsar Merger Rates. II. Neu-tron Star-White Dwarf Binaries∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.1 Merging NS−WD Binaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.1.1 Lifetime of a NS−WD binary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.1.2 Probability Density Function of the Galactic NS−WD Merger Rate Estimates 65

5.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.3 Gravitational Wave Background due to NS−WD Binaries . . . . . . . . . . . . . . . . 68

5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

Chapter 6 Neutron Star−White Dwarf Binaries in Eccentric Orbits∗ . . . . . . . . 78

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

6.2 Formation of Eccentric NS−WD Binaries . . . . . . . . . . . . . . . . . . . . . . . . . 79

6.3 Empirical NS−WD Formation Rate Estimates . . . . . . . . . . . . . . . . . . . . . . 80

6.4 Comparison with Rates from Binary Evolution . . . . . . . . . . . . . . . . . . . . . . 83

Chapter 7 Upper Limits on NS−BH Binaries . . . . . . . . . . . . . . . . . . . . . . 86

Chapter 8 Constraining Population Synthesis Models via the Binary Neutron StarPopulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

8.2 Empirical Rate Constraints from the NS−NS Galactic Sample . . . . . . . . . . . . . . 91

8.2.1 Merging NS−NS Binaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

8.2.2 Wide NS−NS Binaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

8.3 Estimates for Merger Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

8.3.1 Population Synthesis Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

8.3.2 Mapping Population Synthesis Rates versus Parameters . . . . . . . . . . . . . 97

8.4 Constraints from NS−NS Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

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8.4.1 Advanced LIGO Detection Rates . . . . . . . . . . . . . . . . . . . . . . . . . . 100

8.5 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

A Observed Properties of Pulsar Binaries. . . . . . . . . . . . . . . . . . . . . . . . . . . 113

B Note on Bayes’ Theorem and Confidence Intervals . . . . . . . . . . . . . . . . . . . . 114

C Combined P(R) for Three Binary Systems . . . . . . . . . . . . . . . . . . . . . . . . . 116

D Calculating DCO Event Rates with Population Synthesis . . . . . . . . . . . . . . . . 118

D.1 How Rates were Estimated in Ch. 8 . . . . . . . . . . . . . . . . . . . . . . . . 118

D.2 Understanding Errors in Rate Estimates . . . . . . . . . . . . . . . . . . . . . . 119

E Sample Fits to Merger Rates in Ch. 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

F Characterizing the Consistent Region of Population Synthesis Models . . . . . . . . . 123

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List of Tables

2.1 Simulated pulsar surveys. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.2 Model parameters and estimates for Rtot and Rdet in various Bayesian credible regionsfor different pulsar population models . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.2 Model parameters and estimates for Rtot and Rdet in various Bayesian credible regionsfor different pulsar population models . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.1 Estimates for Galactic merger rates and predicted LIGO detection rates in a 95% cred-ible region based on different population models . . . . . . . . . . . . . . . . . . . . . . 51

5.1 Observational properties of merging NS–WD binaries. From left to right, the columnsindicate the pulsar name, spin period Ps, spin-down rate P , orbital period Pb, mostprobable mass of the WD companion mwd, orbital eccentricity e, characteristic age τc,spin-down age τsd, time to reach the death line τd, and references (1) Lundgren, Zepka,& Cordes (1995); (2) Nice, Splaver, & Stairs (2004) (3) Edwards, & Bailes (2001) ; (4)Kaspi et al. (2000) ; (5) Bailes et al. (2003). . . . . . . . . . . . . . . . . . . . . . . . . 63

5.2 Estimates for the Galactic merger rate (Rtot) of NS−WD binaries in 68% and 95%credible regions for all models considered. Model number is same with Table 2.2. Weshow the most likely value of Rtot at 68% and 95% credible regions. . . . . . . . . . . 76

5.2 Estimates for the Galactic merger rate (Rtot) of NS−WD binaries in 68% and 95%credible regions for all models considered. Model number is same with Table 2.2. Weshow the most likely value of Rtot at 68% and 95% credible regions. . . . . . . . . . . 77

6.1 Observational properties of eccentric NS−WD binaries. The columns indicate the pulsarname, spin period P , spin-down rate P , orbital period Pb, the estimated mass of theWD companion mc, orbital eccentricity e, characteristic age τc, time to reach the deathline τd. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

6.2 The estimated Glactic formation rate and the most likely value of Ntot of eccentricNS−WD binaries for models with different pulsar luminosity functions. Model numberis same with Table 2.2. We show the most likely value of Rb in 68% and 95% credibleregions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

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8.1 Observational properties of NS−NS binaries. From left to right, the columns indicatethe pulsar name, spin period, spin-down rate, orbital period, companion mass (MNS isassumed to be 1.35M⊙ except PSR B1913+16 (1.44M⊙) and PSR B1534+12 (1.33M⊙)),eccentricity, characteristic age, spin-down age, GW merger timescale, death-time, mostprobable value of the total number of pulsars in a model galaxy estimated for the ref-erence model (model 6 in KKL), beaming correction factor, parameter in rate equationused in Eq. (8.2), references: (1) Hulse & Talor (1975); (2) Wex, Kalogera, & Kramer(2000); (3) Wolszczan (1991); (4) Stairs et al. (2002) (5) Burgay et al. (2003) (6) Coro-ngiu et al. (2004) (7) Nice, Sayer, & Taylor (1996) (8) Hobbs et al. (2004) (9) Championet al. (2004) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

8.2 Classes of runs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

8.3 Classes used for specific rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

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List of Figures

2.1 Average signal-to-noise degradation factor in pulsar search code versus survey integra-tion time for PSR B1913+16 and PSR B1534+12. . . . . . . . . . . . . . . . . . . . . 28

2.2 The Poisson-distribution fits of P(Nobs) for three values of the total number Ntot ofPSR B1913+16-like pulsars in the Galaxy (results shown for model 1). Points and errorbars represent the counts of model samples in our calculation. Dotted lines representthe theoretical Poisson distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.3 The linear correlation between λ ≡<Nobs> and Ntot is shown for model 1. Solid anddashed lines are best-fit lines for PSR B1913+16-like and PSR B1534+12-like popula-tions, respectively. Points and error bars represent the best-fit values of λ for differentvalues of Ntot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.4 The probability density function of merger rates in both a logarithmic and a linearscale (small panel) is shown for model 1. The solid line represents P (Rtot) and the longand short dashed lines represent P (R) for PSR B1913+16-like and PSR B1534+12-likepopulations, respectively. We also indicate the credible regions for P (Rtot) by dottedlines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.5 Left panel: The correlation between Rpeak and the cut-off luminosity Lmin with differentpower indices p of the luminosity distribution function. Right panel: The correlationbetween Rpeak and the power index of the luminosity distribution function p. . . . . . 40

2.6 The correlation between Rpeak and the radial scale length R0. Rpeak is not sensitive toR0 in the range between 4 − 8 kpc. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.1 Probability density function that represents our expectation that the actual NS−NSbinary merger rate in the Galaxy (bottom axis) and the predicted initial LIGO rate(top axis) take on particular values, given the observations. The curves shown arecalculated assuming our reference model parameters (see text). The solid line showsthe total probability density along with those obtained for each of the three binarysystems (dashed lines). Inset: Total probability density, and corresponding 68%, 95%,and 99% credible regions, shown in a linear scale. . . . . . . . . . . . . . . . . . . . . . 48

4.1 Probability density function of the predicted number of observed NS−NS binary systemsNobs for the PMPS, for our reference model (model 6 in KKL). The mean value isestimated to be 〈Nobs〉 ∼4.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

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4.2 The global Pg(R) on a linear scale (lower panel) and the assumed intrinsic distributionsfor Lmin and p (upper panels). Dotted lines represent the lower (SNL) and upper (SNU)bounds on the observed SN Ib/c rate scaled by 1/10 and 1/100 (see text). The empiricalSN Ib/c rates range over ∼ 600 − 1600Myr−1, where the average is at ∼ 1100 Myr−1

(Cappellaro, Evans, & Turatto 1999), beyond the range shown here. . . . . . . . . . . 59

5.1 The PDFs of the Galactic merger rate estimation in both a logarithmic and a linearscale (inset) are shown for the reference model. The solid line represents P (Rtot).Other curves are P (R) for PSRs J1757−5322 (dot-dash), J0751+1807 (short-dash),and J1141−6545 (long-and-short dash)-like populations, respectively. Dotted lines cor-respond to 68%, 95%, and 99% credible regions for P (Rtot). . . . . . . . . . . . . . . . 68

5.2 The effective GW amplitude hrms for merging NS−WD binaries overlapped with theLISA sensitivity curve. The curve is produced with the assumption of S/N=1 for 1 yrof integration. Dotted lines are results from all models we consider except the referencemodel, which is shown as a solid line (see text for details). We also show the expectedconfusion noise from Galactic WD–WD binaries for comparison (dashed line). . . . . 74

6.1 The probability density function of Galactic formation rate estimates for eccentricNS−WD binaries (solid line) for our reference model. Dashed and dot-dashed linesrepresent the individual probability density functions of the formation rates for sub-population of binaries similar to either PSR B2303+46 or J1141−6545. No correctionsfor pulsar beaming have been applied. . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

6.2 Comparison between the empirical and theoretical rate estimates. Error bars with filledtriangles indicate results from StarTrack, open squares and a solid line are adaptedfrom the literature, and filled circles with error bars are obtained in this work (see textfor details). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

8.1 Empirically-deduced probability distributions for merging (right) and wide (left) NS−NSbinaries; see §, 8.2. The solid vertical lines are at (i) log10 R = −4.5388, and −3.49477,the 95% credible region for the merging NS−NS merger rate; and at (ii) log10 R =−6.7992,−5.7313, in the 95% credible region for the wide NS−NS formation rate. . . 96

8.2 log10 of the Galactic rate versus our fit to the rate, shown for NS−NS, NS−BH, andBH−BH sample points (all superimposed). The shaded region is offset by a factor1 ± 1/

√10. This region estimates the error expected due to random fluctuations in

the number of binary merger events seen in a given sample. (See the appendix for adiscussion of the number of sample points actually present in various runs.) . . . . . . 98

8.3 The a priori probability distribution for the NS−NS (right), NS−BH (center), andBH−BH (left) merger rates, versus the log10 of the rate. These distributions weregenerated from the population synthesis code (dashed line) and fits (solid lines) assumingall parameters in the population synthesis code were chosen at random in the allowedregion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

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8.4 The a priori probability distributions for the visible merging (top) and visible wide(bottom) NS−NS formation rates produced from population synthesis. The solid curvesdenotes the result deduced from artificial data generated from a multidimensional fit tothe visible wide and merging NS−NS rate data. The vertical lines are the respective95% CI bounds presented in Figure 8.1. . . . . . . . . . . . . . . . . . . . . . . . . . . 100

8.5 The a priori probability distribution for the NS−NS (bottom), NS−BH (center), andBH−BH (top) merger rates per Milky Way equivalent galaxy. As in Figure 8.3, thedashed curves show the results obtained from our population synthesis calculations(i.e., our raw code results, smoothed); the thick solid curves show the results after weimpose both our observational constraints (i.e., consistency with the observed numberof visible wide and visible merging NS−NS binaries). . . . . . . . . . . . . . . . . . . 101

8.6 Probability distributions for LIGO’s detection rates for merging NS−NS (dotted line),NS−BH (dashed line), and BH−BH (solid line) binaries, assuming all binaries are pro-duced in the field. This plot was obtained directly from Figure 8.5 using Eq. (8.9). . . 102

8.7 Cumulative probability distributions Pk(X) defined so Pk(X) is the fraction of all modelsconsistent with the two constraints imposed in the text (i.e., the formation rates ofwide and merging NS−NS binaries correspond adequately to observations) and thathave xk < X. The left panel shows the distributions for the 3 kick-related parametersx3, x4, x5; in this panel, the bottom (dashed) curve denotes P3, the middle (solid) curvedenotes P4, and the top dotted curve denotes P5. The right panel shows the distributionsfor P1 (solid top line), P2 (dashed), P6 (dotted), and P7 (solid, bottom curve). . . . . 123

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Chapter 1

Introduction

1.1. Pulsar Binaries as Gravitational-Wave Sources

Interest on compact binaries, consisting of two compact objects orbiting one another, derives from an

intrinsic motivation of understanding their origin and evolution in various astrophysical environments.

In this thesis, I focus on pulsar binaries, which is a subclass of compact binaries, in the context of

gravitational wave (GW) detection. Depending on their progenitor masses, there are three variations of

pulsar binaries1: neutron star-white dwarf (NS−WD) binaries, double-neutron-star (NS−NS or DNS)

systems, and neutron star-black hole (NS−BH) binaries (e.g. Bhattacharya & van den Heuvel 1991 for

the standard scenario to form a double-degenerate binary). Pulsar binaries contain, by definition, at

least one neutron star in the form of an active radio pulsar, and therefore, it is possible to detect these

systems with radio observations. However, searching for pulsars, particularly for those in binaries, is

not a trivial task. Pulsars are intrinsically faint objects2, and moreover, radio pulses can be easily

broadened, or smeared out due to interactions with the interstellar medium or due to the orbital

acceleration of the pulsar in a relativistic orbit (see §2.4 for a brief summary on selection effects of

pulsar searches.). However, the situation has been greatly improved during the last decade. The

1Here, I consider only those with two compact objects. Note that there are several known pulsars in binaries with

main-sequence companions, or in a triple system (see Lorimer 2001; Stairs 2004 for more details).2The unit of a pulsar flux density is Jy (Jansky, 1 Jy≡ 10−26 W m−2 Hz−1). The flux densities measured from known

pulsars are found between 20 µJy − 5Jy (Lorimer & Kramer 2005, based on the ATNF pulsar catalogue). The typical

flux density measure from pulsars considered in this thesis are of the order of ∼mJy. For instance, the measured radio

flux of B1913+16 at 400MHz is ∼ 4 mJy.

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extensive campaigns of large-scale pulsar surveys such as Parkes multibeam pulsar survey (PMPS; e.g.

Manchester et al. 2001), recent developments of effective search algorithms for the most relativistic

pulsar binaries (e.g., Faulkner et al. 2003) have significantly increased the number of known pulsars,

and more importantly, the number of relativistic pulsar binaries (e.g., see the ATNF pulsar catalogue:

http://www.atnf.csiro.au/research/pulsar/psrcat/). As of March 2006, there are 8 NS−NS binaries

observationally confirmed (7 systems are in the Galactic disk, and 1 in M15, a globular cluster), and

∼ 40 NS−WD binaries are known in our Galaxy and 75% of them are found in the Galactic field

(see Stairs 2004 for a detailed list of currently known pulsar binaries). In Table A.1 (Appendix A), I

summarize the most up-to-date observed properties of pulsar binaries considered in this thesis.

Pulsar binaries are strong sources of GWs, and have provided us with indirect evidence of GW

emission from astrophysical systems; the most well-established example is the Hulse-Taylor pulsar

(PSR B1913+16), which has been observed for about 30 yrs since its discovery (Hulse & Taylor 1975).

The observed advance of periastron in this NS−NS binary is in a good agreement with the prediction

from general relativity within 0.03% accuracy (Taylor & Weisberg 1989, Weisberg & Taylor 2003).

Moreover, pulsar binaries are suggested to be one of the practically accessible astrophysical labora-

tories to test general relativity in the strong-field regime (e.g. Damour & Taylor 1991; Kramer et al.

2004). Among pulsar binaries, those in close orbits have drawn attention particularly from the GW

community in view of their implications for current and future GW detectors. Based on state-of-

the-art technology, large-scale interferometers target direct GW detections from the ground and from

space within the coming decade. There are several ground-based interferometers being constructed or

already operational around the world, such as LIGO (Laser Interferometer Gravitational-wave Obser-

vatory; Abramovici et al. 1992) in the United States, GEO600 (Danzmann et al. 1995) in Germany,

VIRGO (Bradaschia et al. 1991) in Italy, and TAMA300 (Tsubono 1995; Ando et al. 2001) in Japan.

These ground-based interferometers are typically sensitive to frequency bands between a few tens of

Hz and a few kHz, and are optimized for detecting GWs emitted during the last few minutes of the

inspiral phase of NS−NS binaries before their final plunge (Abott et al. 2004). NS−WD binaries emit

GWs in much lower frequencies, and are relevant to a space-borne detector, LISA (Laser Interferom-

eter Space Antenna; Bender 1998). LISA is a joint mission between NASA (National Aeronautics

and Space Administration) and ESA (European Space Agency). The LISA pathfinder, a technology

17

demonstration mission, is scheduled for a launch in ∼2009, and LISA itself is pected to be launched in

2015. Based on the current design, LISA will be sensitive to GW signals in the frequency range from

∼ 0.1mHz to 0.1Hz. Potential GW sources for LISA include all known types of Galactic compact bi-

naries, including NS−WD binaries. More exotic systems such as NS or stellar-mass BH (∼ 1− 10M⊙)

binaries with intermediate-mass BHs (& 100 − 104M⊙) are also suggested theoretically. Within the

next decade, LISA will join the network of ground-based detectors, which will be fully operational

and even upgraded by then. Currently, GW astronomy is still in the early stages of development, and

astrophysical understanding of GW sources is one of the prerequisites for improving detector design

and for effective detector assessments. Therefore, event rate predictions for a given type of source are

important for the development of GW interferometers. Such predictions can be inferred by formation

or merger rates relevant to the sources (e.g., Thorne & Cutler 2002).

The merger rates of pulsar binaries have been obtained using two very different methods. One is

purely theoretical and uses models of binary evolution often calibrated to the observationally deter-

mined supernova (SN) rate for the Galaxy. Typically, this population synthesis method has uncer-

tainties of several orders of magnitude in the rate estimates, mainly due to the large parameter space

associated with the details of the stellar binary evolution. However, theoretical approach is useful to

study compact binaries in general, particularly when observed data are not available (Portgies Zwart

& Yungelson, 1998; Bethe & Brown 1999; Nelemans, Yungelson, & Portegies Zwart 2001; Schneider

et al. 2001; Belczynski, Kalogera, & Bulik 2002; Dewi, Podsiadlowski, & Sena 2006 and many more).

The other, more empirical, approach is based on the physical properties of the close NS−NS binaries

known in the Galactic disk and modeling of radio pulsar survey selection effects. For a review and

details of both these approaches, see Kalogera et al. (2001, hereafter KNST) and references therein.

The empirical method has generally provided us with better constraints on the merger rate (KNST),

although the uncertainty still exceeds two orders of magnitude. This is primarily due to (1) the very

small number (only two until recently) of close NS−NS binaries known in the Galactic disk with merger

times shorter than a Hubble time3, and (2) the implicit assumption that this small sample is a good

3τH ≡ 1/H0, where H0 is the present value of Hubble constant (H0 = 100h km s−1 Mpc−1). The most recent

Wilkinson Microwave Anisotropy Probe (WMAP) results presented that h ≃ 0.71. This implies a Hubble time of ≃ 13.8

Gyr (Bennet et al. 2003).

18

representation of the total Galactic population (KNST).

In this thesis, I introduce a novel statistical method to calculate the probability density function

(PDF) of rate estimates for pulsar binaries. In contrast to previous studies, having a PDF of rate

estimates at hand allows us to assign a statistically preferred range containing the true value of

the Galactic merger rate (or formation rate) for a given pulsar binary population at at any desired

statistical significance. Based on the rate estimates and their PDFs, I will discuss the implications of

known pulsar binaries for GW detection.

The organization of this thesis is as follows4. In Ch. 2−4, I consider NS−NS binaries. The GW

signals from NS−NS inspirals are targets for ground-based inteferometers such as LIGO (Finn 2001).

Until 2003, there were only two systems available for empirical studies, PSRs B1913+16 and B1534+12

(Wolszczan 1991). Then there was a breakthrough in 2003 with the discovery of PSR J0737−3039A5

(Burgay et al. 2003). In Ch. 2, I derive the PDF for NS−NS merger rates based on the previously known

systems (B1913+16 and B1534+12), and compare the results with the revised rates including PSR

J0737−3039 in following chapter. In Ch. 4, I derive a global probability distribution of merger rates

that incorporates the presently known systematics from the radio pulsar luminosity function. Based

on the global PDF, I discuss the constraints from the observed Type Ib/c SN rate on the Galactic

NS−NS merger rate. In Ch. 5−6, I consider NS−WD binaries. First, I study those in close orbits

and will merge within a Hubble time, and discuss their implications and contribution to the LISA

noise curve. Secondly, I consider an interesting sub-population of NS−WD binaries that experienced

a SN explosion after the formation of a white dwarf. As the outcome of this evolution scenario, these

NS−WD binaries end up in rather eccentric orbits (e > 0.1). I calculate the formation rate of eccentric

NS−WD binaries considering the two systems known in the Galactic disk. In Ch. 7, I briefly address

the exotic population of NS−BH binaries. Based on the absence of any detection, I try to set an

upper limit on the merger rate of NS−BH binaries with the statistical method described in earlier

4All results shown in this thesis are obtained with the correct unit conversion on the pulse scatter-broadening times

(details can be found in the erratum, The Astrophysical Journal, 614, pp. L137–L138, by Kalogera et al. 2004).5After the discovery of its pulsar companion, the two pulsars in this “double-pulsar” system are now labeled as PSR

0737−3039A (recycled), and PSR J0737-3039B (normal, non-recycled), respectively. In this thesis, I only consider the A

pulsar, and omit the label A for simplicity.

19

chapters. Finally, in Ch. 8, I combine the empirical rates of NS−NS binaries with the population

synthesis method, and constrain theoretical model parameters. This is motivated by the prospects of

detecting GWs from stellar mass BH binaries. BH binaries in close orbits are considered to be the

most promising sources to be detected by ground-based GW interferometers. Given the absence of

detection, however, it is impossible to handle these systems directly with the empirical method. In this

chapter, I compare empirical rate estimates obtained from vaious pulsar binaries with the theoretical

calculations, and show how to constrain the theoretical predictions for merger rates of BH binaries

(e.g., NS−BH, BH−BH).

20

Chapter 2

The Probability Distribution of BinaryPulsar Merger Rates. I. DoubleNeutron Star Systems in the GalacticField∗

2.1. Introduction

The detection of the NS−NS prototype PSR B1913+16 as a binary pulsar (Hulse & Taylor 1975)

and its orbital decay due to emission of GWs (Taylor, Fowler, & McCulloch 1979; Weisberg & Taylor

2003) has inspired a number of quantitative estimates of the merger rate of NS−NS binaries (Clark,

van den Heuvel, & Sutantyo 1979; Narayan, Piran, & Shemi 1991; Phinney 1991, Curran & Lorimer

1995). In general, the merger rate of NS−NS binaries can be calculated based on: (a) our theoretical

understanding of their formation (see Belczynski & Kalogera 2001 for a review and application of this

approach); (b) the observational properties of the pulsars in the binary systems and the modeling

of pulsar survey selection effects (see e.g. Narayan 1987). Interest in these mergers derives from an

∗This chapter is adapted with style changes from “The Probability Distribution of Binary Pulsar Coalescence Rates,

I. Double Neutron Star Systems in the Galactic Field” by C. Kim, V. Kalogera, & D.R. Lorimer that appeared in The

Astrophysical Journal, 584, pp. 985 − 995, February 2003. c©The American Astronomical Society. The results shown

in this chapter are revised using the corrected unit conversion of the the scatter-broadening time as discussed in The

Astrophysical Journal, 614, pp. L137–L138, by V. Kalogera, C. Kim et al. October 2004. c©The American Astronomical

Society.

21

intrinsic motivation of understanding their origin and evolution and their connections to other NS

binaries. However, significant interest derives from their importance as GW sources for the upcoming

ground-based laser interferometers (such as LIGO) and their possible association with γ-ray burst

events (Popham et al. 1998 and references therein). The traditional way of calculating the merger rate

based on observations involves an estimate of the scale factor, an indicator for the number of pulsars

in our Galaxy with the same spin period and luminosity (Narayan 1987). Corrections must then be

applied to these scale factors to account for the faint end of the pulsar luminosity function, the beamed

nature of pulsar emission, and uncertainties in the assumed spatial distribution. The estimated total

number in the Galaxy can then be combined with estimates of their lifetimes to obtain a merger

rate, R. This method was first applied by Narayan, Piran, & Shemi (1991) and Phinney (1991) and

other investigators who followed (Curran & Lorimer 1995; van den Heuvel & Lorimer 1996). Various

correction factors were (or were not) included at various levels of completeness. Summaries of these

earlier studies can be found in Arzoumanian, Cordes, & Wasserman (1999) and KNST. The latter

authors examined all possible uncertainties in the estimates of the merger rate of NS−NS binaries in

detail, and pointed a small-number bias that introduces a large uncertainty (more than two orders

of magnitude) in the correction factor for the faint-pulsar population that must be applied to the

rate estimate. They obtained a total NS−NS rate estimate in the range R = 10−6 − 5 × 10−4 yr−1,

with the uncertainty dominated by the small-number bias. Earlier studies, which made different

assumptions about the pulsar properties (e.g. luminosity and spatial distributions and lifetimes), are

roughly consistent with each other (given the large uncertainties). Estimated ranges of values until now

were not associated with statistical significance statements and an “all-inclusive” estimated Galactic

merger rate lies in the range ∼ 10−7 − 10−5 yr−1.

The motivation for this work is to update the scale factor calculations using the most recent pulsar

surveys, and present a statistical analysis that allows the calculation of Bayesian credible regions as-

sociated with rate estimates. We consider the two binaries found in the Galactic disk: PSR B1913+16

and PSR B1534+12. Following the arguments made by Phinney (1991) and KNST, we do not include

the globular cluster system PSR B2127+11C (Prince et al. 1991). Radio-pulsar-survey selection ef-

fects are taken into account in the modeling of pulsar population. As described in what follows, the

small-number bias and the effect of a luminosity function are implicitly included in our analysis, and

22

therefore a separate correction factor is not needed. For each population model of pulsars, we derive

the probability distribution function of the total Galactic merger rate weighted by the two observed

binary systems. In our results we note a number of important correlations between Rpeak and model

parameters that are useful in generalizing the method. We extrapolate the Galactic rate to cover the

detection volume of LIGO and estimate the detection rates of NS−NS inspiral events for the initial

and advanced LIGO.

The plan for the rest of this chapter is as follows. In §2.2, we describe our analysis method in a

qualitative way. Full details of the various pulsar population models and survey selection effects are

then given in §2.3 and §2.4 respectively. In §2.5, we derive the probability distribution function for

the total Galactic merger rate and calculate the detection rate of LIGO. In §2.6, we summarize our

results and discuss a number of intriguing correlations between various physical quantities. Finally, in

§2.7, we discuss the results and compare them with previous studies.

2.2. Basic Analysis Method

Our basic method is one of “forward” analysis. By this we mean that we do not attempt to invert

the observations to obtain the total number of NS−NS binaries in the Galaxy. Instead, using Monte

Carlo methods, we populate a model galaxy with NS−NS binaries (that match the spin properties of

PSR B1913+16 and PSR B1534+12) with pre-set properties in terms of their spatial distribution and

radio pulsar luminosity function. Details about these “physical models” are given in §2.3.

For a given physical model, we produce synthetic populations of different total numbers of objects

(Ntot). We then produce a very large number of Monte Carlo realizations of such pulsar populations

and determine the number of objects (Nobs) that are observable by all large-scale pulsar surveys carried

out to date by detailed modeling of the detection thresholds of these surveys. This analysis utilizes code

to take account of observational selection effects in a self-consistent manner, developed and described

in detail by Lorimer et al. (1993; hereafter LBDH) which we summarize in §2.4. Performing this

analysis for many different Monte Carlo realizations of the physical model allows us to examine the

distribution of Nobs. We find, as expected and assumed by other studies, that this distribution closely

23

follows Poisson statistics1, and we determine the best-fit value of the mean of the Poisson distribution

λ for each population model and value of Ntot,1913.

The calculations described so far are performed separately for each of PSRs B1913+16 and B1534+12

so that we obtain separate best-fit λ values for the Poisson distributions. Doing the analysis in this way

allows us to calculate the likelihood of observing just one example of each pulsar in the real-world sam-

ple. Given the Poissonian nature of the distributions this likelihood is simply: P (1;λ) = λ exp(−λ).

We then calculate this likelihood for a variety of assumed Ntot values for each physical model.

The probability distribution of the total merger rate Rtot is derived using the Bayesian analysis and

the calculated likelihood for each pulsar (described in detail in §2.5). The derivation of this probability

distribution allows us to calculate the most probable rate as well as determine its ranges of values in

various Bayesian credible regions. Finally, we extrapolate the Galactic rate to the volume expected to

be reached by LIGO and calculate the detection rate, Rdet (see §2.6).

2.3. Models for the Galactic Pulsar Population

Our model pulsar populations are characterized by a Galactocentric radius (R), vertical distance (Z)

from the Galactic plane and radio luminosity (L). Assuming that the distributions of each of these

parameters are independent, the combined PDF of the model pulsar population can be written as:

f(R,Z,L) dR dZ dL = ψR(R) 2π RdR ψZ(Z) dZ φ(L) dL, (2.1)

where ψR(R), ψZ(Z) and φ(L), are the individual PDFs of R, Z and L, respectively. In all models

considered, we assume azimuthal symmetry about the Galactic center.

The spatial distribution of pulsars is rather loosely constrained, but we find that it does not affect

the results significantly for a wide range of models. For the radial and the vertical PDFs, we consider

Gaussian and exponential forms with different values of the radial R0 and the vertical Z0 scale. In

1The nature of pulsar detection is actually expected to follows a Binomial distribution (‘detection’ versus ‘non-

detection’). However, in the case of a small number of sampling, namely Nobs = 1 in our case, Poission distribution is a

good approximation.

24

our reference model, we assume a Gaussian PDF for the radial component and an exponential PDF

for the vertical component. Hence, the combined spatial PDF is given by:

f(R,Z) ∝ exp

(

− R2

2R20

− |Z|Z0

)

, (2.2)

We set R0 = 4.0 kpc and Z0 = 1.5 kpc as standard model parameters. Following Narayan, Piran,

& Shemi (1991), these and other values considered reflect the present-day spatial distribution of the

NS−NS binary population after kinematic evolution in the Galactic gravitational potential.

Having assigned a position of each pulsar in our model galaxy, for later computational conve-

nience, we store the positions as Cartesian x, y, z coordinates, where the Galactic center is defined as

(0.0,0.0,0.0) kpc and the position of the Earth is assumed to lie 8.5 kpc from the center along the x−y

plane, i.e. (8.5,0.0,0.0). From these definitions, the distance d to each pulsar from the Earth can be

readily calculated, as well as the apparent Galactic coordinates l and b.

For the luminosity PDF, we follow the results of Cordes & Chernoff (1997) and adopt a power-law

function of the form

φ(L) = (p− 1)Lp−1minL

−p, (2.3)

where L ≥ Lmin and p > 1. The cut-off luminosity, Lmin, and the exponent p are the model parameters.

Cordes & Chernoff (1997) found Lmin = 1.1+0.4−0.5 mJy kpc2 and p = 2.0 ± 0.2 in a 68% credible region.

We set p = 2.0 and Lmin = 1.0 mJy kpc2 for our reference model. Throughout this chapter, luminosities

are defined to be at the observing frequency ν = 400 MHz.

Having defined the position and luminosity of each pulsar in our model Galaxy, the final step in

defining the model population is to calculate a number of derived parameters required to characterize

the detection of the model pulsars: dispersion measure (DM), scatter-broadening time (τ) and sky

background temperature (Tsky). To calculate DM and τ , we use the software developed by Taylor &

Cordes (1993) to integrate their model of the free-electron column density along the line of sight to

each pulsar defined by its model Galactic coordinates l and b out to its distance d. Frequency scaling of

τ to different survey frequencies is done assuming a Kolmogorov turbulence spectrum with a spectral

25

index of –4.42. Finally, given the model Galactic coordinates of each pulsar, the sky background noise

temperature at 408 MHz (Tsky) is taken from the all-sky catalog of Haslam et al. (1981). Scaling Tsky

to other survey frequencies assumes a spectral index of –2.8 (Lawson et al. 1987).

2.4. Pulsar Survey Selection Effects

Having created a model pulsar population with a given spatial and luminosity distribution, we are

now in a position to determine the fraction of the total population which are actually detectable by

current large-scale pulsar surveys. To do this, we need to calculate, for each model pulsar, the effective

signal-to-noise ratio it would have in each survey and compare this with the corresponding detection

threshold. Only those pulsars which are nominally above the threshold count as being detectable. After

performing this process on the entire model pulsar population of size Ntot, we are left with a sample

of Nobs pulsars that are nominally detectable by the surveys. Repeating this process many times, we

can determine the probability distribution of Nobs which we then use to constrain the population and

merger rate of NS−NS binaries. In this section we discuss our modeling of the various selection effects

which limit pulsar detection.

2.4.1. Survey Parameters

The main factors affecting the signal-to-noise ratio (σ) of a pulsar search can be summarized by the

following expression

σ ∝ SνG

T

√

P∆νt

we, (2.4)

where Sν is the apparent flux density at the survey frequency ν, G is the gain of the telescope, T is

the effective system noise temperature (which includes a contribution Tsky from the sky background

described in the previous section), P is the pulse period, ∆ν is the observing bandwidth, t is the

integration time and we is the pulse width. More exact expressions are given in the detailed description

of the survey selection effects in §2 of LBDH (in particular see their Eqs. 14–18) which we adopt in

2Although recent studies suggest a variety of spectral indices for τ (Lohmer et al. 2001), the effects of scattering turn

out to be negligible in this study since the detections of NS−NS binaries are limited by luminosity to nearby systems.

26

this work. In what follows, we describe the salient points relevant to this study.

For each model pulsar, with known 400-MHz luminosity L and distance d, we calculate the apparent

400-MHz flux density S400 = L/d2. Since not all pulsar surveys are carried out at 400 MHz, we need

to scale S400 to take account of the steep radio flux density spectra of pulsars. Using a simple power

law of the form Sν ∝ να, where α is the spectral index, we can calculate the flux Sν at any frequency

ν as:

Sν = S400

( ν

400MHz

)α, (2.5)

Following the results of Lorimer et al. (1995), in all simulations, spectral indices were drawn from a

Gaussian PDF with a mean of –1.6 and standard deviation 0.4.

The telescope gain, system noise, bandwidth and integration time are well-known parameters for

any given survey and the detailed models we use take account of these. In addition to the surveys

considered by LBDH, we also model surveys listed by Curran & Lorimer (1995), and more recent

surveys at Green Bank (Sayer, Nice & Taylor 1997) and Parkes (Lyne et al. 2000; Manchester et

al. 2001; Edwards et al. 2001). A complete list of the surveys considered, and the references to the

relevant publications is given in Table 2.1.

Up to this point in the simulations, the model parameters are identical for both PSRs B1913+16

and B1534+12. Since we are interested in the individual contributions each of these systems make to

the total Galactic merger rate of NS−NS binaries similar to these systems, we fix the assumed spin

periods P and intrinsic pulse widths w to the values of each pulsar and perform separate simulations

over all physical models considered. The assumed pulse widths are 10 ms and 1.5 ms respectively

for PSRs B1913+16 and B1534+12. The effective pulse width we required for the signal-to-noise

calculation must take into account pulse broadening effects due to the interstellar medium and the

response of the observing system. The various contributions are summarized by the quadrature sum:

w2e = w2 + τ2 + t2samp + t2DM + t2∆DM, (2.6)

where τ is the scatter-broadening timescale calculated from the Taylor & Cordes (1993) model, tsamp

is the data sampling interval in the observing system, tDM is the dispersive broadening across an

27

individual frequency channel and t∆DM is the pulse broadening due to dedispersion at a slightly

incorrect dispersion measure. All of these factors are accounted for in our model described in detail

in LBDH.

2.4.2. Doppler Smearing

For binary pulsars, we need to take account of the reduction in signal-to-noise ratio due to the Doppler

shift in period during an observation. This was not considered in LBDH since their analysis was

concerned only with isolated pulsars. For observations of NS−NS binaries, however, where the orbital

periods are of the order of 10 hours or less, the apparent pulse period can change significantly during

a search observation causing the received power to be spread over a number of frequency bins in the

Fourier domain. As all the surveys considered in this analysis search for periodicities in the amplitude

spectrum of the Fourier transform of the time series, a signal spread over several bins can result in

a loss of signal-to-noise ratio. To take account of this effect in our survey simulations, we need to

multiply the apparent flux density of each model pulsar by a “degradation factor”, F .

To calculate the appropriate F values to use, we generate synthetic pulsar search data containing

signals with periods and duty cycles similar to PSR B1913+16 and PSR B1534+12. These data are

then passed through a real pulsar search code which is similar to those in use in the large-scale surveys

(see Lorimer et al. 2000 for details). For each of the two pulsars, we first generate a control time series

in which the signal has a constant period and find the resulting signal-to-noise ratio, σcontrol, reported

by the search code. We then generate a time series in which the pulses have identical intensity but are

modulated in period according to the appropriate orbital parameters of each binary pulsar. From the

resulting search signal-to-noise ratio, σbinary, the degradation factor F = σbinary/σcontrol. Significant

degradation occurs, therefore, when F ≪ 1. Since accumulated Doppler shift, and therefore F , is a

strong function of the orbital phase at the start of a given observation, for both binary systems, we

calculate the mean value of F for a variety of starting orbital phases appropriately weighted by the

time spent in that particular part of the orbit.

A similar analysis was made by Camilo et al. (2000) for the millisecond pulsars (MSPs) in 47 Tu-

canae. In this chapter, where we are interested in the degradation as a function of integration time,

28

Figure 2.1 Average signal-to-noise degradation factor in pulsar search code versus survey integrationtime for PSR B1913+16 and PSR B1534+12.

we generate time series with a variety of lengths between 1 minute and 1 hour using sampling intervals

similar to those of the actual surveys listed in Table 2.1. The results are summarized in Figure 2.1,

where we plot average F versus integration time for both sets of orbital parameters. As expected,

surveys with the longest integration times are most affected by Doppler smearing. For the PMPS,

which has an integration time of 35 min, mean values of F are 0.7 and 0.3 for PSR B1913+16 and

PSR B1534+12 respectively3. The greater degradation for PSR B1534+12 is due to its mildly ec-

centric orbit (e ∼ 0.3 versus 0.6 for PSR B1913+16) which results in a much more persistent change

in apparent pulse period when averaged over the entire orbit. For the Jodrell Bank and Swinburne

surveys (Nicastro et al. 1995; Edwards et al. 2001), which both have integration times of order 5 min,

we find F ∼ 0.9 for both systems. For all other surveys, which have significantly shorter integration

times, no significant degradation is seen, and we take F = 1.

3In order to improve on the sensitivity to binary pulsars, the PMPS data are now being reprocessed using various

algorithms designed to account for binary motion during the integration time (Faulkner et al. 2003)

29

Table 2.1. Simulated pulsar surveys.

Telescope Year ν1 ∆ν2 t3obs t4samp S5min Detected6 Refs7

Lovell 76 m 1972 408 4 660 40 10 51/31 1,2Arecibo 305 m 1974 430 8 137 17 1 50/40 3,4

Molonglo 1977 408 4 45 20 10 224/155 5Green Bank 100 m 1977 400 16 138 17 10 50/23 6,7Green Bank 100 m 1982 390 16 138 17 2 83/34 8Green Bank 100 m 1983 390 8 132 2 5 87/20 9

Lovell 76 m 1983 1400 40 524 2 1 61/40 10Arecibo 305 m 1984 430 1 40 0.3 3 24/5 9

Molonglo 1985 843 3 132 0.5 8 10/1 11Arecibo 305 m 1987 430 10 68 0.5 1 61/24 12

Parkes 64 m 1988 1520 320 150 0.3 1 100/46 13Arecibo 305 m 1990 430 10 40 0.5 2 2/2 14

Parkes 64 m 1992 430 32 168 0.3 3 298/101 15,16Arecibo 305 m 1993 430 10 40 0.5 1 56/90 17–20

Lovell 76 m 1994 411 8 315 0.3 5 5/1 21Green Bank 100 m 1995 370 40 134 0.3 8 84/8 22

Parkes 64 m 1998 1374 288 265 0.1 0.5 69/170 23Parkes 64 m 1998 1374 288 2100 0.3 0.2 ∼900/600 24,25

1Center frequency in MHz.

2Bandwidth in MHz.

3Integration time in seconds.

4Sampling time in milliseconds.

5Sensitivity limit in mJy at the survey frequency for long-period pulsars (calculated for eachtrial in the simulations).

6Total number of detections and new pulsars

7References: (1,2) Davies, Lyne & Seiradakis (1972,3); (3,4) Hulse & Taylor (1974,5); (5)Manchester et al. (1978); (6,7) Damashek et al. (1978,1982); (8) Dewey et al. (1985); (9)Stokes et al. (1986); (10) Clifton et al. (1992); (11) D’Amico et al. (1988); (12) Nice, Taylor& Fruchter (1995); (13) Johnston et al. (1992); (14) Wolszczan (1991); (15) Manchester etal. (1996); (16) Lyne et al. (1998); (17) Ray et al. (1996); (18) Camilo et al. (1996); (19) Fosteret al. (1995); (20) Lundgren, Zepka & Cordes (1995); (21) Nicastro et al. (1995); (22) Sayer,

30

Nice & Taylor (1997); (23) Edwards et al. (2001); (24) Lyne et al. (2000); (25) Manchester etal. (2002)

31

2.5. Statistical Analysis

In this section we describe in detail the derivation of the probability distribution of the Galactic

merger rate R. The analysis method makes use of Bayesian statistics and takes into account the rate

contributions of both observed NS−NS binaries. At the end of the section we derive the associated

detection rates for LIGO.

2.5.1. The Rate Probability Distribution for Each Observed NS−NS Binary

As already mentioned in § 2.2, for each of the two observed NS−NS binaries (PSR B1913+16 or

PSR B1534+12) we generate pulsar populations in physical and radio luminosity space with pulse

periods and widths fixed to the observed ones and with different absolute normalizations, i.e., total

number Ntot of pulsars in the Galaxy. We generate large numbers of “observed” pulsar samples by

modeling the pulsar survey selection effects (see § 2.4) and applying them on these model populations

of PSR B1913+16-like and PSR B1534+12-like pulsars separately (see § 2.3). For a fixed value of Ntot,

we use these “observed” samples to calculate the distribution of the number of objects in the samples.

One might expect that the number of observed pulsars in a sample Nobs follows very closely a Poisson

distribution:

P (Nobs;λ) =λNobs e−λ

Nobs!, (2.7)

where, by definition, λ ≡<Nobs>. We confirm our expectation by obtaining excellent formal fits to

the Monte Carlo data using such a distribution and calculating the best-fitting value of λ. We vary

Ntot in the range 10 − 104 and find that λ is linearly correlated with Ntot:

λ = αNtot, (2.8)

where α is a constant that depends on the properties (space and luminosity distributions and pulse

period and width) of the Galactic pulsar population. Examples of the Poisson fits and the linear

correlations are shown in Figures 2.2 and 2.3, respectively, for our reference model 1 (see Table 2.2

and § 2.6).

32

Figure 2.2 The Poisson-distribution fits of P(Nobs) for three values of the total number Ntot of PSRB1913+16-like pulsars in the Galaxy (results shown for model 1). Points and error bars represent thecounts of model samples in our calculation. Dotted lines represent the theoretical Poisson distribution.

The main step in deriving a rate probability distribution for each of the observed systems is to first

derive the probability distribution of the total number of pulsars like the observed ones in the Galaxy.

We obtain the latter by applying Bayes’ theorem:

P (H|DX) = P (H|X)P (D|HX)

P (D|X), (2.9)

where P (H|DX) is the probability of a model hypothesis H given data D and model priors X,

P (D|HX) is the likelihood of the data given a model hypothesis and priors, P (H|X) is the proba-

bility of a model hypothesis in the absence of any data information, and P (D|X) is the model prior

probability, which acts as a normalization constant.

In the present work, we make following identifications:

D : is the real observed sample

H : is λ proportional to Ntot

X : is the population model (space and luminosity distributions and pulse period and width) With these

identifications, P (D|HX) is the likelihood of the real observed sample (one “PSR B1913+16-like” and

33

one “PSR B1534+12-like” pulsar) and is obtained by the best-fitting Poisson distribution:

P (D|HX) = P (1;λ(Ntot),X) = λ(Ntot) e−λ(Ntot). (2.10)

In the absence of any data information, the absolute normalization of the model population, i.e., the

total pulsar number Ntot and hence λ is expected to be independent of the shape of the population dis-

tributions and properties represented by X. Therefore, the probability of λ given a set of assumptions

X for the model Galactic population is expected to be flat:

P (H|X) = P (λ(Ntot)|X) = constant, (2.11)

and is essentially absorbed by the model prior probability P (D|X) as a normalization constant. The

probability distribution of λ then is given by:

P (λ|DX) =P (λ|X)

P (D|X)P (1;λ,X) = constant × P (1;λ,X). (2.12)

We impose the normalization constraint:∫∞

0 P (λ|DX)dλ = 1 and find that P (H|X)/P (D|X) = 1

and

P (λ|DX) = P (1;λ,X) = λ e−λ. (2.13)

Note that based on the above expression, the maximum value of P (λ) equal to e−1 always occurs at

λ = 1 or at Ntot = α−1 (see Eq. (2.8)). It is straightforward to calculate P (Ntot):

P (Ntot) = P (λ)∣

∣

∣

dλ

dNtot

∣

∣

∣

= α2Ntote−αNtot (2.14)

For a given total number of pulsars in the Galaxy, we can calculate their rate using estimates of

the associated pulsar beaming correction factor fb and lifetime τlife:

R =Ntot

τlifefb. (2.15)

34

Figure 2.3 The linear correlation between λ ≡<Nobs> and Ntot is shown for model 1. Solid and dashedlines are best-fit lines for PSR B1913+16-like and PSR B1534+12-like populations, respectively. Pointsand error bars represent the best-fit values of λ for different values of Ntot.

Note that this estimate is equivalent to previous studies (KNST and references therein) where the

concept of the scale factor is used instead of our calculated Ntot. We can write equivalently for our

calculation:

Ntot

Nobs=

∫ ∫

VGf(R,Z,L)dV dL

∫ ∫

VDf(R,Z,L)dV dL

, (2.16)

where f(R,Z,L) is the probability distribution of the pulsar population (§ 2.3), VG and VD are the

Galactic volume and the detection volume (for pulsars with pulse period and width similar to each of

the two observed pulsars), respectively. Note that we do not fix the luminosity of the pulsar population

to the observed values, but instead, we estimate Ntot for a distribution of radio luminosities. Since we

consider separately PSR B1913+16-like and PSR B1534+12-like populations, Nobs= 1.

For pulsar beaming fractions, we adopt the estimates obtained by KNST: 5.72 for PSR B1913+16

and 6.45 for PSR B1534+12. For the lifetime estimates we also follow KNST. Our adopted values for

the pulsar lifetimes are 3.65 × 108 yr for PSR B1913+16 and 2.9 × 109 yr for PSR B1534+12.

35

Using Eq. (2.15) we calculate P (R) for each of the two observed pulsars:

P (R) = P (Ntot)∣

∣

∣

dNtot

dR∣

∣

∣

=(ατlifefb

)2R e

−(ατlifefb

)R. (2.17)

2.5.2. The Total Galactic Merger Rate

Once the probability distributions of the rate contributions of the two observed pulsars are calculated,

we can obtain the distribution functions of the total merger rate Rtot. We define the following two

coefficients for each observed system:

A ≡(

ατlifefb

)

1913

and B ≡(

ατlifefb

)

1534

(2.18)

and rewrite the merger rate for each binary system:

P1913(R) = A2Re−AR and P1534(R) = B2Re−BR. (2.19)

One can confirm that each distribution function satisfies the normalization condition

∫ ∞

0P (R)dR = 1. (2.20)

We then define two new variables: R+ ≡ R1913 + R1534 and R− ≡ R1913 − R1534. Since R1913 ≥ 0

and R1534 ≥ 0, we have −R+ ≤ R− ≤ R+. For convenience, we rename R1913 = R1 and R1534 = R2

and perform a two-dimensional probability distribution transformation:

P (R+,R−) = P (R1,R2)

∣

∣

∣

∣

∣

dR1dR+

dR2dR−

dR1dR−

dR2dR+

∣

∣

∣

∣

∣

=1

2P (R1,R2). (2.21)

The probability distribution of the total rate Rtot ≡ R+ is obtained after integrating P (R+,R−) over

R−:

P (R+) =

∫

R−

P (R+,R−) dR− =1

2

∫

R−

P (R1,R2) dR−. (2.22)

36

Since the probability distributions for the rate contributions of each of the two observed pulsars are

independent of each other, their two-dimensional distribution is simply the product of the two.

P (R1,R2) = P (R1)P (R2). (2.23)

After we replace all variables to R+ and R−, we have

P (R+ ≡ Rtot) =A2B2

23e−(A+B

2)R+

[

R2+

∫ +R+

−R+

dR−e(B−A

2)R− −

∫ +R+

−R+

dR−R2−e

(B−A2

)R−

]

=( AB

B −A

)2[

R+(e−AR+ + e−BR+) −( 2

B −A

)

(e−AR+ − e−BR+)]

. (2.24)

We have confirmed that the above function satisfies the normalization∫∞

0 P (Rtot)dRtot = 1.

Having calculated the probability distribution of the Galactic merger rate, we can take one step

further and also calculate ranges of values for the rate Rtot at various credible regions (C.R.). The

lower (Ra) and upper (Rb) limits to these ranges are calculated using:

∫

Rb

Ra

P (Rtot)dRtot = C.R. (2.25)

and

P (Ra) = P (Rb). (2.26)

In all our results, we quote merger rates for 68%, 95% and 99% C.R.

2.5.3. The Detection Rate for LIGO

NS−NS binary systems are expected to emit strong GWs during their inspiral phase, the late stages

of which may be detected by the ground-based GW detectors. In this chapter, we estimate expected

detection rates for LIGO (initial and advanced) using the derived P (R) and peak values of the dis-

tribution, for different physical models of pulsar properties. To calculate the detection rate, we need

to extrapolate the Galactic rate to the volume detectable by LIGO. We use the ratio between the

B-band luminosity density of the Universe and the B-band luminosity of our Galaxy as the scaling

37

99%

95%

68%

Figure 2.4 The probability density function of merger rates in both a logarithmic and a linear scale(small panel) is shown for model 1. The solid line represents P (Rtot) and the long and short dashedlines represent P (R) for PSR B1913+16-like and PSR B1534+12-like populations, respectively. Wealso indicate the credible regions for P (Rtot) by dotted lines.

38

factor (Phinney 1991; KNST). This is based on the assumptions that (i) the B-band luminosity (cor-

rected for dust absorption) correlates with the star–formation rate in the nearby universe and hence

the merger rate, (ii) the B-band luminosity density is constant in the nearby universe. The detection

rate, Rdet, is calculated by the following equation:

Rdet = ǫRtotVdet, (2.27)

where ǫ is the scaling factor assumed to be ≃ 10−2 Mpc−3 (for details see KNST). Vdet is the detection

volume defined as a sphere with a radius equals to the maximum detection distance Dmax for the initial

(≃ 20 Mpc) and advanced LIGO (≃ 350 Mpc) (Finn 2001).

2.6. Results

We have calculated the probability distribution of the Galactic merger rate Rtot, its most likely value

Rpeak and ranges in different statistical credible regions, and the most likely expected detection rates

for the initial and advance LIGO for a large number of model pulsar population properties (see Table

2.2). We have chosen one of them to be our reference model based on the statistical analyses and

results presented by Cordes & Chernoff (1997).

For our reference model (Lmin = 1.0 mJy kpc2, p = 2.0, Ro = 4.0 kpc, and Z0 = 1.5 kpc), we find

the most likely value of Ntot, to be ≃ 185 pulsars for the “PSR B1913+16-like” population, and ≃ 140

pulsars for the “PSR B1534+12-like” population. Using Eq. (2.24), we evaluate the total Galactic

merger rate of NS−NS binaries for this reference case (model 1 in Table 2.2). The most likely value

of the merger rate is ≃ 3.6 Myr−1 and the ranges in different Bayesian credible regions are: ∼ 1 − 8

Myr−1 (68%), ∼ 0.5 − 15 Myr−1 (95%), and ∼ 0.3 − 20 Myr−1 (99%).

In Figure 2.4, P (Rtot) along with P (R1913) and P (R1534) are shown for the reference model. It is

evident that the total rate distribution is dominated by that of PSR B1913+16. At first this appears

to be in contradiction to most other studies of the NS−NS merger rate (Narayan, Piran, & Shemi

1991; Phinney 1991; Curran & Lorimer 1995; van den Heuvel & Lorimer 1996; Stairs et al. 1998;

39

Arzoumanian, Cordes, & Wasserman 1999; KNST). However, it turns out that this difference is due

to the fact that earlier studies restricted the scale factor calculation to the actual observed pulsar

luminosity and the rate estimates were dominated by the low luminosity of, and hence large scale

factor for, PSR B1534+12. Any corrections to the rate estimate which take into account the range of

pulsar luminosities were applied as subsequent upward corrections. However, this effect is eliminated

in our case because we calculate the two rate contributions having relaxed the luminosity constraint

and instead having allowed for a range in luminosity for the pulsars. In this case any differences in

the two separate rate contributions depend only on differences in pulse periods, and widths. Given

that the latter are rather small, it makes sense that, for example, the most likely values of Ntot

for the two pulsars come out to be very similar (e.g. ≃ 185 and ≃ 140, for PSR B1913+16 and

PSR B1534+12, respectively, in model 1). Consequently any difference in the rate contributions from

the two populations is due to the difference in lifetimes (about a factor of 10) for the two observed

pulsars (note that the two do not only have similar Ntot estimates, but also similar beaming correction

factors). Since the lifetime estimate for PSR B1913+16 is much smaller, the total rate distribution is

dominated by its contribution.

In Table 2.2, we list the population parameters and results of Galactic merger and LIGO detection

rates for a large number of models (but still a subset of the models we have investigated in detail).

Model 1 is our reference model defined above and the rest of the models are used to explore the

sensitivity of our results on the pulsar population properties in luminosity and space distributions. The

variation of the luminosity function parameters is within ranges that correspond to a 68% credible

region from the statistical analysis of Cordes & Chernoff (1997). The variation of the space scale

lengths R0 and Z0 extends to values that are not favored by our current understanding, only to allow

us to examine the presence of any correlations. Scrutiny of the results presented in Table 2.2 reveals

that rate estimates are modestly sensitive (and in some cases, e.g., Z0, insensitive) to most of these

variations, except in cases of very low values of Lmin, down to 0.3 mJy kpc2, and unphysically low

values of R0, down to 2 − 3 kpc. Assumptions about the shape of the space distribution (exponential

or Gaussian) are not important. The most important model parameter seems to be the slope of the

luminosity function. The most likely values for the rates are found in the range ≃ 0.7 − 13 Myr−1 for

all models with luminosity-function parameters consistent with a 68% credible region estimated by

40

Figure 2.5 Left panel: The correlation between Rpeak and the cut-off luminosity Lmin with differentpower indices p of the luminosity distribution function. Right panel: The correlation between Rpeak

and the power index of the luminosity distribution function p.

Cordes & Chernoff (1997) and R0 in the range 4 − 8 kpc.

Within a given model, the range of estimated values at 68% credible region is typically broad by

a factor of 5. While at the 95% credible region, the ranges broaden by another factor of about 5,

reaching an uncertainty of ∼ 25 for the rate estimates.

In terms of qualitative variations, it is clear that models with increasing fraction of low-luminosity

or very distant pulsars lead to increasing rate estimates, as expected. A demonstration of this kind

of dependence is shown in Figures 2.5. We have found that there is a strong correlation between the

peak value of the total Galactic merger rate Rpeak estimated by Eq. (2.24) and the cut-off luminosity

Lmin and the power index p. As seen in these Figures Rpeak increases rapidly with decreasing Lmin

or with increasing p (left and right panels shown in Figure 2.5, respectively). Scale lengths of the

spatial distribution (either R0 or Z0) do not show a correlation with Rpeak as strong as that of the

luminosity-function parameters. In Figure 2.6 the most likely rate estimate is plotted as a function of

the radial scale length R0. It is evident that the rate is relatively insensitive to R0 variations unless

R0 becomes very small (. 3 kpc). Small values of the radial scale for the Galactic distribution imply

a large fraction of very distant (and hence undetectable) pulsars in the Galaxy and lead to a rapid

increase of the estimated merger rate. However, such small R0 values are not consistent with our

41

Figure 2.6 The correlation between Rpeak and the radial scale length R0. Rpeak is not sensitive to R0

in the range between 4 − 8 kpc.

current understanding of stellar populations in the Galaxy (Lyne et al. 1985).

2.7. Discussion

In this chapter, we present a new method of estimating the total number of pulsars in our Galaxy and

we apply it to the calculation of the merger rate of double neutron star systems in the Galactic field.

The method implicitly takes into account the small number of pulsars in the observed double-neutron-

star sample as well as their distribution in luminosity and space in the Galaxy. The modeling of pulsar

survey selection effects is formulated in a “forward” way, by populating the Galaxy with model pulsar

populations and calculating the likelihood of the real observed sample. This is in contrast to the

“inverse” way of the calculation of scale factors used in previous studies. The formulation presented

here allows us to: (a) calculate the probability distribution of merger rates; (b) assign statistical

significance to these estimates; and (c) quantify the uncertainties associated with them.

As originally shown by KNST, the most important uncertainties originate from the combination of

a small-number observed pulsar and a pulsar population dominated by faint objects. The probability

42

distribution covers more than 2 orders of magnitude in agreement with the uncertainties in excess of

two orders of magnitudes asserted by KNST. However, for the reference model, even taking a 99%

credible region, the uncertainty is reduced to a factor of ∼ 70. In credible regions of 95% and 68%, the

uncertainty is further reduced to just ∼ 25 and ∼ 5, respectively. We use our results to estimate the

expected detection rates for ground-based interferometers, such as LIGO. The most likely values are

found in the range ∼ (0.3 − 13) × 10−3 yr−1 and ∼ 1.5 − 70 yr−1, for the initial and advanced LIGO.

The statistical method developed here can be further extended to account for distributions of

pulsar populations in pulse periods, widths, and orbital periods. Most importantly the method can

be applied to any type of pulsar population with appropriate modifications of the modeling of survey

selection effects. Pulsar binaries with white dwarf companions are investigated in Chapters 5 and 6.

43

Table 2.2. Model parameters and estimates for Rtot and Rdet in various Bayesian credible regionsfor different pulsar population models

Model1 Parameters Rtot (Myr−1) Rdet of LIGO (yr−1)L2

min p3 R40 Z5

0 peak7 68%8 95%8 initial (×10−3) advanced(mJy kpc2) (kpc) (kpc) peak7 68%8 peak7 68%8

1 1.0 2.0 4.0 (G6) 1.5 (E6) 3.6 +4.4−2.2

+11.0−3.1 1.5 +1.8

−0.9 8.1 +9.9−4.9

2 1.0 2.0 4.0 (G) 0.5 (E) 4.5 +5.2−2.6

+12.9−3.8 1.9 +2.2

−1.1 10.2 +11.6−5.9

3 1.0 2.0 4.0 (G) 2.0 (E) 5.1 +5.9−3.0

+14.7−4.3 2.2 +2.5

−1.3 11.6 +13.2−6.7

4 1.0 2.0 4.0 (E) 1.5 (E) 5.3 +60.1−3.1

+15.2−4.4 2.2 +2.5

−1.3 12.0 +13.7−7.0

5 1.0 2.0 4.0 (G) 1.5 (G) 4.9 +5.6−2.9

+14.1−4.1 2.1 +2.4

−1.2 11.1 +12.7−6.5

6 0.3 2.0 4.0 (G) 1.5 (E) 13.1 +8.0−16.2

+40.6−11.2 5.5 +6.8

−3.3 29.5 +36.4−18.0

7 0.7 2.0 4.0 (G) 1.5 (E) 5.0 +6.2−3.1

+15.5−4.3 2.1 +2.6

−1.3 1.3 +13.9−6.9

8 1.5 2.0 4.0 (G) 1.5 (E) 2.4 +2.9−1.5

+7.3−2.0 1.0 +1.2

−0.6 5.4 +6.6−3.3

9 3.0 2.0 4.0 (G) 1.5 (E) 1.2 +1.5−0.7

+3.7−1.0 0.5 +0.6

−0.3 2.7 +3.3−1.6

10 0.3 1.8 4.0 (G) 1.5 (E) 4.0 +4.8−2.4

+11.9−3.3 1.7 +2.0

−1.0 8.9 +10.7−5.3

11 0.7 1.8 4.0 (G) 1.5 (E) 2.0 +2.4−1.2

+6.1−1.7 0.9 +1.0

−0.5 4.6 +5.5−2.7

12 1.0 1.8 4.0 (G) 1.5 (E) 1.5 +1.8−0.9

+4.6−1.3 0.6 +0.8

−0.4 3.4 +4.1−2.1

13 1.5 1.8 4.0 (G) 1.5 (E) 1.1 +1.3−0.7

+3.4−0.9 0.5 +0.6

−0.3 2.5 +3.0−1.5

14 3.0 1.8 4.0 (G) 1.5 (E) 0.7 +0.8−0.4

+1.9−0.5 0.3 +0.3

−0.2 1.5 +1.8−0.9

15 0.3 2.2 4.0 (G) 1.5 (E) 31.3 +39.5−19.4

+99.3−26.9 13.1 +16.5

−8.1 70.5 +88.9−43.8

16 0.7 2.2 4.0 (G) 1.5 (E) 11.1 +13.9−6.9

+35.0−9.5 4.7 +5.8

−2.9 25.0 +31.4−15.5

17 1.0 2.2 4.0 (G) 1.5 (E) 7.5 +9.3−4.6

+23.5−6.4 3.1 +3.9

−1.9 16.8 +21.0−10.4

18 1.5 2.2 4.0 (G) 1.5 (E) 4.6 +5.7−2.8

+14.2−3.9 1.9 +2.4

−1.2 10.2 +12.7−6.3

19 3.0 2.2 4.0 (G) 1.5 (E) 2.1 +2.7−1.3

+6.7−1.8 0.9 +1.1

−0.6 4.8 +6.0−3.0

20 1.0 2.5 4.0 (G) 1.5 (E) 14.3 +18.7−9.2

+47.1−12.5 6.0 +7.8

−3.8 32.3 +42.1−20.6

21 1.0 2.0 2.0 (G) 1.5 (E) 6.7 +8.0−4.0

+20.2−5.6 2.8 +3.4

−1.7 15.0 +18.1−9.0

22 1.0 2.0 3.0 (G) 1.5 (E) 5.2 +6.4−3.1

+16.0−4.4 2.2 +2.7

−1.3 11.7 +14.4−7.1

23 1.0 2.0 5.0 (G) 1.5 (E) 3.7 +4.7−2.3

+11.7−3.2 1.6 +2.0

−1.0 8.4 +10.5−5.2

24 1.0 2.0 6.0 (G) 1.5 (E) 3.8 +4.8−2.3

+12.0−3.2 1.6 +2.0

−1.0 8.5 +10.7−5.3

25 1.0 2.0 7.0 (G) 1.5 (E) 3.9 +4.9−2.4

+12.4−3.4 1.6 +2.1

−1.0 8.8 +11.1−5.5

26 1.0 2.0 8.0 (G) 1.5 (E) 4.3 +5.4−2.7

+13.6−3.7 1.8 +2.3

−1.1 9.7 +12.2−6.0

44

Table 2.2—Continued

Model1 Parameters Rtot (Myr−1) Rdet of LIGO (yr−1)L2

min p3 R40 Z5

0 peak7 68%8 95%8 initial (×10−3) advanced(mJy kpc2) (kpc) (kpc) peak7 68%8 peak7 68%8

27 1.0 2.0 9.0 (G) 1.5 (E) 4.7 +6.0−3.0

+15.1−4.1 2.0 +2.5

−1.2 10.7 +13.5−6.6

1Model No.

2Minimum luminosity Lmin in mJy kpc2.

3Power index of the luminosity function p.

4Radial scale length Ro in kpc.

5Vertical scale height Zo in kpc.

6Gaussian (G), and exponential (E) functions for spatial distributions.

7Peak value from P(Rtot).

8Credible region.

45

Chapter 3

Revised Merger Rates for NS−NSSystems: Implications for PSRJ0737−3039∗

Two recent developments make it appropriate to revisit the merger rate calculations. First, the

discovery of the 2.45-hr NS−NS binary PSR J0737−3039 in a large-area survey using the Parkes

radio telescope brings the number of known NS−NS systems to merge in the Galactic field to three

(Burgay et al. 2003). With an orbital period of only 2.45-hr, PSR J0737−3039 will merge in only 85

Myr, a factor of 3.5 shorter than the merger time of PSR B1913+16. This immediately hints towards

a possible significant increase of the merger rate (Burgay et al. 2003). Second, a novel statistical

method has been developed by KKL that automatically takes into account statistical biases inherent

in small-number samples, like the relativistic NS−NS binaries, and in addition allows us to quantify

our expectation that the actual NS−NS binary merger rate has a particular value, given the current

observations. In this chapter, we use this statistical method and investigate in detail the effect of this

new discovery on the estimates of Galactic NS−NS merger rates and its implications for GW detection

in this decade.

∗This chapter is adapted with style changes from “The Cosmic Coalescence Rates for Double Neutron Star Binaries”

by V. Kalogera, C. Kim et al. that appeared in The Astrophysical Journal, 601, pp. L179−L182, February 2004, and the

relevant erratum shown in The Astrophysical Journal, 614, pp. L137−L138, October 2004. c©The American Astronomical

Society.

46

3.1. Method for Rate Calculation

Until recently, estimates of NS−NS merger rates provided a range of possible values without any

information on the likelihood of these values. KKL presented a newly developed statistical analysis

that allows the calculation of a probability distribution for rate estimates and the determination of

credible regions associated with the rate estimates. The method can be applied to any radio pulsar

population. Here we update the results of KKL, taking into account the recent discovery of the new

NS−NS binary PSR J0737−3039 (Burgay et al. 2003).

The method is described in detail in KKL, but we briefly summarize the main elements here. The

method involves the simulation of selection effects inherent in all relevant radio pulsar surveys and a

Bayesian statistical analysis for the probability distribution of the merger rate estimates. The small-

number bias and the effect of the faint end of the pulsar luminosity function, previously identified as

the main sources of uncertainty in rate estimates (KNST), are implicitly included in this analysis.

For a model Galactic pulsar population with an assumed spatial and luminosity distribution, we

determine the fraction of the total population which are actually detectable by current large-scale pulsar

surveys. In order to do this, we calculate the effective signal-to-noise ratio for each model pulsar in

each survey, and compare this with the corresponding detection threshold. Only those pulsars which

are nominally above the threshold count as detectable. After performing this process on the entire

model pulsar population of size NPSR, we are left with a sample of Nobs pulsars that are nominally

detectable by the surveys. By repeating this process many times, we can determine the probability

distribution of Nobs, which we then use to constrain the population and with a Bayesian analysis derive

the probability expectation that the actual Galactic NS−NS merger rate takes on a particular value,

given the observations. More details are given in § 2 of LBDH and in KKL.

When this method was first developed by KKL, it was shown that, although the shape of the

probability distribution of rate estimates is very robust, the rate value at peak probability systemat-

ically depends primarily on the characteristics of the radio pulsar luminosity function: its slope and

the physical minimum luminosity of pulsars. Both of these are constrained by the general pulsar pop-

ulation (see Cordes & Chernoff 1997), but we explore the dependence of our results on the assumed

47

values.

Here we consider the same set of pulsar population models as in KKL, but we choose model 6

as our reference model in view of the recent discovery of very faint pulsars (for a review, see Camilo

2003). With the addition of the new NS−NS binary PSR J0737−3039, our calculations differ from

those in KKL in two main ways: (1) the latest Parkes survey that led to the discovery of the new

system (Burgay et al. 2003) is included, and (2) we calculate and account for the effects of Doppler

smearing for NS−NS binaries akin to PSR J0737−3039 by creating fake time series for a variety of

orbital phases (see KKL for details). Even for a ≃ 4.5−min integration, this effect alone reduces the

average signal-to-noise ratio of a 2.45-hr NS−NS binary by 35% (Burgay et al. 2003).

The statistical analysis presented in KKL has been extended to account for three systems (see

Appendix C; also Kim et al. 2004). The calculation of summarized in Appendix C. In our calculations,

we adopt a total lifetime for PSR J0737–3039, defined as the sum of the current age and the remaining

lifetime until the final merge, equal to 100 + 85 = 185 Myr (Burgay et al. 2003). In the absence of

detailed beam observations for the new binary, we adopt a beaming factor of ≃ 6 equal to the average

of the two observationally constrained beams of the previously known NS−NS binaries (KNST; see

also Burgay et al. 2003). We do note, however, that studies of other known recycled pulsars (the

majority of them strongly recycled, spinning faster compared to the NS−NS binaries by about an

order of magnitude) have shown that beaming fractions can vary significantly (Kramer et al. 1998).

It is important to keep in mind that any uncertainties in the beaming factor proportionally affect the

rate estimates, but not the rate increase factors derived here.

3.2. Galactic NS−NS Merger Rate

For our reference pulsar model (with a radio luminosity function consistent with current pulsar ob-

servations; Cordes & Chernoff 1997; Camilo 2003), we find the most likely value of the total merger

rate to be R = 83 Myr−1. The ranges of values in 68% and 95% credible regions are 40−170 and

20−290 Myr−1, respectively. The width of these ranges are somewhat smaller than previous estimates

(the ratio between upper and lower limits in a 68% credible region is 4.4; cf. 5.6 found by KKL),

48

Figure 3.1 Probability density function that represents our expectation that the actual NS−NS binarymerger rate in the Galaxy (bottom axis) and the predicted initial LIGO rate (top axis) take on par-ticular values, given the observations. The curves shown are calculated assuming our reference modelparameters (see text). The solid line shows the total probability density along with those obtained foreach of the three binary systems (dashed lines). Inset: Total probability density, and corresponding68%, 95%, and 99% credible regions, shown in a linear scale.

49

confirming the expectation that a bigger observed sample would reduce the uncertainty in the rate

estimates (KNST). The new value for R is a factor of 6.4 higher than found by KKL. From the result-

ing probability distribution shown in Figure 3.1, it is clear that PSR J0737−3039 dominates the total

rate over the other two systems. This is due to two separate factors: (1) The estimated total number

of NS−NS binaries similar to PSR J0737−3039 (∼ 1700) is far higher than those of each of the other

two systems (∼ 700 for PSR B1913+16 and ∼ 500 for PSR B1534+12). This is mainly due to the

shorter pulsar spin and binary orbital period of PSR J0737−3039, which results in a significant Doppler

smearing and efficiently “hides” them in the Galaxy. (2) The total lifetime of PSR J0737−3039 (185

Myr) is significantly shorter than those of the other two (365 Myr for PSR B1913+16 and 2.9 Gyr for

PSR B1534+12).

We now explore our results for all other models considered in Ch. 2. Our main results are shown

in Table 3.3, where we have included a subset of models that reflect the widest variations of the rates

(as shown in Ch. 2, variations in the space distribution of pulsars are not important). The main

conclusions that can be easily drawn are:

• The increase factor on the merger rate is highly robust against all systematic variations of the

assumed pulsar models and is strongly constrained in the range 5–7; this is consistent with but

somewhat lower than the simple estimate presented in Burgay et al. (2003).

• The shape of the rate probability distribution also remains robust, but the peak values depend

on the model assumptions on different pulsar populations in the same way as described in detail

in Ch. 2 (see Figures 2.5,2.6).

3.3. NS−NS Inspiral Event Rates and Conclusions

Estimates of NS−NS merger rates have suffered from the small number of relativistic binaries known

in our Galaxy, mainly because of the implicit assumption in all methods used so far that the observed

sample represents the Galactic NS−NS population. Here we show that the recent discovery of the third

relativistic binary in the Galactic field with binary properties (pulse-profile and orbital characteristics)

significantly different from those of systems previously known reveals a new sub-population in the

50

Galaxy. Consequently, it leads to a significant increase (by factors of 5–7) of the merger rate estimates.

We now consider the implications of our revised rate estimates for the detection of these events

by LIGO and the other upcoming GW interferometers. Since these instruments can detect NS−NS

merger out to ≃ 20 Mpc for initial LIGO (≃ 350 Mpc for advanced LIGO; Finn 2001), it is necessary

to extrapolate our Galactic event rate out to the Local Group. Using the standard extrapolation of

our reference model out to extragalactic distances (Phinney 1991; KNST), we find the most probable

event rates for our reference model are one per 30 yr and one per 2 day, for initial and advanced

LIGO, respectively. In a 95% credible region, the most optimistic predictions for the reference model

are one event per 8 yr and two events per day for initial and advanced LIGO, respectively. However,

considering the full set of 27 models in a 95% credible region indicates that the respective rates can

reach up to one event per 3 yr and five events per day, respectively. These results are quite encouraging,

since, for initial LIGO in particular, this is the first time that NS−NS merger rate estimates are within

an astrophysically relevant regime. Within a few years of LIGO operations, it should become possible

to test these predictions directly and, in turn, place better constraints on the properties of binary

radio pulsars and the cosmic population and evolution of NS−NS binaries.

51

Table 3.1. Estimates for Galactic merger rates and predicted LIGO detection rates in a 95%credible region based on different population models

Modela R IRFb 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

aModel numbers correspond to KKL. Model 1 was usedas a reference model in KKL (also see Ch. 2). Model 6 isour reference model in this study (see text).

bIncrease rate factor compared to previous rates reportedin KKL; IRF≡ Rpeak,1913+1534+0737

Rpeak,1913+1534.

52

Chapter 4

A Few Remarks on the NS−NS MergerRate Estimates∗

In this chapter, we summarize the current results for Galactic NS−NS merger rates, and briefly address

the uncertainties in the age of J0737−3039, which has a major contribution to the total rate. Then,

we describe results on the possible new detections with PMPS, the global PDF calculation of rate

estimates, and the upper limits on the Galactic merger rate incorporating supernovae constraints. At

the end of this chapter, we discuss the implications of the most recently discovered close pulsar binary

PSR J1906+0746 on the Galactic merger rate estimates.

4.1. The Galactic NS−NS merger rate

In Ch. 2, we introduced a statistical method to calculate P (R) considering merging NS−NS binaries.

After the discovery of PSR J0737−3039, we derived a combined P (R) considering the three observed

NS−NS systems in the Galactic disk (for details see Appendix C). To calculate the merger rate of

∗This chapter contains text adapted with style changes from two manuscripts: (i)“The Galactic Double-Neutron-Star

Merger Rate: Most Current Estimates” by C. Kim et al. that appeared in Binary Radio Pulsars, ASP Conference Series,

328, pp. 83 − 90, proceedings of Aspen winter conference held in Aspen, USA, January 11 − 17, 2004, Edited by F.A.

Rasio and I.H. Stairs (San Francisco: Astronomical Society of the Pacific), January 2005. c©Astronomical Society of the

Pacific; (ii)“Effects of PSR J0737-3039 on the Merger Rate of DNSs and Implications for GW Detection” by C. Kim

et al., proceedings of the conference A Life with Stars, a meeting in honor of Ed van den Heuvel, held in Amsterdam,

Netherlands, August 22–26, 2005 (in press). c©Elsevier.

53

NS−NS systems in our Galaxy, we need to estimate: (i) the number Ntot of Galactic pulsars with pulse

and orbital characteristics similar to those in the observed sample; (ii) the lifetime τlife of each observed

system; (iii) an upward correction factor fb for pulsar beaming. We calculate Ntot by modeling in

detail the pulsar-survey selection effects for a number of pulsar population models described in KKL.

The model assumptions for the pulsar luminosity function dominate the systematic uncertainties of

our overall calculation. The lifetime of a binary pulsar is defined by τlife ≡ τsd + τmrg, where τsd is

the spin-down age of a recycled pulsar (Arzoumanian, Cordes, & Wasserman 1999) and τmrg is the

remaining lifetime until the two neutron stars merge (Peters & Mathews 1963; Peters 1964). We note

that the lifetime of PSR J0737−3039 is estimated to be 185 Myr, which is the shortest among the

observed systems. The beaming correction factor fb is defined as the inverse of the fractional solid

angle subtended by the pulsar beam. Its calculation requires detailed geometrical information on the

beam. Following KNST, we adopt fb = 5.72 for PSR B1913+16 and 6.45 for PSR B1534+12. Without

good knowledge of the geometry of PSR J0737−3039, we adopt the average value of the other two

systems (≃ 6.1). In Figure 3.1, we show P (R) for our chosen reference model that allows for a low

minimum pulsar luminosity (Lmin = 0.3 mJy kpc2; Model 6 in KKL). The most likely value of R turns

out to be 83Myr−1, larger by a factor of ≃ 6.4 compared to the rate estimated before the discovery

of PSR J0737−3039. We find the same increase factor for all pulsar population models examined.

This revised merger rate implies an increase in the detection rate of NS−NS inspirals for ground-based

GW interferometers such as LIGO. Using the standard extrapolation of our reference model out to

extragalactic distances (see KNST), we find that the most probable event rates are 1 per 29 yrs and 1

per 2 days, for initial and advanced LIGO, respectively. In a 95% credible region, the most optimistic

predictions for the reference model are 1 event per 8 yrs and 2 events per day for initial and advanced

LIGO, respectively.

The revised NS−NS merger rate is dominated by PSR J0737−3039. Therefore, if the estimated

lifetime of this system is revised in the future, it will directly affect our rate estimation. Lorimer et

al. (2005) calculated the spin-down age of the system with various spin-down models and suggested

an age in the range 30 − 70 Myr. This is shorter than the value we adopted for our calculation

(τsd =100 Myr). For our reference model, the peak values of rate estimates corresponding to this age

uncertainties span in a range between R ≃ 90 − 115Myr−1.

54

4.2. Predictions for Future Discoveries

As mentioned earlier in Ch. 2, long integration times combined with very short binary orbital periods

strongly select against the discovery of new binary pulsars. Specifically, in the large-scale PMPS with

an integration time of 35 min, the signal-to-noise ratio is severely reduced by Doppler smearing due to

the pulsars’ orbital motion. Acceleration searches of the PMPS archival data significantly improved

the detection efficiency of NS−NS binaries (Faulkner et al. 2003). Although the data analysis is

on-going, acceleration searches already led to the discovery of PSR J1756−2251 (see Faulkner et al.

2005; Lyne 2005). Here, we calculate the probability distribution for the number of NS−NS binaries

detectable by the PMPS (P(Nobs)), assuming that the reduction in flux due to Doppler smearing is

corrected perfectly. To illuminate the effect of the Doppler smearing, we calculate the average number

of expected new discoveries akin to each of the three known NS−NS binaries.

Following Kalogera, Kim, & Lorimer (2003), we can write the probability distribution of the

expected observed numberN iobs for each NS−NS pulsar sub-population i (PSRs B1913+16, B1534+12,

and J0737−3039):

Pi(Nobs) =βi

2

(1 + βi)2(Nobs + 1)

(1 + βi)Nobs, (4.1)

where the constants βi ≡ αiαi,PMPS

, and α is defined in Eq. (2.8). β shows how less likely it is to

detect pulsars without acceleration searches relative to with acceleration searches for PMPS. For each

sub-population, the mean values of Nobs can be calculated and we find them to be:

〈Nobs〉1913 = 0.9, 〈Nobs〉1534 = 1.2, 〈Nobs〉0737 = 1.9. (4.2)

As expected, it is evident that the discovery of NS−NS pulsars in tight binaries like PSR J0737−3039

would be most favored with acceleration searches. Furthermore, we can also calculate the combined

probability distribution of the expected number of NS−NS pulsars that can be detected with PMPS

acceleration searches in the future. The result is shown for our reference model in Figure 4.1. The

average combined number is ∼ 4 and the discovery of up to 2 (or 4) NS−NS systems has a probability

equal to ≃ 30% (≃ 63%). We conclude that, if acceleration searches can correct for the Doppler

smearing effect perfectly, then the PMPS could be expected to detect an average of ∼4 NS−NS

55

Figure 4.1 Probability density function of the predicted number of observed NS−NS binary systemsNobs for the PMPS, for our reference model (model 6 in KKL). The mean value is estimated to be〈Nobs〉 ∼4.0.

pulsars with pulse profile and orbital properties similar to any of the three already known systems.

The increase of the observed sample is very important for the reduction of the uncertainties as-

sociated with the merger rate estimates. We note, however, that the discovery of new systems that

are similar to the three already known does not necessarily imply a significant increase in the rate

estimates. Significant changes are expected in the case that new systems with pulse profiles or binary

properties significantly different are discovered, as it is such systems that will reveal a new NS−NS

sub-population in the Galaxy. For example, we calculate the contribution of PSR J1756−2251 on the

Galactic NS−NS merger rate. In order to calculate the exact merger rate including PSR J1756−2251,

a detailed simulation is necessary to calculate the effects of the PMPS acceleration searches. How-

ever, the approximate contribution of PSR J1756−2251 to the Galactic merger rate can be easily

obtained. In what follows, we calculate the P(R) considering all four discovered NS−NS binaries;

PSRs B1913+16, B1534+12, J0737−3039, and J1756−2251. First, we estimate the number of pulsars

similar to PSR J1756−2251. Adapting the observed properties, and assuming different degradation

factors for PSR J1756−1756 for the PMPS survey. For example, with a degradation factor for PSR

J1756−2251 of F = 0.7, we obtain N1756 ∼ 500 (cf., N1534 ∼ 400, and N1913 ∼ 700). Then, we calculate

56

a total PDF for the case of four systems1, and compare the peak values of R with and without PSR

J1756−2251. We find the total rate increases by only ∼ 4% due to the new discovery, i.e. the increase

rate factor R1913+1534+0737+1756/R1913+1534+0737 = 1.04. This is expected because PSR J1756−2251

can be identified as a member of the B1913+16-like population, which has already been taken into

account in the calculation. As we mentioned earlier, only future detections of pulsars from a signifi-

cantly different population (compared to the known systems), or from the most relativistic systems,

will result in a non-trivial contribution to the rate estimates.

4.3. Global P(R) and Supernova Constraints on NS−NS Merger

Rates

In Ch. 2, we showed that estimated Galactic NS−NS merger rates are strongly dependent on the

assumed luminosity distribution function for pulsars (Figure 2.5). So far, we have reported results for

each set of population model assumptions. Here we describe how we can incorporate the systematic

uncertainties from these models and calculate, Pg(R), a global PDF of rate estimates. However, we

stress that the information needed for such a calculation is currently not up to date; therefore, specific

quantitative results could change when constraints on the luminosity function are derived from the

current pulsar sample.

We calculate Pg(R) using the prior distributions of the two model parameters for the pulsar

luminosity function: the cut-off luminosity Lmin and power-index p. We calculate these priors by

fitting the marginal PDFs of Lmin and p presented by Cordes & Chernoff (1997). We obtain the

following analytic formulae for f(Lmin) and g(p):

f(Lmin) = α0 + α1Lmin + α2L2min and (4.3)

1We do not show the derivation of P(R) with four systems in this thesis. The calculation is straightforward as shown

in Appendix C for three systems.

57

g(p) = 10β0+β1p+β2p2, (4.4)

where αi and βi (i = 0, 1, 2) are coefficients we obtain from the least-square fits and the functions are

defined over the intervals Lmin = [0.0, 1.7] mJy kpc2 and p = [1.4, 2.6]. Although Cordes and Chernoff

(1997) obtained f(Lmin) over Lmin ≃ [0.3, 2] mJy kpc2 centered at 1.1 mJy kpc2, we consider f(Lmin)

with a peak at ∼ 0.8mJy kpc2. This is motivated by the discoveries of faint pulsars with L1400 below

1 mJy kpc2 (Camilo 2003).

Now, we use the above priors to calculate Pg(R):

Pg(R) =

∫

pdp

∫

Lmin

dLminP (R)f(Lmin)g(p) . (4.5)

In Figure 4.2, we show the distributions of Lmin and p adopted (top panels) and the resulting global

distribution of Galactic NS−NS merger rate estimates (bottom panel). We find that Pg(R) is strongly

peaked at only around 15 Myr−1. We note that this is a factor ≃ 5.5 smaller than the revised rate

from the reference model (R = 83 Myr−1). In the 95% credible region, we find that the Galactic

NS−NS merger rates lie in the range ∼ 1 − 170 Myr−1. These imply LIGO event rates in the range

∼ (0.4 − 70) × 10−3 yr−1 (initial) and ∼ 2 − 380 yr−1 (advanced). Given these implications, it is clear

that up-to-date constraints on Lmin and p and their PDFs (a follow-up on Cordes & Chernoff 1997)

are urgently needed.

4.4. Rate Constraints from Type Ib/c Supernovae and Binary

Evolution Models

Based on our current understanding of NS−NS formation, the progenitor of the second neutron star

is expected to form during a Type Ib/c SN (Bhattacharya & van den Heuvel 1991). Therefore, the

empirical estimates for the Type Ib/c SN rate in our Galaxy can be used to provide upper limits on the

NS−NS merger rate estimates. From Cappellaro, Evans, & Turatto (1999) we adopt RSN Ib/c ≃ 1100±

500Myr−1 (for Sbc–Sd galaxies). Here, we assume H0 = 71km/s/Mpc and LB,gal = 9 × 109 LB,sun

58

(KNST).

In order to find the fraction of SN Ib/c actually involved in the formation of NS−NS, we have

examined population synthesis models calculated with the code StarTrack (Belczynski, Kalogera,

& Bulik 2002; Belczynski et al. 2005) and find very low rate ratios: Γ ≡ R/RSN Ib/c ∼ 0.001 −

0.005. Several models with He-star winds consistent with current observations (weaker than previously

thought) lead to Γ ≃ 0.005. We note that systematic overestimation of RSN Ib/c relative to RSN II rates

has already been pointed out (Belczynski, Kalogera, & Bulik 2002; this is related to the assumption of

complete removal of H-rich envelopes). However, we find that this discrepancy would raise the value

of Γ by just a factor of a few. As an approximate constraint, we adopt the empirical RSN Ib/c and scale

it by 1/10 and 1/100, reflecting the results from population synthesis calculations. We overlay these

scaled values in Figure 4.2 (dotted lines in the bottom panel) using the ranges for SN Ib/c reported

by Cappellaro, Evans, & Turatto (1999). We note that our most optimistic NS−NS merger rate is

R = 224+594−181 Myr−1 in a 95% credible region (Model 15 in KKL). We obtain Γ for SN Type Ib/c to

be ∼0.8 with the upper limit of R at the 95% credible region. This corresponds to Γ ∼ 0.1 with a SN

Type II rate, which is factor 6.1 larger than that of SN Type Ib/c. In both cases, the most optimistic

model is lower than the current empirical SN rate estimates, but not really consistent with the results

of population synthesis calculations. If we consider the global distribution, with the upper limit of R

at the 95% credible region, we obtain Γ ∼ 0.15 and 0.025 for SN Type Ib/c and II, respectively.

4.5. Comments on PSR J1906+0746

In this section, we discuss the implications of a recently discovered relativistic system, PSR J1906+0746,

on the pulsar binary merger rates. PSR J1906+0746 is a young pulsar in a highly relativistic pulsar

binary with an eccentric orbit. It was discovered by the ALFA (Arecibo L-band Feed Array) pulsar

survey (Lorimer et al. 2006)2. This system is remarkable because of the extremely young age of the

pulsar (characteristic age of ∼ 112 kyr; Lorimer et al. 2006). The nature of the companion of this pulsar

is not yet known. Based on the total mass of the system and the mass-mass diagram, the companion

2The observed properties of this pulsasr are shown in Table A.1 in Appendix A.

59

Figure 4.2 The global Pg(R) on a linear scale (lower panel) and the assumed intrinsic distributions forLmin and p (upper panels). Dotted lines represent the lower (SNL) and upper (SNU) bounds on theobserved SN Ib/c rate scaled by 1/10 and 1/100 (see text). The empirical SN Ib/c rates range over∼ 600 − 1600Myr−1, where the average is at ∼ 1100 Myr−1 (Cappellaro, Evans, & Turatto 1999),beyond the range shown here.

60

is presumably a relatively light neutron star or a heavy (O-Mg-Ne) white dwarf (& 0.9M⊙). If the

companion is another neutron star, J1906+0746 would be the first discovery of a non-recycled pulsar

in a NS−NS system. Assuming J1906+0746 is a NS−NS system, we calculate its contribution to the

Galactic merger rate. This system’s lifetime (∼82Myr) is set by the death-time of PSR J1906+0746

(τlife = τc + τd ∼ τd, where τc ≪ τd). We adapt the degradation factor for PSR J1906+0746 from PSR

J1141−6545 (Kaspi et al. 2000; Bailes et al. 2003). This is motivated by their similarity in pulsar prop-

erties and orbital characteristics. We calaulate P (R) considering the four systems: PSRs B1913+16,

B1534+12, J0737-3039, and J1906+07463. Here, we ignore PSR J1756−2251 for simplicity, and this

can be justified by the small contribution of this pulasr on the total rate. Following the definition

shown in Ch. 3, we calculate IRF, and calculate the contribution of PSR J1906+0746 to the Galactic

NS−NS merger rate. We find that, if PSR J1906+0746 is indeed the 5-th merging NS−NS binary in

the Galactic disk, it will increase the Galactic NS−NS merger rate by about a factor 2. This implies

that the current estimated NS−NS merger rate including J0737−3039 can still be doubled! If PSR

J1906+0746 is an eccentric NS−WD system, such as PSR J1141−6545 or PSR B2303+46 (Stokes,

Taylor, & Dewey 1985; van Kerkwijk & Kulkarni 1999), it will be as important as PSR J1141−6545

that currently dominates the formation rate of eccentric NS−WD binaries (see Ch. 6 for full discus-

sions). Optical follow-up observations as well as long-term timing and polarization observations will

be essential to reveal the nature of PSR J1906+0746.

3We assume a beaming correction factor of 5 for PSR J1906+0746.

61

Chapter 5

The Probability Distribution Of BinaryPulsar Merger Rates. II. NeutronStar-White Dwarf Binaries∗

There are now more than 40 neutron star-white dwarf (NS−WD) binary systems known in the Galactic

disk (see e.g. Lorimer (2001) for a review). Here, we consider the subset of NS−WD binaries which will

merger due to GW emission within a Hubble time, and consider the implications of NS−WD for the

GW detection with LISA. There are currently three such merging binaries known: PSR J0751+1807

(Lundgren, Zepka, & Cordes 1995), PSR J1757−5322 (Edwards & Bailes 2001, Jacoby et al. 2006),

and PSR J1141−6545. We calculate the Galactic merger rate of NS−WD binaries based on their

observed properties using the method introduced in KKL. The GW frequencies emitted by these

systems fall within the ∼ 0.1 − 100 mHz frequency band of the LISA (Bender et al. 1998). Using

our rate estimates we calculate the GW amplitude due to the NS−WD binaries out to cosmological

distances and compare it to the sensitivity curve of LISA (Larson, Hiscock, & Hellings 2000) as well as

the Galactic confusion noise estimates from white dwarf binaries (Bender & Hils 1990,1997; Nelemans,

Yungelson, & Portegies Zwart 2001; Schneider et al. 2001).

The organization of the rest of this Chapter is as follows. In §5.1, we consider the lifetimes of

NS−WD binaries and summarize the techniques we use to calculate their merger rate. The results of

∗This chapter is adapted with style changes from “The Probability Distribution Of Binary Pulsar Coalescence Rates.

II. Neutron Star-White Dwarf Binaries” by C. Kim et al. that appeared in The Astrophysical Journal, 616, pp. 1109–1117,

December 2004. c©The American Astronomical Society.

62

these calculations are presented in §5.2. In §5.3 we use our rate results to calculate the expected GW

backgound produced by NS−WD binaries Finally, in §5.4, we discuss the implications of our results.

5.1. Merging NS−WD Binaries

In general, the merger rate of a binary system containing an observable radio pulsar is defined by

R =NPSR

τlife× fb , (5.1)

where NPSR1 is the estimated number of pulsars in our Galaxy with pulse profiles and orbital charac-

teristics similar to those of the known systems, fb is a correction factor for pulsar beaming and τlife

is the lifetime of the binary system. In the following subsections, we calculate the total lifetime of a

pulsar binary and derive the PDF of the Galactic merger rate, P (R), for NS−WD binaries.

5.1.1. Lifetime of a NS−WD binary

In Table 5.1, we summarize the observational properties and relevant lifetimes for the 3 pulsar systems

considered in this work. We define the lifetime of a merging pulsar binary τlife to be the sum of

the current age of the observable pulsar and the remaining lifetime of the system. Assuming the

pulsar spins down from an initial period P0 to the currently observed value P (both in sec) due to

a non-decaying magnetic dipole radiation torque (see e.g. Manchester & Taylor (1977) ), its current

“spin-down” age

τsd =P

2P

(

1 −[

P0

P

]2)

, (5.2)

where P (in ss−1) is the observed period derivative. For young pulsars like PSR J1141−6545, it is

usually assumed that P0 ≪ P so that τsd reduces to the familiar characteristic age τc = P/(2P ) ≃ 1.5

Myr. For the older recycled pulsars, however, Arzoumanian, Cordes, & Wasserman (1999) pointed out

that this assumption is usually not appropriate, since the weaker magnetic fields of these objects mean

that their present spin periods are only moderately larger than the periods produced during accretion.

1This is equivalent to the so-called ‘scale factor’ (89).

63

Table 5.1: Observational properties of merging NS–WD binaries. From left to right, the columnsindicate the pulsar name, spin period Ps, spin-down rate P , orbital period Pb, most probable mass ofthe WD companion mwd, orbital eccentricity e, characteristic age τc, spin-down age τsd, time to reachthe death line τd, and references (1) Lundgren, Zepka, & Cordes (1995); (2) Nice, Splaver, & Stairs(2004) (3) Edwards, & Bailes (2001) ; (4) Kaspi et al. (2000) ; (5) Bailes et al. (2003).

PSRs Ps P Pb mwd e τc τsd τd Ref.(ms) (s s−1) (hr) (M⊙) (Gyr) (Gyr) (Gyr)

J0751+1807 3.479 8.08×10−21 6.315 0.18 < 10−4 6.8 6.7 n/a 1,2J1757−5322 8.870 2.78×10−20 10.879 0.67 10−6 5.1 4.9 n/a 3J1141−6545 393.9 4.29×10−15 4.744 0.986 0.172 1.5×10−3 n/a 0.104 4,5

Adopting the spin-up line from Arzoumanian, Cordes, & Wasserman (1999), we may write

P0 =

(

P

1.1 × 10−15

)3/4

s . (5.3)

Using the above two equations we calculate τsd for the two recycled pulsars J0751+1807 and J1757−5322

to be 6.7 and 4.9 Gyr, respectively (see Table 5.1).

The remaining lifetime of a pulsar binary is defined by the shorter of the merger time of the binary

due to the emission of GWs, τmrg, or the time that the pulsar will reach the “death line”, τd (109).

For young pulsars like PSR J1141−6545 which have relatively short radio lifetimes, τd<τmrg. Recycled

pulsars, on the other hand, have far smaller spin-down rates than young pulsars so that it is likely

that close binaries containing a recycled pulsar will merge before the pulsar reaches the death-line

(τmrg<τd). For circular orbits, the results of Peters (1964) calculations for the merger time of a binary

system of two point masses m1 and m2 with orbital period Pb can be written as simply:

τmrg = 9.83 × 106 yr

(

Pb

hr

)8/3 ( µ

M⊙

)−1(m1 +m2

M⊙

)−2/3

, (5.4)

where the reduced mass µ = m1m2/(m1 +m2). For the eccentric binary PSR J1141–6545, we use the

more detailed calculations of Peters (1964) to calculate τmrg. Most of observed merging NS−WD bi-

naries as well as the DNS systems have τmrg∼ 108−9 yr.

Our understanding of pulsar emission is rather poor and therefore it is not clear how to calculate

64

an accurate time associated with the termination of pulsar emission and hence τd. Here we assume the

spin-down torque is dominated by magnetic-dipole radiation with no evolution of the magnetic field.

The surface magnetic field of a neutron star, Bs, can be estimated from the current spin period P (s)

and spin-down rate P (ss−1):

Bs = 3.2 × 1019(PP )1/2 G . (5.5)

Chen & Ruderman (1993) comprehensively discussed the evolution of a pulsar period based on different

magnetic field structures. Their results are consistent with previous studies (109; 126). We adopt their

case C (Eq. 9 in their paper) according to which the radio emission terminates when the “death-period”

Pd =( Bs

1.4 × 1011 G

)7/13s (5.6)

is reached. Assuming that the surface magnetic field remains constant, we can integrate Eq. 5.5 to

calculate the time for the pulsar period to reach Pd. We find that

τd =

(

P 2d − P 2

2PP

)

. (5.7)

For PSR J1141−6545, we use Eqs. 5.6 and 5.7 to find the remaining observable lifetime τd ∼ 104

Myr. This is significantly less than τmrg for this binary system (∼ 600 Myr). Including the modest

contribution from the characteristic age of PSR J1141−6545, we take the observable lifetime of the

binary system to be τlife=τc+τd∼105 Myr. We note in passing that Edwards & Bailes (2001) estimated

the remaining lifetime of PSR J1141−6545 to be only ∼10 Myr. Although no details of their calculation

were presented in their paper, they probably assumed some decay of the magnetic field which led to

their lower value τd and hence τlife.

In the cases of the recycled pulsars J0751+1807 and J1757−5322, which have lower magnetic field

strengths and hence longer radio lifetimes, both binaries will merge before the pulsars stop radiating

(i.e. τmrg≪ τd), so we calculate their lifetime using τlife=τsd+τmrg. The estimated lifetimes are 14.3

Gyr (J0751+1807) and 12.7 Gyr (J1757−5322), about two orders of magnitude longer than for the

young NS−WD PSR J1141−6545.

65

5.1.2. Probability Density Function of the Galactic NS−WD Merger Rate

Estimates

The basic strategy we use to calculate P (Rtot) is described in detail in KKL. In brief, using a detailed

Monte Carlo simulation, for each observed NS−WD binary, we determine the fraction of the popula-

tion that is actually detectable by careful modeling of all large-scale pulsar surveys. We include the

selection effects that reduce the detectability of short-period binary systems when integration times

are significant in comparison. Our pulsar population model takes into account the distribution of

pulsars in the Galaxy and their luminosity function. Treating each pulsar seperately, our simulations

effectively probe the specific pulsar sub-populations with pulse and orbital characteristics similar to

those of PSRs J0751+1807, J1757−5322, or J1141−6545.

From the simulations we obtain Nobs pulsars detected by the surveys out of a Galactic population

of Ntot in each model. We calculate Nobs repeatedly for a fixed Ntot. As shown in detail in KKL,

the distribution of Nobs follows a Poisson function, P (Nobs;Nobs). We calculate the best-fit value of

Nobs, which is the mean number of observed pulsars in a given sample, for a given Ntot. Since we

consider each observed pulsar separately, we set Nobs = 1. For example, one PSR J0751+1807 and no

other pulsar similar to this (in terms of spin and orbital properties of the pulsar) have been observed.

The likelihood of detecting one pulsar similar to the observed one from the given pulsar population

with Ntot samples is simply P (1;Nobs). We vary Ntot and calculate P (1;Nobs) to determine the most

probable value of Ntot. Also, we found Nobs is directly proportional to Ntot. We calculate α, which is

the slope of the function Nobs= αNtot for each observed system for a pulsar population model.

Then, using Bayes’ theorem, we calculate P (Nobs) from the likelihood P (1;Nobs) and eventually

calculate P (R) using a change of variables. We repeat the whole procedure for all three observed

merging NS−WD binaries, and combine the three individual PDFs to obtain a total PDF of Galactic

merger rate of NS−WD binaries, P (Rtot).

In KKL, we showed that a normalized PDF of the merger rate for an individual pulsar binary

system can be written as follows:

Pi(R) = C2i R e−CiR , (5.8)

66

where Ci is a coefficient determined by properties of the ith pulsar:

Ci ≡(

ατlifefb

)

i

. (5.9)

Here, the beaming correction factor fb is the inverse of the fraction of 4π sr covered by the pulsar

radiation beam during each rotation. In the case of the two DNS systems, PSRs B1913+16 and

B1534+12, KNST adopted fb ∼ 6 based on pulse profile and polarization measurements of two pulsars.

The lack of such observations for the current sample of NS−WD binaries means that it is difficult to

estimate reliable values of fb. Therefore, in this chapter, we do not correct for pulsar beaming (i.e. fb =

1). As a result, all our values should be considered as lower limits.

In KKL, we calculated P (Rtot) considering two observed DNSs systems (labeled by the subscripts

1 and 2). We defined the total rate R+ ≡ R1 + R2 and showed that

P (R+) =( C1C2

C2 − C1

)2[

R+

(

e−C1R+ + e−C2R+

)

−( 2

C2 − C1

)(

e−C1R+ − e−C2R+

)]

, (5.10)

where C1 < C2. In Appendix C, we show that this can be extended for the current case of interest

where we have three binary systems such that R+ ≡ R1 + R2 + R3. This leads to

P (R+) =C2

1C22C

23

(C2 − C1)3(C3 − C1)3(C3 − C2)3

[

(5.11)

(C3 − C2)3e−C1R+

[

−2(−2C1 + C2 + C3) + R+[−C1(C2 + C3) + (C21 + C2C3)]

]

+ (C3 − C1)3e−C2R+

[

2(C1 − 2C2 + C3) + R+[C2(C3 +C1) − (C22 + C3C1)]

]

+ (C2 − C1)3e−C3R+

[

−2(C1 + C2 − 2C3) + R+[−C3(C1 + C2) + (C23 + C1C2)]

]

]

,

where the coefficients Ci (i = 1, 2, 3) are defined by Eq. 5.9 and C1 < C2 < C3. This result was

already used in our recent rate estimation for DNS systems to include the newly discovered pulsar,

PSR J0737-3039 (Burgay et al. 2003; Kalogera et al. 2004). As before, the credible region (CR) and

the lower and upper limits (RL and RU) of the merger rate estimates are defined in the same way we

67

described in KKL, i.e.∫

RU

RL

P (R+) dR+ = CR , (5.12)

and

P (RL) = P (RU) . (5.13)

5.2. Results

In Figure 5.1, we show the resulting P (Rtot) for NS−WD binaries along with the individual PDFs for

each observed merging binaries. The figure shown here is obtained from our reference model (model

6 in KKL). As we found for the DNS systems in KKL, P (Rtot) is highly peaked and dominated by a

single object. In this case, PSR J1141−6545 dominates the results by virtue of its short observable

lifetime (τlife ∼105 Myr). This is in spite of the fact that the estimated total number of binaries similar

to PSR J0751+1807 (N0751 ≃ 2900) is the largest among the observed systems.

We summarize our results for different pulsar population models in Table 5.2. The model parame-

ters are identical to those described in KKL. We note however, following Kalogera et al. (2004), that

our reference model is now model 6 (Lmin = 0.3 mJy kpc2) rather than model 1 (Lmin = 1.0 mJy kpc2).

This choice reflects the recent discoveries of faint pulsar with 1400-MHz radio luminosities below than

1.0 mJy kpc2 (Camilo 2003). The peak values of P (Rtot) lie in the range between ∼ 0.2 − 10 Myr−1

where the reference model shows a peak around 4 Myr−1. For the reference model, the uncertainties

in the rates, (defined by RU/RL) are estimated to be ∼ 6, 27 and 62 with 68%, 95%, and 99% CR,

respectively. Comparing this to results from Kalogera et al. (2004), we find that the uncertainties at

different CR of the merger rate of NS−WD binaries are typically larger by factor of ∼1.4 than those

of the DNS systems. This result is robust for all models we consider.

The correlations between the peak value of the total Galactic merger rate Rpeak and the model

parameters (e.g. the cut-off luminosity Lmin and the power index p) seem to be similar to those of

DNSs we observed in KKL. As we found for the DNS systems, Rpeak values strongly depend on a

pulsar luminosity function rather than a spatial distribution of pulsars in the Galaxy, in other words,

Rpeak values rapidly increase as the fraction of faint pulsars increases.

68

99%

95%

68%

68%

95%

99%

Figure 5.1 The PDFs of the Galactic merger rate estimation in both a logarithmic and a linear scale

(inset) are shown for the reference model. The solid line represents P (Rtot). Other curves are P (R) for

PSRs J1757−5322 (dot-dash), J0751+1807 (short-dash), and J1141−6545 (long-and-short dash)-like

populations, respectively. Dotted lines correspond to 68%, 95%, and 99% credible regions for P (Rtot).

5.3. Gravitational Wave Background due to NS−WD Binaries

Close binaries consisting of compact objects (e.g. NS−WD binaries) are suggested as important GW

sources in a frequency range below 1 mHz. In this range, due to the large number of sources, LISA

69

would not be able to resolve each source within a given frequency band. Hence the Galactic binaries

are expected to establish a confusion noise level (or “background”) dominated by WD−WD binaries

(Bender & Hils 1990,1997; Nelemans, Yungelson, & Portegies Zwart 2001; Schneider et al. 2001)..

In this work, we consider the contribution from NS−WD binaries to the predicted confusion noise

level. Using our results from the previous section, we calculate the amplitude of GW signals from

NS−WD binaries in the nearby Universe and compare it with the LISA sensitivity curve2. In this

work, we assume that the three observed systems represent the whole population of NS−WD binaries

in our Galaxy.

We calculate the characteristic strain amplitude of GWs (hc) from NS−WD binaries using the

results given by Phinney (2001) for circular binaries. In general, binaries with an eccentricity e emit

GWs at frequencies f = nν, i.e. the n-th harmonic of the orbital frequency ν. In the case of circular

binaries, n = 2 due to the orbital symmetry and the quadrupole nature of GWs. The eccentricity

of PSRs J0751+1807 and J1757−5322 are e ∼ 10−4 and ∼ 10−6, respectively. Hence, it is safe to

consider them as circular binaries. PSR J1141−6545 has an appreciable eccentricity (e = 0.17), but

for simplicity, we consider only the n = 2 harmonic as if it were a circular binary3.

For an observation of length Tobs with a GW detector, the contribution from background sources

(NS−WD binaries in this work) depends on the number of sources within the frequency resolution,

∆f = 1/Tobs. Following Schneider et al. (2001),, we define an effective GW amplitude hrms(f) ≡

hc(f)(∆f/f)1/2, where hc(f) is a characteristic strain amplitude. Phinney (2001) showed a simple

analytic formula to calculate hc(f) for a population of in-spiraling circular-orbit binaries with a given

2We use the online sensitivity curve generator to calculate the sensitivity curve of LISA

(http://www.srl.caltech.edu/ shane/sensitivity/MakeCurve.html).3We calculate the power distribution in various harmonics for this eccentricity based on the result of Peters & Mathews

(1963). We note that the GW amplitude calculated for PSR J1141−6545 in this work corresponds to ∼70% of the total

power of the gravitational radiation emitted from this binary. The remaining power is contributed from the higher

harmonics.

70

number density in the nearby Universe. We use Eq. 16 in his paper to calculate hc(f)4, and find

hrms(f) ≃ 1.7 × 10−26(M

M⊙

)5/6( f

mHz

)−7/6( No

Mpc−3

)1/2(Tobs

yr

)−1/2, (5.14)

where M is the “chirp mass” of a NS−WD binary defined by

M ≡ (MNSMWD)3/5

(MNS +MWD)1/5, (5.15)

and No is the comoving number density of NS−WD, i.e. the number of sources per Mpc3.

We calculate the GW amplitude of NS−WD binaries in a frequency range fmin < f < fmax.

Estimated GW frequencies of three NS−WD binaries based on their current separations are all less

than ∼ 0.1 mHz. In our calculation, however, we set the minimum frequency fmin to be 1 mHz taking

into account the fact that the confusion noise level is mainly dominated by Galactic WD–WD binaries

at lower frequency range f < 1mHz (Nelemans, Yungelson, & Portegies Zwart 2001).The maximum

GW frequency fmax is calculated by fmax = 2/Pb,min, where Pb,min is the minimum orbital period of

the binary at the WD Roche-lobe overflow.

Following Eggleton (1983), we calculate the minimum possible separation of the binary using

amin = RWD/rL, where rL is the effective Roche lobe radius. We estimate the radius of a white dwarf

companion RWD adopting the results given by Tout, Aarseth, & Pols (1997). We show the estimated

mass of white dwarf companions in Table 5.15. Converting amin to fmax based on the Kepler’s 3rd

law, we find that

fmax ≃ 0.16 rL3/2(MWD +MNS

M⊙

)1/2[( Mch

MWD

)2/3−(MWD

Mch

)2/3]−3/4Hz , (5.16)

4We assume a case that the last term in Phinney’s Eq. 16,

„

<(1+z)−1/3>0.74

«1/2

becomes unity. The calculation is not

significantly affected by different assumptions on cosmological models and comoving number density functions of merging

binaries (Phinney 2001).5We calculate mwd with the following assumptions on the NS mass mNS and inclination angle i: [mPSR, i]=[2.2M⊙,

78◦] (PSR J0751+1807), [1.35M⊙,60circ] (PSR J1757−5322). We adopt mwd for PSR J1141−6545 from Bailes et al.

(2003)

71

where Mch = 1.44M⊙ is the Chandrasekhar limit. We note that fmax = fmax(MNS,MWD). Based on

Eq. 5.16, we define three frequency regions: (a) fmin < f < fmax, 0751, (b) fmax, 0751 < f < fmax, 1757,

and (c) fmax, 1757 < f < fmax, 1141. In the region (a), for example, all three observed NS−WD systems

contribute to the GW background. However, for frequencies larger than fmax, 0751, PSR J0751+1807-

like populations have already reached the Roche lobe overflow and we can not apply Eq. 5.14 to these

systems. Therefore, in a frequency region (b), we consider PSR J1141−6545-like and J1757−5322-like

populations. Similarly, for the highest frequency range (region (c)), we consider the contribution to

the GW signals from PSR J1141−6545-like population only.

In order to calculate hrms(f), we need the chirp mass M and present-day comoving number density

of NS−WD binaries No. Following Farmer & Phinney (2003), we define the “flux-weighted” averaged

chirp mass:

<M >≡∑Fgw,i Mi∑Fgw,i

=

∑Npeak,i Mi13/3

∑Npeak,i Mi10/3

, (5.17)

where Fgw is the GW flux (Fgw ∝ f10/3M10/3 Npeak) and Npeak is the peak value of P (Ntot) of the

each sub-population of NS−WD binaries in our Galaxy. The subscript i represents each pulsar sub-

population. For example, in region (b), <M>= (N1757M175713/3 + N1141M1141

13/3)/

(N1757M175710/3+N1141M1141

10/3). Because Npeak is a constant and independent of the GW frequency,

it follows that <M> is independent of frequency. As a result, the evolution of orbital characteristics

and hence the GW frequency of a binary are solely determined by the in-spiral process. This is true

regardless of the inital distribution of orbital characteristics.

We now calculate the comoving number density of NS−WD binaries

No =

∫ ∞

0N(z)dz , (5.18)

where N(z)dz is the number of NS−WD binaries per unit comoving volume between redshift z and

z + dz. Noting that the number density of NS−WD binaries is proportional to the total number of

systems, we may write

No = ǫ NPSR , (5.19)

where ǫ is the star formation rate density per unit comoving volume (ρ) normalized to the Galactic

72

star formation rate (r) i.e. ǫ = ρ/r (in Mpc−3). NPSR is the most likely value of the total number

of pulsars for each frequency range. (e.g. NPSR = N0751 + N1757 + N1141 in region (a)). We derived

P (Ntot) for individual systems in KKL. In a similar fashion to the merger rate estimation described

in Appendix C, the combined PDF of Ntot can be calculated from individual PDFs of the observed

systems. Then NPSR can be obtained from the peak value of the PDF we calculate for each frequency

range we discussed above.

Using the results of Cole et al. (2001), we find

ǫ =ρ(0)

r

∫ zmax

0

(1 + baz)

(1 + (zc )

d)dz Mpc−3 , (5.20)

where (a, b, c, d) = (0.0166, 0.1848, 1.9474, 2.6316) are parameters which take into account dust-extinction

corrections (see Cole et al. (2001) for further details). Assuming a Hubble constant Ho = 65 km s−1

Mpc−1, we calculate the Galactic star formation rate density ρ(0) ≃ 0.01M⊙ yr−1 Mpc−3. Following

Cappellaro, Evans, & Turatto (1999), assuming the Salpeter initial mass function, we convert the

Galactic SNII+Ib/c rate6 to the star formation rate finding r ∼ 0.7M⊙ yr−1. Numerically integrating

Eq. 5.20 out to zmax = 5, which is considered to be the onset of the galaxy formation Schneider et al.

(2001), we find ǫ ∼ 0.6 Mpc−3. The number density of NS−WD binaries No then can be calculated

by Eq. 5.19 for a given NPSR for each model.

In Figure 5.2, we plot the GW amplitude hrms against the simulated LISA sensitivity curve calcu-

lated for a signal-to-noise ratio S/N=1. All dotted lines correspond to the GW amplitude calculated

from the full set of pulsar population models we consider and the solid line is the result from our

reference model. The range of the GW amplitude for all models spans about an order of magnitude.

We find that the GW background amplitude from NS−WD binaries is about 1–2 orders of magnitude

smaller than the expected sensitivity curve of LISA at GW frequencies larger than 1mHz and it is

unlikely that this population will be detected with LISA. In the lower frequency region (below ∼

1mHz), the GW background amplitude from NS−WD binaries increases as f decreases. However, the

6Cappellaro, Evans, & Turatto (1999) assumed a Hubble constant Ho = 75 km s−1 Mpc−1. Since we adopt Ho = 65

km s−1 Mpc−1, we have multiplied their results by a factor (65/75)2. We also note that we consider Sbc-Sd type galaxies

only, which would be relevant for active star-forming regions.

73

contribution from NS−WD binaries to the GW background noise level would still be less than ∼ 10%

of the GW amplitude from WD–WD binaries (dashed line in 5.2). We note that, however, we have

not considered any beaming corrections, so the NS−WD background curves should be viewed as lower

limits. This possibility is discussed briefly in the next section.

74

WD-WD

Figure 5.2 The effective GW amplitude hrms for merging NS−WD binaries overlapped with the LISA

sensitivity curve. The curve is produced with the assumption of S/N=1 for 1 yr of integration. Dotted

lines are results from all models we consider except the reference model, which is shown as a solid line

(see text for details). We also show the expected confusion noise from Galactic WD–WD binaries for

comparison (dashed line).

75

5.4. Discussion

We have used detailed Monte Carlo simulations to calculate the Galactic merger rate of NS−WD bi-

naries. From the reference model, the most probable value of Rtot is estimated to be 4.11+5.25−2.56 Myr−1

in a 68% credible region. We find that the merger rate of NS−WD binaries is about factor of 20

smaller than those of DNS for all pulsar population models we consider. As mentioned above, we did

not take into account any beaming correction for NS−WD binaries. If we assume a beaming fraction

of pulsars in NS−WD binaries similar to that of pulsars found in DNS, fb ∼ 6, then the discrepancy

beween Rpeak (DNS) and Rpeak (NS−WD) is signicantly reduced. As a simple estimate, if we assume

fb,1141 ∼ 5, but keeping fb = 1 for the other two binaries, the estimated Galactic merger rate increases

to 18.06+26.05−12.74 Myr−1 in a 68% credible region. Hence the ratio between DNS and NS−WD merger

rate decreases to about 5. Because the contribution from PSRs J0751+1807 and J1757−5322 is an

order of magnitude smaller than that of PSR J1141–6545, moderate values of beaming fraction for

those recycled pulsars do not change the result significantly.

Based on the number of sources of NS−WD binaries in our Galaxy, we estimate the effective

GW amplitude from the cosmic population of these systems. We find that the GW background from

NS−WD binaries is too weak to be detected by LISA for the nominal beaming correction. Only by

adopting an unreasonably large beaming correction factor, fb > 10, could these systems be detectable

by LISA in the mHz range. These results are in good agreement with an independent study by Cooray

(2004) based on statistics of low mass X-ray binaries. We finally note that combining the results from

KKL and this work can give us strong constraints on the population synthesis models. The preferred

models, which are consistent with both RNS−WD and RDNS, can then be used for estimating the

merger rate of NS−BH binaries, which will be discussed in the remaining chapters.

76

Table 5.2. Estimates for the Galactic merger rate (Rtot) of NS−WD binaries in 68% and 95%credible regions for all models considered. Model number is same with Table 2.2. We show the most

likely value of Rtot at 68% and 95% credible regions.

Model No. Rtot (Myr−1)peak 68% 95%

1 1.23 +1.57−0.77

+3.97−1.04

2 1.00 +1.27−0.62

+3.20−0.83

3 1.32 +1.69−0.83

+4.27−1.12

4 1.53 +1.99−0.97

+5.02−1.31

5 1.19 +1.53−0.75

+3.87−1.01

6 4.11 +5.25−2.56

+13.23−3.47

7 1.73 +2.21−1.08

+5.57−1.46

8 0.83 +1.07−0.52

+2.71−0.71

9 0.43 +0.55−0.27

+1.39−0.36

10 1.42 +1.88−0.92

+4.76−1.23

11 0.72 +0.96−0.47

+2.42−0.62

12 0.55 +0.72−0.35

+1.82−0.47

13 0.40 +0.53−0.26

+1.33−0.34

14 0.23 +0.30−0.15

+0.76−0.20

15 10.03 +12.46−6.09

+31.38−8.34

16 3.68 +4.57−2.25

+11.53−3.07

17 2.45 +3.06−1.50

+7.71−2.04

18 1.55 +1.94−0.95

+4.90−1.29

19 0.72 +0.91−0.44

+2.29−0.60

77

Table 5.2—Continued

Model No. Rtot (Myr−1)peak 68% 95%

20 5.48 +6.69−3.29

+16.86−4.52

21 1.96 +2.37−1.17

+5.96−1.61

22 1.55 +1.97−0.96

+4.96−1.3

23 1.13 +1.47−0.71

+3.72−0.97

24 1.14 +1.49−0.73

+3.76−0.98

25 1.21 +1.56−0.76

+3.95−1.03

26 1.33 +1.72−0.84

+4.35−1.13

27 1.46 +1.90−0.93

+4.79−1.25

78

Chapter 6

Neutron Star−White Dwarf Binaries inEccentric Orbits∗

6.1. Introduction

Binary pulsars with white dwarf companions (NS−WD binaries) in eccentric orbits have been revealed

with binary pulsar and optical observations in recent years. This sub-population is considered to be

rather special because, based on our standard understanding of binary evolution, NS−WD binaries

are expected to be circular: a neutron star forms first from the original binary primary and the white

dwarf formation follows mass transfer episodes that are expected to circularize the binary orbit. For

a review of this scenario, see Lorimer (2001). The neutron star produced by this evolutionary path is

expected to be ‘recycled’, i.e. spun-up by mass accretion from its companion. However, the existence

of eccentric NS−WD binaries such as PSR B2303+46 and PSR J1141–6545 implies that a different

evolutionary path from the standard scenario is also be possible (Tutukov & Yungelson 1993; Portgies

Zwart & Yungelson 1999; Tauris & Sennels 2000; Nelemans, Yungelson, & Portegies Zwart 2001;

Brown et al. 2001; Davies, Ritter, & King 2002). The non-zero eccentricity of the binary orbit is

introduced by a SN explosion of the secondary companion that occurs after the original primary has

∗This chapter is adapted with style changes from a manuscript with title “The Galactic Formation Rate of Eccentric

Neutron Star−White Dwarf Binaries” by V. Kalogera, C. Kim et al. that appeared in Binary Radio Pulsars, ASP

Conference Series, 328, pp. 261 − 267, proceedings of Aspen winter conference held in Aspen, USA, January 11 − 17,

2004, Eds. F.A. Rasio and I.H. Stairs (San Francisco: Astronomical Society of the Pacific), January 2005. c©Astronomical

Society of the Pacific.

79

already evolved into a white dwarf.

Our motivation for this study is to estimate the Galactic formation rate of eccentric binaries based

on the observed pulsars. We apply our statistical analysis developed to estimate the Galactic DNS

merger rate (KKL) and derive a PDF of the Galactic formation rate for eccentric NS−WD binaries.

This method provides us with rate estimates that are independent from those obtained using binary

evolution calculations. We compare our empirical rate estimates to results from population synthesis

calculations and conclude that, despite the large uncertainties, the results are indeed consistent.

6.2. Formation of Eccentric NS−WD Binaries

In our classical understanding of binary evolution, we expect that the formation process of NS−WD bi-

naries in close orbits involves the circularization of the binary orbit, even for systems with massive

white dwarfs: the neutron star forms first and possibly induces an eccentricity, but subsequent mass

transfer from the white dwarf progenitor is expected to circularize the orbit (Verbunt & Phinney

1995). Indeed PSR J0751+1807 (Lundgren, Zepka, & Cordes 1995; Nice, Splaver, & Stairs 2004) and

PSR J1757−5322 (Edwards & Bailes 2001) have very small eccentricities (e ≤ 10−4). However, the

discovery of the white dwarf companion to PSR B2303+46 led various groups to consider a different

evolutionary path that could explain the observed eccentricity of e = 0.658 (Portgies Zwart & Yungel-

son 1999; Tauris & Sennels 2000; Nelemans, Yungelson, & Portegies Zwart 2001; Brown et al. 2001;

Davies, Ritter, & King 2002).

In these modified scenarios, the original primary star, which is massive enough to form a white

dwarf but not a neutron star, evolves first and transfers its mass to the secondary star. After the

primary star loses its envelope it becomes a white dwarf, while the secondary continues its evolution

but at a higher mass than its initial mass (high enough to form a neutron star) due to the mass

transfer phase from the original primary. The primary eventually fills its Roche lobe and a common

envelope phase ensues: the white dwarf and the helium core of the secondary spiral together as the

common envelope of the two stars is ejected from the system. The outcome is a tight binary with

a white dwarf and a helium star that is massive enough to explode in a supernova. This introduces

80

Table 6.1: Observational properties of eccentric NS−WD binaries. The columns indicate the pulsarname, spin period P , spin-down rate P , orbital period Pb, the estimated mass of the WD companionmc, orbital eccentricity e, characteristic age τc, time to reach the death line τd.

PSRs P P Pb mc e τc τd(ms) (s s−1) (hr) (M⊙) (Myr) (Gyr)

J1141−6545 393.9 4.29×10−15 4.744 0.986 0.172 1.45 0.104B2303+46 1066 5.69×10−16 296.2 1.24 0.658 29.7 0.140

an eccentricity to the orbit of the NS−WD binary that forms. Hence eccentric NS−WD binaries are

formed because of two main facts relevant to the early evolution of the binary progenitor: (1) the

initial binary components are both massive enough to form only a white dwarf, but still close to being

massive enough to form a neutron star; (2) the initial mass transfer phase from the primary to the

secondary increases the mass of the secondary enough that it can now form a neutron star eventually,

but only after the initial primary ends its evolution as a white dwarf.

6.3. Empirical NS−WD Formation Rate Estimates

We calculate Ntot by modeling in detail the pulsar survey selection effects associated with the discovery

of these systems for a number of parent pulsar population models described in KKL. The model

assumptions for the pulsar luminosity function dominate the systematic uncertainties of our overall

calculation. For our reference model (model 6 in KKL), we obtain the most likely value of Ntot for

PSRs J1141-6545 and B2303+46 to be N1141 ≃370 and N2303 ≃240, respectively.

The lifetime of the system is defined as τlife ≡ τc + τd, where τc is the characteristic age. τd is

the time which a pulsar will reach the “death line” (Ruderman & Sutherland 1975). We calculated

τd following (Chen & Ruderman 1993; eq. (9) in their paper). The calculated PDF of the formation

rate is dominated by PSR J1141-6545-like population due to both the shorter lifetime and number

abundance of this population (Figure 6.2). The estimated formation rate for our reference model is

Rb= 6.8+5.6−3.5

+13.7−5.6 Myr−1 at 68% and in a 95% credible region, respectively. The most likely values

of Rb lie in the range 0.5 − 16Myr−1 for all the PSR population models we consider (see Table 6.2

for the rate estimates for a number of models with different pulsar luminosity functions). In the

81

Figure 6.1 The probability density function of Galactic formation rate estimates for eccentricNS−WD binaries (solid line) for our reference model. Dashed and dot-dashed lines represent theindividual probability density functions of the formation rates for sub-population of binaries similarto either PSR B2303+46 or J1141−6545. No corrections for pulsar beaming have been applied.

82

absence of observational constraints on the geometry of both pulsars, we decided to not include any

such upward correction for pulsar-beaming. Therefore, our estimated rates should be considered as

lower limits. We can also compare these estimates with those for the Galactic DNS rate. The most

likely values of DNS rates are found in the range 4− 224 Myr−1 for the models considered with pulsar

beaming correction (Kim et al. 2005; Kalogera et al. 2004). Since the beaming correction factor for

young pulsars is presumably larger than those for recycled pulsars, the true Galactic formation rate

for eccentric NS−WD binaries are expected to be comparable to that of DNS systems we estimated.

Population synthesis calculations have been widely used to study the formation of various types of

compact object binaries including eccentric NS−WD binaries. The range of details in the evolutionary

calculations as well as the extent (if any) of the parameter studies vary significantly. In this section we

summarize the main results from theoretical studies in literature and compare them to the empirical

rates derived in the previous section.

Portegies Zwart & Yungelson (1999) obtained a formation rate for eccentric NS−WD binaries

comparable, but somewhat larger than that of DNS systems (Rb=44 and 34 Myr−1 for eccentric

NS−WD and DNS systems respectively). Tauris & Sennels (2000) presented similar results (Rb∼ 57

Myr−1) and concluded that the formation rate of eccentric NS−WD binaries is ∼18 times higher

than that of DNS systems. Davies, Ritter, & King (2002) estimated the Galactic formation rate of

systems like PSR J1141−6545 or B2303+46. They obtained formation rates in the wide range of

∼ 10−5 − 10−3 yr−1 when considering both PSR J1141−6545−like and B2303+46−like systems with

different evolutionary histories. Finally, Nelemans, Yungelson, & Portegies Zwart (2001) derived 240

Myr−1 for the formation rate of all types of NS−WD binaries, which is comparable to other studies.

We consider this rate to be an upper limit of eccentric NS−WD (shown as an open square with a

downward arrow in Figure 6.1).

In addition to the above results from then literature, we have performed population synthesis

calculations using the StarTrack population synthesis code (Belczynski, Kalogera, & Bulik 2002;

Belczynski et al. 2005). We have explored a selection of models (more than any of the other studies)

that promise to give us the widest variations of rate estimates. We derive a range ∼ 30 − 100 Myr−1

(using an absolute normalization of models based on empirical supernova rate estimates for our Galaxy;

83

Table 6.2: The estimated Glactic formation rate and the most likely value of Ntot of eccentric NS−WDbinaries for models with different pulsar luminosity functions. Model number is same with Table 2.2.We show the most likely value of Rb in 68% and 95% credible regions.

Model Rb (Myr−1) N1141 N2303

peak 68% 95%

1 2.1 +1.7−1.0

+4.0−1.6 110 70

6 6.8 +5.6−3.5

+13.7−5.4 370 240

9 0.7 +0.6−0.4

+1.4−0.6 40 30

10 2.6 +2.1−1.3

+5.1−2.0 140 90

12 1.0 +0.8−0.5

+1.9−0.8 50 40

14 0.5 +0.4−0.2

+0.8−0.4 20 20

15 15.7 +13.1−8.1

+31.9−12.4 870 530

17 3.9 +3.3−2.0

+8.1−3.1 220 130

19 1.1 +0.9−0.6

+2.3−0.9 60 40

20 8.3 +7.2−4.3

+17.7−6.6 490 260

Cappellaro, Evans, & Turatto 1999).

The different results from various population studies are mainly attributed to varying assumptions

about the initial mass function, the initial mass-transfer process, the assumed star-formation or super-

nova rate used as a normalization factor, the initial binary fraction, and NS kicks. We also note that

the lifetime of the system is a free parameter in the theoretical rate estimation, which is attributed to

at least an order of magnitude of the uncertainty in the calculation.

6.4. Comparison with Rates from Binary Evolution

In Figure 6.2, we overlap the empirical formation rate estimates for eccentric NS−WD binaries (filled

circles with error bars labed by KKL) with previous studied, open squares and a think solid line

labeled by N01 (Nelemans, Yungelson, & Portegies Zwart 2001), TS00 (Tauris & Sennels 2000), and

PZY99 (Portgies Zwart & Yungelson 1999), and D02 (Davies, Ritter, & King 2002), respectively, as

well as results from StarTrack (thin solid lines with filled triangles). It is encouraging that the overall

rate estimates from different methods appear to be consistent with one another. If we consider the

most likely values of the empirical rate estimates (filled circles), they are somewhat smaller than the

theoretical values. However, in a 95% credible region, the empirical and theoretical estimates become

84

comparable. We also emphasize that the upward correction for pulsar-beaming has not been applied,

and therefore the empirical rate estimates should be considered as lower limits.

In conclusion, we find that our empirical estimates for the Galactic formation rate of eccentric

NS−WD binaries are overall consistent with the estimates derived based on binary population synthesis

models. Despite the large uncertainties we consider this consistency as evidence that our current

theoretical understanding for the formation of eccentric NS−WD is reasonable. However, the extent

of the range covered by the empirical estimates and the population synthesis studies that attempt

even a minimal parameter study (e.g., Davies, Ritter, & King (2002) and our results from StarTrack)

indicates once again the necessity for careful parameter studies of rate calculations. It also indicates

that the empirical rates could in principle be used to constrain binary evolution calculations. In Ch.

8, we will discuss more on this possibility.

85

Figure 6.2 Comparison between the empirical and theoretical rate estimates. Error bars with filledtriangles indicate results from StarTrack, open squares and a solid line are adapted from the literature,and filled circles with error bars are obtained in this work (see text for details).

86

Chapter 7

Upper Limits on NS−BH Binaries

Among pulsar binaries, those with BH companions have the largest possible chirp masses (see Eq.

(5.15) for a definition), and therefore the strongest GW amplitudes, which makes this population

interesting for ground-based interferometers1. According to the standard stellar binary evolution

scenario, those pulsars belong to NS−BH binaries may or may not have been recycled depending on

the details of binary evolution. None of these two types of NS−BH binaries (recycled PSR-BH and

non-recycled PSR-BH) have been observed, yet. Due to the absence of any such discoveries, only the

purely theoretical approach has been used to estimate the merger rate of this population. Theoretical

studies suggest that the merger rates of NS−BH binaries are in a range between ∼ 10−8 − 10−5 yr−1

(e.g., Portegies Zwart & Yungelson 1998; Bethe & Brown 1999; Belczynski, Kalogera, & Bulik 2002;

Voss & Tauris 2003; Sipior, Portegies Zwart, & Nelemans 2004; Pfahl, Rappaport, & Podsiadlowski

2005; Dewi, Podsiadlowski, & Sena 2006). Note that there are significant uncertainties in the rate

estimates of the NS−BH binaries due to the large parameter space incorporated in the theoretical

models. These results consider a whole NS−BH binary population, but theoretical estimates on the

NS−BH merger rates are likely to be dominated by non-recycled PSR−BH binaries (see next chapter

for further discussion). Recently, Pfahl, Podsiadlowski, & Rappaport (2005) introduced a semi-analytic

analysis on the merger rate of NS−BH binaries considering only those contain recycled pulsars. They

conclude that the upper limit on the merger rate of NS−BH is likely to be ∼ 10−7 yr−1 or even less.

1For example, a NS (1.4M⊙)−BH (10M⊙) binary has M∼ 3M⊙, which is three times bigger than that of a binary

containing two 1.4M⊙ NSs. This implies about an order of magnitude large detection volume for NS−BH binaries

compared with that of NS−NS binaries for a GW interferometer such as LIGO (see §8.4).

87

In principle, the empirical method described in earlier chapters can deal with any type of pulsar

binaries, if there is an observation. In the case of NS−BH binaries, which has no available data,

we can still calculate an upper limit on the merger rate of NS−BH binaries. Here, we focus on

recycled PSR−BH binaries. The reason we are more interested in recycled pulsars are because of its

long lifetime to be observable (typically ∼Gyr timescale versus 10 − 100Myr for those non-recycled

ones), and relatively large beaming opening angle. Both factors can enhance the chance of a discovery

compared to non-recycled pulsars. In addition, fast-spinning pulsars with narrow pulse shapes are

observationally preferred for a timing purposes; Recycled pulsars, particularly those with narrow pulse

shapes, are considered to be good targets to measure the post-Newtonian parameters with currently

available radio telescopes. Therefore, it is important to constrain the abundance of this population

(Kramer et al. 2004; Lorimer & Kramer 2005).

Following §2.5, we calculate an upper limit on the Galactic merger rate of recycled PSR−BH

binaries. Here, we assume that those (recycled) pulsars belong to NS−BH binary population would

not be very different from those discovered in NS−NS binaries. For simplicity, we set a spin period

and duty cycle of our model pulsar to be the average value obtained from the three known NS−NS

binaries (PSRs B1913+16, B1534+12, and J0737-3039): Ps = 50ms, duty cycle = 0.15. Then we try

different degradation factors (F ) and lifetimes adapted from the known NS−NS binaries separately

(e.g, F = 0.15, 0.3, 0.7 for PMPS) to model different orbital characteristics. Generally, it is expected

that the orbital acceleration in the PSR−BH binaries would be larger than that of NS−NS binaries due

to the heavier mass of the BH companion, and this can severely smear out pulse signal over different

frequency bins. However, we simply adapt the known degradation factors from NS−NS binaries in

this exercise, which can be valid for systems with a few M⊙ BH companions. In order to reflect the

absence of data for NS−BH binaries, we set Nobs = 0 in Eq. (2.13), and calculate the data likelihood:

P (0;λ) = e−λ. Then we obtain the PDF of rate estimates as follows: P(R)= τe−Rτ (cf. Eq. (2.17)).

We note that we assume no beaming correction factor because we are interested in the upper limits.

The P(R) has essentially same with the likelihood function, only weighted by the system’s lifetime.

By integrating the combined P(R) over a range [0,RU), we obtain the upper limit to the recycled

PSR−BH merger rate to be RU ∼ 10−3 yr−1 in a 95% credible region. The main uncertainty in our

empirical calculation comes from the assumption on the pulsar lifetime as well as degradation factors

88

actually relevant to the recycled PSR−BH binaries.

This exercise is rather crude and does not provide strong constraints for our purposes compared

to the theoretical results. However, it demonstrates an application of the statistical method for rate

estimates to the case of an upper limit calculation with assigned statistical significances. This can be

most useful when recycled PSR−BH binaries are eventually discovered. In next chapter, we discuss

how to constrain theoretical model parameters using empirical rate estimates. At the end of Ch. 8,

we will come back to the merger rate of BH and non-recycled pulsar binaries.

89

Chapter 8

Constraining Population SynthesisModels via the Binary Neutron StarPopulation

8.1. Introduction

Interest in the formation channels and rates of double compact objects (DCOs) has increased in recent

years partly because, at the late stages of their inspiral through the emission of GWs, they can be

strong enough sources to be detected by the many presently-operating ground-based GW detectors

(i.e., LIGO, GEO600, VIRGO, TAMA300). But with the notable exception of NS−NS mergers (KKL),

the merger rates for DCOs with black holes have not been constrained empirically. The only route

to rate estimates for NS−BH and BH−BH binaries is through population synthesis models. These

involve a Monte Carlo exploration of the likely life histories of binary stars, given statistics governing

the initial conditions for binaries and a method for following the behavior of single and binary stars (see,

e.g., Belczynski, Kalogera, & Bulik 2002) Unfortunately, our understanding of the evolution of single

and binary stars is incomplete, and we parameterize that uncertainty with a great many parameters

(∼ 30), many of which can cause the predicted DCO merger rates to vary by more than an order

of magnitude when varied independently through their plausible range. To arrive at more definitive

0*Adapted with style changes from a manuscript with title “ Constraining Population Synthesis Models via the

Binary Neutron Star Population” appeared in The Astrophysical Journal, 633, pp. 1076−1084, November 2005, by R.

O’Shaughnessy, C. Kim, et al. c©The American Astronomical Society

90

answers for DCO merger rates, we must substantially reduce our uncertainty in the parameters that

enter into population synthesis calculations through comparison with observations.

Clues to the physics underlying the formation of tight compact binaries can be obtained through

a study of each individual known DCO system. Some authors have followed this path, for example

examining the potential evolutionary and kinematic histories of each individual binary to deduce the

pulsar kicks needed to reproduce their evolutionary path (e.g., Willem et al. 2004). However, the

simplest and most direct way to constrain the parameters of a given population synthesis code is to

compare several of its many predictions against observations. For example, the empirically estimated

formation rates derived from the six known Galactic NS−NS binaries – half of which are tight enough to

merge through the emission of GWs within 10 Gyr – should be reproduced by any physically reasonable

combination of model parameters for population synthesis.

In this chapter, we describe the constraints the observed NS−NS population places upon the

most significant parameters that enter into one population synthesis code, StarTrack (Belczynski,

Kalogera, & Bulik 2002; Belczynski et al. 2005). Furthermore, we use the set of models consistent

with this observational constraint to revise our population-synthesis-based expectations for various

DCO merger rates.

In §8.2 we describe the observational constraints from NS−NS: reviewing and extending the work

of KKL, we briefly summarize the observed sample of NS−NS binaries, the surveys which detected

them, and the implications of the (known) survey selection effects for the expected NS−NS formation

rate. In §8.3 we describe our population synthesis models and their predictions for NS−NS binary

(and other compact binary) formation rates. Since a comprehensive population synthesis survey of all

possible models is not computationally feasible, we describe an efficient approximate fitting technique

(used previously in O’Shaughnessy, Kalogera, & Belzcynski (2005); hereafter OKB) we developed to

accurately approximate the results of complete population synthesis calculations. Finally, in §8.4 we

select from our family of possible models those predictions which are consistent with the observational

constraints. We then employ that sample to generate refined predictions for the expected NS−NS,

NS−BH, and BH−BH merger rate through the emission of GWs.

We find that the observed NS−NS population can provide a tight constraint (albeit a complicated

91

one to interpret) on the many parameters entering into population synthesis models. With this work

serving as an outline of the general method, we propose to impose in the future several additional

constraints, including notably the lack of any observed NS−BH systems, the empirical supernova

rates as well as formation rates of binary pulsars with white dwarf companions.

8.2. Empirical Rate Constraints from the NS−NS Galactic Sample

Seven NS−NS binaries have been discovered so far in the Galactic disk. Recently KKL developed

a statistical method to calculate the probability distribution of rate estimates derived using the ob-

served sample and modeling of survey selection effects. Four of the known systems will have merged

within 10 Gyr (i.e., “merging” binaries: PSRs J0737−3039, B1913+16, B1534+12, and J1756−2251)

and three are wide with much longer merger times (PSRs J1811-1736, J1518+4904, and J1829+2456).

PSR J1756−2251 was discovered recently (Faulkner et al. 2005) and has not been included in the

current calculations. We do not expect, however, that this system will significantly change our expec-

tations of the merger rate. (see Kalogera et al. 2004 for details). This fourth system is sufficiently

similar to PSR B1913+16 and was discovered with pulsar acceleration searches, the selection effects of

which have already been accounted for (see Table 8.1 for the properties of the six systems used here).

In what follows we use observational constraints based on the rate probability distribution derived

by Kalogera et al. (2004) for merging binaries and the equivalent results for the three wide binaries

(presented for the first time in this work). In what follows we use the index k = 1 . . . 6 to refer to the

three merging (1,2,3) and the three wide (4,5,6) NS−NS binaries.

8.2.1. Merging NS−NS Binaries

The discovery of NS−NS binaries with radio pulsar searches and our understanding of the selection

effects involved allows us to estimate the total number of such systems in our Galaxy and their

formation rate. KKL developed a statistical analysis designed to account for the small number of

known systems and associated uncertainties. Specifically, they found that the posterior probability

distribution function Pk for NS−NS formation rates Rk for each sub-population k of pulsars similar

92

to the kth known binary pulsar is given by

Pk(R) = A2kRe−AkR. (8.1)

The parameter Ak depends on some of the properties of the pulsars in the observed NS−NS sample

[see KKL Eq. (17)]:

A = τlife/(fbNPSR) (8.2)

where f−1b is the fraction of all solid angle the pulsar beam subtends; τlife is the total binary pulsar

lifetime

τlife = τsd + τmrg (merging) , (8.3)

where τsd is the pulsar spindown age (Arzoumanian, Cordes, & Wasserman 1999) and τmrg is the

time remaining until the pulsar merges through the emission of GW (Peters & Mathews 1963; Peters

1964); and NPSR is the total estimated number of systems similar to each of the observed one (i.e.,

N−1PSR is effectively a volume-weighted probability that a pulsar with the same orbit and an optimally

oriented beam would be seen with a conventional survey; this factor incorporates all our knowledge

of pulsar survey selection effects as well as the pulsar space and luminosity distributions. Table 8.1

lists for each merging NS−NS binary several intrinsic parameters (i.e., the best known values for fb;

several lifetime-related parameters, such as τsd and τmgr) and two key quantities which depend on our

analysis of selection effects: NPSR and the deduced A, i.e., via Eq. (8.3). Results are shown for our

preferred model for binary pulsar space and luminosity distribution; see model 6 and details in KKL.

The total NS−NS posterior density of the combined rate represented by the observed samples can be

computed by a straightforward convolution,

P(Rtot) =

∫

dR1dR3dR2δ(Rtot −R1 −R2 −R3)

× P1(R1)P2(R2)P3(R3) (8.4)

described in detail in §5.2 of KKL and presented in detail for the three-binary case in Eq. (A8) of Kim

et al. (2004).

93

KKL also demonstrated that the resulting rate distributions depend only weakly on the spatial

distribution of NS−NS locations (see their Figure 7 and the end of their §6). Thus the NS−NS rate

distribution effectively depends on only one model assumption, the choice of the intrinsic radio pulsar

luminosity function – which, in the KKL approach is given by KKL Eq. (3), following Cordes &

Chernoff (1997),

φ(L)dL = (p− 1)(L/Lmin)−pdL/Lmin. (8.5)

Thus it is controlled by two parameters, the minimum allowed pulsar luminosity (Lmin) and the power

law p > 1 governing their relative luminosity probabilities.

KKL did not complete their calculation for a comprehensive posterior probability distribution for

the NS−NS rate estimates, however, because up-to-date empirical probability constraints for p and

Lmin are not available (cf., Kalogera et al. 2004). Instead, they presented results for a few selected

models, emphasizing one model (model 6) whose properties (Lmin = 0.3mJy kpc2 and p = 2) are close

to the median values they expect will be found when all present observations are taken into account.

For this particular model, the empirical parameters Ak which describe the posterior densities are given

in Table 8.1.

8.2.2. Wide NS−NS Binaries

The same general technique outlined above can be applied to the formation rate of wide NS−NS

binaries: the same form of distribution function P(R) (Eq. 8.1) applies and it depends on the same

parameter A (Eq. 8.2). The main change is the relevant lifetime. Since these binaries do not merge,

their detectable lifetime is now the sum of the time remaining before the pulsar spins down (τsd,

described earlier) and the length of time the pulsar will remain visible (the “death time” of a pulsar;

see Chen & Ruderman 1993). However, since τsd estimates are somewhat uncertain, we require that

they do not exceed the current age of the Galactic disk (10 Gyr). To summarize, then, the only change

94

Table 8.1. Observational properties of NS−NS binaries. From left to right, the columns indicate the pulsar name, spin period,spin-down rate, orbital period, companion mass (MNS is assumed to be 1.35M⊙ except PSR B1913+16 (1.44M⊙) and PSR

B1534+12 (1.33M⊙)), eccentricity, characteristic age, spin-down age, GW merger timescale, death-time, most probable value of thetotal number of pulsars in a model galaxy estimated for the reference model (model 6 in KKL), beaming correction factor,

parameter in rate equation used in Eq. (8.2), references: (1) Hulse & Talor (1975); (2) Wex, Kalogera, & Kramer (2000); (3)Wolszczan (1991); (4) Stairs et al. (2002) (5) Burgay et al. (2003) (6) Corongiu et al. (2004) (7) Nice, Sayer, & Taylor (1996) (8)

Hobbs et al. (2004) (9) Champion et al. (2004)

PSRs P as Pb

s P cb Md

c ee τ fc τ g

sd τhmrg τ i

d NjPSR fk

b Al Refsm

(ms) (10−18s s−1) (hr) (M⊙) (Gyr) (Gyr) (Gyr) (Gyr) (Myr)

(1) merging NS−NS

B1913+16 59.03 8.63 7.752 1.39 0.617 0.11 0.065 0.3 4.34 617 5.72 0.103 6,7B1534+12 37.90 2.43 10.098 1.35 0.274 0.25 0.19 2.7 9.55 443 6.45 1.014 8,9J0737−3039 22.70 1.74 2.454 1.25 0.088 0.16 0.10 0.085 13.5 1621 6.085 0.018 10

(2) wide NS−NS

J1811−1736 104.182 0.916 450.7 1.66 0.828 1.8 1.8 n/a 7.8 606 6 2.64 13J1518+4904 40.935 0.02 207.216 1.35 0.25 32.4 32.3 n/a 54.2 282 6 32.9 14,15J1829+2456 41.0098 ∼ 0.05 28.0 1.15 0.139 13.0 12.9 n/a 43.7 272 6 37.9 16

95

from the previoius approach is to replace the previous expression for the lifetime, Eq. (8.3), with

τlife = min (τsd, 10Gyr) + τd(wide) . (8.6)

Table 8.1 lists pulsar parameters and deduced quantities for the three wide NS−NS binaries used in

this study. Current pulsar observations do not provide us with any estimates of the beaming fractions

relevant to the pulsars in these wide systems. Guided by the beaming fraction distribution for merging

pulsars, we adopt a value of 6 for the beaming factor for all the wide NS−NS pulsars. For simplicity, we

present the results for only the preferred luminosity model (i.e., for the specific choice for p and Lmin

mentioned above). These distributions again follow Eq. (8.1), with parameters Ak given by Table 8.1,

where Ak is determined for each pulsar class k from physical parameters presented in the table. For

each class separately (merging and wide binaries) we use Eq. (A8) of Kim et al. (2004) to generate

a composite probability distributions for the formation rate of binaries in that class: Pm (merging)

and Pw (wide). Thus we arrive at the two estimates shown in Figure 8.1 for the empirical probability

distribution

p(logR) = P(R)R ln 10

for the formation rates Rm and Rw of these two classes of binary. From these distributions we derive a

95% credible region for each formation rate; for example, the upper and lower rate limits Rw,± satisfy

∫ Rw,−

0dRPw(R) =

∫ ∞

Rw,+

dRPw(R) = 0.025 . (8.7)

8.3. Estimates for Merger Rates

8.3.1. Population Synthesis Estimates

We estimate formation and merger rates for several classes of double compact objects using the

StarTrack code first developed by Belczynski, Kalogera, & Bulik (2002) and recently updated and

tested significantly as described in detail in Belczynski et al. 2005. In this code, seven parameters

96

-8 -7 -6 -5 -4 -3log10 HR yrL

0

0.5

1

1.5

2

-8 -7 -6 -5 -4 -3

0

0.5

1

1.5

2

Figure 8.1 Empirically-deduced probability distributions for merging (right) and wide (left) NS−NSbinaries; see §, 8.2. The solid vertical lines are at (i) log10 R = −4.5388, and −3.49477, the 95%credible region for the merging NS−NS merger rate; and at (ii) log10 R = −6.7992,−5.7313, in the95% credible region for the wide NS−NS formation rate.

strongly influence compact object merger rates: the supernova kick distribution (3 parameters), the

massive stellar wind strength (1), the common-envelope energy transfer efficiency (1), the fraction of

mass accreted by the accretor in phases of non-conservative mass transfer (1), and the binary mass

ratio distribution described by a negative power-law index (1). To allow for an extremely broad range

of possible models, we used the specific parameter ranges quoted in §, 9.2 of OKB.

We randomly choose model parameters in this space and evaluate their implications, by progres-

sively examining the evolution of binary after binary. We then extract from our simulations predictions

for several DCO formation rates (NS−NS, NS−BH, and BH−BH) by scaling up the ratio of DCO for-

mation events we obtain in each simulation (n) to the total number of binaries studied in the simulation

(N) by a factor proportional to the expected ratio between N and the number of stars formed in the

Milky Way. We set this scaling factor by assuming a constant star-formation rate of M ≈ 3.5M⊙yr−1,

as described in the Appendix of OKB.

Extracting predictions for the “visible” NS−NS formation rates: To compare the predictions of

population synthesis calculations against the empirical rate constraints derived for the pulsar samples,

we must determine the formation rates of NS−NS binaries that could be “visible” as pulsars. Since

we do not follow the detailed pulsar evolution with StarTrack (due to major uncertainties related to

pulsar magnetic field evolution), we choose a minimal criterion for identifying NS−NS binaries that

97

possibly contain a recycled pulsar: if the first NS in the binary has experienced any accretion episode

(through either Roche-lobe overflow and disk accretion or a common-envelope phase), then the binary

is identified as a potential binary recycled pulsar and is included in the calculation of the NS−NS

“visible” pulsar formation rate.

Practical complications in merger rate calculations: We would have preferred to proceed as in

OKB and perform, for each separate DCO type (e.g., BH−BH binaries), a sequence of Monte Carlo

computations tailored to determine this type’s merger rate to some fixed accuracy (say, 30%) as a

function of all population synthesis parameters. Instead, owing to computational limitations, we had

to extract multiple types of information from each population synthesis run; Appendix D describes in

greater detail the collection of population synthesis runs we performed and the manner in which these

runs were used to estimate various DCO formation rates.

8.3.2. Mapping Population Synthesis Rates versus Parameters

In order to constrain population synthesis parameters based on rate measurements, we must be able

to invert the relation between rate and model parameters to find all possible models consistent with

a given rate. In other words, we must fit the rates over all seven parameters.

OKB first demonstrated that, even using sparse data in a high-dimensional space of population

synthesis parameters, an effective fit could be found for formation rates of DCOs (see OKB Figure 2

and their §4). We constructed separate polynomial least-squares fits to each of the five rate functions

we need (i.e., for NS−BH, BH−BH, visible merging NS−NS, and visible wide NS−NS binaries we

performed a cubic least-squares fit; and for the overall NS−NS merger rate – including all merging

NS−NS binaries, whether we expect them to be electromagnetically visible or not – we used a quartic

least squares fit). Figure 8.2 demonstrates that the fit is good: the errors are on the limiting scale

we would expect, given the uncertainties in the input (i.e., the standard deviation of the logarithmic

rate errors, log10 Rfit/Rtrue are 0.086 (NS−NS), 0.22 (NS−BH), and 0.167 (BH−BH) are comparable

with the minimum possible uncertainty we would expect given a perfect fit, log10(1+ 1/√

10) ≈ 0.119;

see Appendix D for a detailed discussion of the minimal uncertainties expected for each rate). The

fits are sufficiently good that for our purposes we can replace population synthesis calculations with

98

-7 -6 -5 -4 -3log10 HRfit yrL

-7-6-5-4-3

log 10HR

yrL

-8 -7 -6 -5 -4

-8-7-6-5-4

-7 -6 -5 -4 -3log10 HRfit yrL

-7-6-5-4-3

log 10HR

yrL

-8 -7 -6 -5 -4

-8-7-6-5-4

Figure 8.2 log10 of the Galactic rate versus our fit to the rate, shown for NS−NS, NS−BH, andBH−BH sample points (all superimposed). The shaded region is offset by a factor 1 ± 1/

√10. This

region estimates the error expected due to random fluctuations in the number of binary merger eventsseen in a given sample. (See the appendix for a discussion of the number of sample points actuallypresent in various runs.)

evaluations of our fits. In particular and by way of example, in Figure 8.3 we generate a histogram for

various DCO formation rates predicted by population synthesis, using (i) the actual outputs deduced

from the population synthesis code, sampled at random Monte Carlo points (dashed line), and (ii) the

outputs obtained from fits to the dataset from (i), sampled at a much larger number of data points (to

insure smoothness; solid line). The two methods produce strikingly similar histograms, demonstrating

that the fit will be adequate for our purposes.

8.4. Constraints from NS−NS Observations

In §. 8.2 we constructed two straightforward empirical constraints (i.e., credible regions for the forma-

tion rate of “visible” merging and wide NS−NS binaries) we could place on the output of population

synthesis calculations. In this section, we apply these two constraints, individually and together, and

determine their effect on rate predictions of population synthesis calculations.

Bounding merging NS−NS rate: Figure 8.4 superimposes the observational bounds taken from the

observed merging NS−NS distribution (i.e., the 95% credible region; see Figure 8.1) on top of the

distribution of “visible” NS−NS merger rates obtained from unconstrained population synthesis. We

limit attention only to those population synthesis models consistent with our constraint; specifically,

99

-8 -7 -6 -5 -4 -3 -2log10 HR yrL

0.2

0.4

0.6

0.8

-8 -7 -6 -5 -4 -3 -2

0

0.2

0.4

0.6

0.8

Figure 8.3 The a priori probability distribution for the NS−NS (right), NS−BH (center), and BH−BH(left) merger rates, versus the log10 of the rate. These distributions were generated from the populationsynthesis code (dashed line) and fits (solid lines) assuming all parameters in the population synthesiscode were chosen at random in the allowed region.

we randomly choose population synthesis models, evaluate the “visible” merging NS−NS rate using

our fit, and retain the model only if the rate lies within these two bounds. By applying the NS−NS

constraints, we reject 72% of models that initially considered plausible. For each of the small residual

of consistent models, we can evaluate the NS−NS, NS−BH, and BH−BH merger rates (again using

our fits). We find that the merger rates increase slightly on average: the mean merger rate increases

by a factor ×2.2 for NS−NS, ×2 for NS−BH, and ×1.2 for BH−BH.

Bounding wide NS−NS rate: Figure 8.4 also shows the 95% credible region for the wide visible

NS−NS formation rate (Figure 8.1) on top of our a priori population synthesis distribution for the

wide visible NS−NS formation rate. We find that with this constraint we have to exclude 80% of a

priori plausible models. Using only models which satisfy this second constraint, we find DCO merger

rates have dropped relative to our a priori predictions: the average NS−NS, NS−BH, and BH−BH

merger rates are reduced by a factor 0.22, 0.26, and 0.65, respectively.

Both constraints simultaneously : Very few population synthesis models (less than 2%) satisfy both

constraints simultaneously. Since the set of consistent models is much smaller than initially permitted

a priori, we have less uncertainty in our predictions: the standard deviation in the log of the merger

rates has changed from our initial a priori uncertainty of 0.55, 0.63, and 0.60 (i.e., plus or minus a

factor of 3.5, 4.3, and 4) for NS−NS, NS−BH, and BH−BH mergers, respectively, to 0.44, 0.48, and

100

Figure 8.4 The a priori probability distributions for the visible merging (top) and visible wide (bottom)NS−NS formation rates produced from population synthesis. The solid curves denotes the resultdeduced from artificial data generated from a multidimensional fit to the visible wide and mergingNS−NS rate data. The vertical lines are the respective 95% CI bounds presented in Figure 8.1.

0.61 (i.e., plus or minus a factor of 2.8, 3, and 4). Further, since the wide constraint proves slightly

more restrictive, the mean merger rates have dropped slightly from our prior expectations: the average

predicted merger rates are 6.7/Myr [NS−NS] (down by a factor 0.25), 1.4/Myr [NS−BH] (down by a

factor 0.24), and 1.1/Myr [BH−BH] (down by a factor 0.43).

8.4.1. Advanced LIGO Detection Rates

While we have presented the number of mergers occurring per Milky Way equivalent galaxy, advanced

LIGO’s inspiral detection range depends on the masses of the component objects. Specifically, one

4 km advanced LIGO detector (see Harry 2005) is expected to detect a binary with chirp mass Mc =

(m1m2)3/5/(m1 +m2)

1/5 (at signal-to-noise ratio 8) out to a distance

d = 191Mpc (Mc/1.2M⊙)5/6 . (8.8)

101

0.2

0.4

0.6

0.8

0

0.2

0.4

0.6

0.8

-7 -6 -5 -4 -3

log10

(R yr)

0.2

0.4

0.6

0.8

0

0.2

0.4

0.6

0.8

0.2

0.4

0.6

0.8

0

0.2

0.4

0.6

0.8

Figure 8.5 The a priori probability distribution for the NS−NS (bottom), NS−BH (center), andBH−BH (top) merger rates per Milky Way equivalent galaxy. As in Figure 8.3, the dashed curves showthe results obtained from our population synthesis calculations (i.e., our raw code results, smoothed);the thick solid curves show the results after we impose both our observational constraints (i.e., consis-tency with the observed number of visible wide and visible merging NS−NS binaries).

Thus, if the real merger rate and chirp mass distribution for type α are Rα and pα(Mc), respectively,

then LIGO will on average detect α merger events at a rate

Rα,LIGO = 0.042 Rα

⟨

(Mc/M⊙)15/6⟩

α(8.9)

where⟨

(Mc)15/6

⟩

α=∫

dMcpα(Mc)M15/6c , and where for simplicity we assume a uniform distribution

of Milky Way equivalent galaxies with density 0.01/(Mpc)3 (see Nutzman et al. (2004) for a discussion

of short-scale corrections to this distribution in the case of short-range interferometers, like initial

LIGO).

Since Figure 8.5 was produced using merger rate fits, we do not have the chirp mass information

needed to translate that figure into a corresponding distribution for the LIGO detection rate. In what

102

-1 0 1 2 3 4log10 HR yrL

0.20.40.60.8

1-2 -1 0 1 2 3

00.20.40.60.8

Figure 8.6 Probability distributions for LIGO’s detection rates for merging NS−NS (dotted line),NS−BH (dashed line), and BH−BH (solid line) binaries, assuming all binaries are produced in thefield. This plot was obtained directly from Figure 8.5 using Eq. (8.9).

follows we adopt mean chirp masses as derived from all our models in the archives. We find:

⟨

(Mc)15/6

⟩

NS−NS= 2M

15/6⊙ (8.10)

⟨

(Mc)15/6

⟩

NS−BH= 5.8M

15/6⊙ (8.11)

⟨

(Mc)15/6

⟩

BH−BH= 111M

15/6⊙ . (8.12)

These are to be compared to 224M15/6⊙ for two 10 M⊙ BH, to 15.5M

15/6⊙ for a 10 M⊙ BH and a 1.4 M⊙

NS, and to 1.2M⊙ for two 1.4 M⊙ NS. Figure 8.6 presents our preliminary estimates for the advanced

LIGO detection rate distribution. Note this figure provides the same information as Figure 8.5, except

that the merger rates for each species have been rescaled according to Eq. (8.9).

8.5. Summary and Conclusions

In the context of matching theory with observations of the binary NS population, we have described

how to constrain the predictions for DCO merger rates from population synthesis codes such as

StarTrack by using two specific observational constraints. We find that to be consistent with the

rate statistics of the observed NS−NS population (in the 95% credible region), we must exclude at

least 98% of all the models we think a priori likely. We do not focus on explicitly describing the seven-

dimensional region of StarTrackmodel parameters consistent with our constraints, both because (i) we

103

lack a compact way to describe a seven-dimensional region, and (ii) our region has meaning specifically

for the StarTrack code. Other codes have different parameterizations of the same physical phenom-

ena, leading to potentially quite different representations of the same constraint region. However, in

Appendix F we give some information about the mean constraints on population synthesis parameters

but also the strong variance around these mean values. As described in §8.4 and particularly via

Figure 8.5, to extract a physically meaningful statement about the effect of imposing constraints (as

opposed to a describing information about parameters of one particular code), we have described how

these constraints have improved our understanding of three DCO merger rates [NS−NS, NS−BH, and

BH−BH]. We find that, using these two initial constraints, (i) the most probable merger rates (i.e., at

peak probability density) are systematically lower than we would expect a priori, at least by a factor

2; and (ii) we reduce the uncertainty in the NS−NS and NS−BH merger rates by moderate factors

(i.e., the standard deviation of logR drops by 0.16 and 0.03, respectively).

This work only outlines the beginning of a large program we have undertaken to better constrain

our understanding of the evolution of single and binary stars and the associated predictions for GW

sources. We intend to add a few additional empirical constraints of DCOs (e.g., NS−WD binaries) and

the lack of observations of certain binary compact objects (notably, NS−BH binaries). Apart from

rate constraints, the observed properties of DCOs (mass ratios, orbital separations and eccentricities)

could also be used as constraints. Further constraints (such as observations of pulsar kicks, which

constrain the supernova kick distribution) can be added by other means, as prior distributions on the

space of model parameters. We fully expect to have much stronger constraints on our understanding

of population synthesis in the near future.

Stronger constraints, however, will require a considerably more systematic approach than the

straightforward presentation we have used here. As the expected uncertainties decrease, greater care

must be taken to include every uncertainty, no matter how minor, many of which for clarity we have

neglected. For example, in future calculations we expect to include uncertainties in Lmin and p, our

fit and rate estimates, and even the star formation rate we use to convert simulation results into rate

estimates. Additionally, we will self-consistently choose the constraint confidence intervals in order to

construct a meaningful posterior confidence interval on each merger rate.

104

Note on PSR−BH rates: Recently, Pfahl, Podsiadlowski, & Rappaport (2005) have estimated that

millisecond PSR−BH systems should be exceedingly rare: they find an upper bound of 10−7 yr−1 on

the formation rate of binary systems containing a BH and a recycled pulsar. In our own computations,

we have never seen such a system form, even though we have seen ≈ 6× 104 merging NS−NS binaries

form. This result suggests the branching ratio for PSR−BH:NS−NS formation is ≪ 10−4. If we

use a conservative value for the merging NS−NS formation rate, 10−4/yr/galaxy, then we expect the

formation rate of PSR−BH binaries to be significantly less than 10−8/yr/galaxy, entirely consistent

with their constraint.

105

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Wolszczan, A. 1991, Nature, 350, 688

113

A. Observed Properties of Pulsar Binaries.

PSRs Ps Ps Pb MPSR (Mc) e τc τsd τmrg τd Refs

(ms) (s s−1) (hr) (M⊙) (Gyr) (Gyr) (Gyr) (Gyr)

NS−WD

J0751+1807 3.48 7.79×10−21 6.31 2.1 (0.191) ∼ 10−7 7.08 6.90 4.40 > τkH 1,2

J1757−5322 8.87 2.78×10−20 10.88 1.35 (0.67) ∼ 10−6 5.06 4.93 7.98 > τkH 3,4

J1141−6545 393.90 4.29×10−15 4.74 1.30 (0.986) 0.172 0.0015 n/a 0.58 0.104 5,6

B2303+46 1066.37 5.69×10−16 295.20 1.34 (1.3) 0.658 0.03 n/a > τH 0.14 7,8

NS−NS

(1) merging NS−NS

B1913+16 59.03 8.63×10−18 7.75 1.4408 (1.3873) 0.617 0.11 0.065 0.30 4.34 9,10,11

B1534+12 37.90 2.42×10−18 10.09 1.3332 (1.3452) 0.274 0.25 0.19 2.74 9.55 12,13

J0737-3039 22.70 1.74×10−18 2.45 1.337 (1.25) 0.088 0.21 0.15 0.085 > τH 14,15

J1756-2251 28.46 1.02×10−18 7.67 1.40 (1.18) 0.181 0.44 0.38 1.65 > τH 16

J2127+11C 30.53 4.99×10−18 8.05 1.35 (1.36) 0.681 0.097 0.046 0.22 7.67 17(2) wide NS−NS

J1518+4904 40.94 <4 ×10−20 207.22 1.56 (1.05) 0.249 <16.2 <16.1 > τH > τH 18

J1811-1736 104.18 9.16×10−19 451.20 1.62 (>0.94) 0.828 1.80 1.75 > τH 7.84 19,20

J1829+2456 41.00 ∼5×10−20 28.22 1.14 (1.36) 0.139 ∼12.99 ∼12.90 > τH > τH 21

unclear

J1906+0746 144.07 2.03×10−14 3.98 1.71 (>0.9) 0.085 1.1×10−4 n/a 0.31 0.082 22

Table A.1 Summary of observed properties of pulsar binaries. Values shown in this table are adopted from themost recent literatures available. PSR J2127+11C, a merging NS−NS in M15, is also included for comparison tothe disk population. From left to right, the columns indicate spin period, spin-down rate, orbital period, estimatedmass of a companion, eccentricity, characteristic age of a pulsar (τc), spin-down age of a pulsar (τsd; we calculatedτsd only for recycled pulsars), merging time of a binary system due to the emission of GWs (τmrg) where τH is aHubble time, death time of a pulsar (τd), most probable value of the total number of pulsars in a model galaxyestimated for the reference model (model 6 in KKL), references: (1) Lundgren, Zepka, & Cordes (1995); (2) Niceet al. (2005); (3) Edwards, & Bailes (2001); (4) Jacoby et al. (2006); (5) Kaspi et al. (2000); (6) Bailes et al.(2003); (7) Stokes, Taylor, & Dewey (1985); (8) van Kerkwijk, & Kulkarni (1999); (9) Hulse & Talor (1975); (10);Taylor & Weisberg (1989); (11) Weisberg & Taylor (2003); (12) Wolszczan (1991); (13) Stairs et al. (2002); (14)Burgay et al. (2003); (15) Lyne et al. (2004); (16) Faulkner et al. (2005); (17) Prince et al. (1991); (18) Nice,Sayer, & Taylor (1996); (19) Lyne et al. (2000); (20) Corongiu et al. (2004); (21); Champion et al. (2004); (22)Lorimer et al. (2006)

114

B. Note on Bayes’ Theorem and Confidence Intervals

In this work, we adopt a Bayesian approach that is best suited when handling statistical inferences

from small data samples. Traditional frequentist tests address the question of how probable an event

of interest is, in other words, how frequently a specific event occurs, as one repeats the observation.

On the other hand, Bayesian inference provides a tool for the assessment of how good a suggested

hypothesis is, given previously known information or assumptions about the data (D). For a given

prior (or intrinsic) distribution of an unknown quantity (or parameter) r, we can calculate a posterior

distribution P(r|D) by weighting the prior distribution with the likelihood of the data, L(D). This is

Bayes’ theorem written as follows

P (r|D) ≡ P (r) × L(D)

P (D), (B1)

where P(D) is the prior distribution of the data, which acts as a normalization constant. One can

calculate a probability by integrating the posterior distribution over the range of parameters. In the

context of Bayesian inference, the range of parameters corresponding to a given probability is called a

credible region. Often, the credible region is confused with a confidence interval, which is based on the

context of the frequentist method, and therefore has a rather different meaning. A confidence interval

indicates that one would observe a value within this interval at a given freqeuncy (or probability). For

instance, if a confidence interval for the length of a desk is given at 95% probability, one should expect

that 95 out of 100 measurements will have values within the given confidence interval. A credible

region, however, represents a range of parameters within which the true value is included at the given

probability.

This work attempts to answer the question,“what is the most likely value of the Galactic merger

rates of certain types of pulsar binaries?”, and hence, the merger rate R is our parameter. In this

thesis, we describe how to calculate the PDF of R by means of the Bayesian inference, and assign

credible regions at 68% or 95% probability. As an example, in Ch. 3, the Galactic NS−NS merger

rates corresponds to a 95% credible region range in between 17 − 292 Myr−1 based on a reference

pulsar population model (Table 3.3). This statement should be read as such: “at 95% probability, the

115

true value of R is included in the range between 17 − 292 Myr−1”.

116

C. Combined P(R) for Three Binary Systems

In Ch. 2, we derived expressions for P (Rtot) for one and two merging binaries. Here, we extend

this PDF to the case of three systems. Following Eq. 2.18, we define a coefficient for each observed

NS−WD:

A ≡(ατlifefb

)

1141, B ≡

(ατlifefb

)

0751, and C ≡

(ατlifefb

)

1757, (C1)

where A < B < C. Recall that α is the slope of the function Nobs= αNtot and is determined for each

pulsar population model for each NS−WD system. By definition, the total Galactic merger rate is the

sum of all three observed systems:

R+ ≡ R1 + R2 + R3 . (C2)

Redefining R+ ≡ Ra + Rb, where Ra ≡ R1 + R2 and Rb ≡ R3, we transform Ra and Rb to new

variables R+ and R− ≡ Ra −Rb

P (R+,R−) = P (Ra,Rb)

∣

∣

∣

∣

∣

dRadR+

dRbdR−

dRadR−

dRbdR+

∣

∣

∣

∣

∣

=1

2P (Ra,Rb) . (C3)

Since both R+ and R− are positive, −R− ≤ R+ ≤ +R−.

The PDF of the total rate R+ is obtained after integrating P (R+,R−) over R−:

P (R+) =

∫

R−

P (R+,R−) dR− =1

2

∫

R−

P (Ra,Rb) dR− , (C4)

where

P (Ra,Rb) = P (Ra)P (Rb) . (C5)

Here, P (Ra = R1 + R2) is given by eq. (5.10) and one can rewrite the formula with appropriate

coefficients defined earlier:

P (Ra) =( AB

B −A

)2[

Ra

(

e−ARa + e−BRa

)

−( 2

B −A

)(

e−ARa − e−BRa

)]

. (C6)

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The individual PDF P (Rb) ≡ P (R3) (eq. (5.8)) is also rewritten as follows:

Pi(Rb) = C2 Rb e−CRb . (C7)

Replacing Ra = (1/2)(R+ + R−) and Rb = (1/2)(R+ −R−) in eq. (C4), the normalized P (R+) can

be obtained by integration. After some algebra, one find:

P (R+) =A2B2C2

(B −A)3(C −A)3(C −B)3(C8)

[

(C −B)3e−AR+

[

−2(−2A+B + C) + R+[−A(B + C) + (A2 +BC)]

]

+ (C −A)3e−BR+

[

2(A− 2B + C) + R+[B(C +A) − (B2 + CA)]

]

+ (B −A)3e−CR+

[

−2(A+B − 2C) + R+[−C(A+B) + (C2 +AB)]

]

]

.

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D. Calculating DCO Event Rates with Population Synthesis

Ch. 8 relies upon formation rates extracted from a large sequence of archived population synthesis

calculations. This appendix describes how these archives were generated and used to produce formation

rates. It also explains the expected uncertainties in each merger rate estimate.

D.1. How Rates were Estimated in Ch. 8

Principles of archive generation: For a given combination of population synthesis parameters, we

generate a large collection N of binaries. To add a binary to the archive, we first generate progenitor

binary parameters (m1,m2, a, e) [i.e., the two progenitor masses (m1,2) and the initial semimajor axis

(a) and eccentricity (e)] which are (i) drawn from the distribution functions presented in OKB and

which (ii) satisfy any conditions we impose to reject binaries irrelevant to the study at hand – e.g.,

m1,2 > 4, or more elaborate conditions described in OKB. (The latter conditions offer a significant

speed improvement, but rely upon the experience gained in previous runs to insure that the conditions

imposed do not reject physically relevant systems.) Given satisfactory initial conditions, binaries are

assigned a randomly-chosen formation time, then evolved from whenever they form until the present

day. Binaries are successively added to the archive until some termination condition is reached –

typically, that the number n of a given class of binaries, such as merging NS−NS binaries, has crossed

a threshold (e.g., n = 10).

Archive classes: Our population synthesis runs are summarized in Table 8.2. Each run of the

population synthesis code falls into a certain class, depending on what choices were made for (i) the

target systems on which the termination threshold was set (i.e., stop when we get n merging BH−BH

binaries; see column 2 of Table 8.2), (ii) the specific threshold n chosen (i.e., which insures that the

formation rate of the target system type is determined to an accuracy roughly ∼ 1/√n; see column 3

of Table 8.2), and (iii) the combination of conditions applied to filter progenitor binary systems (see

the last column of Table 8.2). In Table 8.2, the filters B, and S correspond to using the partitions

presented in listed in Table 2 of OKB for BH−BH binaries (B) and NS−NS binaries (S); the ‘W’ filter

uses only the first NS−NS partition listed in Table 2 of OKB, which filters out WD progenitors. Note

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the first column of Table 8.2 merely provides a label for the archive class.

Applying archives: From the ratio of the number of binaries of a given type seen to the number

of binaries in a run, modulo a normalization factor presented in OKB, we can calculate the formation

rates for any binary type of interest. However, to avoid extreme biases associated with a poor choice

of filter or stopping condition, we use only certain archives to estimate merger rates, as described in

Table 8.3. [In this table, all rates are total merger rates, with the exception of the last two rows, which

correspond to the visible merging (v) and visible wide (vw) NS−NS binaries.]

D.2. Understanding Errors in Rate Estimates

Tables 8.2 and 8.3 provide the information needed to understand errors in our formation rate estimates.

Example: The BH−BH formation rate estimate is produced from a single archive (b). Archive b is

a collection of runs which stop when 10 merging BH−BH binaries are found; while the filters in this

archive can prevent the formation of binaries involving NS, they do not significantly limit BH−BH

binary formation. Therefore, archive b is ideally suited to estimate the BH−BH merger rate to an

accuracy of order 1/√

10 ≈ 30%.

Example: The NS−NS formation rate estimate is produced from an amalgam of archives (a′, a′′,

b′, and c). None of these archives applies filters which prevent NS−NS formation, though one (a′)

applies filters which prevent the formation of nearly anything else. However, these archives do involve

different termination criteria: the first two terminate when a large number of NS−NS binaries have

Table 8.2. Classes of runs

Type Target Number of runs n Filters

a NS−NS 488 10 (none)a’ NS−NS 137 100 Sa” NS−NS 408 300 Wb BH−BH 306 10 Bb’ BH−BH 285 10 Wc NS−BH 357 10 W

120

formed, whereas the last two terminate when only 10 BH−BH or NS−BH archives have formed. If

only the first two were used, we could guarantee the NS−NS merger rate to be known to within 10%

accuracy. However, to augment our statistics, we additionally included binaries from b′ and c; while

these archives should usually have many NS−NS binaries (cf. Figure 8.3), we cannot guarantee any

minimum number a priori. To simplify error estimates, we selected only those elements of b′ and c

with more than 30 merging NS−NS binaries. Thus, we expect the NS−NS rate to be known to within

20% accuracy.

Example: The estimate for the visible wide NS−NS formation rate is the least accurate and most

challenging calculation performed in this work. Since visible wide NS−NS systems were rare and since

we did not (unlike the BH−BH case) have a simulation dedicated to discovering them, we had to

scavenge through all the archives which could have produced them (i.e., not b) in sufficient numbers

(i.e., not a) to permit a moderately accurate rate estimate. In practice, we selected those runs which

formed more than 8 wide visible NS−NS binaries, which should in principle give us an accuracy of

order (few)×35%. [In practice, we found a roughly 75% accuracy, comparable to the 65% accuracy of

our next-most-accurate estimate (the NS−BH rate).]

Table 8.3. Classes used for specific rates

Type a a’ a” b b’ c

BH−BH xNS−BH x x xNS−NS x x x xNS−NS(v) x x x x xNS−NS(vw) x x x x

121

E. Sample Fits to Merger Rates in Ch. 8

Ch. 8 and in OKB rely upon fits to 7-dimensional functions obtained from the StarTrack popula-

tion synthesis code. In this section, we provide an example of an explicit formula for a quadratic-order

polynomial fit to the BH−BH merger rate. We express this fit in terms of the following dimension-

less parameters xk ∈ [0, 1]: x1 = r/3 where r ∈ [0, 3] is a negative power-law index describing the

mass ratio distribution assumed for binaries; x2 = w characterizes the strength of stellar winds; the

kick velocity distribution consists of two Maxwellian distributions with 1-D dispersions of σ1 and

σ2, which are varied within [0,200 km s−1] and (200, 1000 km s−1], respectively, using two parameters:

x3 = σ1/(200 km/s) and x4 = −1/4 + σ2/800 km/s); a third parameter x5 = s is used as the relative

weight between low and high kick magnitudes; x6 = αλ is the effective common-envelope efficiency;

and x7 = fa is the fraction of mass accreted by the accretor in phases of non-conservative mass transfer.

[These parameters are discussed more thoroughly in OKB and in the original StarTrack paper (Bel-

czynski, Kalogera, & Bulik 2002).]. In terms of these parameters, we find that the following quadratic

fit to the BH−BH merger rate:

log10 [RBH−BHyr] = −5.84517 + 1.30448x1 − 0.406066x12

−0.310686x2 − 0.407175x1 x2 + 0.0142072x22

+0.717803x3 − 0.367487x1 x3 − 0.48743x2 x3

+0.29931x32 − 0.770174x4 + 0.242792x1 x4

−0.0811259x2 x4 − 0.0954582x3 x4 + 0.113668x42

+0.460929x5 − 0.114934x1 x5 + 0.490873x2 x5

−0.691954x3 x5 + 0.588787x4 x5 − 0.810968x52

−3.27367x6 − 0.214978x1 x6 + 0.674502x2 x6

−0.779896x3 x6 + 0.0891919x4 x6 + 1.37719x5 x6

+2.30296x62 + 1.68227x7 − 0.289592x1 x7

−0.19047x2 x7 − 1.18196x3 x7 − 0.0281177x4 x7

+0.517042x5 x7 + 0.67596x6 x7 − 0.454039x72 (E1)

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Relation of this fit to those used in Ch. 8: The fit presented above is substantially less accurate

than those actually used in Ch. 8 or in OKB: it is accurate only to within a factor 1.8±1 (i.e., when

we evaluate this fit at all of our trial points, we find that the standard deviation between our results

and the fit to be⟨

(logRBH−BH,fit − logRBH−BH)2⟩1/2

= 0.26). The fits actually used in Ch. 8 are

typically cubic (120 parameters) and quartic (330 parameters) order. Given the large number of these

parameters we chose to just provide the quadratic-order fit as an example above.

The rate functions are demonstrably not separable: we cannot fit the rate functions well with a

function of form X1(x1)X2(x2) . . . X7(x7). Even this toy fit contains strong off-diagonal terms.

123

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

P

X X

Figure 8.7 Cumulative probability distributions Pk(X) defined so Pk(X) is the fraction of all modelsconsistent with the two constraints imposed in the text (i.e., the formation rates of wide and mergingNS−NS binaries correspond adequately to observations) and that have xk < X. The left panel showsthe distributions for the 3 kick-related parameters x3, x4, x5; in this panel, the bottom (dashed) curvedenotes P3, the middle (solid) curve denotes P4, and the top dotted curve denotes P5. The right panelshows the distributions for P1 (solid top line), P2 (dashed), P6 (dotted), and P7 (solid, bottom curve).

F. Characterizing the Consistent Region of Population Synthesis Models

Using our monte-carlo method to select models compatible with our constraints, we have found a

relatively small seven-dimensional volume consistent with observations that corresponds to about 2%

of all the runs we performed. Unfortunately – with some exceptions – this volumetric constraint does

not translate to easily-understood and strong constraints on the individual parameters xk (using the

notation of Appendix E). On the one hand, because the dimension is high, weak constraints on each

parameter can correspond to very strong volumetric constraints. On the other hand, as demonstrated

in Figure 8.7, because the consistent region is extended through our high-dimensional model space in

a inhomogeneous anisotropic fashion, wide ranges of values of each parameter are still allowed, even

after applying the constraints.

To provide the reader with a global view of the parameter values associated with the models that

turn out to be consistent with our constraints, in Figure 8.7 we show that the cumulative distributions

of the consistent model parameter values for each of the seven parameters.

It is evident that the full ranges of values ([0,1]) are covered by the model parameters xk for

the set of models consistent with the constraints. However, certain qualitative conclusions can be

124

drawn: about 80% of the consistent models have kick relative weights (parameter x5) below 0.3; about

50% of the consistent models have mass-ratio power-law indices smaller (in absolute value) than 0.6

(x1 < 0.2; fractions of mass lost from the binary during non-conservative mass transfer phases in the

range 20%-60% are not favored.

Given the above, any simple attempt to describe the consistent region will necessarily be a crude

approximation. Nonetheless, for completeness we attempt to characterize the extended consistent

region through its mean values. The mean model consistent with our constraints is given by:

x = (x1, x2, . . . , x7) = (0.33, 0.46, 0.61, 0.53, 0.18, 0.43, 0.57) (F1)

These mean values correspond to a model with: fairly flat mass ratio distribution (power-law of

≃ −1); moderate stellar winds (strengths reduced by factors of ≃ 2); moderate kicks drawn from

Mawellians with σ1 ≃ 120 km s−1, σ2 ≃ 625 km s−1, and with relative weights of ≃ 20% (favoring σ2

over σ1); moderate values for an effective common-envelope efficiency (αλ ≃ 0.4 including the central

concentration parameter λ); and moderately non-conservative mass transfer phases (≃ 40% of the

mass is lost from the binary). It is very important though to keep in mind the broad ranges of these

parameters shown in Figure 8.7.