Progenitors of Low-redshift Gamma-Ray Bursts

The bimodal distribution of the observed duration of gamma-ray bursts (GRBs) has led to the identification of two distinct progenitors; compact star mergers, comprising either two neutron stars (NSs) or an NS and a black hole, for short GRBs (SGRBs), and the so-called collapsars for long GRBs (LGRBs). It is therefore expected that formation rate (FR) of LGRBs should be similar to the cosmic star formation rate (SFR), while that of SGRBs to be delayed relative to the SFR. The localization of some LGRBs in and around the star-forming regions of host galaxies and some SGRBs away from such regions support this expectation. Another distinct feature of SGRBs is their association with gravitational-wave (GW) sources and kilonovae. However, several independent investigations of the FRs of long and short bursts, using the Efron–Petrosian non-parametric method, have shown the presence of a mild luminosity evolution, and an LGRB FR that is significantly larger than SFR at low redshift, and similar to the FR of SGRBs. In addition, the recent discovery of association of two low-redshift LGRB 211211A and LGRB 230307A with a kilonova cast doubt about their collapsar origin. In this Letter we review these results and show that our results predict that about 60% ± 5% of LGRBs with redshift less than 2 could have compact star merger as progenitors increasing the expected rate of the GW sources and kilonovae significantly. The remaining 40% ± 5% have collapsars as progenitors, with some having associated supernovae.


INTRODUCTION
Since the discovery of GRBs by Vela satellites several instruments on board multiple satellites (CGRO, Beppo-Sax, Fermi-GBM, Swift, Konus-Wind and others) have detected more than thousand GRBs, majority of which are classified as LGRBs and most of the rest as SGRBs.A good fraction of these GRBs have measured spectroscopic or photometric redshifts, or redshifts based on host galaxies.The bimodality of the distribution of observed duration, T 90 , of GRBs was established by the BATSE instrument of CGRO (Mazets et al. 1981;Kouveliotou et al. 1993), dividing them into two classes of SGRBs and LGRBs separated at T 90 = 2 s.This feature has stayed robust and remained valid for the subsamples with rest frame durations.However, the value of dividing T 90 is somewhat uncertain.A different classification in collapsar and not collapsar events of Swift GRBs by Bromberg et al. (2013), obtains a dividing T 90 ∼ 0.8s.There are other features that separate the two classes.LGRBs on average have softer spectra than SGRBs, as measured either by their spectral hardness ratio or the value of E p , the photon energy at the peak of the νf (ν) energy spectrum.1 Localized LGRBs are often associated with star forming galaxies and regions while SGRBs are found in older systems and away from star forming regions (see, review by Berger (2014)).These dichotomies have led to two distinct progenitors.LGRBs are believed to have been produced by supernova explosions of massive Wolf-Rayet stars, so that their FR is expected to follow the cosmic SFR.SGRBs, on the other hand, are believed to have been produced during NS-NS or NS-BH mergers, thus their FR is expected to follow a delayed form of the SFR.Such mergers are also expected to lead to kilonovae and produce GWs.Recent association of a GW source with SGRB 170817 (Troja et al. 2017;Abbott et al. 2017;Pian et al. 2017) has firmed this belief.The left panel of Figure 1 shows a recent determination of the cosmic SFR by Madau & Dickinson (2014), and examples of the delayed-SFR from Paul (2018) assuming a power law distribution of the delay time.
These dichotomies can be further tested by comparing the distributions of their intrinsic characteristics, mainly the luminosity function (LF), and luminosity and FR evolutions, L(Z) = L 0 g(Z) and ρ(Z), all of which can be described by the bi-variate distribution Ψ(L, Z).2In particular, a clearer picture of FRs of GRBs can provide a better estimation of rate of occurrence of low mass merger GW sources, and kilonovae, especially at low redshifts that are within the reach of current GW detectors.
The main goal of this letter is to determine the FR of GRBs, ρ(Z), at low redshifts, Z < 3.In the next section we describe the method we use to this end, and in §3 we review results obtained for FR of GRBs using this method, and present evidence which indicates that low redshift LGRBs could also have compact star mergers as progenitors, which would increase expected rate of GW detection.A brief summary and conclusions are presented in §4.

THE BASIC PROBLEM AND PROCEDURES
To achieve the above mentioned goal we need an accurate description of the general LF, Ψ(L, Z).This of course requires samples of GRBs with known redshifts,3 and a knowledge of all observational selection effects, which truncate the data and introduce bias.The attempts to correct for this bias, known as the Eddington or Malmquist Bias, has had a long history.Early attempts, including a majority of current ones dealing with GRBs, have used parametric forward fitting methods, whereby a set of functional forms are fit to the data to determine the "best fit values" of the parameters, using χ 2 or maximum likelihood algorithms.These methods assume parametric forms for several functions; spectrum, light curves, g(Z) and ρ(Z), each with three or more parameters, raising serious questions about the uniqueness of the results.They also require binning and thus large samples, which is not always the case, especially for SGRBs.There are several non-parametric, non-binning methods (e.g. the V /V max , Schmidt (1968); C − , Lynden-Bell ( 1971)) that are more appropriate for small samples.However, a major drawback of these methods is the ad hoc assumption of independent or uncorrelated variables, which for extragalactic sources means no luminosity evolution, g(Z) =const.For more details, see a review by Petrosian (1992).To overcome this shortcoming, Efron & Petrosian (1992, 1999), (hereafter called the EP method) developed new methods, whereby one first determines whether L and Z are correlated or not.If correlated, then it introduces a new variable L 0 ≡ L/g(Z) and finds the luminosity evolution function, g(Z), that yields an uncorrelated L 0 and Z, whose distributions, LF ψ(L 0 ) and FR ρ(Z), can be obtained using, e.g. the Lynden-Bell C − method.The advantage of this combined EP-L method is that the two evolutions are obtained directly from the data, non-parametrically and without binning. 4In the past, these methods have been proven to be very useful for studies of GRBs (Lloyd & Petrosian 1999;Lloyd-Ronning & Ramirez-Ruiz 2002;Kocevski & Liang 2006;Yonetoku et al. 2004;Dainotti et al. 2013Dainotti et al. , 2015Dainotti et al. , 2017bDainotti et al. ,a, 2020Dainotti et al. , 2021bDainotti et al. ,a, 2022b)), and AGNs (Maloney & Petrosian 1999;Singal et al. 2011Singal et al. , 2016Singal et al. , 2019;;Petrosian et al. 2022;Dainotti et al. 2022a;Singal et al. 2022;Lenart et al. 2023).

RESULTS ON FORMATION RATES
In this section we present results from more recent works on the FR of GRBs using the non-parametric EP-L method.As emphasized above this method requires a sample with measured redshift and well defined observational selection effects; samples referred to as "complete".Most instruments have well defined γ-ray peak flux threshold and hence provide complete samples.However, securing a redshift is a complicated process and includes several other observational selection effects that are not as easily quantified as the peak flux limit.Swift has been most successful in attempting to quantify these effects.After the γ-ray trigger the on board X-ray instrument attempts a more accurate localization which is more likely for brighter sources.This introduces addition selection effect based on the X-ray flux limit.Then, the optical-UV telescope attempts to produces a spectrum, and thus possibly redshift, and to obtain a higher resolution localization that is sent to ground based telescopes for identification of a host galaxy and further redshift measurement.This third step is more complex and hard to quantify.

LGRBs
The right panel of Figure 1 shows five results from five independent analyses, four of which use different samples of Swift LGRB data and one (in green) using Konus-WIND LGRB data (Tsvetkova et al. 2017).It should be emphasized that none of the samples used in these works are strictly speaking complete.Yu et al. (2015) and Tsvetkova et al.   2009)), which yields significantly different FR histories than using the γ-ray threshold alone, especially at mid range redshifts.To account for more unknown optical selection bias Petrosian et al. (2015) use a γ-ray threshold 10 times larger than the nominal Swift threshold.As evident all of these estimates of FR of LGRBs deviate significantly, though at different degrees, from the cosmic SFR at Z < 3 or z < 2. It should be noted that Pescalli et al. (2015) compare their result with an earlier determination of the cosmic SFR which is flatter at low redshift than the more recent Madau & Dickinson (2014) one shown above, and conclude concordance with SFR.However, as shown in Figure 1, their estimate, though to lesser degree than all others, shows a significant (about factor of 3; ∼ 5σ) deviation at Z < 1.5. 5n the left panel of Figure 2 we compare the average value of the logarithms of the above 5 FR estimates with the cosmic SFR.6This average, not too different from the estimate by Petrosian et al. (2015), which includes both gamma-ray and X-ray observational biases, clearly deviates from the SFR by more than 10σ's at the lowest redshift bin.Thus, we are compelled to suggest the presence of two LGRB components; one following the cosmic SFR and another low redshift component shown by the solid and dashed blue curves obtained from subtracting the first component from the observed average and Petrosian et al.FR redshift distribution, respectively (again normalized at the peak of the cosmic SFR).This second FR component has an almost identical shape to that of SGRB FR, shown by the magenta curve (described in the next section), and is similar to the delayed SFR rates shown in the left panel Figure 1.

SGRBs
The discoveries of gravitational wave sources and the associations of some with SGRBs and kilonovae, mentioned above, has made the determination of the FR of SGRBs a critical issue, and has led to increased activity on this front These are all based on parametric forward fitting method described above.More recently, Dainotti et al. ( 2021b) have used the non-parametric EP-L method, for determining the FR od SGRBs, which we now described briefly.There are in general smaller numbers of SGRBs than LGRBs and even smaller samples with known redshift.The size of the sample is important.For larger sample one can obtain more reliable and detailed result no matter what method is used.But as emphasized above the observational selection that affect ones ability to determine the redshift is even more critical.The most important selection criteria is having a complete sample with well defined γ-ray peak flux threshold.In absence of X-ray observations and due to small numbers, the more complete and conservative method used by Petrosian et al. (2015) cannot be used for SGRBs.Instead, Dainotti et al. (2021b) use a method of finding the lowest possible flux threshold, hence the largest sample, which statistically indicates that the sample with redshift was drawn from the complete parent sample including all SGRBs.We have repeated this procedure with a slightly larger SGRB sample.The results from this work are presented in the right panel of Figure 2. In general, the EP-L method yields cumulative distributions of the variables, in this case the rate, σ(Z), of sources with redshift > Z, related to the differential rate as where V (Z) is the co-moving volume up to redshift Z.
The result are shown by the blue points and are fit by a broken power law, parameters of which are given in the figure.The black line shows the RE obtained from differentiation of equation ( 1) and the analytic form for σ(Z).This RE is shown by the magenta curve in the left panel normalized to the others at the lowest redshift.As evident it has a form very similar to that of the low redshift component of LGRB FR.
Thus, it is reasonable to conclude that these result provide a strong evidence that both long and short low redshift GRBs have a common progenitor.The fact that these FRs are similar to delayed SFR lead us to also conclude that mergers of compact stars (NSs and BHs) are the progenitor of all low redshift GRBs.This conjecture has received strong support from a new observation showing coincidence in time and space of a optical/NIR kilonova with a low redshift (z=0.076)LGRB (GRB211211A; Rastinejad et al. (2022)).Fermi-LAT observations show a transient (duration 20 ks) starting about 1 ks after the trigger that is interpreted to be due to inverse Compton scattering of kilonova photons by the GRB jet accelerated electrons (Mei et al. 2022).These evidences complement the results presented here based on population studies.

SUMMARY AND CONCLUSIONS
In this letter we have explored how the FRs of long and short GRBs compare with cosmic SFR.The general expectation is that LGRBs produced by collapsars (supernova explosion of massive Woolfe-Rayet stars) should follow the SFR, while FR of SGRBs, produced by merger of compact stars (NSs and BHs), should be delayed relative to SFR (Fig. 1, left).
1. We describe methods used for obtaining the FR history of extragalactic sources and the advantages of the nonparametric, non-binning Efron-Petrosian-LyndenBell method, which obtains the cosmological distributions and evolution of sources, yeilding the rate of luminosity evolution, the luminosity function, and the FR evolution, especially for small sample of sources with complex observational selection biases.
2. We present results on five independent determination of the FR of LGRBs, all using the EP-L method and showing significant deviations, though at different levels, from SFR at low redshifts (Fig. 1, right).
3. Taking the difference between the logarithmic average FR of 5 results and SFR, we show presence a distinct low redshift LGRB component that decreases rapidly with increasing redshift in a manner expected from delayed SFR (Fig. 2, left).
4. We also present a new determination of FR of SGRBs, again using EP-L method and data selection procedure described in Dainotti et al. (2021b) (Fig. 2, right).This FR is almost identical to that of the low reshift component of LGRBs (Fig. 2, left).
In summary, we have presented evidence for significant, and almost identical, deviations of the FRs of both long and short GRBs from SFR at low redshifts, having a form expected from delayed SFR.The latter similarity is qualitative only but the assumption of power law distributions of delayed time used in Paul ( 2018) is questionable.For example, Wanderman & Piran (2015), in addition to power law, test log-normal distributions.There are other possibilities not explored yet.
Our main conclusion is that the low redshift component of LGRBs, like the SGRBs, have compact merger progenitors.The evidence presented here in support of this comes from population study.An independent observation supporting this scenario comes from recent discovery of association of a low redshift LGRBs with Kilonova discussed above.
For the future it will be useful to explore more quantitative comparison of the FR of low redshift GRBs with different delayed scenarios to determine the correct delay mechanism.Discovery of more association of low redshift LGRBs and Kilonovae will also be very helpful.

Figure 1 .
Figure 1.Left: Cosmic SFR from most comprehensive compilation by Madau & Dickinson (2014) and two examples of Delayed-SFR with power law distribution of delay times, and indexes 1.0 and 3.0; from Paul (2018) Right: Results of 5 independent determination of LGRB FR, of somewhat different samples, using the EP-L method, all normalized at the peak value of the observed SFR (black solid line with selected error bars): From top (a)-mganta Lloyd-Ronning et al. (2019); (b)-cyan Yu et al. (2015); (c)-green Tsvetkova et al. (2017) (d)-red Petrosian et al. (2015); (e)-bluePescalli et al. (2015).With different degrees, all are showing higher LGRB FR than the SFR at low redshift (see text for more detailed comparison of these works).

(
2017) use the nominal γ-ray peak flux threshold of respective instruments.Lloyd-Ronning et al. (2019) use fluence and energy ǫ iso , instead of peak flux and luminosity, and several observed correlations.Pescalli et al. (2015) use a higher γ-ray threshold and obtain a sample with 80% redshift completion.Petrosian et al. (2015) use both the γ-ray and X-ray thresholds (based on X-ray flux compilation by Nysewander et al. (

Figure 2 .
Figure 2. Left: Same as Figure 1 left but showing the average value of 5 independent estimates plus Petrosian et al. results in red.The blue lines show the two low redsiuft components obtained bt subtracting the cosmic SFR from the observed ones.The magenta is the FR of SGRBs described in the next section.Right: FR of SGRBs using most recent Swift sample of SGRBs with known redshift, using the procedures described inDainotti et al. (2021b).Here σ(z) is the cumulative (> Z) redshift distribution corrected for observational selection effects (blue dots), with arbitrary normalization.Blue curve is a broken power law fit (with indicated form anf fit parameters), from which we obtain the FR, ρ(Z) analytically as described in the text.