Deducing the Lifetime of Short Gamma-Ray Burst Progenitors from Host Galaxy Demography

a r X i v :a s t r o -p h /0601622v 2 7 M a y 2007

Draft version February 5,2008

Preprint typeset using L A T E X style emulateapj v.10/09/06

DEDUCING THE LIFETIME OF SHORT GAMMA-RAY BURST PROGENITORS FROM HOST GALAXY

DEMOGRAPHY

Zheng Zheng 1,2and Enrico Ramirez-Ruiz 1,3

Draft version February 5,2008

ABSTRACT

The frequency of short gamma-ray bursts (GRBs)in galaxies with distinct star formation histories can be used to constrain the lifetime of the progenitor systems.As an illustration,we consider here the constraints that can be derived from separating the host galaxies into early and late types.On average,early-type galaxies have their stars formed earlier than late-type galaxies,and this di?erence,together with the time delay between progenitor formation and short GRB outburst,leads to di?erent burst rates in the two types of hosts.Presently available data suggest,but not yet prove,that the local short GRB rate in early-type galaxies may be comparable to that in late-type galaxies.This suggests that,unlike Type Ia supernovae,at least half of the short GRB progenitors that can outburst within a Hubble time have lifetimes greater than about 7Gyr.Models of the probability distribution of time delays,here parametrized as P (τ)∝τn ,with n ?1are favored.This apparent long time delay and the fact that early-type galaxies in clusters make a substantial contribution to the local stellar mass inventory can explain the observed preponderance of short GRBs in galaxy clusters.Subject headings:gamma rays:bursts –stars:formation –cosmology:observations –galaxies:formation

1.INTRODUCTION

The progenitors of short duration,hard spectrum,gamma-ray bursts (GRBs)are not yet well identi?ed.Even with the recent localizations of a handful of short-hard GRBs (Bloom et al.2006;Gehrels et al.2005;Villasenor et al.2005;Berger et al.2005;Fox et al.2005;Berger et al.2006;Berger 2007),no transient emission has been found that directly constrains the na-ture of the progenitor system.The current view of most researchers is that GRBs arise in a very small fraction of stars that undergo a catastrophic energy release event to-ward the end of their evolution (e.g.,Mochkovitch et al.1993;Lattimer &Schramm 1976;Eichler et al.1989;Narayan,Paczy′n ski &Piran 1992;Ru?ert &Janka 1999;Klu′z niak &Lee 1998;Rosswog &Ramirez-Ruiz 2003;Rosswog,Ramirez-Ruiz &Davies 2003;Setiawan et al.2004;Lee et al.2004;Aloy et al.2005;MacFadyen et al.2005;Levan et al.2006;Lee &Ramirez-Ruiz 2007).Much of the current e?ort is dedicated to understanding the di?erent progenitor scenarios and trying to determine how the progenitor and the burst environment can a?ect the observable burst and afterglow characteristics (e.g.,Lee et al.2005).The lifetime of progenitors of short bursts can be meaningfully constrained by properties of their host galaxies (e.g.,Fox et al.2005;Gal-Yam et al.2005).We suggest here a method to constrain the lifetime of the progenitors by making use of the star formation histories of their host galaxies.

Based on the short burst afterglows localized so far (Berger 2007),two of the three relatively nearby (z 0.3)events (GRB 050509b and GRB 050724)are plau-sibly associated with galaxies exhibiting characteristic early-type spectra (Berger et al.2005;Bloom et al.2006;

1

Institute for Advanced Study,Einstein Drive,Princeton,NJ 08540;zhengz,enrico@https://www.sodocs.net/doc/2c5268363.html, 2Hubble Fellow 3Chandra Fellow

Gehrels et al.2005;Prochaska et al.2006).In the other case (GRB 050709),the host galaxy shows signature of a dominant stellar population with age of ～1Gyr (Covino et al.2006)and it also exhibits strong emission lines that indicate ongoing star formation (Covino et al.2006;Fox et al.2005;Prochaska et al.2006).There is also independent support that,at least two of the nine short bursts localized so far (Berger 2007)are associated with clusters of galaxies (Bloom et al.2006;Gladders et al.2005;Pedersen et al.2005).In contrast to what is found for long-soft GRBs,for which all of the con?rmed host galaxies are actively forming stars (e.g.,Trentham et al.2002;Christensen et al.2004),these ob-servations clearly signify that,like Type Ia supernovae,short GRBs are triggered in galaxies of all types.What is more,it indicates that there is a time delay between short burst occurrence and the main epoch of star formation activity in the hosts,as determined by the progenitor’s lifetime.

In this paper,we present a method to deduce the life-time of short GRB progenitors from the burst rates in host galaxies of di?erent types that have distinct star formation histories.Limited by the accuracy of how well we can separate their contribution to the star formation history,we show,with this constraint in mind,how a large sample host galaxies could be used to determine the lifetime distribution of short GRB progenitors.Here we chose to separate the formation history of stars re-siding in early-and late-type galaxies,but,as shown in §5,our study can be easily generalized to any set of sub-populations of galaxies possessing distinct star formation histories.The layout is as follows.In §2,we review the local stellar budgets in early-and late-type galaxies.The total star formation history is subsequently decom-posed in §3into a sum of the stars that assembled in today’s early-and late-type galaxies.With the decom-position in place,in §4,we show how the frequency of short GRBs in a well-de?ned sample of host galaxies of di?erent types can be used to constrain the lifetime of

2Zheng &

Ramirez-Ruiz

Fig.1.—The local stellar mass density from galaxies of di?erent types.The histogram with errorbars is for galaxies of all types while the dashed curve is a Schechter function ?t to the stellar mass function of the entire galaxy sample.Red and blue histograms are for early-and late-type galaxies,respectively.The early-and late-type galaxies are divided according to their color (dotted )or light concentration (solid ).This plot is based on the galaxy stellar mass functions derived by Bell et al.(2003).

the progenitors.Finally,in §5we summarize our re-sults and outline future prospects.Throughout this pa-per,we assume a spatially-?at ΛCDM cosmology with ?m =1??Λ=0.3and the Hubble constant h =0.7in units of 100km s ?1Mpc ?1.

2.THE LOCAL STELLAR MASS INVENTORY

The focus of this paper is the relatively local (z ～0)host galaxy population,but our study can be easily gen-eralized to any redshift.The main idea is to use the di?erence in the star formation history of galaxy sub-populations to probe the lifetime distribution of short GRB progenitors.As an example,here we use the event rate of burst residing in local early-and late-type galax-ies.On average,early-type galaxies have their stars formed earlier than late-type galaxies,and this di?er-ence,together with the delay time for short GRB out-burst,leads to di?erent burst rates in these two types of galaxies.As shown in §3,our method of inferring the star formation history of a given population relies on tracing the assembly’s history of the stellar mass in galaxies at z =0.For this reason,in this section,we brie?y review the local stellar mass budgets in galaxies of di?erent types.

With a universally applicable stellar initial mass func-tion (IMF),the stellar mass function (MF)of galaxies can be estimated from well-de?ned samples of galax-ies.Galaxy MFs have been measured based on a few large galaxy redshift surveys (e.g.,Kau?mann et al.2003;Bell et al.2003).For the calculations presented here,we adopt the galaxy MF measured by Bell et al.(2003)using a large sample of galaxies from the Two Mi-cron All Sky Survey (2MASS;Skrutskie et al.2006)and the Sloan Digital Sky Survey (SDSS;York et al.2000).The Bell et al.catalog provides well-de?ned samples of galaxies of di?erent types as de?ned by either the light concentration or the color of galaxies.

In Figure 1,we show the stellar MF,φ,multiplied by the stellar mass,M star ,for galaxies of di?erent types.The quantity ρstar (M star )=M star φ(M star )is the stellar mass density contributed by galaxies with stellar mass M star .The peak of ρstar for galaxies of all types comes from M ?galaxies,where M ?～1011M ⊙is the charac-teristic stellar mass derived from the Schechter function ?t.The stellar mass density distribution for late-type galaxies peaks at masses slightly less than M ?and has a tail extending to low stellar masses.Early-type galaxies have a narrow stellar mass density distribution around M ?and dominate the local stellar mass budget above a few times 1010M ⊙.Although early-type galaxies are far less in total number than late-type galaxies,they are on average more massive and thus make a larger contribu-tion to the local stellar mass density.The total stellar mass from early-type galaxies is about 1.3(2.3)times that from late-type galaxies,if the classi?cation of galaxy types is based on light concentration (color).

If early-and late-type galaxies had similar star forma-tion histories,then the ratio of the short GRB rates in galaxies of di?erent types would simply be given by the ratio of the stellar mass density in these galaxies.How-ever,the two types of galaxies have distinct star forma-tion histories –on average,stellar populations in early-type galaxies are older than those in late-type galaxies.For this reason,the delay time between formation and the short GRB outburst plays an important role in de-termining the burst rates in these two types of galaxies.

3.THE STAR FORMATION HISTORY OF GALAXIES

The cosmic star formation history (SFH),which in-cludes stars forming in both early-and late-type galax-ies,can be probed by di?erent techniques,including the rest-frame UV continuum emitted by young stars,the in-frared or sub-millimeter reprocessed radiation,and line emission from star-forming regions (e.g.,Madau et al.1996;Giavalisco et al.2004;P′e rez-Gonz′a lez et al.2005;Hopkins et al.2000;see Hopkins 2004and Fardal et al.2006for a summary and a compilation of SFHs from di?erent techniques).

Analytic ?ts to the SFH based on compiled star forma-tion rate (SFR)at various redshifts have been derived by many authors (e.g.,Hopkins 2004;Hopkins &Beacom 2006;Cole et al.2001;Fardal et al.2006).Here,as an illustration,we use the analytic ?t to the overall SFH given by Fardal et al.(2006),

SFR all (z )=

p 1p 2p 3ρ0(1+z )p 2

The Lifetime of Short Gamma-Ray Burst Progenitors3 In addition to the total SFH,we are also interested in

knowing the SFH for stars residing in galaxies of di?erent

types.More speci?cally,given the z～0early-and late-

type galaxy populations,our aim is to understand how

the stars in these two di?erent types of galaxies were

pieced together.We,of course,only need to know the

SFH for galaxies of one type,as the SFH for the other

type can be easily derived by subtracting the known con-

tribution from the total SFH in equation(1).

Ideally,to determine the SFH of a given population

of z～0galaxies,one would like to empirically derive

their assembly history by identifying their progenitor sys-

tems at di?erent redshifts.Alternatively,we can study

the stellar population contents of z～0galaxies to infer

their SFH.The former is not trivial,while considerable

e?orts have been made to the latter through spectral syn-

thesis modeling(e.g.,Heavens et al.2004;Panter et al.

2006;Cid Fernandes et al.2005,2006),which decom-

poses the observed spectra of galaxies into stellar popula-

tions of a range of ages and metallicities.Heavens et al.

(2004)?rst used this technique to infer the global SFH

and that as a function of stellar mass.More recently,

Panter et al.(2006)improved it to yield a better agree-

ment with other empirical determinations of the global

SFH,which peaks at systematically higher redshifts than

that in Heavens et al.(2004).Obviously,this method

would improve with a better understanding of the sys-

tematic e?ects.

A more readily available method is to use galaxy for-

mation models to separate the SFH as a function of

galaxy type.The accuracy of galaxy formation models

crucially depends on our understanding of the gas physics

and star formation as well as feedback processes.Cur-

rent galaxy formation models,although far from provid-

ing a perfect description,can successfully explain many

aspects of the observed properties of galaxies.Hope-

fully,future improvements in galaxy formation modeling

would help provide a better description of the SFH as a

function of galaxy type.

With this in mind,we proceed separating the global

SFH using a particular galaxy formation model(with

model uncertainties roughly accounted for).It should

be noted here that our method is applicable to any two

or more galaxy populations provided they have distinct

SFHs.This is clearly illustrated in§5,in which,for

comparison,an observationally motivated decomposition

based on galaxy masses is used instead.

De Lucia et al.(2006)studied the formation history of

elliptical galaxies using a galaxy formation model based

on the Millennium Simulation of the concordanceΛCDM

cosmology(Springel et al.2005).They calculated the

average SFH of z～0elliptical galaxies of various stellar

masses.If early-type galaxies are identi?ed here as ellip-

ticals,we can then compute their average star formation

rate(M⊙yr?1Mpc?3)as a function of redshift using

SFR early(z)= dM star dF(M star,z)

4Zheng &

Ramirez-Ruiz

Fig.2.—Left panel:Star formation histories for z ～0galaxies of di?erent types.The total star formation history (black )is decomposed into two parts —formation histories of stars that are now in the z ～0early-(red )and late-type galaxies (blue ).Solid,dotted and dashed red and blue curves result from di?erent treatments of the SFH of early-type galaxies (see text),which give a clear representation of the uncertainties associated with the assumed SFH decomposition.The curves shown here are for early-and late-type galaxies as de?ned by the concentration of their light pro?le (corresponding to the solid histograms in Fig.1).Right panel:The ratio of short GRB rates in early-and late-type galaxies at z ～0as a function of the index n of the progenitor lifetime distribution P (τ)∝τn .The thick solid,dotted and dashed curves correspond to the di?erent SFH splits shown in the left panel.The thin curves,on the other hand,are for the same SFH decomposition but for galaxy types classi?ed based on color instead of light concentration.The two dot-dashed curves give a fair representation of the uncertainties in the determination of the burst ratio,which arise primarily from the di?erent methods and adjustments used here to decompose the SFH (see text).

ambiguity of the earl-type de?nition criteria.These un-certainties are propagated self consistently when estimat-ing the lifetime distribution of short GRB progenitors (§4).The region enclosed by the dashed-dotted curves in the right panel of Fig.2shows the combined e?ects on the derived lifetime distributions of short GRB progen-itors when adopting the three variants discussed above,allowing a 0.2dex change in the overall SFH,and con-sidering the two de?nitions of early-type galaxies.

As a representative example,in the left panel of Fig-ure 2,we plot the SFH in equation (1)and the SFH of the z =0early-type galaxies de?ned through color,as black and red curves,respectively.The di?erent line types for the SFH of early-type galaxies are for the various ad-justments listed above.In general,the derived average SFHs of the z =0early-type galaxies peak at a look-back time of about ～10–12Gyr and subsequently decline to-ward low redshifts.The shape of this early-type SFH is quite consistent with that of the fossil bulge model ?t in Nagamine et al.(2006)and that of SFH of galaxies with stellar mass above 1011M ⊙inferred in Panter et al.(2006).

The average SFH for the z =0late-type galaxies,computed by subtracting (2)from (1),is shown as blue curves in the left panel of Figure 2.The average SFR for the z =0late-type galaxies has a steep rising starting at lookback times of ～11Gyr,subsequently peaking at ～8–9Gyr,and slowly declining thereafter.The SFR be-comes increasingly dominated by late-type galaxies from lookback times of ～9Gyr to the present epoch.

We note here that the overall SFH in equation (1)has a slightly higher amplitude than the predicted early-type SFR at lookback times greater than 12Gyr,which gives rise to a modest second peak in the late-type SFR at high redshifts.This may be interpreted as a contribution from

local late-type galaxies to the SFR at these early epochs,which could be related to the formation of the bulges of these galaxies.It may also be an artifact associated with the inaccuracy of the model,which we consider to be more likely.We ?nd that even if taken at face value,this early contribution to the formation of stars in local late-type galaxies leads only to a less than 10%increase in the inferred short GRB burst ratio in the two types of galaxies (§4).For this reason,we leave the SFR for late-type galaxies at these epochs as those given by the decomposition and do not perform any correction.

4.CONSTRAINTS ON THE LIFETIME OF SHORT GRB

PROGENITORS

The short GRB rate is a convolution of the SFR and the distribution of time delays between formation and outburst (e.g.,Piran 1992)

R i (z )=C t (z )

dτSFR i (t ?τ)P (τ),(3)

where the subscript i denotes the type of galaxies in

consideration,P (τ)is the probability distribution of the time delay τ,and C is a normalization constant.In prin-ciple,details of the star formation process may be related to the assembly history of galaxies.Therefore,the dis-tribution P (τ)and the normalization constant C could be di?erent for di?erent types of galaxies.We make the simple assumption here that this dependence is weak so that the same distribution and normalization is used for all types of galaxies.

The distribution P (τ)of the time delay for short GRB is not yet well understood.It can be constrained us-ing,for example,the burst rate as a function of red-shift (e.g.,Nakar et al.2006;Guetta &Piran 2005;Ando 2004).The luminosity function of short GRBs and its

The Lifetime of Short Gamma-Ray Burst Progenitors

5

redshift evolution used in this method are not well deter-mined at present.To avoid such complications,here we propose to use the local rates of short GRBs in di?erent types of galaxies as an alternative and complementary method to constrain the probability distribution P (τ).As an illustration,we adopt a simple parameterization,P (τ)∝τn ,and calculate the ratio of rates of short GRBs at z ～0in early-and

late-type galaxies as a function of n .The right panel of Figure 2shows the results of such exercise.The thick solid,dotted,and dashed curves are for three di?erent treatments of the early-type SFR (see §3),corresponding to the three di?erent curves in the left panel.Here galaxy types are de?ned by their light concentration.The three thin curves in the right panel have been calculated using a similar treatment but are for early-type galaxies as de?ned by their color.The two dot-dashed curves give the maximum range of un-certainties in the resultant burst rate ratio,as they give a fair representation of all the uncertainties involved in separating the SFH as discussed in §3.

For larger n ,the distribution P (τ)is weighted more to-ward longer time delays,so that early-type z ～0galax-ies,which on average form their stars earlier than late-type z ～0galaxies,are more likely to host short GRBs.For smaller (negative)n ,short GRB progenitor systems with shorter time delays would dominate and one would be more likely to ?nd them in late-type galaxies.For n ?3/2,short GRB progenitors would be those with very short time delays so that the short GRB rate ratio is basically determined by the ratio of the z ～0SFRs in early-and late-type galaxies.However,the amplitude of the plateau at n ?3/2in the right panel of Figure 2should be taken cautiously,because it depends on how one interpolates the SFR in early-type galaxies derived by De Lucia et al.(2006)to z ～0.For this reason,the method proposed here becomes less useful for time delays that are short when comparing to the mean age of stars residing in early-type galaxies.Clearly,as we argue in what follows,current observational constraints for short GRBs are such that uncertainties in the determination of the amplitude of the plateau at n ?3/2are irrelevant.Our calculation is based on z ～0galaxy populations.Of the nine well localized short bursts,only three hosts,two elliptical galaxies and one star-forming galaxy,re-side at z 0.3.Therefore,the face value of the burst ratio in local early-and late-type galaxies is ～2provided there are no selection e?ects that make short GRBs more likely to be detected in a galaxy of a given type (e.g.,because of the low density of the interstellar medium one may expect the GRB afterglow in elliptical galaxies to be faint,e.g.,Belczynski et al.2006).The error-bar on this observed ratio is not readily assigned because of the small number statistics.However,even with a conservative estimate of the ratio as ～2±1.5,the anal-ysis presented here alludes to a probability distribution with n ?1,thus favoring long delay times.This is in contrast to what is found in Type Ia supernovae,for which the relatively low frequency in early-type galaxies (e.g.,van den Bergh 1990;Mannucci et al.2005)yields n ?1.

If the distribution P (τ)is assumed to have a log-normal form,there are a series of combinations of the mean and the width that can explain the observed burst rate ratio in early-and late-type galaxies at z ～0.We

Fig.3.—Illustration of di?erent functional forms of P (τ)that give rise to the same observed short burst rate ratio in early-and late-type galaxies.If P (τ)is assumed to have a log-normal form,there are a series of combinations of the mean and the width that can explain a given observed burst rate ratio,here assumed to be either 1(bottom panel )or 2(top panel ).A narrow and a wide log-normal distributions are shown.

illustrate the underlying powerlaw and two cases of log-normal P (τ)distributions for burst ratio of one and two,respectively,in Figure 3,using the SFH split correspond-ing to the top curve in the right panel of Figure 2.For a narrow log-normal distribution,it just peaks at τ0when the ratio of the SFRs in two types of galaxies at a look-back time τ0equals the burst ratio.A wide log-normal distribution and the corresponding powerlaw distribution have a general similarity with each other,but they di-verge from each other at small time delays if the power-law index is negative,which indicates that the method has a poor constraint in P (τ)at the low τend.For a burst ratio of unity,we ?nd n >?0.45.If we take 13.7Gyr as an upper limit for the delay time,which allows short GRB progenitors to outburst within a Hubble time,a distribution P (τ)with n =?0.45implies that about half of progenitors of short GRBs should have lifetimes longer than 3.9Gyr.If at lower τ,P (τ)does not rise as steeply as a powerlaw,this number becomes higher —for example,with a cuto?in P (τ)at τ<2Gyr,the number becomes 6.7Gyr.For a wide (narrow)log-normal distri-bution,a burst ratio of order unity implies that about half of progenitors of short GRBs should have lifetimes longer than 7.0(8.5)Gyr.

Obviously,the above calculation is only sketchy and should be taken with a grain of salt at present,al-though,given the uncertainties assigned to the modeling,the lower bound of n seems rather robust.Constraints should improve as more host galaxies of short GRBs are detected and the SFH separation method is improved.In order to reach a 10%accuracy in the the burst ratio,observations of some 100local short GRB host galaxies are needed.If the split of the SFHs could be well deter-mined,an associated uncertainty ?n ～0.3in the time delay distribution could be easily achieved.If the real-istic SFH separation lead to a 20%uncertainty in the predicted burst rate ratio,the constraints on n would be degraded to ?n ～0.9.On the other hand,as suggested

6Zheng&Ramirez-Ruiz

by our analysis,it is possible to set a very reliable lower bound on n.In brief,for a more precise constraint to be derived on the time delay of short GRB progenitors using the method proposed here,one needs a larger sample of local short GRB host galaxies to be observed,the split of the SFH to be better determined,and the classi?cation scheme of GRB host galaxies to be consistent with the assumed SFH split.

5.DISCUSSION

In the local universe,based on the stellar mass func-tions of galaxies,about55–70%of the stellar mass is in early-type galaxies,and,from the SFH we inferred, the corresponding stars mainly formed about9Gyr ago. Two of the three host galaxies of local short GRBs (z 0.3)are associated with old and massive galax-ies with little current or recent star formation,which makes it unlikely that short bursts are associated with massive stars.Presently available data suggests,but not yet prove,a long time delay between the formation of the progenitor system and the short GRB outburst—for progenitors that can outburst within a Hubble time, about half of them have lifetime longer than～7Gyr.It is fair to conclude that,based on the current host galaxy sample,the progenitors of short GRBs appears to be longer lived than those of Type Ia supernovae.Fox et al. (2005)also reach similar conclusion by arguing that Type Ia supernovae occur more frequently in late-type,star-forming galaxies.Simply based on a comparison of Hub-ble types between short GRBs and Type Ia supernovae host galaxies,Gal-Yam et al.(2005)argue that the de-lay time of short GRBs should be several Gyr even if the Type Ia supernovae delay time is as short as～1Gyr. The lifetime of the progenitor systems is estimated here by using the SFH of elliptical galaxies from a galaxy for-mation model.This allow us to separate the early-and late-type galaxy contributions to the overall cosmic SFH. It would,however,be more self-consistent to infer SFHs of di?erent types of galaxies by modeling the observed spectra with stellar population synthesis models.In ei-ther case,the uncertainty in the derived SFHs should be folded into the errors derived by this method for the distribution of time delays of short GRB progenitors.

In our calculation,di?erent de?nitions of early-type galaxies(by color or by light concentration)introduce uncertainties in the resultant SFH.This,in principle, would not be a problem since we can choose to use the same de?nition for identifying the short GRB host galaxy type.The method proposed here is not limited to a sepa-ration between early and late galaxies.As long as galax-ies are divided into two(or more)sub-populations that have distinct star formation histories,they can be used to constrain the lifetime distribution of GRB(or Type Ia supernova)progenitors,following the same reasoning presented here.For example,Shin&Berger(2006)re-cently applied our approach to?eld and cluster elliptical galaxies.Another example would be to divide galax-ies according to their stellar mass.Panter et al.(2006) present SFH of local galaxies as a function of galaxy stel-lar mass by modeling their spectra.Low mass galaxies on average form their stars later than high mass galaxies, which can be used to constrain P(τ)if the stellar mass of GRB hosts can be obtained.As an illustration,in Fig-ure4,we show the separation of the SFR of galaxies with low and high stellar masses(below or above1011M⊙)us-ing results by Panter et al.(2006).The shape of the over-all SFH used in Panter et al.(2006)is slightly di?erent from that in Figure2,which may re?ect some systemat-ics in the derivation of the SFH based on the fossil record. If the systematics can be well controlled,this approach can become even more powerful than the one used here by inferring SFHs of individual GRB host galaxies.A maximum likelihood method can then be used to con-strain P(τ)based solely on the SFH of host galaxies, which puts our proposed method to an extreme—divid-ing the galaxy populations into individual galaxies.Fur-thermore,instead of assuming a functional form of P(τ), such an application to a large number of individual host galaxies may allow constraints on a non-parametric form of the distribution[i.e.,constraining P(τ)in di?erentτbins].

In this paper,we limit our study to z～0galaxies, but the method can be easily generalized to galaxy pop-ulations at any redshift provided that one can accurately infer their SFHs(e.g.,through spectral synthesis).The observational task can be minimized by focusing on the observations of host galaxies of individual short GRBs. However,for applications at high redshift to be useful, the luminosity function of short GRBs needs to be un-derstood.

Throughout this paper we have assumed the same life-time distribution of short GRB progenitors in both early-and late-type galaxies.However,star formation pro-cesses in these two types of galaxies may not be identical. For example,elliptical galaxies can form by the merging of two gas-rich galaxies(e.g.,Mihos&Hernquist1994). Many globular clusters can form in the merging process (Schweizer2003),which could enhance,for example,the fraction of binary progenitors and also change the life-time distribution(e.g.,Grindlay et al.2006).The mag-nitude of this kind of e?ect on the GRB progenitors is a formidable challenge to theorists and to computational techniques.It is,also,a formidable challenge for ob-servers,in their quest for detecting minute details in ex-tremely faint and distant sources.

At least two short GRB host galaxies are found in clus-ter environments.There may exist a selection bias of detecting short GRBs in a dense medium(Bloom et al. 2006).To study the association of short GRBs with clusters,it would be useful to separate the stellar mass function into that for?eld galaxies and that for clus-ter galaxies in addition to early-and late-type galaxies. More promising for the immediate future,the prepon-derance of cluster environments can be investigated ob-servationally.Important information may be gained by studying the local stellar mass inventory shown in Fig-ure1.Approximately50%of the stellar mass contents in early-type galaxies are in galaxies with M star>1011M⊙that typically reside in clusters.Since it is likely that in clusters galaxies shut o?their star formation process early on,a long progenitor lifetime further increases the tendency for short GRBs to happen in cluster galaxies. It is fair to conclude that the observed preponderance of cluster environments for short GRBs is consistent with an old stellar population that preferentially resides in early-type galaxies.

Detailed observations of the astrophysics of individual GRB host galaxies may be essential before stringent con-

The Lifetime of Short Gamma-Ray Burst Progenitors

7

Fig. 4.—Similar to Fig.2,but the SFH is decomposed into those for galaxies with di?erent stellar masses.This decomposition is based on the results in Panter et al.(2006),where the SFHs of local,individual galaxies are inferred from modeling their spectra and are then subsequently grouped according to the present stellar mass content.In the left panel,the solid curve shows a spline ?t to the overall SFH inferred by Panter et al.(2006),and the dashed and dotted curves give the spline ?ts to the SFRs of galaxies with stellar mass below and above 1011M ⊙,respectively.For lookback time larger than ～12.5Gyr,a cuto?pro?le is introduced.In the right panel,the ratio of short GRB rates in galaxies with stellar mass above and below 1011M ⊙is plotted as a function of the index n of the progenitor lifetime distribution P (τ)∝τn .The purpose of the ?gure is to illustrate that the method presented in this paper can be generalized to any sub-populations of galaxies that have distinct SFHs (see text).

straints on the lifetime of short GRB progenitors can be placed.If con?rmed with further host observations,this tendency of short GRB progenitors to be relatively old can help di?erentiate between various ways of forming a short GRB.

We thank J.Bloom,N.Dalal,W.Lee,D.Pooley,and

J.Prochaska for helpful conversations and John Bea-com for useful comments.We thank the referee for de-tailed comments that improved this paper.We acknowl-edge the support of NASA through a Hubble (ZZ)and Chandra (ER-R)Postdoctoral Fellowship awards HF-01181.01-A and PF3-40028,respectively.

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