RECONCILING THE GAMMA-RAY BURST RATE AND STAR FORMATION HISTORIES

Gamma-ray bursts (GRBs) are short and intense pulses of soft gamma rays. It is generally accepted that long-duration GRBs (>2 s, but see Bromberg et al. 2012) arose during the collapse of massive stars, the so-called collapsars (MacFadyen & Woosley 1999). This understanding is largely based on the observed association of several long bursts with type Ibc supernovae (SNe; Hjorth & Bloom 2012).⁴ In such cases, it is only natural to expect that GRBs will follow the cosmic star formation rate (SFR) and that like SNe, their positions within their host galaxies will correspond to the SFR within these galaxies. Surprisingly, both expectations are not satisfied by observations.

During the last eight years, the Swift satellite has routinely provided us with accurately localized GRBs. From these data it is possible to construct the luminosity function and cosmic GRB rate. Recently, Wanderman & Piran (2010, WP hereafter) estimated the rate and the luminosity function of long-duration GRBs using a novel method that simultaneously solves for the GRB rate and the GRB luminosity function. One of their most interesting findings was that, assuming that the GRB luminosity function does not depend on the cosmic time, the GRB event rate does not follow the SFR of the typical galaxy population (see their Figure 9), showing deviations both at low (z < 3) and high (z > 3) redshifts. A lot of attention (Daigne et al. 2006; Le Floc'h et al. 2006; Guetta & Piran 2007; Guetta & Della Valle 2007; Kistler et al. 2009; Wanderman & Piran 2010; Virgili et al. 2011; Robertson & Ellis 2012) has been paid to the deviation of the GRB rate from the directly measured SFR obtained using various methods, e.g., the UV luminosities of galaxies at high redshifts (Bouwens et al. 2009). While the deviation at high redshifts is clear, measurements at both high and low redshifts suffer from poor statistics and possible observational biases. Here, we focus on the less noticed but statistically significant and easily verified deviation of the GRB rate from the well-documented SFR at low (z < 3) redshifts. At the same time, using Hubble Space Telescope (HST) imaging of GRB and SN host galaxies, Fruchter et al. (2006) showed that while the likelihood of finding SNe in a given position within their host galaxies is linearly proportional to the SFR at that point, GRBs are much more concentrated within the regions of the highest SFRs (see also Svensson et al. 2010). Thus, GRBs do not follow the star formation in either time or space.

An obvious question that arises is how these two issues can be reconciled with the idea that long GRBs arise from the collapse of massive stars. A related question is of course what kind of galaxies host GRBs and whether the SFRs within these galaxies are similar to the cosmic ones. In a new attempt to address the first question, we ignore all available information on GRB hosts and attack this question using a different approach. Our way to answer the question of which galaxies host GRBs is to have a complete census of star formation histories of all galaxies in the universe and extract the ones that match the observed GRB rate. Here, we do so by exploiting a large local SDSS sample of galaxies and analyzing their fossil record in detail. This allows us to effectively have a complete sample of galaxies up to z ∼ 3 and compare the SFRs in low-mass low-metallicity galaxies with the inferred GRB rate.

Once we address this question, we turn to the second question: why are GRBs more concentrated in high-SFR regions than SNe? To address this issue we make the reasonable assumption that GRBs arise from more massive stars than SNe (Östlin et al. 2008). Following this assumption we compare the positions, within low-mass galaxies, of massive (>20 M_☉) stars with the positions of less massive (>9 M_☉) stars. Finally, we suggest a simple theoretical argument concerning star formation that explains this trend.

The spectra of galaxies encode information about the histories of the stellar population components, dust, and star formation. Various tools have been developed to extract this information (e.g., Heavens et al. 2000; Tojeiro et al. 2007) and to compare the resulting extracted information with both extrinsic and intrinsic galaxy properties. The MOPED (Heavens et al. 2000) algorithm implements the general process of reforming a complex data set (e.g., a galaxy spectra) into a set of parameters (e.g., SFR, metallicity) and parameter combinations, assuming uncorrelated noise, such that the data compression does not experience loss.

An easily accessible, robust code is the VErsatile SPectral Analysis⁵ (hereafter VESPA, see Tojeiro et al. 2007, 2009, for more details) package, which recovers star formation and metallicity histories based on the galactic spectra using synthetic stellar-population models. The software recovers histories in adaptive age bins according to the signal-to-noise of the galaxy spectrum on a case-by-case basis and addresses the age–metallicity relation. Two popular synthetic stellar population models are included in the VESPA output—those of Bruzual & Charlot (2003), and Maraston (2005) and Maraston et al. (2009)—which differ in their respective resolutions, and the use of empirical libraries to model the thermally pulsating asymptotic giant branch. Furthermore, VESPA corrects for Galactic extinction using the dust maps of Schlegel et al. (1998) and fits for the dust in each galaxy using a dust model with either one or two parameters. We exploit the VESPA database to compare the observed GRB rate with those of different types of SDSS galaxies. In particular, we address the question: is there any subset of galaxies from the Sloan Digital Sky Survey (SDSS) whose SFR matches the observed GRB rate? If so, what are the physical properties of these galaxies?

The ∼10⁶ spectroscopically selected galaxies in this work were drawn from the SDSS Data Release 7 (York et al. 2000; Abazajian et al. 2009, and references therein, hereafter SDSS DR7). We selected both "Main Galaxy Sample" and luminous red galaxies. Galaxy spectra were reprocessed using VESPA and we followed Tojeiro et al. (2009) in adopting the two-parameter dust model, but note that our results are insensitive to the choice of dust model.

We extract the star formation histories of the galaxies from the VESPA database for the highest resolution output (16 bins) in lookback-time bins as specified in Table 1. VESPA also returns the total stellar mass, both at present and at formation, of the galaxy and the metallicity for each bin. We explore the total mass and the metallicity in the VESPA database searching for a match with the WP data (their Figure 2, upper panel). Note that the derived WP GRB rate shows two characteristic features that do not match the global star formation history of the general population: the peak is shifted to z ∼ 3, and below that redshift, the GRB rate distribution is flatter than the SFR. At high redshift the GRB rate is higher than the SFR rate inferred using, e.g., the UV luminosities of the galaxies (see also Jakobsson et al. 2012). From previous experience, we know that this behavior is a characteristic of low-mass galaxies (Heavens et al. 2004). Figure 2 in Heavens et al. (2004) shows how the shape and peak of the star formation history change as a function of the galactic mass; this already points to low-mass galaxies as promising candidates. This idea is also supported by the non-detection of extreme-redshift GRB host galaxies in very deep HST imaging (Tanvir et al. 2012). However, in order to find a good match with the WP GRB rate, a cut in metallicity is also needed. In particular, we find that if we select SDSS galaxies with present stellar masses <10¹⁰ M_☉ and metallicities Z < 0.1 Z_☉, we obtain a good match between the GRB rate and the SFR. This means that we need to exclude about half of the dwarfs to obtain such a metallicity cutoff (see Figure 6 in Panter et al. 2008). Any other population produces a steeper SFR with redshift and thus cannot be matched to the GRB rate.

Our reported metallicities correspond to the whole stellar population of the galaxy, as derived by VESPA. These are different from those typically inferred for local GRB hosts. For example, recently, Graham & Fruchter (2012) measured the metallicity of the star-forming regions (H ii regions) in identified GRB hosts. They measured oxygen abundances from emission lines. Their derived metallicities are somewhat larger than ours (by a factor of 50%). This is due to the fact that we measure the whole stellar population, including the old stellar population in the irregulars.⁶ This lowers the overall metallicity (see the blue line of Figure 3 in Panter et al. 2008). So reporting only the metallically of the latest and most recent burst in star formation would bring our metallicity into agreement with the metallicity reported by Graham & Fruchter (2012).

Figure 1 depicts the WP GRB rate distribution with redshift, where the solid line marks the result of our fit. First, note that we could not go beyond z > 3 as the VESPA catalog does not contain information on these high-redshift bins. The reason for this is that we are reconstructing the star formation history from the fossil record at a medium redshift of z ∼ 0.2 and going back so far in time is very difficult. Therefore, we limit our analysis to redshifts below 3, not covering the peak of the observed GRB rate and the high-redshift region beyond it.

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We find that, based only on matching the observed SFR in the specific population of galaxies with the WP GRB rate, the hosts of long GRBs are dwarf galaxies with very low metallicities. This result is in good agreement with direct observations of those hosts (see, e.g., Savaglio et al. 2009, and subsequent discussion in Section 4), but these samples are limited to a few dozen objects. In fact, the inconsistency between the GRB rate and typical star formation could have been predicted, given that most GRB hosts are low-metallicity dwarf galaxies with a different SFR than the typical star-forming galaxies and hence the overall SFR. The agreement between the metallicity cut in our analysis and the observational evidence that most GRB hosts have low metallicity further supports the WP results on the GRB rate and suggests that the specific nature of GRB hosts rather than a luminosity evolution (as suggested by numerous authors) is responsible for observed distribution of GRBs redshifts and peak fluxes.

We now turn to the second puzzle, concerning the origin of the discrepancy between the GRB and SN positions within their hosts. To do so we explore the properties of massive stars in these dwarf low-metallicity galaxies. Specifically, we compare the population of massive stars >9 M_☉ with one of the very massive stars >20 M_☉. The basic idea is that while the former lead to SNe, the latter are GRB progenitors and if the two populations are distributed differently this explains the different distribution of GRBs versus SNe.

The most detailed study of the population of massive stars in dwarfs in the local group is that by Bianchi et al. (2012), who used GALEX observations to determine the number of massive stars (>9 M_☉) in six dwarf galaxies (Phoenix, Pegasus, Sextans A and B, WLM, and NGC 6822), although for the last one they only have lower limits. We have used the masses for these dwarfs derived by Karachentsev et al. (2004, Column 4 in their Table 4) to plot in Figure 2 the numbers of massive stars as a function of the dwarf galaxy mass. Inspection of Figure 2 reveals that there is a clear correlation between the number of massive stars and the mass of the dwarf galaxy. Additionally, the ratio ${(\#> 20)}/{(\#>9)}$ of number of very massive to massive stars increases with the mass of the dwarf galaxy. The lines show fits to the data points in Bianchi et al. (2012) and are well fitted by power laws with power 0.7 (for >9 M_☉) and 1.2 (for >20 M_☉), i.e., there is nearly a factor of two difference in the power. This empirical observation implies that very massive stars (>20 M_☉) are more abundant in more massive dwarfs. This plot can be compared directly with Figure 4 in Svensson et al. (2010), which plots the same quantity but for number of SNe and GRBs in hosts galaxies. Our results agree very well with the Svensson et al. (2010) trend. Given that larger dwarf galaxies are denser than lower mass ones, we can take this relation as a proxy to the relation according to which very massive stars are more abundant in denser regions in such galaxies. This assertion, which we will shortly explore further, now serves as the basis for the rest of the analysis.

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We can use the data in Bianchi et al. (2012) to produce a plot similar to Figure 3 in Fruchter et al. (2006), which shows the number of SNe/GRBs as a function of the total light in the galaxy. Because we do not have pixel-by-pixel information for the dwarfs, we simply integrate the light in radial rings. This is equivalent to what is done in Fruchter et al. (2006) if the galaxies do not show angular asymmetries, which is the case for the dwarfs we are considering. The results are shown in Figure 3. The solid thick lines are fits to the Fruchter et al. (2006) results, while the dashed line is the number of stars >9 M_☉ and the dotted line corresponds to >20 M_☉ stars, where we have averaged over the five local dwarfs. There is good agreement between the different curves.

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Because dwarf galaxies have very low surface densities, the outer edges of dwarf galaxies will contain very little mass as compared with the inner part (Tolstoy et al. 2009). For concreteness let us denote as "outer edges" all mass beyond one equivalent radius and as the "inner part" all mass inside it. For dwarf galaxies this means that there is less than a factor of 10 mass in the outer edges than in the inner part. Using Figure 2 this implies that the outer edges will have a factor of three fewer very massive (>20 M_☉) stars relative to massive stars (>9 M_☉) than the inner part. If we now assume that GRBs can only originate on very massive stars then this is in agreement with the observations of the spatial distribution of SNe and GRBs by Fruchter et al. (2006). Thus, the fact that GRBs are mostly hosted in dwarf galaxies, when combined with the dependence of the initial mass function on the density in these galaxies, also explains the spatial distribution discrepancy between SNe that follow the local SFR linearly and GRBs that follow the local SFR much stronger than linearly.

It is interesting to explore whether there are any theoretical hints that would suggest this observed "Fruchter et al. law" of a differential initial mass function. The current consensus (McKee & Ostriker 2007) is that massive stars form because of the turbulent nature of the interstellar medium. Padoan & Nordlund (2002) give a detail theory of star formation due to magnetized turbulence. In particular, they predict that the maximal mass of the initial mass function is (their Equation (16))

$$\begin{equation} m_{\rm max} \approx \frac{\rho L^3}{{\cal M}_A}, \end{equation} \tag{ 1 }$$

where ρ is the density in the star-forming region, L is the size of the largest scale on which turbulence is driven, and ${\cal M}_A$ is the Alfvén Mach number. It is clear from the above equation that lower density implies a lower maximal mass. This is the case for dwarf galaxies, in which the density beyond the equivalent radius is a factor of a few below that in the inner part (Tolstoy et al. 2009). The size of the largest turbulent scale will remain the same, while the ${\cal M}_A$ will be similar or slightly larger because of the lower density. Thus, the maximal mass in the outer lower mass region will be lower as observed (see Figure 2). Note also that the same authors predict that, at lower densities, the peak of the mass function will move to larger masses, thus increasing the number of SNe.

While the above theoretical argument and the observations of massive stars in local dwarf galaxies would suggest that the number of GRBs should equal the number of SNe as the mass of the host galaxy increases beyond 10¹⁰ M_☉, it is the metallicity cutoff that prevents this from happening.

The first implication of our results is that low metallically is an important part of the collapsar model. While some special cases show GRBs in high-metallicity regions, low metallicity is clearly an important factor for most of the population. This has numerous implications on models of GRB progenitors and on the operation of their inner engines. This result is not new. In fact, ample information is available on the nature of GRB hosts. Inspection of the hosts (Fruchter et al. 1999, 2006; Chary et al. 2002; Bloom et al. 2002; Le Floc'h et al. 2003; Tanvir et al. 2004; Castro Cerón et al. 2006; Savaglio et al. 2009; see also Fynbo et al. 2012; Levesque 2013 for recent reviews) reveals that usually they are low-mass irregular galaxies and have low metallicity (Prochaska et al. 2004; Sollerman et al. 2005; Fruchter et al. 2006; Modjaz et al. 2006; Stanek et al. 2006; Thöne et al. 2007; Wiersema et al. 2007; Margutti et al. 2007; Graham & Fruchter 2012; Savaglio et al. 2009; Thöne et al. 2013). These findings are consistent with theoretical modeling that suggests that low metallicity is essential to produce high angular momentum and high stellar mass needed for GRB progenitors (Yoon & Langer 2005; Woosley & Heger 2006; Wolf & Podsiadlowski 2007), although these constraints may be avoided by placing GRB progenitors in binary systems (Fryer & Heger 2005; Podsiadlowski et al. 2010) or uncoupling the evolution of the core and atmosphere of the GRB progenitors (Ekström et al. 2012; Georgy et al. 2012).

Recent observations have revealed a population of more massive and dusty GRB hosts (Castro Cerón et al. 2006; Savaglio et al. 2009), in particular when darker (i.e., those with less luminous optical afterglow) GRBs are targeted (Perley et al. 2013). As the Wanderman & Piran (2010) rate concerns only unobscured GRBs it is the only one we are able to model. Clearly, our study only applies to visible GRB hosts and we are unable to exclude the possibility that some GRB hosts have solar metallicity values (see also Robertson & Ellis 2012). Nevertheless, it is important to stress that the fraction of GRBs found in dusty and massive galaxies is not high enough to reconcile the GRB rate with the typical SFR, at least out to z = 1 (Perley et al. 2013; Kocevski et al. 2009). Namely, given that the fraction of dark GRBs is about 50%, it is clear that obscured GRB hosts do not contribute to the GRB rate more than that and therefore they cannot sufficiently modify the overall GRB rate and make it compatible with the overall SFR. Thus, GRBs do prefer galaxies with lower mass with respect to the typical star-forming galaxy population. As for the metallicity, while GRB hosts typically have low metallicities, a handful of GRB absorbers show roughly solar abundance (Savaglio et al. 2003; Prochaska et al. 2004; Savaglio et al. 2012) or metal enhancement (De Cia et al. 2012).

Furthermore, Perley et al. (2013) present an extensive compilation of obscured GRBs finding massive and luminous host galaxies at z > 2. They also point out that at lower redshifts they can only find hosts in low-mass, low-metallicity galaxies. What is the metallicity of a luminous, massive galaxy at a lookback time of 9–10 Gyr? In Perley et al. (2013), Figure 10 shows that the hosts of obscured GRBs have masses in the range 10⁹–10¹¹ M_☉. Inspection of Figure 3 in Panter et al. (2008) shows that for the most massive of these galaxies, 9–10 Gyr ago the metallicity of the gas (i.e., stars formed at that time) will be below solar, with most galaxies in the Perley et al. (2013) sample below 1/3 the solar value. Maiolino et al. (2008) have measured gas metallicities at z > 2. Figure 9 in their paper shows the evolution of gas metallicity as a function of mass. Indeed the metallicity is below solar for all masses at z > 2 and about a dex below solar for most galaxies in the Perley et al. (2013) sample. It seems that the conclusion is that even for galaxies where the hosts are massive the metallicity of the gas that will form stars is below solar.

There are, however, a few systems of GRB hosts at high-z (z > 3) for which the metallicity has been measured directly, that show supersolar values (Savaglio et al. 2012; De Cia et al. 2012). If it turns out that these are not outliers but represent a common population, then one would have to investigate the GRB host population further. In particular, what is the internal metallicity distribution at high-z in galaxies? It can be very patchy (e.g., Jimenez & Haiman 2006) and how can one then reconcile the observed GRB rate with the SFR inferred from galaxies?

The fact that low-metallicity plays an important role in the formation of GRBs also has far reaching implications concerning the use of GRBs to explore the early universe. Properly utilizing GRBs as probes of the early universe requires a thorough understanding of their formation and the host environments that they sample (Levesque 2013). Galaxies exhibit a strong mass–metallicity relation (see, e.g., Figure 6 in Panter et al. 2008) and massive galaxies (>10¹¹ M_☉) do not have metallicities as low as those of GRB hosts. Thus, the GRBs host galaxy population is biased relative to the overall galaxy population and even relative to the population of dwarf galaxies as we had to impose a metallicity cut of ≈1/10 the solar value. This excludes about half of the dwarf galaxies as possible GRB hosts.

This last point suggests that as we move to the early universe, where metallicity was lower, the GRB rate was much higher and a significantly larger fraction of stars resulted in GRBs. This is consistent with the observation in WP that the GRB rate at high redshifts is significantly flatter than the SFR. Note, however, that a massive galaxy will enrich its gas extremely fast (in a dynamical time), thus even if the original gas is of low metallicity, only galaxies that have low rates of star formation will be able to host GRBs as they will be able to keep the gas at low metallicity. As we move to lower redshifts and the overall metallicity of the gas increases, even dwarf galaxies will have difficulty hosting GRBs if they already have a high enough metallicity of the available gas to form stars. This explains why the GRB rate decreases at lower redshifts. However, because dwarf galaxies dominate the SFR at low-z, the GRB rate decreases slower than the SFR of the overall galaxy population. If we use the metallicity of the damped Lyα systems as a tracer for the metallicity of the gas where galaxies form, we see that at redshift z ≈ 2–3 the metallicity of the gas decreases below 1/10 the solar value (see Figure 12 in Rafelski et al. 2012). This same conclusion is obtained if one looks at the metallicity histories of SDSS galaxies (see Figure 3 in Panter et al. 2008), thus the GRB rate should increase at that redshift in agreement with WP.

After having identified GRB hosts as low-metallicity dwarf galaxies, we can now explore properties of this population within the SDSS data. First, we examine the location in the sky as a function of R.A. and decl. of this population. Figure 4 depicts the low-metallically dwarf galaxies as blue dots. For reference we have also plotted the locus of luminous red galaxies (i.e., galaxies with larger masses (∼L_*) with older stellar population). It is apparent that the low-metallicity dwarfs are located in the filaments—voids of the cosmic web. The bottom panel of Figure 4 shows the positions at z = 0.1 while the top panel depicts the position at z = 3 (for this case we have used the current local positions but identified the galaxies according to their star formation history at z = 3).

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In order to quantify the location in the sky of low-metallicity dwarfs we compute their correlation function, weighted by the star formation history. Thus, we compute the so-called mark correlation, using star formation as a mark. As can be seen in Figure 5, where the notation WW/DD indicates that the mark statistic is the ratio of weighted pair counts to unweighted pair counts (in this notation, the traditional unweighted correlation function would be DD/RR, where RR is the number of unweighted pair counts in a random distribution.). The low-metallicity dwarfs are less correlated than the mean population at scales smaller than 20 Mpc. The shaded region shows the 2σ confidence region from all low-metallicity dwarfs in the SDSS. Our prediction then is that GRBs are to be found in underdense regions of the dark matter density field, i.e., they will be anti-biased. Most GRBs inhabit regions that show the lowest rates of merging and are undisturbed, despite evidence that some systems do show signs of interaction (e.g., Fruchter et al. 2006).

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It is curious to point out that planet formation requires exactly the opposite environment: high metallicity. Therefore, overall, (long) GRBs seem to inhabit regions where no planets form, thus presenting no risk of life extinction. While this is generally correct all over the universe, it is particularly relevant concerning life extinctions on Earth. The Milky Way is a rather massive galaxy with very few low-metallicity regions. In fact, using the star formation and metallicity histories of Milky-Way-type galaxies (Panter et al. 2008) in the SDSS sample we find that only 2% of the Milky Way has metallicity below 1/10 of solar, which we found as a typical upper limit on the metallicity of galaxies that host GRBs. This implies that the expected rate of GRBs in the Milky Way is much more than the one simply expected from an estimate based on the average current cosmic GRB rate per unit time and unit volume (∼1.3 Gpc⁻³ yr⁻¹).

We thank the anonymous referee for useful comments on the manuscript. R.J. thanks Paolo Padoan for discussions on star formation theory. R.J. and T.P. are most grateful to Annalisa de Cia for many useful suggestions and discussions on a previous version of this manuscript. R.J. and T.P. also thank the Einstein Cafe in Bern for hospitality while part of this work was performed. T.P. acknowledges support from an ERC advanced grant: GRB. Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS Web site is http://www.sdss.org/.