Short Gamma-Ray Bursts and Binary Mergers in Spiral and Elliptical Galaxies: Redshift Distribution and Hosts
aa r X i v : . [ a s t r o - ph ] N ov Draft version October 28, 2018
Preprint typeset using L A TEX style emulateapj v. 11/26/03
SHORT GAMMA-RAY BURSTS AND BINARY MERGERS IN SPIRAL AND ELLIPTICAL GALAXIES:REDSHIFT DISTRIBUTION AND HOSTS
R. O’Shaughnessy
Northwestern University, Department of Physics and Astronomy, 2145 Sheridan Road, Evanston, IL 60208, USA
K. Belczynski
Tombaugh Fellow, New Mexico State University, Las Cruces, New Mexico, 88003, USA
V. Kalogera
Northwestern University, Department of Physics and Astronomy, 2145 Sheridan Road, Evanston, IL 60208, USA
Draft version October 28, 2018
ABSTRACTTo critically assess the binary compact object merger model for short gamma ray bursts (GRBs) –specifically, to test whether the short GRB rates, redshift distribution and host galaxies are consis-tent with current theoretical predictions – it is necessary to examine models that account for thehigh-redshift, heterogeneous universe (accounting for both spirals and ellipticals). We present an in-vestigation of predictions produced from a very large database of first-principle population synthesiscalculations for binary compact mergers with neutron stars (NS) and black holes (BH), that samplea seven-dimensional space for binaries and their evolution. We further link these predictions to (i)the star formation history of the universe, (ii) a heterogeneous population of star-forming galaxies,including spirals and ellipticals, and (iii) a simple selection model for bursts based on flux-limiteddetection. We impose a number of constraints on the model predictions at different quantitativelevels: short GRB rates and redshift measurements, and, for NS-NS, the current empirical estimatesof Galactic merger rates derived from the observed sample of close binary pulsars. Because of therelative weakness of these observational constraints (due to small samples and measurement uncer-tainties), we find a small, but still substantial, fraction of models are agreement with available shortGRB and binary pulsar observations, both when we assume short GRB mergers are associated withNS-NS mergers and when we assume they are associated with BH-NS mergers. Notably, we do notneed to introduce artificial models with exclusively long delay times. Most commonly models producemergers preferentially in spiral galaxies, in fact predominantly so, if short GRBs arise from NS-NSmergers alone. On the other hand, typically BH-NS mergers can also occur in elliptical galaxies (forsome models, even preferentially), in agreement with existing observations. As one would expect,model universes where present-day BH-NS binary mergers occur preferentially in elliptical galaxiesnecessarily include a significant fraction of binaries with long delay times between birth and merger(often O (10Gyr)); unlike previous attempts to fit observations, these long delay times arise naturallyas properties of our model populations. Though long delays occur, almost all of our models (both apriori and constrained) predict that a higher proportion of short GRBs should occur at moderate tohigh redshift (e.g., z >
1) than has presently been observed, in agreement with recent observationswhich suggest a strong selection bias towards successful follow-up of low-redshift short GRBs. Finally,if we adopt plausible priors on the fraction of BH-NS mergers with appropriate combination of spinsand masses to produce a short GRB event based on Belczynski et al. (2007), then at best only a smallfraction of BH-NS models could be consistent with all current available data, whereas NS-NS modelsdo so more naturally.
Subject headings:
Stars: Binaries: Close; Gamma-ray bursts
1. INTRODUCTION
The afterglows of several short-hard gamma ray bursts(GRBs) have recently been localized on the sky, al-lowing reasonably precise determination of their hosts,redshifts, and energetics (see, e.g., Berger et al. 2006;Berger 2006; Barthelmy et al. 2005; Fox et al. 2005).The energetics, presence in both old and star form-ing host galaxies, absence of supernovae afterglow char-acteristics, and in some cases apparent host offsetsof these bursts seem qualitatively consistent with the
Electronic address: [email protected] merger hypothesis (MH): the notion that most short-hard bursts arise from the disruption of a neutronstar (NS) in either a NS-NS or black hole-NS binary(see, e.g, Lee et al. (2005) and references therein; thisGRB model was first presented by Paczynski (1986)).While other populations, such as bursts from magne-tars, may contribute to the total short GRB eventrate, the fraction of GRBs produced by nearby mag-netars is believed to be small (see, e.g., Nakar et al.2006; Popov & Stern 2006; Lazzati et al. 2005). Further-more, empirical and theoretical estimates for compactobject merger rates based on studies of the Milky Way(see, e.g., Kim et al. 2003; O’Shaughnessy et al. 2005,2007b; Belczynski et al. 2002b, 2006b; Kalogera et al.2004; Nagamine et al. 2006) are roughly consistent withBATSE and Swift observations (Guetta & Piran 2006,2005; Ando 2004).The number of observed radio pulsars with neutronstar companions can provide a robust quantitative testof the MH. For example, using well-understood selec-tion effects and fairly minimal population modeling (i.e.,a luminosity function and a beaming correction factor),Kim et al. (2003) developed a statistical method to de-termine which double neutron star coalescence rates wereconsistent with NS-NS seen in the Milky Way. However,in distinct contrast to NS-NS in the Milky Way, little isknown directly about the short GRB spatial or luminos-ity distribution.Short GRBs are still detected quite infrequently (i.e, ahandful of detections per year for Swift); sufficient statis-tics are not available for a robust nonparametric estimateof their distribution in redshift z and peak luminosity L . To make good use of the observed ( z, L ) data, wemust fold in fairly strong prior assumptions about thetwo-dimensional density d N/dtdLdz ( L, z ). Typically,these priors are constructed by convolving the star for-mation history of the universe with a hypothesized dis-tribution for the “delay time” between the short GRBprogenitor’s birth and the GRB event, as well as withan effective (detection- and angle-averaged) luminositydistribution for short GRBs. Observations are thus in-terpreted as constraints on the space of models, ratherthan as direct measurements of the z, L distribution(Ando 2004; Guetta & Piran 2005, 2006; Gal-Yam et al.2005). A similar technique has been applied with con-siderable success to long GRB observations (see,e.g.,Porciani & Madau 2001; Guetta & Piran 2005; Schmidt1999; Che et al. 1999, and references therein): as ex-pected from a supernovae origin, the long GRB rate isconsistent with the star formation history of the uni-verse. And within the context of specific assumptions about the merger delay time distribution and star forma-tion history of the universe (i.e., dn/dt ∝ /t and homo-geneous through all space, respectively), Gal-Yam et al.(2005) and Nakar et al. (2005) have compared whethertheir set of models can produce results statistically con-sistent with observations. Among other things they con-clude that, within these conventional assumptions, themerger model seems inconsistent with the data.These previous predictions assume homogeneous starforming conditions throughout the universe, with rateproportional to the observed time-dependent star for-mation rate (as given by, for example, Nagamine et al.(2006) and references therein). In reality, however,the universe is markedly heterogeneous as well as time-dependent; for example, large elliptical galaxies formmost of their stars early on. Similarly, predictions for thedelay time distribution and indeed the total number ofcompact binaries depend strongly on the assumptions en-tering into population synthesis simulations. These simu-lations evolve a set of representative stellar systems usingthe best parameterized recipies for weakly-constrained(i.e., supernovae) or computationally taxing (i.e., stellarevolution) features of single and binary stellar evolution.By changing the specific assumptions used in these re-cipies, physical predictions such as the NS-NS merger rate can vary by a few orders of magnitude (see,e.g.Kalogera et al. 2001, and references therein). In partic-ular, certain model parameters may allow much betteragreement with observations.In this study we examine predictions based on a largedatabase of conventional binary population synthesismodels: two sets of 500 concrete pairs of simulations( § In order to make predic-tions regarding the elliptical-to-spiral rate ratio for bi-nary mergers, we adopt a two-component model for thestar formation history of the universe ( § §
2. GAMMA RAY BURSTS: SEARCHES ANDOBSERVATIONS
Emission and detection models
To compare the predictions of population synthesis cal-culations with the observed sample of short GRBs, wemust estimate the probability of detecting a given burst.We therefore introduce (i) a GRB emission model con-sisting of an effective luminosity function for the isotropicenergy emitted, to determine the relative probabilityof various peak fluxes, and a spectral model, for K-corrections to observed peak fluxes, and (ii) a detectionmodel introducing a fixed peak-flux detection threshold.Overall we limit attention to relatively simple modelsfor both GRB emission and detection. Specifically, weassume telescopes such as BATSE and Swift detect allsources in their time-averaged field of view ( ≈ π and1 . f d given by 1 /f d = 1 / . / π ) with peak fluxes at the detector F d greater thansome fixed threshold of F d = 1ph sec − cm − in 50 to 300keV (see,e.g. Hakkila et al. 1997). We note that Swift’striggering mechanism is more complex (Gehrels, privatecommunication) and appears biased against detections ofshort bursts; for this reason, BATSE results and detec-tion strategies will be emphasized heavily in what follows.Similarly, though observations of short gammaray bursts reveal a variety of spectra (see,e.g.Ghirlanda et al. 2004, keeping in mind the observed peakenergy is redshifted), and though this variety can have Because simulations that produce many BH-NS mergers neednot produce many NS-NS mergers and vice-versa, we perform twoindependent sets of 500 pairs of simulations, each set designed toexplore the properties of one particular merger type (i.e, BH-NS orNS-NS). The statistical biases motivating this substantial increasein computational effort are discussed in the Appendix.
Fig. 1.—
Characteristic distance to a source q ˙ N/ πF d ver-sus its comoving distance. Points: Short bursts with well-definedredshifts (SH1; see Table 1). Solid line: The critical character-istic distance r c ( z ) ≡ q ˙ N d ( z ) / πF d = r ( z ) p (1 + z ) k ( z ) versuscomoving distance r ( z ), for our simple power-law spectral model F ν ∝ ν . . Given our assumptions, systems with fluxes ˙ N corre-sponding to points above this curve can be seen at the earth witha band-limited detector in 50 −
300 keV with peak flux ≥ F d . significant implications for the detection of moderate-redshift ( z >
1) bursts, for the purposes of this paper weassume all short gamma ray bursts possess a pure power-law spectrum F ν ∝ ν − α with α = − .
5. Though severalauthors such as Ando (2004) and Schmidt (2001) haveemployed more realistic bounded spectra, similar purepower-law spectra have been applied to interpret low-redshift observations in previous theoretical data analy-sis efforts: Nakar et al. (2005) uses precisely this spectralindex; Guetta & Piran (2006) use α = − . Because our spectral model is manifestly unphysicaloutside our detection band (50 −
300 keV), we cannotrelate observed, redshifted fluxes to total luminosity. In-stead, we characterize the source’s intrinsic photon lumi-nosity by the rate ˙ N = dN/dt e at which it appears toproduce B = 50 −
300 keV photons isotropically in itsrest frame, which we estimate from observed fluxes F inthis band via a K-correction:˙ N ≡ F (4 πr )(1 + z ) k ( z ) (1) k ( z ) ≡ R B F ν dν/ν R B (1+ z ) F ν dν/ν = (1 + z ) − . (2)where r ( z ) is the comoving distance at redshift z . Togive a sense of scale, a luminosity L/ (10 erg − s − )corresponds to a photon luminosity ˙ N / (4 × s − );similarly, the characteristic distance to which a photonflux can be seen is r c ≡ p N/ πF d ≃ N / × s − ) / ( F d / − s − ) − / .Finally, we assume that short GRBs possess an intrin-sic power-law peak flux distribution: that the peak fluxesseen by detectors placed at a fixed distance but randomorientation relative to all short GRBs should either (i) beprecisely zero, with probability 1 − /f b or (ii) collectively In reality, however, a break in the spectrum is often observed,redshifted into the detection band. Under these circumstances, theK-correction can play a significant role in detectability. be power-law distributed, from some (unknown) mini-mum peak flux to infinity, with some probability 1 /f b .[This defines f b , the beaming correction factor, in termsof the relative probabilities of a visible orientation.] Forconvenience in calculation, we will convert this power-law peak-flux distribution into its equivalent power-lawphoton rate ˙ N distribution P ( > ˙ N ) ≡ (cid:26) f − b ( ˙ N / ˙ N min ) − β if ˙ N > ˙ N min f − b if ˙ N ≤ ˙ N min (3)where we assume β = 2; this particular choice of thepower-law exponent is a good match to the observedBATSE peak-flux distribution (see, e.g. Guetta & Piran2006; Nakar et al. 2005; Ando 2004; Schmidt 2001, andreferences therein). The fraction of short bursts that arevisible at a redshift z is thus P ( z ) ≡ P ( > ˙ N d ), where ˙ N d is shorthand for 4 πr (1+ z ) k ( z ) F d . Once again, these as-sumptions correspond approximately to those previouslypublished in the literature; elementary extensions (for ex-ample, a wider range of source luminosity distributions)have been successfully applied to match the observedBATSE flux distributions and Swift redshift-luminositydata [e.g., in addition to the references mentioned previ-ously, Guetta & Piran (2005)].2.2. GRB Observations
While the above discussion summarizes the most crit-ical selection effects – the conditions needed for GRBdetection – other more subtle selection effects can signif-icantly influence data interpretation. Even assigning aburst to the “short” class uses a fairly coarse phenomeno-logical classification [compare, e.g., the modern spec-tral evolution classification of Norris & Bonnell (2006),the machine-learning methods of Hakkila et al. (2003),and the original classification paper Kouveliotou et al.(1993)]; alternate classifications produce slightly but sig-nificantly different populations (see,e.g. Donaghy et al.2006, for a concrete, much broader classification scheme).Additionally, short GRB redshift measurements can beproduced only after a second optical search, with its ownstrong selection biases toward low-redshift hosts (see,e.g.,Berger et al. 2006).To avoid controversy, we therefore assemble our listof short GRBs from four previously-published compila-tions: (i) Berger et al. (2006) (Table 1), which providesa state-of-the-art Swift-dominated sample with relativelyhomogeneous selection effects; (ii) Donaghy et al. (2006)(Table 8), a broader sample defined using an alternativeshort-long classification; and finally (iii) Berger (2007)and (iv) Gehrels et al. (2007) which cover the gaps be-tween the first two and the present. [We limit attentionto bursts seen since 2005, so selection effects are fairlyconstant through the observation interval. For similarreasons, we do not include the post-facto IPN galaxy as-sociations shown in Nakar et al. (2005) (Table 1).] Thiscompilation omits GRB 050911 discussed in Page et al.(2006) but otherwise includes most proposed short GRBcandidates. As shown in Table 1, the sample consistsof 21 bursts; though most (15) have some redshift in-formation, only 11 have relatively well-determined red-shifts. However, even among these 12 sources, some dis-agreement exists regarding the correct host associationsand redshifts of GRBs 060505 and 060502B (see,e.g.,Berger et al. 2006).
Table 1. Short Gamma Ray Bursts
GRB a Det b z c T90 d P e Id f OA g Type h Usage i Refs j > < a Gamma-ray burst index b Detector in which the GRB was initially detected; S denotes Swift, H denotes HETE-II. c Redshift of the host, if well identified. d Duration of the burst. e Peak photon flux of the burst (ph/cm /s). f Whether the host was optically identified. g Whether the burst produced a visible optical afterglow. h Morphology of the host: elliptical (E) or spiral (S). i Summary of the previous columns: S1 bursts were initially seen by Swift and have a well-defined redshift; S2 bursts were seen by Swiftand have some uncertain redshift information; S3 bursts include all bursts seen by Swift only. Similarly, SH1 includes all bursts seen bySwift or HETE-2 with a well-defined redshift. j References: (1) Donaghy et al. (2006) (2) Gehrels et al. (2005) (3) Lee et al. (2005) (4) Bloom et al. (2006b) (5) Berger et al. (2005a)(6) Berger (2007) (7) Barthelmy et al. (2005) (8) Fox et al. (2005) (9) Villasenor et al. (2005) (10) Pedersen et al. (2005) (11) Covino et al.(2006) (12) Gehrels et al. (2007) (13) Berger et al. (2005b) (14) Prochaska et al. (2006) (15) Campana et al. (2006) (16) Grupe et al.(2006a) (17) Berger (2006) (18) Nakar (2007) (19) La Parola et al. (2006) (20) Dietz (2006) (21) Berger et al. (2006) (22) Soderberg et al.(2006a) (23) Levan et al. (2006) (24) de Ugarte Postigo et al. (2006) (25) Roming (2005) (26) Bloom et al. (2006a) (27) Thoene et al.(2007) (28) Ofek et al. (2007)
To make sure the many hidden uncertainties and se-lection biases are explicitly yet compactly noted in sub-sequent calculations, we introduce a simple hierarchicalclassification for bursts seen since 2005: S n represent thebursts detected only with Swift; SH n the bursts seen ei-ther by Swift or HETE-II; n = 1 corresponds to burstswith well-determined redshifts; n = 2 corrresponds tobursts with some strong redshift constraints; and n = 3includes all bursts.Starting in May 2005, Swift detected 9 short GRBsin a calendar year. For the purposes of comparison,we will assume the Swift short GRB detection rate tobe R D, Swift = 10yr − ; compensating for its partial skycoverage, this rate corresponds to an all-sky event rateat earth of f d, Swift R D, Swift ≃
90 yr − . However, in or-der to better account for the strong selection biases ap-parently introduced by the Swift triggering mechanismagainst short GRBs (Gehrels, private communication),we assume the rate of GRB events above this thresholdat earth to be much better represented by the BATSEdetection rate R d, BATSE when corrected for detectorsky coverage, namely f d, BATSE R D, BATSE = 170 yr − (Paciesas et al. 1999) . For similar reasons, in this paper Section 2 of Guetta & Piran (2005) describes how this rate can we express detection and sensitivity limits in a BATSEband (50-300 keV) rather than the Swift BAT band.2.3.
Cumulative redshift distribution
As Nakar et al. (2005) demonstrated and as describedin detail in §
4, the cumulative redshift distribution de-pends very weakly on most parameters in the short GRBemission and detection model (i.e., f b , f d , ˙ N , and F d ).When sufficiently many unbiased redshift measurementsare available to estimate it, the observed redshift distri-bution can stringently constrain models which purportto reproduce it. At present, however, only 11 reliableredshifts are available, leading to the cumulative redshiftdistribution shown in Figure 2 (thick solid line). Wecan increase this sample marginally by including moreweakly-constrained sources. In Figure 2 (shaded region)we show several distributions consistent with SH2, choos-ing redshifts uniformly from the intersection of the regionsatisfying any constraints and 0 < z < be extracted from the BATSE catalog paper, taking into accounttime-dependent changes in the instrument’s selection effects. Fig. 2.—
Cumulative redshift distribution of detected shortGRBs. The thick solid curve provides the cumulative distributionof well-constrained GRBs (i.e., the class SH1). The shaded re-gion indicates the range of cumulative distributions produced byassigning redshifts to the weakly-constrained (i.e., the class SH2)in a manner consistent with the constraints. When only an upperor lower limit is available, we pick redshifts using a uniform priorfor redshifts between 0 and 5. mulative redshift distributions still agree at very low red-shifts.The small sample size seriously limits our ability to ac-curately measure the cumulative distribution: given thesample size, a Kolmogorov-Smirnov 95% confidence in-terval includes any distribution which deviates by lessthan 0 .
375 from the observed cumulative distribution.Rather than account for all possibilities allowed by ob-servations, we will accept any model with maximum dis-tance less than 0 .
375 from the cumulative redshift dis-tribution for the well-known bursts (i.e., from the solidcurve in in Figure 2).By performing deep optical searches to identify hostsfor unconstrained bursts, Berger et al. (2006) havedemonstrated that recent afterglow studies are biased to-wards low redshift – nearby galaxies are much easier todetect optically than high-redshift hosts – and that a sub-stantial population of high-redshift short bursts shouldexist. This high-redshift population becomes more ap-parent when a few high-redshift afterglows seen withHETE-II before 2005 are included; see Donaghy et al.(2006) for details.2.4.
Comparison with prior workShort GRB interpretation : Several previous efforts havebeen made to test quantitative MH-based predictionsfor the host, redshift, luminosity, and age distribu-tions [Meszaros et al. (2006); Guetta & Piran (2006);Nakar et al. (2005); Gal-Yam et al. (2005); Bloom et al.(1999); Belczynski et al. (2006c); Perna & Belczynski(2002)]. However, many authors found puzzlingdiscrepancies; most notably, as has been empha-sized by Gal-Yam et al. (2005); Nakar et al. (2005)and by Guetta & Piran (2006) (by comparing redshift-luminosity distributions to models) and as has seeminglybeen experimentally corroborated with GRB 060502BBloom et al. (2006a), typical observed short GRBs ap-pear to occur ≈ (1 − few) × Gyr after their progenitors’birth. By contrast, most authors expect population syn-thesis predicts a delay time distribution dp/dt ∝ /t (e.g., Piran 1992), which was interpreted to imply that short delay times dominate, that DCO mergers occurvery soon after birth, and that mergers observed on ourlight cone predominantly arise from recent star forma-tion. Additionally, on the basis of the observed redshift-luminosity distribution alone, Guetta & Piran (2006)and Nakar et al. (2005) conclude short GRB rates tobe at least comparable to observed present-day NS-NSmerger rate in the Milky Way. They both also notethat a large population of low-luminosity bursts (i.e., L < erg) would remain undetected, a possibilitywhich may have some experimental support: post-factocorrelations between short GRBs and nearby galaxiessuggests the minimum luminosity of gamma ray bursts( L min ) could be orders of magnitude lower (Nakar et al.2005; Tanvir et al. 2005). Such a large population wouldlead to a discrepancy between the two types of inferredrates. In summary, straightforward studies of the ob-served SHB sample suggest (i) delay times and (ii) toa lesser extent rate densities are at least marginallyand possibly significantly incongruent with the observedpresent-day Milky Way sample of double NS binaries,and by extension the merger hypothesis (cf. Sections3.2 and 4 of Nakar et al. 2005). A more recent studyby Berger et al. (2006) suggests that high-redshift hostsmay be significantly more difficult to identify optically.Using the relatively weak constraints they obtain regard-ing the redshifts of previously-unidentified bursts, theyreanalyze the data to find delay time distributions con-sistent with dP/dt ∝ /t , as qualitatively expected fromdetailed evolutionary simulations.In all cases, however, these comparisons were basedon elementary, semianalytic population models, with noprior on the relative likelihood of different models: amodel with a Gyr characteristic delay between birth andmerger was a priori as likely as dP/dt ∝ /t . For this rea-son, our study uses a large array of concrete populationsynthesis simulations, in order to estimate the relativelikelihood of different delay time distributions. Population synthesis : Earlier population synthesis stud-ies have explored similar subject matter, even in-cluding heterogeneous galaxy populations (see, e.g.Belczynski et al. 2006c; de Freitas Pacheo et al. 2005;Perna & Belczynski 2002; Bloom et al. 1999; Fryer et al.1999; Belczynski et al. 2002a). These studies largely ex-plored a single preferred model, in order to produce whatthese authors expect as the most likely predictions, suchas for the offsets expected from merging supernovae-kicked binaries and the likely gas content of the circum-burst environment. Though preliminary short GRB ob-servations appeared to contain an overabundance of shortGRBs (Nakar et al. 2005), recent observational analysessuch as Berger et al. (2006) suggest high-redshift burstsare also present, in qualitative agreement with the de-tailed population synthesis study by Belczynski et al.(2006c). The present study quantitatively reinforces thisconclusion through carefully reanalyzing the implicationsof short GRB observations, and particularly throughproperly accounting for the small short GRB sample size.The extensive parameter study included here, however,bears closest relation to a similar slightly smaller studyin Belczynski et al. (2002a), based on 30 population syn-thesis models. Though intended for all GRBs, the rangeof predictions remain highly pertinent for the short GRBpopulation. In most respects this earlier study was muchbroader than the present work: it examined a much widerrange of potential central engines (e.g., white dwarf-blackhole mergers) and extracted a wider range of predictions(e.g., offsets from the central host). The present paper,however, not only explores a much larger set of popu-lation synthesis models ( ≃
3. OTHER RELEVANT OBSERVATIONS
Multicomponent star formation history
The star formation history of the universe has beenextensively explored through a variety of methods: ex-traglactic background light modulo extinction (see,e.g.,Nagamine et al. 2006; Hopkins 2004, and referencestherein); direct galaxy counts augmented by massestimates (see,e.g. Bundy et al. 2005, and referencestherein); galaxy counts augmented with reconstructedstar formation histories from their spectral energy distri-bution (e.g. Heavens et al. 2004; Thompson et al. 2006;Yan et al. 2006; Hanish et al. 2006); and via more gen-eral composite methods (Strigari et al. 2005). Since allmethods estimate the total mass formed in stars fromsome detectable quantity, the result depends sensitivelyon the assumed low-mass IMF and often on extinction.However, as recently demonstrated by Hopkins (2006)and Nagamine et al. (2006), when observations are inter-preted in light of a common Chabrier IMF, observationslargely converge upon a unique star-formation rate perunit comoving volume ˙ ρ = dM/dV dt bridging nearbyand distant universe, as shown in Figure 3.Less clearly characterized in the literature are the com-ponents of the net star formation history ˙ ρ : the historyof star formation in relatively well-defined subpopula-tions such as elliptical and spiral galaxies. For mostof time, galaxies have existed in relatively well-definedpopulations, with fairly little morphological evolutionoutside of rare overdense regions (see, e.g. Bundy et al.2005; Hanish et al. 2006, and references therein). Dif-ferent populations possess dramatically different histo-ries: the most massive galaxies form most of their starsvery early on (see,e.g. Feulner et al. 2005) and henceat a characteristically lower metallicity. Further, ashas been extensively advocated by Kroupa (see, e.g.Kroupa & Weidner 2003; Fardal et al. 2006, and refer-ences therein) the most massive structures could con-ceivably form stars through an entirely different collapsemechanism (“starburst-mode”, driven for example bygalaxy collisions and capture) than the throttled moderelevant to disks of spiral galaxies (“disk-mode”), result-ing in particular in a different IMF.Both components significantly influence the present-day merger rate. For example, the initial mass func-tion determines how many progenitors of compact bina-ries are born from star-forming gas and thus are avail-able to evolve into merging BH-NS or NS-NS binaries.Specifically, as shown in detail in § Short GRBs have been associated with more refined morpho-logical types, such as dwarf irregular galaxies. For the purposes ofthis paper, these galaxies are sufficiently star forming to be “spiral-like”.
Fig. 3.—
Star formation history of the universe used in thispaper. Solid line: Net star formation history implied by Eq. (4).Dashed, dotted line: The star formation history due to ellipticaland spiral galaxies, respectively. via Figure 4, elliptical galaxies produce roughly threetimes more high mass binaries per unit mass than theirspiral counterparts. Additionally, as first recognized byde Freitas Pacheco et al. (2006), even though ellipticalgalaxies are quiescent now, the number of compact bina-ries formed in ellipticals decays roughly logarithmically with time (i.e., dn/dt ∝ /t ). Therefore, due to the highstar formation rate in elliptical-type galaxies ∼
10 Gyrago, the star forming mass density δρ e born in ellipticalsroughly t e ∼
10 Gyr ago produces mergers at a rate den-sity ∼ δρ e /t e that is often comparable to or larger thanthe rate density of mergers occurring soon after theirbirth in spiral galaxies ∼ dρ s /dt . Standard two-component model
As a reference model we use the two-component starformation history model presented by Nagamine et al.(2006). This model consists of an early “elliptical” com-ponent and a fairly steady “spiral” component, with starformation rates given by˙ ρ = ˙ ρ e + ˙ ρ s (4)˙ ρ C = A C ( t/τ C ) e − t/τ C (5)where cosmological time t is measured starting fromthe beginning of the universe and where the twocomponents decay timescales are τ e,s = 1.5 and4.5 Gyr, respectively (see Section 2 and Table 2of Nagamine et al. 2006). These normalization con-stants A e,s = 0 . , . M ⊙ yr − Mpc − were chosenby Nagamine et al. (2006) so the integrated amountof elliptical and spiral star formation agrees with (i)the cosmological baryon census [Ω ∗ ≈ . R ˙ ρ e /ρ c = Ω ∗ / . × . R ˙ ρ e /ρ c =Ω ∗ / . × .
4, respectively.Each component forms stars in its own distinctive con-ditions, set by comparison with observations of the MilkyWay and elliptical galaxies. We assume mass convertedinto stars in the fairly steady “spiral” component is doneso using solar metallicity and with a fixed high-mass IMFpower law [ p = − . p ∈ [ − . , − .
06] and metallicity Z within0 . < Z/Z ⊙ <
2. These elliptical birth conditionsagree with observations of both old ellipticals in the lo-cal universe (see Li et al. 2006, and references therein) aswell as of young starburst clusters (see Fall et al. 2005;Zhang & Fall 1999, and references therein).3.2.
Binary pulsar merger rates in the MilkyWay
If binary neutron stars are the source of short GRBs,then the number of short GRBs seen in spirals shouldbe intimately connected to the number of binary pul-sars in the Milky Way that are close enough to mergethrough the emission of gravitational radiation. Fourunambiguously merging double pulsars have been foundwithin the Milky Way using pulsar surveys with well-understood selection effects. Kim et al. (2003) developeda statistical method to estimate the likelihood of doubleneutron star formation rate estimates, designed to ac-count for the small number of known systems and theirassociated uncertainties. Kalogera et al. (2004) summa-rize the latest results of this analysis: taking into accountthe latest pulsar data, a standard pulsar beaming correc-tion factor f b = 6 for the unknown beaming geometry ofPSR J0737–3037, and a likely distribution of pulsars inthe galaxy (their model 6), they constrain the rate tobe between r MW = 16 . − and 292 . − . (95%confidence interval) .Assuming all spiral galaxies to form stars similarly toour Milky Way, then the merger rate density in spiralsat present R [ s ] ( T ) must agree with the product of theformation rate per galaxy r MW and the density of spiralgalaxies n s . Based on the ratio of the blue light den-sity of the universe to the blue light attributable to theMilky Way, the density of Milky Way-equivalent galaxieslies between 0 . × − Mpc − and 2 × − Mpc − (seePhinney (1991), Kalogera et al. (2001), Nutzman et al.(2004), Kopparapu et al. (2007) and references therein).We therefore expect the merger rate density due tospirals at present to lie between 0 . − Mpc − and5 . − Mpc − (with better than 95% confidence).
4. PREDICTIONS FOR SHORT GRBS
Population synthesis simulations
We study the formation of compact objects with the
StarTrack population synthesis code, first developed byBelczynski et al. (2002b) and recently significantly ex-tended as described in detail in Belczynski et al. (2006a); The range of binary neutron star merger rates that we ex-pect to contains the true present-day rate has continued to evolveas our knowledge about existing binary pulsars and the distri-bution of pulsars in the galaxy has changed. The range quotedhere reflects the recent calculations on binary pulsar merger rates,and corresponds to the merger rate confidence interval quoted inO’Shaughnessy et al. (2007b) (albeit with a different conventionfor assigning upper and lower confidence interval boundaries). Inparticular, this estimate does not incorportate conjectures regard-ing a possibly shorter lifetime of PSR J0737-3037, as described inKim et al. (2006). The properties of this pulsar effectively deter-mine the present-day merger rate, and small changes in our under-standing of those properties can significantly change the confidenceinterval presented. see § ∼ StarTrack population synthesiscode, in addition to the IMF and metallicity (which varydepending on whether a binary is born in an ellipti-cal or spiral galaxy), seven parameters strongly influ-ence compact object merger rates: the supernova kickdistribution (modeled as the superposition of two in-dependent Maxwellians, using three parameters: oneparameter for the probability of drawing from eachMaxwellian, and one to characterize the dispersion ofeach Maxwellian), the solar wind strength, the common-envelope energy transfer efficiency, the fraction of an-gular momentum lost to infinity in phases of non-conservative mass transfer, and the relative distributionof masses in the binary. Other parameters, such asthe fraction of stellar systems which are binary (here,we assume all are, i.e., the binary fraction is equalto 1) and the distribution of initial binary parameters,are comparatively well-determined (see e.g..Abt (1983),Duquennoy & Mayor (1991) and references therein). Even for the less-well-constrained parameters, some in-ferences have been drawn from previous studies, eithermore or less directly (e.g., via observations of pulsarproper motions, which presumably relate closely to su-pernovae kick strength; see, e.g., Hobbs et al. (2005),Arzoumanian et al. (2002), Faucher-Gigu`ere & Kaspi(2006) and references therein) or via comparison ofsome subset of binary population synthesis resultswith observations (e.g., § plausible range of each of these parameters. More specifically, despitethe Milky Way-specific studies of O’Shaughnessy et al.(2005, 2007b) (which apply only to spirals, not the ellip-tical galaxies included in this paper), in this study we willcontinue to assume all seven parameters are unknown,drawn from the plausible parameter ranges described inO’Shaughnessy et al. (2007b).As noted previously in § Z = Z ⊙ and a high-mass IMF slope Particularly for the application at hand – the gravitational-wave-dominated delay between binary birth and merger – the de-tails of the semimajor axis distribution matter little. For a similarbut more extreme case, see O’Shaughnessy et al. (2007c). of p = − .
7, and “elliptical” conditions, with a muchflatter IMF slope and a range of allowed metallicities0 . < Z/Z ⊙ < Archive selection : Our collection of population synthesissimulations consists of roughly 3000 and 800 simulationsunder spiral and elliptical conditions, respectively. Ourarchives are highly heterogeneous, with binary samplesizes N that spread over a large range. A significantfraction of the smaller simulations were run with pa-rameters corresponding to low merger rates, and haveeither no BH-NS or no NS-NS merger events . There-fore, though the set of all population synthesis simula-tions is unbiased, with each member having randomlydistributed model parameters, the set of all simulationswith one or more events is slightly biased towards simu-lations with higher-than-average merger rates. Further,the set of simulations with many events, whose proper-ties (such as the merger rate) can be very accurately es-timated, can be very strongly biased towards those mod-els with high merger rates. Fortunately, as discussed atmuch greater length in the Appendix, the set of simula-tions with nN ≥ × and n >
20 has small selectionbias and enough simulations (976 and 737 simulationsNS-NS and BH-NS binaries under spiral-like conditions,as well as 734 and 650 simulations under elliptical condi-tions, respectively) to explore the full range of populationsynthesis results, while simultaneously insuring each sim-ulation has enough events to allow us to reliably extractits results. 4.2.
Results of simulations
From each population synthesis calculation ( α ) per-formed under elliptical or spiral conditions ( C = e, s )and for each final result ( K ), we can estimate: (i) thenumber of final K events per unit mass of binary merg-ers progenitors, i.e., the mass efficiency ( λ C,α,K ); and(ii) the probability P c,α,K ( < t ) that given a progenitorof K the event K (e.g., a BH-BH merger) occurs by time t since the formation of K . Roughly speaking, for eachsimulation we take the observed sample of n binary pro-genitors of K , with M ...n and delay times t ...n , andestimate λ = nN f cut h M i (6) P m ( < t ) = X j Θ( t − t j ) (7)where Θ( x ) is a unit step function; N is the total num-ber of binaries simulated, from which the n progenitorsof K were drawn; h M i is the average mass of all possiblebinary progenitors; and f cut is a correction factor ac-counting for the great many very low mass binaries (i.e.,with primary mass m < m c = 4 M ⊙ ) not included in oursimulations at all. Expressions for both h M i and f cut interms of population synthesis model parameters are pro-vided in Eqs. (1-2) of O’Shaughnessy et al. (2007a). Inpractice, P m ( t ) and dP m /dt are estimated with smooth-ing kernels, as discussed in Appendix B. Given the char-acteristic sample sizes involved (e.g., n >
200 for NS-NS), In practice, the sample size is often chosen to insure a fixednumber of some type of event. As a result, usually the sample size N and the number of any type of event n are correlated. Fig. 4.—
Smoothed histograms of the mass efficiency λ [Eq. (6)]of simulations used in our calculations, shown for spiral (solid)and elliptical (dotted) birth conditions. As expected given thedifferences in the IMF, elliptical galaxies produce BH-NS binariesroughly three times more efficiently than spirals. However, appar-ently because our population synthesis sample involves highly cor-related choices for n and N (see the Appendix and Figure A13),our distribution of NS-NS mass efficiencies remains biased, pro-ducing identical distributions for both elliptical and spiral birthconditions. we expect P m to have absolute error less than 0.05 at eachpoint (95% confidence) and dP m /dt to have rms relativeerror less than 20% (95% confidence). Since these errorsare very small in comparison to uncertainties in otherquantities in our final calculations (e.g., the star forma-tion rate of the universe), we henceforth ignore errors in P and dP/dt .Figures 4, 5, and 6 show explicit results drawn fromthese calculations. From these figures, we draw the fol-lowing conclusions: Uncertainties in binary evolution significantly affect re-sults : As clearly seen by the range of possiblities allowedin Figures 4 and 5, our imperfect understanding of binaryevolution implies we must permit and consider modelswith a wide range of mass efficiencies λ and delay timedistributions P m ( < t ). The merger time distribution is often well-approximatedwith a one parameter family of distributions, dP m /dt ∝ /t : As suggested by the abundance of near-linear dis-tributions in Figure 5, the delay time distribution P m isalmost always linear log t . Further, from the relative in-frequency of curve crossings, the slope of P m versus log t seems nearly constant. As shown in the bottom panels ofFigure 6, this constant slope shows up as a strong corre-lation between the times t (5%) and t (50%) at which P m reaches 0.05 and 0.5 when log t (5%) /M yr > . t (50%) ≈ (cid:26) log t (5%) + 2 . t (5%) > .
510 log t (5%) −
11 if log t (5%) < . The merger time distribution is at most weakly correlatedwith the mass efficiency : Finally, as seen in the top pan-els of Figure 6, a wide range of efficiencies are consistentwith each delay time distribution. The maximum andminmum mass efficiency permitted increase marginallywith longer median delay times t (50%) – roughly an or-der of magnitude over five orders of magnitude of t (50%).But to a good approximation, the mass efficiencies anddelay times seem to be uncorrelated.4.3. Converting simulations to predictions
Each combination of heterogeneous population synthe-sis model, star formation history, and source model leadsto a specific physical signature, embedded in observablessuch as the relative frequencies with which short GRBsappear in elliptical and spirals, the average age of progen-itors in each component, and the observed distributionof sources’ redshifts and luminosities. All of these quan-tities, in turn, follow from the two functions R C ( t ), themerger rate per unit comoving volume in C = ellipticalor spiral galaxies.The rate of merger events per unit comoving volume isuniquely determined from (i) the SFR of the componentsof the universe dρ { C } /dt ; (ii) the mass efficiency λ [ C ] atwhich K mergers occur in each component C ; and (iii)the probability distribution dP m { C } /dt for merger eventsto occur after a delay time t after star formation: R ( t ) = X C R [ C ] (9) R { C } ( t ) = Z dt b λ { C,K } dP m { C } dt ( t − t b ) dρ { C } dt ( t b )(10)Though usually R [ C ] ( t ) is experimentally inaccessible,because our source and detection models treat ellipticaland spiral hosts identically, the ratio uniquely deter-mines the fraction f s of spiral hosts at a given redshift: f s ( z ) = R [ s ] / ( R [ e ] + R [ s ] ) (11)Additionally, as described in § R [ s ] ( T universe )), if thoseprogenitors are double neutron star binaries.Unfortunately, the relatively well-understood physical merger rate distributions R [ C ] are disguised by strongobservational selection effects described in §
2, notablyin the short GRB luminosity function. Based on ourcanonical source model, we predict the detection rate R D of short GRBs from to be given by R D = X C R D [ C ] (12) R D [ C ] = f − d Z R [ C ] P [ C ] ( z )4 πr cdt (13) ≈ ˙ N min f d f b F d Z cdt R [ C ] ( t )(1 + z ) k ( z ) In practice, gas-poor elliptical hosts should produce weakerafterglows. Since afterglows are essential for host identification andredshifts, elliptical hosts may be under-represented in the observedsample. where the latter approximation holds for reasonable˙ N min /f b < s − (i.e., below values corresponding toobserved short bursts). While the detection rate dependssensitively on the source and detector models, within thecontext of our source model the differential redshift dis-tribution p ( z ) p ( z ) dz ∝ dtdz X C,K R [ C ] ( t ( z ))(1 + z ) k ( z ) ˙ Nf b (14)and the cumulative redshift distribution P ( < z ) ≡ R z p ( z ) dz do not depend on the source or detector model(Nakar et al. 2005). Detected luminosity distribution : In order to be self-consistent, the predicted luminosity distribution shouldagree with the observed peak-flux distribution seen byBATSE. However, so long as ˙ N min is small for all popula-tions, our hypothesized power-law form ˙ N − for the GRBluminosity function immediately implies the detected cu-mulative peak flux distribution P ( > F ) is precisely con-sistent: P ( > F ) = ( F d /F ), independent of source pop-ulation; see for example the discussion in Nakar et al.(2005) near their Eq. (5). While more general sourcepopulation models produce observed luminosity func-tions that contain some information about the redshiftdistribution of bursts – for example, Guetta & Piran(2006) and references therein employ broken power-lawluminosity distributions; alternatively, models could in-troduce correlations between brightness and spectrum –so long as most sources remain unresolved (i.e., small˙ N min ), the observed peak flux distribution largely reflectsthe intrinsic brightness distribution of sources. SinceNakar et al. (2005) demonstrated this particular bright-ness distribution corresponds to the observed BATSEflux distribution, we learn nothing new by comparingthe observed peak flux distribution with observations andhenceforth omit it.4.4. Predictions for short GRBs
Given two sets of population synthesis model param-eters – each of which describe star formation in ellipti-cal and spiral galaxies, respectively – the tools describedabove provide a way to extract merger and GRB de-tection rates, assuming all BH-NS or all NS-NS mergersproduce (possibly undetected) short GRB events . Ratherthan explore the (fifteen-dimensional: seven parametersfor spirals and eight, including metallicity, for ellipticals)model space exhaustively, we explore a limited array of 500 “model universes” by (i) randomly selecting We draw our two-simulation “model universe” from two collec-tions of order 800 simulations that satisfy the constraints describedin the Appendix, one for ellipticals and one for spirals. Computa-tional inefficiencies in our postprocessing pipeline prevent us fromperforming a thorough survey of all ∼ possible combinationsof existing spiral and elliptical simulations. At present, our population synthesis archives for elliptical andspiral populations were largely distributed independently; we can-not choose pairs of models with similar or identical parameters for,say, supernovae kick strength distributions. The results of binaryevolution from elliptical and spiral star forming conditions, if any-thing, could be substantially more correlated than we can presentlymodel. We note however that there is no a priori expectation norevidence that evolutionary parameters should indeed be similar inelliptical and spiral galaxies. Fig. 5.—
Cumulative probabilities P m ( < t ) that a NS-NS binary (left panels) or BH-NS binary (right panels) will merge in time less than t , for twenty randomly-chosen population synthesis models, given spiral (top) and elliptical (bottom) star forming conditions. A verticaldashed line indicates the age of the universe. For the sample sizes involved, these distributions are on average accurate to within 0.05everywhere (with 90% probability); see Figure B14. Fig. 6.—
Scatter plots to indicate correlations between results of various simulations. Top panels: Scatter plot of mass efficiency log λ and average delay time log t (50%). Bottom panels: Scatter plot of log t (5%) versus log t (50%); also shown is the analytic estimate of Eq.(8). Left panels indicate spiral conditions; right panels indicate elliptical conditions. Despite the differences between these two types ofsimulations (i.e., metallicity and initial mass function), the range of delay time distributions and mass efficiencies are largely similar (i.e.,the corresponding left and right panels resemble one another). two population synthesis simulations, one each associ-ated with elliptical ( e ) and spiral ( s ) conditions; (ii) esti-mating for each simulation the mass efficiency ( λ e,s ) anddelay time distributions ( P e,s ); (iii) constructing the netmerger rate distribution R using Eqs. (9,10); and (iv)integrating out the associated expected redshift distribu-tion p ( z ) [Eq. (14)].The results of these calculations are presented in Fig-ures 7, 8, 9, 10, and 11. [These figures also compareour calculations’ range of results to observations of shortGRBs (summarized in Table 1) and merging Milky Waybinary pulsars (Kim et al. 2003); these comparisons willbe discussed extensively in the next section.] More specifically, these five figures illustrate the distribution ofthe following four quantities that we extract from each“model universe”: Binary merger rates in present-day spiral galaxies : Toenable a clear comparison between our multi-componentcalculations, which include both spiral and ellipticalgalaxies, and other merger rate estimates that incorpo-rate only present-day star formation in Milky Way-likegalaxies, the two solid curves in Figure 7 show the distri-butions of present-day NS-NS and BH-NS merger ratesin spiral galaxies seen in the respective sets of 500 simu-lations.In principle, the BH-NS and NS-NS merger rates1
Fig. 7.—
The distribution of merger rate densities in spiral-typegalaxies R s for BH-NS mergers (top) and NS-NS mergers (bot-tom); the solid curve includes all simulations, the dashed curveonly those simulations reproducing the observed redshift distribu-tion; and the dotted curve (bottom panel only) only those simula-tions reproducing the NS-NS merger rate in spiral galaxies derivedfrom an analysis of binary pulsars (also shown on the bottom panel,in gray; see § should be weakly correlated, as the processes (e.g., com-mon envelope) which drive double neutron star to mergealso act on low-mass BH-NS binaries, albeit not alwayssimilarly; as a trivial example, mass transfer processesthat force binaries together more efficiently may depletethe population of NS-NS binaries in earlier evolution-ary phases while simultaneously bringing distant BH-NSbinaries close enough to merge through gravitational ra-diation. Thus, a simulation which contains enough BH-NS binaries for us to estimate its delay time distribution dP/dt need not have produced similarly many NS-NS bi-naries. For this reason, we constructed two independentsets of 500 “model universes”, one each for BH-NS orNS-NS models. However, as a corollary, the randomly-chosen simulations used to construct any given BH-NS“model universe” need not have enough merging NS-NSto enable us to calculate the present-day merger rate,and vice-versa. In particular, we never calculate thedouble neutron star merger rates in the BH-NS modeluniverse. Thus, though the BH-NS and NS-NS mergerrates should exhibit some correlation, we do not exploreit here. In particular, in the next section where we com-pare predictions against observations, we do not requirethe BH-NS “model universes” reproduce the present-dayNS-NS merger rate. Short GRB detection rate : As described in detail in § not seen depends strongly but simply on unknownfactors such as the fraction of bursts pointing towardsus, which we characterize by 1 /f b where f b is the beam-ing factor, and the fraction of bursts luminous enoughto be seen at a given distance, which we characterize by P ( > N d ) where N d = 4 πr k ( z )(1 + z ) F d is the mini-mum photon luminosity visible at redshift z . The shortGRB detection rate also depends on the detector , includ-ing the fraction of sky it covers (1 /f d ) and of coursethe minimum flux F d to which each detector is sensi-tive. To remove these significant ambiguities, in Figure8 we use solid curves to plot the distribution of detectionrates found for each of our 500 “model universes” (toppanel and bottom panel correspond to the BH-NS andNS-NS model universes, respectively), assuming (i) thatno burst is less luminous than the least luminous burstseen, namely, GRB 060505, with an apparent (band-limited) isotropic luminosity ˙ N min seen ≃ × s − , or L ≃ × erg s − (see Table 1); (ii) that beaming hasbeen “corrected”, effectively corresponding to assumingisotropic detector sensitivity and source emission; and(iii) that the detector has a peak flux detection thresh-old of F d = 1 cm − s − , corresponding roughly to thetrue BATSE and Swift peak flux thresholds presented in §
2. These choices are conservative; therefore, our rateestimate for each model universe is an upper bound . Short GRB redshift distribution : The short GRB detec-tion rate depends on the ability of gamma-ray detec-tors to distinguish burst events from background noiseand the GRB luminosity function. Its selection effectstherefore depend only on the properties of gamma-raytelescopes. On the other hand, the short GRB redshiftdistribution implicitly contains several other selection ef-fects which have not been factored into our analysis (e.g.,regarding the availability and reliability of an associationof each burst to a host and the ability of optical telescopeto extract a spectrum and redshift of that host). Lackingthe ability to characterize these selection effects, we per-form the most straightforward prediction for the distribu-tion of short GRB redshifts: using Eq. (14), we assumeall detected short GRB events have accurate and un-ambiguous optical follow-up and redshifts. Based uponthat assumption, we generate two sets of 500 candidateredshift distributions for short GRBs, assuming either(i) the set of merging BH-NS binaries or (ii) the set ofmerging NS-NS binaries is identical to (iii) the set of allshort GRBs. The results are shown in the top left panelsof Figures 9 (for NS-NS binaries) and 10 (for BH-NS bi-naries). Rather than provide all 500 cumulative redshiftdistributions, we have sorted these distributions by theirvalue at z = 0 . Fig. 8.—
Distribution of predicted all-sky, beaming-corrected short GRB detection rates log R D f b f d , if bursts arise from BH-NS(top) and NS-NS (bottom) mergers and all bursts produce a higher peak flux of 50-300 keV photons than the least-luminous burst inour sample (with ˙ N ≃ × s − ), compared to the observedBATSE (dashed vertical line) and Swift (dotted) all-sky detectionrate. In other words, a plot of the most optimistic predictionsof each two-component “model universe” for the burst detectionrate: assuming a detector has all sky visibility, that all burst emitisotropically with no beaming, and that no bursts are faint enoughto be missed. Almost all models produce at least as many burstsas are observed; all models can therefore reproduce observations bychoosing some low minimum peak photon luminosity ˙ N min , typi-cally 10 -10 times lower than the one of the faintest burst seen. Asin Figure 7, the solid curves include all 500 “model universes”; thedashed curves include only those “model universes” which repro-duce the short GRB redshift distribution; the dotted curve (bottompanel only) includes only models which are reasonably consistentwith the present-day double neutron star merger rate in the MilkyWay; and the dot-dashed curve (top panel only) includes only thosesimulations which, under the most optimistic assumptions, predictshort GRBs should occur at least as frequently as has been seen. Fraction of mergers in spiral galaxies : Finally, in Figure11 (solid curves) we show the a priori probability thata fraction f s of binary mergers occur in spiral galaxies,using the scaled variable X = arctanh(2 f s − f s that are very near 1 or 0;to give a sense of scale, X = 1(2) corresponds to 88%(98%) of all mergers occurring in spiral galaxies.As seen in Figure 11, a priori we cannot say whetherelliptical or spiral galaxies should host most presently-occurring binary mergers. Furthermore, a significantfraction of models have | X | > f s can be traced back directly to thelarge range of mass efficiencies shown in Figure 4. Fora randomly chosen pair of population synthesis modelfor elliptical and spiral galaxies, one will often be signif-icantly larger than the other, leading to a spiral fraction f s near its limits (i.e., f s ≃ ,
5. TESTING PREDICTIONS AGAINST OBSERVATIONS
To summarize, in this paper we are trying to examinewhether either of two very simple hypotheses are consis-tent with what we have heretofore seen of short GRBsand binary mergers are consistent with our understand-ing of the star formation history of the universe and ofbinary stellar evolution. The two simplest hypothesisare, on the one hand, that every short GRB is a binaryBH-NS merger and that every BH-NS merger producesa short GRB; and on the other hand the correspond-ing statement for NS-NS binaries. Further, assumingthat in both cases some fraction of all possible modelsare consistent with observations, we want to understandthe properties of those consistent models and the physi-cal processes that require those properties to be as theyare. Of course, these comparisons are affected by obser-vational selection effects. Notably, Berger et al. (2006)has suggested that the lack of deep optical follow-up onfaint short GRBs has biased the redshift distributionto low redshift. Lacking any ability to control an un-known bias, we proceed with the simplest possible test:we compare existing results to currently known obser-vations. Specifically, we require our predictions for thebinary neutron star merger rate (Figure 7, available onlyfor binary neutron star short GRB “model universes”),for the short GRB detection rate (Figure 8), and for theshort GRB redshift distribution (Figures 9 and 10) bestatistically consistent with the corresponding observa-tions, all of which are shown in their respective plots.We then compare the distribution of constrained quan-tities (in the aforementioned Figures and in Figures 11and 12) with their initial distributions, to understand thephysics implied by observations. Comparisons I: NS-NS as burst source
No definitive evidence exists to determine the fractionof short bursts that are less luminous than the faintestcurrently known; to discover the degree to which, if any,short GRBs are beamed ; and to demonstrate that allmergers must produce bursts. Therefore, we must take We start with the simplest and most tractable pair of hy-potheses. A more realistic model would allow for a fraction ofboth BH-NS and NS-NS mergers to produce short GRBs, alongwith some proportion of young neutron stars (in bursts from softgamma repeaters, commonly abbreviated SGRs). However, withso many degrees of freedom, such a model would be very difficult toconstrain without additional observational inputs (e.g., direct con-firmation of the fraction of mergers which are SGRs in the nearbyuniverse; a reliable measure of the fraction of mergers that occur inelliptical and spiral galaxies; gravitational wave detection of mergerevents; etc.). Because of the limited number of constrained models and thelarge number of underlying parameters (fifteen), we do not attemptto characterize the underlying physical parameters of constraint-satisfying models. Instead, we focus on describing the essentialfeatures of models, such as a long characteristic delay time or highmass efficiency, that allow a model to satisfy observational con-straints. Evidence has been presented to suggest that short GRB emis-sion has a break in one or more frequencies, a result that has been Fig. 9.—
Demonstration that population synthesis models can reproduce short GRB redshift distributions, assuming NS-NS mergersproduce all short GRBs. As in Figure 10, the top right and left panels compare the range of short GRB redshift distributions expectedfrom our two-component model : 1%/99% (dotted), 10%/90% (dashed), and 25%/75% (solid) redshift distributions are overlaid on theobserved cumulative redshift distribution, both for all simulated models (top left) and those models which remain everywhere close to theobserved redshift distribution (bottom left). Top right: As above, but including only those NS-NS models which produce a NS-NS mergerrate in agreement with observations of the Milky Way merger rate (Figure 7). Bottom right: As above, but including only those NS-NSmodels which reproduce the number of binary pulsars seen in the Milky Way and the short GRB redshift distribution. the bottom panel of Figure 8 at face value: if no burstsare less luminous than those seen, every short GRBmodel based on NS-NS produces many more short GRBsthan are seen, and therefore every double neutron star“model universe” can reproduce the present-day shortGRB detection rate by a proper choice of, for example,the minimum luminosity of short GRBs. Therefore, only two of the three available observations – the short GRBredshift distribution and the set of merging Milky Waybinary pulsars – can reject some of the 500 double neu-tron star “model universes.”The statistics of this comparison are summarized inFigure 9. Relatively few of our double neutron starmodels are consistent with observations of merging dou-ble pulsars in the Milky Way : 78 out of 500 models,or ≃ interpreted as a “jet break” produced by a beamed jet (see,e.g.,Soderberg et al. 2006b; Grupe et al. 2006b; Nakar 2007, and ref-erences therein). At present we choose to remain conservativeregarding beaming until several multi-band observations confirmthese breaks exist. Fewer still would be consistent should we require agreementwith more binary pulsar rate constraints, as has been shown inO’Shaughnessy et al. (2005) and O’Shaughnessy et al. (2007b). tom left panel of Figure 9. Therefore 35 out of 500,roughly 7% of all models, appear fully consistent withall observations considered here and with the assumptionthat short GRBs are produced exclusively by all NS-NSmergers.Because observations of the Milky Way suggest a NS-NS merger rate towards the high end of what our sim-ulations produce (see the bottom panel of Fig. 7), andbecause spiral star formation extends through the recentuniverse – that is, precisely into the time intervals duringwhich a significant fraction of short GRBs have been ob-served (see Figure 2) – to an excellent approximationwe can conclude that, if short GRBs are due to NS-NS mergers, the best-fitting models of all observations produce mergers and short GRBs preferentially in spiralgalaxies , with rapid mergers following quickly upon re-cent star formation. More specifically, we can genericallydraw the following explicit conclusions:
High spiral mass efficiency needed : As indicated in Fig-ure 7, only a few populaton synthesis simulations canproduce as many merging NS-NS binaries in spiral galax-ies as does a natural extrapolation of the known MilkyWay NS-NS merger rate. The few “model universes”which reproduce these high merger rates necessarily havenon-typical high mass efficiencies λ s for forming doubleneutron stars in spiral galaxies. High spiral fraction preferred : As a consequence of suchhigh mass efficiencies λ s – that is, because of the highrate at which the spiral galaxies in these “universes”4 Fig. 10.—
Demonstration that population synthesis models can reproduce short GRB redshift distributions, assuming BH-NS mergersproduce all short GRBs. The jagged curve and shaded regions provide the cumulative redshift distribution for observed short GRBs (Fig.2). Top left panel: The smooth curves illustrate the range of short GRB redshift distributions; out of the 500 simulated, the two solidcurves correspond to the curves with the 25th and 75th percentile values of P (0 . .
38 (i.e., a 5% Kolmogorov-Smirnov false alarm probability). Top right panel: Aspreviously, but including only those model which, when given the most favorable assumptions, still predict too few short GRB detections.Bottom right panel: As previously, but requiring both the predicted short GRB rate and redshift distribution be simultaneously consistentwith GRB observations. produce NS-NS systems – most mergers occur in spirals;equivalently, the spiral fraction f s of constraint-satisfyingmodels is often strongly biased towards spiral galaxies( f s > / Low spiral fraction implies long elliptical delays : Con-versely, in order for elliptical galaxies to host a signif-icant fraction of mergers, those elliptical galaxies mustproduce binaries with an unusually long characteristicdelay between birth and merger. For example, in the right two panels of Figure 12 compare the prior (top) andconstrained (bottom) distribution of spiral fraction andmedian delay time t (50%) between birth and merger. Asseen in the top panel, the most common delay betweenbirth and merger in a randomly chosen elliptical popu-lation synthesis simulation is around 8 Gyr; however, asseen in the bottom panel, for the handful of “model uni-verses” which have fewer than 80% of double neutron starmergers born in spiral galaxies, the median characteristicdelay time is at least an order of magnitude larger.5.2. Comparisons II: BH-NS as burst source
On the one hand, for purely technical reasons describedin § do predict too few bi-nary mergers to reproduce observations, even given themost optimistic assumptions regarding beaming and theminimum luminosity of short GRBs; see Figure 8. Fur-ther, as shown in Figure 10, a significant fraction of red-shift distributions (183 out of 500, or ≃ Mergers usually in spirals : Just as with NS-NS models,5
Fig. 11.—
Distribution of the fraction of BH-NS mergers (toppanel) and NS-NS mergers (bottom panel) expected to occur inspiral galaxies at the present day [Eq. (11) evaluated at z = 0].Solid curves include all models; dashed lines include only thosemodels reproducing the redshift distribution; dotted line includesonly those models reproducing the merger rate of NS-NS binariesin spiral galaxies, based on the Milky Way; and the dot-dashedline includes only models which could possibly produce as manyshort GRB events as are observed. To better indicate a very highor very low fraction f s of mergers occurring in spirals, we plot thedistributions above versus X = tanh − (2 f s − though a priori our “model universes” are equally likelyto produce mergers and short GRBs in elliptical and spi-ral galaxies, the set of “model universes” which repro-duce a short GRB redshift distribution that is dominatedby recent mergers is biased towards spiral-dominatedmodels : compare the solid and dashed curves in the toppanel of Figure 11. However, unlike the NS-NS casewhere both the redshift distribution (preferring recentstar formation) and double neutron star merger rate (re-quiring very high merger rates in spiral galaxies) limit usto “model universes” that produce extremely many merg-ing binaries, for BH-NS “model universes” only the red-shift distribution carries any information that can biasresults in favor of spiral galaxies [Fig. 11]. In particu-lar, as illustrated by Figure 7, the distribution of spiral-galaxy BH-NS merger rates remains nearly the same nomatter what constraints have been applied. Ellipticals can be hosts only with significant characteristicdelays : Further, as one expects on physical grounds, el-liptical galaxies can contain a significant fraction of shortGRBs and mergers only when unusually long character-istic delays between birth and merger are involved; seethe left two panels of Figure 12.Unlike the double neutron star case, a significant frac-tion of “model universes” predict low spiral fractions f s < .
8. In order for old elliptical galaxies to host a
Fig. 12.—
A mixed contour and scatter plot of the spiral fraction f s versus the characteristic delay t (50%) between birth and mergerin elliptical galaxies, for BH-NS models (left) and NS-NS models(right). The top panels include all 500 models; the bottom panelsprovide only those models which satisfy both constraints (for BH-NS simulations, the 137 “model universes” which are consistentwith the short GRBs redshift distribution and detection rate; forNS-NS simulations, the 35 “model universes” which are consistentwith the short GRB redshift distribution and the number of merg-ing binary neutron stars in the Milky Way). This figure illustrates(i) that the vast majority of constraint-satisfying models have mostmergers in spirals [also illustrated in Figure 11] and more critically(ii) that the few models which permit most mergers to occur inelliptical galaxies are slightly biased towards longer characteristicdelay times t (50%). number of mergers similar to the young spiral galax-ies, the binaries in old elliptical galaxies must surviveover rather long times – many Gyr after their forma-tion. Comparatively speaking, population synthesis sim-ulations of BH-NS binaries produce more systems withthe required characteristic long delay time than do sim-ulations of NS-NS binaries; see for example the fractionof simulations in the top right panels in either Fig. 6 orthe top left panel of 12 with t (50%) greater than 10 Gyr.5.3. What fraction of mergers produce short GRBs?
So far, we have required our “model universes” to pro-duce when given the most favorable assumptions regard-ing burst luminosity distribution and beaming at leastas frequent short GRB detections as are observed, asshown in Figure 8. This fairly weak constraint is im-posed because, under perfectly plausible but less opti-mistic assumptions, a great many short GRBs could bemissed. However, the difference in Figure 8 between, onthe one hand, the vertical lines indicating the observedshort GRB detection rate and, on the other hand, anoptimistic prediction f d f b R D can be reinterpreted as the fraction of short GRBs that must be missed in order forour predictions to correspond with observations.By way of example, if a NS-NS “model universe” corre-sponds to an optimistic detection rate of order 10 yr − ,then only 1 of every 100 NS-NS mergers could produce6short GRBs brighter than the least luminous seen andaimed in our direction. This factor of 1 /
100 could beproduced by any one or combination of several factors,all of which act to decrease the predicted short GRB de-tection rate below these optimistic predictions: (i) beam-ing, since the predicted detection rate is proportional to f b and our optimistic calculations assume f b = 1; (ii)fainter short GRBs, since the detection rate is also pro-portional to the photon luminosity ˙ N min of the faintestshort burst (i.e., ∝ ˙ N min / ˙ N min seen ); (iii) some intrinsicphysics which prevents all but a select few mergers toproduce detectable bursts; or even (iv) further changes inour population synthesis model, such as assuming fewerthan 100% (our present choice) of all stars are born inbinaries.No incontrovertible evidence exists that requires anyof these factors be less than unity. However, there isgood observational and theoretical motivation for re-evaluating our detection rate constraint using a prioron the product of all these factors: essentially, while allcould be nearly unity, good reasons exist to imagine thatseveral may be significantly less than 1. For example,(i) Soderberg et al. (2006a) and others have argued thatshort GRBs may be beamed. Further, (ii) the least lu-minous burst seen is nearly at our detector’s sensitivitylimit, yet short GRBs peak flux distribution is a fea-tureless power law. Since only a remarkable coincidencecould produce such a fortuitous combination of detectordesign and source strength that this burst is indeed at ornear the intrinsic burst sensitivity limit, we can reason-ably expect the minimum-luminosity burst to be signifi-cantly less luminous than the faintest burst yet seen. Ad-ditionally, as described in Belczynski et al. (2007), (iii)theoretical simulations of BH-NS mergers could possiblyproduce short GRB merger events only for a limited ar-ray of binary component masses and spins; the fractionof mergers which could be short GRBs ranges from 1/3to 1/50, depending on the BH birth spin. Finally, (iv)based on comparison with the present-day Milky Way,the binary fraction could be up to a factor two lowerthan the value we assume; e.g., Belczynski et al. (2007)adopts a binary fraction of 50%. If each of these fourfactors is decreases the fraction of short GRB detectionsbelow our predictions by only factor of roughly 3, thenour expectations for the short GRB detection rate cor-responding to a given “model universe” should be lowerby nearly a factor of 100!Adopting a canonical factor of 1/100 to transform theoptimistic detection rates presented in Figure 8 into a“realistic” expectations leads to a dramatic transforma-tion of our understanding. On the one hand, extremelyfew BH-NS “model universes” would produce short GRBevents as frequently as are observed. On the other, these“realistic” assumptions cause the 35 constraint-satisfyingNS-NS “model universes” to predict roughly as manyshort GRB events as are observed (i.e., imagine shiftingthe dotted peak in Figure 8 to the left by 2 orders of mag-nitude). While this tantalizing correspondance could bea coincidence and while the precise results of our “realis-tic” prior cannot be taken too seriously, they do remindus of two salient features of our predictions: (i) that BH-NS “model universes” are consistent with the data onlygiven relatively optimistic assumptions, and that they quickly become inconsistent with observations if thoseassumptions are relaxed; and (ii) that on the other handbecause NS-NS “model universes” can allow for so manydetections and therefore have much more flexibility inthe fraction of short GRBs that are missed, and becausedouble neutron star models are otherwise entirely consis-tent with all existing observations, double neutron starmodels for short GRBs remain an attractive candidateexplanation for short GRBs. Comparison to related studies : Belczynski et al. (2007)(hereinafter BTRS) also used the
StarTrack popula-tion synthesis code and hydrodynamical simulations ofBH-NS mergers with BH spins (Rantsiou et al. 2007) todetermine whether BH-NS merger events occurred fre-quently enough, with the right combinations of parame-ters, to reproduce short GRB observations. In contrastto the present broad study, which relies only on observa-tional constraints, this study adopts several of the “re-alistic” priors mentioned above to reduce the fractionof merger events that produce short GRBs. Specifically,this paper relies on well-motivated hydrodynamical stud-ies to argue that at best a small fraction of BH-NS merg-ers (1 / /
50, depending on black hole birth spin) willproduce burst events. Additionally, the BTRS simula-tions rely on a single “most-plausible” set of populationsynthesis parameters, including in particular a high com-mon envelope efficiency αλ . As a result, compared tothe wide range of common-envelope efficiencies used inthis study, BTRS find relatively low BH-NS merger rates.We also note that BTRS assume a 50% binary fraction,lower than the 100% used here. Combining these threefactors, BTRS conclude that BH-NS mergers occur tooinfrequently (less than 10 Gpc − yr − ) to explain shortGRB merger rates.
6. CONCLUSIONS
In this paper, we used a large archive of concrete, cur-rent population synthesis calculations to generate mergerrate densities for NS-NS and BH-NS binaries. Using as-sumptions regarding short GRB source luminosities anddetection selection effects, we then compared these ratedensities to short GRB detection rates and redshift dis-tributions, as well as to (when available) the present-dayMilky Way binary neutron star merger rate. Whetherassuming short GRBs arose from either BH-NS mergersor NS-NS mergers, a small but still substantial fractionof models were consistent with existing observations us-ing the most optimistic assumptions for certain priors(i.e., no beaming and all stars born in binaries). Con-trary to an earlier study by Nakar et al. (2005), we havedemonstrated by using a two-component star formationmodel that exceptionally long characteristic delays be-tween binary birth and merger are not uniquely requiredneeded to reproduce the short GRB redshift distributionand detection rate, though they are strongly preferredif we additionally demand a significant fraction of short The common envelope efficiency reflects the fraction of orbitalenergy needed to eject the envelope; since most BH-NS binaries gothrough a common-envelope phase, a low efficiency implies thesebinaries will have a tight final orbit. Additionally, the simulations used in this paper allowed muchless mass accretion onto black holes during the common-envelopephase. Based on studies conducted in our group, this change doesnot produce a dramatic difference in merger rates . spiral-dominated models : f s > .
7. Doubleneutron star models reproduce observations only if spi-ral hosts dominate; BH-NS models can permit a widerange of spiral fractions. Though we have not imposedit as a constraint, the fraction of short GRBs in spiralhosts potentially provides an extremely powerful addi-tional constraint on source models. For example, thesix host identifications shown in Table 1 suggest the ob-served spiral fraction ˆ f s is 50%. However, to impose itreliably requires a careful study of the systematic uncer-tainties in the two-component star formation model usedas well as of the selection effects in host galaxy identifica-tions. For example, more massive elliptical galaxies willmore efficiently retain any strongly-kicked neutron starbinaries but have less gas into which a the short GRBblast wave can collide and produce an afterglow; theseand other selection effectsGenerally speaking, the exceedingly small number ofwell-identified short GRBs strongly limits our ability todraw conclusions based on comparisons to properties ofthat set, such as to the redshift distribution or to the frac-tion of bursts seen in spiral hosts. More well-identifiedhosts are needed; additionally, these hosts should hope-fully be drawn from a less-biased sample than the presentsample appears to be (see Berger et al. 2006). Of course,we encourage deep follow-up searches for afterglows, par-ticularly since our model universes almost always predicta higher proportion of high-redshift short GRBs thanhas yet been observed. We particularly encourage thedevelopment of thorough redshift-limited surveys, sincea determination of the relative proportion of bursts instar-forming or elliptical hosts can be done in the nearbyuniverse has significant potential to improve our under-standing of the formation mechanism.Though the limited sample of well-characterized shortGRBs currently limits our ability to constrain inputphysics, we expect that the significant increase inevent statistics expected over the next few years willmake these events a primary mechanism to constraindouble compact object merger rates and the associ-ated astrophysics. For example, with well-understoodshort GRBs already outnumbering the few known dou- ble neutron stars in our galaxy, short GRBs wouldsoon provide the most precise (nonparametric) obser-vational constraint on compact object merger rates (cfKim et al. 2003), provided the selection effects are quan-titatively understood. In turn, these constraints informus about binary stellar evolution (O’Shaughnessy et al.(2007b),Belczynski et al. (2006a)). Additionally, sinceshort GRB sources would also be copious emit-ters of gravitational waves, Swift and ground-basedgravitational-wave observatories operating in coincidencecould conceivably probe the details of an unusually closemerger event itself (see,e.g., Kobayashi & M´esz´aros 2003;Finn et al. 2004; Nakar et al. 2005). LIGO in particu-lar is both already operating at design sensitivity andconducting triggered searches from GRB observations(Abbott & et al 2005). Given its compelling qualitativeagreement and far-reaching scientific impact, the mergerhypothesis deserves a detailed comparison against thebest possible models, to either corroborate it, definitivelydisprove it, or discover weaknesses in our assumptionsand models.Finally, the current sample does not yet allow usto quantify the minimum luminosity and beaming ofshort GRBs. However assumptions regarding the frac-tion of merger events which do not produce short GRBsbrighter than the weakest burst seen so far and point-ing towards us combined with theoretical priors (see,e.g.Belczynski et al. 2007), indicate that BH-NS mergers donot occur frequently enough to explain all short GRBs.We anticipate that future larger samples will allow us toplace stronger constraints on the merger models. Acknowledgments—
We would like to thank DonLamb, Ehud Nakar, Avishay Gal-Yam, Re’eem Sari,Arieh Konigl, Shri Kulkarni, Neil Gehrels, and the par-ticipants of the Ringberg Short GRB conference of 2007for helpful comments during this paper’s long gestation.This work was partially supported by NSF award PHY-0353111 and a Packard Fellowship in Science and Engi-neering awarded to VK, and grants KBN 1P03D02228and 1P03D00530 to KB.
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APPENDIX
ARCHIVE SELECTION
The predicted short GRB detection rate depends directly on the fraction of star forming mass λ predicted to endup in GRB progenitors. However, the mass efficiency λ can vary substantially depending on model assumptions. Inorder to constrain the range of short GRB detection rates expected (for a fixed minimum short GRB luminosity ˙ N ),0 Fig. A13.—
For spiral (left two panels) and elliptical (right to panels) archives, a scatter plot of the number of NS-NS (top) and BH-NS(bottom) binaries n seen in a simulation of N progenitor binaries (all of m > M ⊙ , with primary mass drawn from the broken-KroupaIMF). The two solid lines show the cutoffs n >
20 and nN > × imposed to insure data quality and reduce sampling bias. Simulationsused in this paper, shown in black, must lie above and to the right of these cutoffs. we require the unbiased distributrion of λ . However, the large variety of simulation sizes in our simulations introducesa bias in our estimate of the mass efficiency distribution – or, equivalently, in the distribution of n/N for n the numberof binaries seen and N the number of binaries simulated. Specifically, not all of our archived population synthesissimulations contain enough of each type of event (indexed by K ) to produce reliable predictions involving it. The set ofsimulations with n greater than any fixed threshold threshold (including n = 0) is biased, over-representing simulationswith high n/N ; see for example Figure A13. Since the mass efficiency λ is directly proportional to n/N [Eq. (6)], wedescribe a filter on n and N which preserves the distribution of n/N (for the distribution of λ ) and simultaneously insures that the average simulation has many binaries ( h n i > dP/dt canbe accurately estimated).Formally, each population synthesis simulation converts some unknown fraction s of progenitor binaries into targetevents n = sN . In practice our population sample size N is roughly randomly chosen, independent of the unknown s .In other words, we expect the distribution of n and N to derive from two independent distributions for N and s , withdensities f and g : dP (ln N, ln n ) = f (ln N ) g (ln( n/N )) d ln N d ln n = f (ln N ) g (ln n/N ) d ln √ N nd ln n/N (A1)The distribution of s = n/N can be quickly extracted from any two-dimensional distribution in n, N by: gd ln s = Z δ (ln( s/ ( n/N ))) dP . (A2)However, the above method assumes the distribution perfectly resolved. In practice simulations are generally not repeated, so “fractional” and small n cannot be resolved by repeated trials; instead, only simulations with s ≫ /N will produce enough events to provide reliable estimates of s , and thus its distribution. This truncation effect biasesour reconstruction of the two-dimensional density and therefore our reconstruction of the mass efficiency distribution.If we had perfect foreknowledge, however, we could have chosen our sample size so nN was constant and large. Thoughwe would choose n and N in a highly correlated fashion, we would guarantee each point was well-resolved; our methodabove would correctly reconstruct the distribution of s . Hence choosing all population synthesis archives with nN greater than any threshold will allow accurate estimation of the distribution of s .Based on the above discussion and the observed distributions in n and N , we introduce cutoffs on n ( >
20) and nN ( > × ) which remove the least resolved simulations from consideration (via the n cutoff), reduce the biasassociated with a minimum n (via the nN cutoff), and insure that most simulations have at least 200 binaries (both).1 Fig. B14.—
Results for errors in estimates of P ( t m ) and dP m /dt , versus the number of points used in the estimate, for the trialdistribution function dP m /dt = . t between 30 and 10 Myr. Left: Plot of the average and 90% supremum-norm error versus n for P m ( t ), along with fits to these two quantities (namely, n − . / . n − . , respectively). Right: Plot of the average and 90% bound onfractional error in dP m /dt versus the number of points available, along with fits to these two quantities (namely, 0 . n − . and 0 . n − . ,respectively). ESTIMATING DELAY TIME DISTRIBUTION
Poisson errors associated with the limited sample size of our population synthesis simulations inhibits our abilityto reconstruct the delay time distribution, whether represented as a cumulative distribution P m ( t ) ≡ P m ( < t ), theprobability that a delay between birth and merger is less than t , or as dP m /dt . Using classical statistical methods –see, e.g., Merritt (1994), Thompson & Tapia (1976), and references therein – we smooth the n observed delay timesto build our estimates for P m ( t ) and dP m /dt . This appendix briefly reviews those methods and the accuracy of theresulting estimates. Estimating cumulative distribution : We estimate the cumulative distribution P ( < t ) for a sample of populationsynthesis events by a function ˆ P ( t ) that smoothes sample dataover a short smoothing length s in l t ≡ log ( t/ Myr):ˆ P s ( l t ) ≡ n X k =1 Θ s ( l t − l t,k ) (B1)Θ s ( x ) = 12 (cid:16) x/s √ (cid:17) (B2) s = [(max k l t,k ) − (min k l t,k )]10 n . (B3)for n the number of events in our population synthesis sample, k = 1 . . . n an index over tat same sample, and l t,k = log( t k / Myr) the logarithm of the delay t k between binary formation and merger for the k th binary. Theextremely short smoothing length used here approximately minimizes the average difference between predictions andresults.To test this approach, for several n we drew many Monte Carlo samples of n delay times from a canonical cumulativedistribution: P m ( l t ) = lt − log(30)7 − log(30) Θ( l t − log(30))Θ(7 − l t )Figure B14 shows the average and 90% distance max l t | P m ( l t ) − ˆ P m ( l t ) | between our function and our fit versus thenumber of points smoothed to estimate P m ( t ).Given the archive selection procedure presented in A, each archive typically contains of order n ≈
100 mergingbinaries. For such a typical archive, our smoothing method will reconstruct P ( t ) almost everywhere to better than 5%,with the largest errors typically arising near the largest and smallest delay times. As expected, the maximum erroragrees perfectly with the distance between observations and data for a Kolmogorov-Smirnov 95% hypothesis test. Estimating differential distribution : Similarly, we estimate dP m /dt by ˆ˙ P m, s , defined using the time derivative of P m ( < t ) after converting from logarithmic to physical time:ˆ˙ P m ≡ t ln 10 d ˆ P m dl t = 1 t ln 10 n X k =1 p π ( s ) e − (log t − l t,k ) / s ) (B4) s ≡ [(max k l t,k ) − (min k l t,k )]1 . √ n (B5)2Compared to the previous case of estimating the cumulative distribution P m ( t ), where a significant fraction of allsampled points affect the estimate at any t , a significantly longer smoothing length s is required to estimate dP m /dt ,since roughly only those few sample points within s of t contribute to our estimate. As a result, we cannot confidentlyestimate dP m /dt more accurately than ≈ dP m /dt , we applied it to the trial problemmentioned above. To estimate the mean relative error I associated with our estimate in the physically pertinent interval of roughly 100 Myr to 15 Gyr by I ≡ "Z . . dl t . | ˆ˙ P − ˙ P | ( ˆ˙ P + ˙ P ) / / , (B6)which generalizes an rms measurement of the relative error in P to allow for large relative errors [i.e., when ˆ˙ P m =˙ P m (1+ δ ( l t )), then I = ( R δ dl t / ∆ l t ) / ]. The second panel of Figure B14 demonstrates that large errors are inevitable,even with many sample points and using a smoothing length s which is approximately optimized for each n . Whilethis slow convergence is a familiar problem for all density estimators (see,e.g. Thompson & Tapia 1976), it severelylimits the ability of any population synthesis simulation to reliable estimate dP m /dt , given the severe computationallimits involved. By way of example, a delay time distribution accurate to 5% would require roughly 6 ≈ Implications for short GRB predictions : The short GRB detection rate, redshift distribution, and elliptical-to-spiralratio all depend on estimates of dP m /dt . However, while our reconstruction of dP m /dt for any fixed populationsynthesis model involves substantial uncertainties, these are not much greater than our corresponding uncertainty inour estimate of that model’s the mass efficiency λ . Further, these uncertainties are vastly less than the systematicuncertainty involved in not knowing the physically appropriate population synthesis model (i.e., roughly two orders ofmagnitude uncertainty in λλ