Does the GRB Duration Depend on Redshift?

Abstract

Several hundred gamma-ray burst (GRB) redshifts have been determined to date. One of the other important properties—besides the distance—of the GRBs is the duration of the burst. In this paper, we analyse these two important quantities of the phenomena. In this paper, we map the two-dimensional distribution and explore some suspicious areas. As it is well known that the short GRBs are closer than the others, we search for parts in the Universe where the GRB duration is different from the others. We also analyse whether there are any ranges in the duration where the redshifts differ. We find some suspicious areas, however, no other significant region was found than the short GRB region.

1. Introduction

Gamma-ray bursts (GRBs) are the most energetic phenomena in the universe, which can be detected up to very high redshifts ($z=8.2$ spectroscopically [1,2] and $z=9$ by photometry [3]). The GRBs have attracted a lot of attention as promising distance indicators complementary to other cosmological probes and, among others, these calibrated GRBs are consistent with that of SNe Ia [4,5]. The GRB data could be combined with other objects to examine the evidence for different cosmological models. Amati et al. [6] showed that the $\Lambda$CDM model is statistically favoured over the wCDM scenario. Moreover, Khadka and Ratra [7] found that the cosmological parameters obtained from GRBs were consistent with baryon acoustic oscillation and Hubble parameter measurements and the GRBs could be used as complementary and outstanding probes to trace dark energy’s evolution in support of other indicators [8,9].

Although there are indications that there are more GRB groups [10,11,12,13,14,15,16,17,18,19,20,21] other than the classical short and long dichotomy [22,23], in this paper we deal only with short (SGRB) and long (LGRB) gamma-ray bursts. The cosmological origin of gamma-ray bursts (GRBs) is well established (e.g., [24]). Assuming that the Universe exhibits large-scale homogeneity and isotropy, the same is also expected for GRBs. GRBs are, to date, the only objects which have been sampled in the observable Universe as a whole, thus the large-scale angular isotropy of the sky distribution of GRBs has been well studied over recent decades. Most of these studies have demonstrated that the sky distribution of GRBs is isotropic [25,26,27,28,29,30,31].

However, there are indications for some large-scale anisotropy in space distribution of quasars [32] and GRBs [33,34,35,36]. Some GRB subsamples appear to deviate significantly from isotropy. Vavrek et al. [30] reported that the angular distributions of short and long GRBs are different, while Řípa [37], Řípa and Shafieloo [38,39] analysed the isotropy of Fermi GRBs according to their properties (duration, fluences, peak fluxes). Cline et al. [40] found that the angular distributions of very short GRBs are anisotropic, and Magliocchetti et al. [29] reported that the short GRB class in general deviates from angular isotropy. Mészáros et al. [28] and Litvin et al. [41] wrote that the angular distribution of intermediate duration ($T_{90}$) GRBs is not isotropic. LGRBs are believed to be produced from core-collapsed supernovae (SNe) [42,43,44] which is supported by observations that some GRBs are associated with SNe (e.g., [45,46]). This model implies that the LGRB event rate should trace the cosmic star formation rate [47,48,49,50,51,52]. They generally occur in faint, blue, low-mass star-forming galaxies [53,54] and also in the bright regions of their hosts [55,56,57].

Before the launch of the Neil Gehrels Swift Observatory, there were a few suggestions for cosmological time dilation in GRB light curves (e.g., [58,59]). However, after the launch of the Neil Gehrels Swift Observatory and the Fermi Gamma-ray Space Telescope, there were no clear signatures of cosmological time dilation in the GRB light curves (e.g., [60,61]).

GRBs are the only astrophysical objects sampled in the whole observable Universe [62]. If we can define distance indicators from high-energy GRB observed parameters (e.g., Amati relation [63], Ghirlanda relation [64], Yonetoku relation [65], Liang–Zhang relation [66], Dainotti relation [67]), we can test the cosmological models even at very high redshifts including the validity of the cosmological principle. The light curves’ spectral dependency and the detectors’ response will further complicate the connection between the intrinsic and observed $T_{90}$s. Therefore, it is interesting to investigate whether the $1+z$ time dilatation factor is detectable in the shape of observed light curves [68]. It is also an interesting problem whether the physics of the GRB central engine at low and high redshifts are the same or not [69,70].

2. Duration versus Redshift Distribution Analyses

Currently, nearly five hundred redshifts have been observed for GRBs. The Caltech GRBOX web-page contains most of them1, therefore, in this analysis we use their data set. In this database, 34 GRBs had photometric redshifts. We have to note that where both photometric and spectroscopic redshifts were available, we used the latter ones. Based on the Swift GRB observations, less then 20% of the photometric redshifts differ significantly from the spectroscopic ones. This means that approximately five to six observations from 34 contain excessive noise, which is approx. 1% of the total data. We believe that this is not a major source of error besides the Poisson noise. Among these 487 GRBs, 474 had $T_{90}$ duration information ($T_{90}$ is the time over which a burst emits from 5% of its total measured counts to 95% [71,72]) as well. SWIFT BAT was used to estimate $T_{90}$ of a burst from 15–150 keV [61], in contrast the FERMI GBM team which computes $T_{90}$ in the 50–300 keV range [73]. Earlier, Rácz et al. [74] and Racz et al. [75] did not find any trace of selection effects in the spectral behaviour of GRBs, using the common GRB observation by both satellites. Pinter et al. [76] have shown a small, but measurable, difference between the two satellites’ $T_{90}$ distribution which appears for the shortest and the very long GRBs. The origin of the difference is that in some cases the Swift detects a soft, long emission tail of the GRB which is not detected by the Fermi GBM. This suggests a question about grouping. We do not deal with the grouping here, as we study only the duration range where this effect is not significant; we will omit the short GRBs for reasons we will discuss later, and will concentrate on the long GRBs while the number of the deviating longest GRBs is small.

In Figure 1 the cumulative distribution of the redshifts of the 474 GRBs can be seen; one can observe that the ∼60% of the GRBs are in the $log(1+z)$ = (0.25–0.6) range.

2.1. Comparison with the Whole Sample Redshifts

Figure 2 shows the redshift vs. duration ($T_{90}$) distribution of the 474 GRBs. To study whether the redshift distribution depends on the duration ($T_{90}$) parameter, one can use several statistical tests. Here, we ordered the GRBs by duration and chose n consecutive ones. This group’s redshift distribution was compared with the complementary $474-n$ GRBs’ redshift distribution using a Kolmogorov–Smirnov test (KS). (Note: in this paper we use 10 base logarithm, therefore $log$ always means $lo{g}_{10}$).

We compared the redshift distributions starting the group at the kth position. We carried out this process for different group sizes from $n=8$ to $n=99$. As an example, Figure 3 shows the KS p value’s dependence of k for $n=20$, 42 and 65, respectively. The green (0.0455) and blue (0.0027) lines show 2 and 3 sigma significance, respectively.

2.2. Comparison with the 421 Non-Short GRBs’ Redshifts

From the KS tests results in Figure 3, one can see that the biggest deviations are for the shortest bursts. It is well known that these short GRBs are typically closer in distance than long GRBs [77,78,79]. Some believe that 2 s is the border between short and long bursts [25], however, many studies show the border is not that obvious [27,28,80,81].

As the different groups’ $T_{90}$ distributions are quite wide, we chose the $T_{90}=5.0$ s for this division to make sure there are no short GRBs in the sample. This means the first 53 GRBs ($T_{90}\le 5$ s) were not used for calculating the distribution of the long GRBs’ redshift distribution. (For the difference in the 421 long GRBs’ distribution and the whole 474 GRB sample distribution, see Figure 1). Then, we repeated the method which was described in Section 2.1. We also carried out this process from $n=8$ to $n=99$. Figure 4 shows the KS p value’s dependence of k for $n=20$, 42 and 65, respectively. The green (0.0455) and blue (0.0027) lines show 2 and 3 sigma significance, respectively.

Figure 5 shows the two-parameter (n, duration) KS p value. Note that the short part was cut from the figures, since p is extremely low in the short duration area (see Figure 3), which also means high significance. This means the short GRB redshift distribution differs with a very high significance level from the long duration GRBs redshift distribution. We call this part of the GRBs’ distribution AREA1.

The p value reaches 0.0027 in three areas, the aforementioned short duration part (AREA1), the $T_{90}$ (16 s, 20 s) interval (AREA2) with n between 12 and 21 and the $T_{90}$ (49 s, 61 s) interval (AREA3) with n between 23 and 36. The p value reaches 0.0455 in only four areas. The three aforementioned areas and the $T_{90}$ (9 s, 21 s) interval (AREA4) with n between 59 and 68. Figure 6 and Figure 7 show the cumulative redshift distribution of AREA3–4.

2.3. The Redshift vs. T₉₀ Method

One can make a similar analysis by swapping the variables: order the GRBs by redshift, select a redshift interval, then compare the duration distribution of this subsample with the duration distribution of the complementary sample. Since there are few short bursts with redshift bigger than one, we omitted the 53 GRBs which had $T_{90}\le 5$ s. Therefore, we analysed the remaining 421 GRBs. Here, we ordered the GRBs by redshift and chose the closest, consecutive n GRBs and compared the n closest GRBs’ duration distribution with the $421-n$ GRBs’ duration distribution, performing the Kolmogorov–Smirnov test (KS). We repeated this process starting from the kth GRB and repeated the process with a block size of n running from 8 to 99.

Figure 8 shows the two-parameter (n, redshift) p value. The p value reaches 0.0027 in two areas, the $1.49<z<1.61$, $19<n<38$ (AREA5) and $2.91<z<3.075$, $11<n<19$ (AREA6). Figure 9 and Figure 10 show the cumulative redshift distribution of AREA5–6.

3. Summary and Conclusions

Several hundred GRB redshifts have been determined to date. Some of the most important properties of the GRBs are the duration and the redshift of the bursts. In this paper, we analysed 474 GRBs which had duration and redshift information as well. The most significant interrelation between these two quantities was that the short GRB redshifts are significantly smaller (we call this AREA1). Therefore, in this research we excluded the short GRBs (to be sure, we chose the $T_{90}>5$ s) and we analysed the remaining 421 GRBs.

We chose a certain number (n = 8–99) of GRBs which had a certain duration in an interval and compared their redshifts with the remaining GRBs’ redshift distribution. We found three intervals where the redshift distribution was different than in the rest of the GRBs (see Figure 6 and Figure 7). AREA2 ($T_{90}$ is between 16 s, 20 s and n between 12 and 21) and AREA4 ($T_{90}$ = (9 s, 21 s), n = (59, 68)) bursts tend to be farther and AREA3 ($T_{90}$ = (49 s, 61 s), n = (23, 36)) GRBs tend to be closer than the average.

It should be noted that the one-block KS significances will be lower than the whole sample because we conducted many non-independent statistical tests. The trial factor can be estimated by $421/n$ (i.e., for $n=65$ it is ≈6.5).

We also analysed the other way, by choosing a redshift interval and checking whether the $T_{90}$ distribution in this interval is different than the remaining GRBs’ duration distribution (see Figure 9 and Figure 10). We found two intervals where the duration distribution tended to be longer than in the overall sample (see AREA5 and AREA6 in Figure 11).

Balázs et al. [82] and Rácz et al. [74] have shown that variance of the observed $T_{90}$ is dominated by the intrinsic variance of the duration. Our results are consistent with these findings as no significant time dilatation effects were found [60,61]. As a consequence, the redshift plays a marginal role in the observed distribution of GRB durations [83].

More precisely, the AREA4 GRBs are relatively far, because there are fewer GRBs closer than $z\sim 1.5$ in the range of $T_{90}$ = (9 s, 21 s) than is typical. The origin of this effect could be either simply a statistical fluctuation or this could be a signature of some selection effect around the overlap of the intermediate [10,11,12,13,14,15,16,17,18,21] and long GRB groups. Another fact is that, in this duration range, there are relatively more Fermi GBM observations [76] which further complicates the selection effects. For the redshift observations, the Swift BAT usually produces arcminute precision positional data, which is not the case for the Fermi GBM. Further detailed studies are needed to answer this question.