Galactic Merger Rates of Pulsar Binaries

Transcription

1 NORTHWESTERN UNIVERSITY Galactic Merger Rates of Pulsar Binaries A DISSERTATION SUBMITTED TO THE GRADUATE SCHOOL IN PARTIAL FULFILLMENT OF THE REQUIREMENTS for the degree of DOCTOR OF PHILOSOPHY Field of Physics and Astronomy By Chunglee Kim EVANSTON, ILLINOIS June 2006

2 2 c Copyright by Chunglee Kim 2006 All Rights Reserved

3 3 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 J , I obtain the most likely values of Galactic NS NS merger rate range between 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 yr 1 and 190 yr 1, respectively. In the case of merging NS WD binaries, the most likely rates are found in the range 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

4 4 to merging binaries, I estimate formation rates of eccentric NS WD binaries, which range 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.

5 5 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. André de Gouvêa and Dr. Fred Radio for their interests and their challenging questions.

6 6 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 Gürkan, 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 Grandclént, 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!

7 7 Table of Contents Chapter 1 Introduction Pulsar Binaries as Gravitational-Wave Sources Chapter 2 The Probability Distribution of Binary Pulsar Merger Rates. I. Double Neutron Star Systems in the Galactic Field Introduction Basic Analysis Method Models for the Galactic Pulsar Population Pulsar Survey Selection Effects Survey Parameters Doppler Smearing Statistical Analysis The Rate Probability Distribution for Each Observed NS NS Binary The Total Galactic Merger Rate The Detection Rate for LIGO Results Discussion Chapter 3 Revised Merger Rates for NS NS Systems: Implications for PSR J Method for Rate Calculation Galactic NS NS Merger Rate NS NS Inspiral Event Rates and Conclusions Chapter 4 A Few Remarks on the NS NS Merger Rate Estimates

8 8 4.1 The Galactic NS NS merger rate Predictions for Future Discoveries Global P(R) and Supernova Constraints on NS NS Merger Rates Rate Constraints from Type Ib/c Supernovae and Binary Evolution Models Comments on PSR J Chapter 5 The Probability Distribution Of Binary Pulsar Merger Rates. II. Neutron Star-White Dwarf Binaries Merging NS WD Binaries Lifetime of a NS WD binary Probability Density Function of the Galactic NS WD Merger Rate Estimates Results Gravitational Wave Background due to NS WD Binaries Discussion Chapter 6 Neutron Star White Dwarf Binaries in Eccentric Orbits Introduction Formation of Eccentric NS WD Binaries Empirical NS WD Formation Rate Estimates Comparison with Rates from Binary Evolution Chapter 7 Upper Limits on NS BH Binaries Chapter 8 Constraining Population Synthesis Models via the Binary Neutron Star Population Introduction Empirical Rate Constraints from the NS NS Galactic Sample Merging NS NS Binaries Wide NS NS Binaries Estimates for Merger Rates Population Synthesis Estimates Mapping Population Synthesis Rates versus Parameters Constraints from NS NS Observations

9 Advanced LIGO Detection Rates Summary and Conclusions A Observed Properties of Pulsar Binaries B Note on Bayes Theorem and Confidence Intervals C Combined P(R) for Three Binary Systems D Calculating DCO Event Rates with Population Synthesis D.1 How Rates were Estimated in Ch D.2 Understanding Errors in Rate Estimates E Sample Fits to Merger Rates in Ch F Characterizing the Consistent Region of Population Synthesis Models

10 10 List of Tables 2.1 Simulated pulsar surveys Model parameters and estimates for R tot and R det in various Bayesian credible regions for different pulsar population models Model parameters and estimates for R tot and R det in various Bayesian credible regions for different pulsar population models Estimates for Galactic merger rates and predicted LIGO detection rates in a 95% credible region based on different population models Observational properties of merging NS WD binaries. From left to right, the columns indicate the pulsar name, spin period P s, spin-down rate P, orbital period P b, most probable mass of the WD companion m wd, orbital eccentricity e, characteristic age τ c, spin-down age τ sd, time to reach the death line τ d, 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) Estimates for the Galactic merger rate (R tot ) 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 R tot at 68% and 95% credible regions Estimates for the Galactic merger rate (R tot ) 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 R tot at 68% and 95% credible regions Observational properties of eccentric NS WD binaries. The columns indicate the pulsar name, spin period P, spin-down rate P, orbital period P b, the estimated mass of the WD companion m c, orbital eccentricity e, characteristic age τ c, time to reach the death line τ d The estimated Glactic formation rate and the most likely value of N tot of eccentric NS WD binaries for models with different pulsar luminosity functions. Model number is same with Table 2.2. We show the most likely value of R b in 68% and 95% credible regions

11 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 (M NS is assumed to be 1.35M except PSR B (1.44M ) and PSR B (1.33M )), eccentricity, characteristic age, spin-down age, GW merger timescale, death-time, most probable value of the total 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) Classes of runs Classes used for specific rates

12 12 List of Figures 2.1 Average signal-to-noise degradation factor in pulsar search code versus survey integration time for PSR B and PSR B The Poisson-distribution fits of P(N obs ) for three values of the total number N tot of PSR B like pulsars in the Galaxy (results shown for model 1). Points and error bars represent the counts of model samples in our calculation. Dotted lines represent the theoretical Poisson distribution The linear correlation between λ <N obs > and N tot is shown for model 1. Solid and dashed lines are best-fit lines for PSR B like and PSR B like populations, respectively. Points and error bars represent the best-fit values of λ for different values of N tot 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(R tot ) and the long and short dashed lines represent P(R) for PSR B like and PSR B like populations, respectively. We also indicate the credible regions for P(R tot ) by dotted lines Left panel: The correlation between R peak and the cut-off luminosity L min with different power indices p of the luminosity distribution function. Right panel: The correlation between R peak and the power index of the luminosity distribution function p The correlation between R peak and the radial scale length R 0. R peak is not sensitive to R 0 in the range between 4 8 kpc Probability density function that represents our expectation that the actual NS NS binary 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 are calculated assuming our reference model parameters (see text). The solid line shows the total probability density along with those obtained for each of the three binary systems (dashed lines). Inset: Total probability density, and corresponding 68%, 95%, and 99% credible regions, shown in a linear scale Probability density function of the predicted number of observed NS NS binary systems N obs for the PMPS, for our reference model (model 6 in KKL). The mean value is estimated to be N obs

13 The global P g (R) on a linear scale (lower panel) and the assumed intrinsic distributions for L min and p (upper panels). Dotted lines represent the lower (SN L ) and upper (SN U ) bounds on the observed SN Ib/c rate scaled by 1/10 and 1/100 (see text). The empirical SN Ib/c rates range over Myr 1, where the average is at 1100 Myr 1 (Cappellaro, Evans, & Turatto 1999), beyond the range shown here 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(R tot ). Other curves are P(R) for PSRs J (dot-dash), J (short-dash), and J (long-and-short dash)-like populations, respectively. Dotted lines correspond to 68%, 95%, and 99% credible regions for P(R tot ) The effective GW amplitude h rms 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) The probability density function of Galactic formation rate estimates for eccentric NS WD binaries (solid line) for our reference model. Dashed and dot-dashed lines represent the individual probability density functions of the formation rates for subpopulation of binaries similar to either PSR B or J No corrections for pulsar beaming have been applied Comparison between the empirical and theoretical rate estimates. Error bars with filled triangles 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) Empirically-deduced probability distributions for merging (right) and wide (left) NS NS binaries; see, 8.2. The solid vertical lines are at (i) log 10 R = , and , the 95% credible region for the merging NS NS merger rate; and at (ii) log 10 R = , , in the 95% credible region for the wide NS NS formation rate log 10 of the Galactic rate versus our fit to the rate, shown for NS NS, NS BH, and BH 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 events seen in a given sample. (See the appendix for a discussion of the number of sample points actually present in various runs.) The a priori probability distribution for the NS NS (right), NS BH (center), and BH BH (left) merger rates, versus the log 10 of the rate. These distributions were generated from the population synthesis code (dashed line) and fits (solid lines) assuming all parameters in the population synthesis code were chosen at random in the allowed region

14 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 result deduced from artificial data generated from a multidimensional fit to the visible wide and merging NS NS rate data. The vertical lines are the respective 95% CI bounds presented in Figure The a priori probability distribution for the NS NS (bottom), NS BH (center), and BH BH (top) merger rates per Milky Way equivalent galaxy. As in Figure 8.3, the dashed 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 we impose both our observational constraints (i.e., consistency with the observed number of visible wide and visible merging NS NS binaries) 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 the field. This plot was obtained directly from Figure 8.5 using Eq. (8.9) Cumulative probability distributions P k (X) defined so P k (X) is the fraction of all models consistent with the two constraints imposed in the text (i.e., the formation rates of wide and merging NS NS binaries correspond adequately to observations) and that have x k < X. The left panel shows the distributions for the 3 kick-related parameters x 3,x 4,x 5 ; in this panel, the bottom (dashed) curve denotes P 3, the middle (solid) curve denotes P 4, and the top dotted curve denotes P 5. The right panel shows the distributions for P 1 (solid top line), P 2 (dashed), P 6 (dotted), and P 7 (solid, bottom curve)

15 15 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 binaries 1 : 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 objects 2, 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 1 Here, 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). 2 The unit of a pulsar flux density is Jy (Jansky, 1 Jy 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 B at 400MHz is 4 mjy.

16 16 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: 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 B ), 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 laboratories 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-ofthe-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 Observatory; 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 Interferometer 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 17 demonstration mission, is scheduled for a launch in 2009, and LISA itself is pected to be launched in 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 binaries, including NS WD binaries. More exotic systems such as NS or stellar-mass BH ( 1 10M ) binaries with intermediate-mass BHs ( M ) 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 determined supernova (SN) rate for the Galaxy. Typically, this population synthesis method has uncertainties 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 time 3, and (2) the implicit assumption that this small sample is a good 3 τ H 1/H 0, where H 0 is the present value of Hubble constant (H 0 = 100h km s 1 Mpc 1 ). The most recent Wilkinson Microwave Anisotropy Probe (WMAP) results presented that h This implies a Hubble time of 13.8 Gyr (Bennet et al. 2003).

18 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 follows 4. 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 B and B (Wolszczan 1991). Then there was a breakthrough in 2003 with the discovery of PSR J A 5 (Burgay et al. 2003). In Ch. 2, I derive the PDF for NS NS merger rates based on the previously known systems (B and B ), and compare the results with the revised rates including PSR J 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 4 All 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). 5 After the discovery of its pulsar companion, the two pulsars in this double-pulsar system are now labeled as PSR A (recycled), and PSR J B (normal, non-recycled), respectively. In this thesis, I only consider the A pulsar, and omit the label A for simplicity.

19 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 20 Chapter 2 The Probability Distribution of Binary Pulsar Merger Rates. I. Double Neutron Star Systems in the Galactic Field 2.1. Introduction The detection of the NS NS prototype PSR B 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 , February 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 c The American Astronomical Society.

21 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 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 = 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 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 associated with rate estimates. We consider the two binaries found in the Galactic disk: PSR B and PSR B Following the arguments made by Phinney (1991) and KNST, we do not include the globular cluster system PSR B C (Prince et al. 1991). Radio-pulsar-survey selection effects 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 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 R peak 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 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 B and PSR B ) 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 (N tot ). We then produce a very large number of Monte Carlo realizations of such pulsar populations and determine the number of objects (N obs ) 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 N obs. We find, as expected and assumed by other studies, that this distribution closely

23 23 follows Poisson statistics 1, and we determine the best-fit value of the mean of the Poisson distribution λ for each population model and value of N tot,1913. The calculations described so far are performed separately for each of PSRs B and B 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 sample. 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 N tot values for each physical model. The probability distribution of the total merger rate R tot 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, R det (see 2.6) 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 = ψ R (R)2π RdR ψ Z (Z)dZ φ(l)dl, (2.1) where ψ R (R), ψ Z (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 R 0 and the vertical Z 0 scale. In 1 The nature of pulsar detection is actually expected to follows a Binomial distribution ( detection versus nondetection ). However, in the case of a small number of sampling, namely N obs = 1 in our case, Poission distribution is a good approximation.

24 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 2R0 2 Z ), (2.2) Z 0 We set R 0 = 4.0 kpc and Z 0 = 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 convenience, 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)L p 1 min L p, (2.3) where L L min and p > 1. The cut-off luminosity, L min, and the exponent p are the model parameters. Cordes & Chernoff (1997) found L min = mjy kpc2 and p = 2.0 ± 0.2 in a 68% credible region. We set p = 2.0 and L min = 1.0 mjy kpc 2 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 (T sky ). 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 25 index of Finally, given the model Galactic coordinates of each pulsar, the sky background noise temperature at 408 MHz (T sky ) is taken from the all-sky catalog of Haslam et al. (1981). Scaling T sky to other survey frequencies assumes a spectral index of 2.8 (Lawson et al. 1987) 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 N tot, we are left with a sample of N obs pulsars that are nominally detectable by the surveys. Repeating this process many times, we can determine the probability distribution of N obs 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 Survey Parameters The main factors affecting the signal-to-noise ratio (σ) of a pulsar search can be summarized by the following expression σ S νg P νt, (2.4) T w e where S ν 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 T sky 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 w e 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 ) which we adopt in 2 Although recent studies suggest a variety of spectral indices for τ (Löhmer 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 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 S 400 = L/d 2. Since not all pulsar surveys are carried out at 400 MHz, we need to scale S 400 to take account of the steep radio flux density spectra of pulsars. Using a simple power law of the form S ν ν α, where α is the spectral index, we can calculate the flux S ν at any frequency ν as: ( ν ) α S ν = S 400, (2.5) 400 MHz 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 B and B 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 B and B The effective pulse width w e 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: w 2 e = w2 + τ 2 + t 2 samp + t2 DM + t2 DM, (2.6) where τ is the scatter-broadening timescale calculated from the Taylor & Cordes (1993) model, t samp is the data sampling interval in the observing system, t DM is the dispersive broadening across an

27 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 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 B and PSR B 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 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, σ control, 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, σ binary, the degradation factor F = σ binary /σ control. 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 Tucanae. In this chapter, where we are interested in the degradation as a function of integration time,