New evidence of multiple channels for the origin of gamma-ray bursts with extended emission

Gamma-ray bursts (GRBs) are the most intense explosions in the universe. GRBs with extended emission (GRB EE) constitute a small subclass of GRBs. GRB EE are divided into EE-I GRBs and EE-II GRBs, according to the Amati empirical relationship rather than duration. We test here if these two types of GRB have different origins based on their luminosity function (and formation rate). Therefore, we use Lynden-Bell's c^- method to investigate the LF and FR of GRBs with EE without any assumption. We calculate the formation rate of two types of GRBs. For EE-I GRBs, the fitting function can be written as \rho (z) \propto {(1 + z)^{ - 0.34 \pm 0.04} for z<2.39 and \rho (z) \propto {(1 + z)^{ - 2.34 \pm 0.24}} for z>2.39. The formation rate of EE-II can describe as \rho (z) \propto {(1 + z)^{ - 1.05 \pm 1.10}} for z<0.43 and \rho (z) \propto {(1 + z)^{ - 8.44 \pm 1.10}} for z>0.43. The local formation rate are \rho (0) = 0.03 Gpc^{-3}yr^{-1} for some EE-I GRBs and \rho (0) = 0.32 Gpc^{-3}yr^{-1} for EE-II GRBs. Based on these results, we provide a new evidence that the origins of EE-I GRBs are different from EE-II GRBs from the perspective of event rate. The EE-I GRB could be produced from the death of the massive star, but EE-II GRB bursts may come from other processes that are unrelated to the SFR. Our findings indicate that the GRB with EE could have multiple production channels.


INTRODUCTION
Gamma-ray bursts (GRBs) are the most violent explosions in the universe and emit high-energy radiations which are produced in an ultra-relativistic jet (Zhang 2007;Gehrels et al. 2009).In the internal shock model, the inner engine will produce shells with comparable energy but a different Lorentz factors Γ.The slower shell, which is followed by a faster shell, catches up with it and collides, which can produce the pulse profile observable in most GRBs (Kobayashi et al. 1997;Daigne & Mochkovitch 1998;Piran 1999).Traditionally, GRBs are divided into long GRBs (lGRB;  90 > 2 s) and short GRBs (sGRB;  90 < 2 s) (Kouveliotou et al. 1993). 90 is the time interval during which the integrated photon counts accumulate from 5% to 95% of the total photon counts in the prompt emission.The measurement of  90 is influenced by different instruments.The dividing line of Swift GRB is 1 s (Zhang et al. 2020b;Deng et al. 2022) which is very close to the 1.27 s value of GBM GRBs (Gruber et al. 2014), and the duration distribution peak at 0.21 and 42.66 for sGRB and lGRB, respectively.lGRBs are generally believed to originate from the death of massive stars.The association between GRB and supernovae provided direct evidence (Hjorth et al. 2003;Stanek et al. 2003).Therefore, lGRBs can be seen as a tool for tracking the star formation rate (Pescalli et al. 2016;Yu et al. 2015;Dong et al. 2022).sGRB are believed to originate from the merger of binary compact objects.The association of sGRB 170817A with transient gravitational waves supports the idea that some sGRB is produced by binary ★ E-mail: sunqibin@ynao.ac.cn † E-mail: zbzhang@gzu.edu.cnneutron stars (BNS) system (GW 170817; Abbott et al. 2017).The more comparison of comprehensive properties of GRB associated with supernovae-kilonovae can be found in our recent work (Li et al. 2023).
However, many authors have also reported a special class of GRBs with extended emission (EE), where the EE is defined as a lowintensity burst following the initial main emission (Lazzati et al. 2001;Connaughton 2002;Burrows et al. 2005;Norris & Bonnell 2006;Lan et al. 2020).Currently, there are several popular speculations about the production of EE: (1) spin-down of a strongly magnetized neutron star (Bucciantini et al. 2012) ; (2) a relativistic wind extracting the rotational energy from a protomagnetar (Metzger et al. 2008); (3) material fallback of the material heated by r-process (Desai et al. 2019).Current research shows that the second peak of GRB 000727 occurs seven seconds after the initial peak (Mazets et al. 2002) .Interestingly, EE components can be identified in both lGRBs and sGRBs from the light curve of prompt emission.Norris et al. (2000) suggested an anti-correlation between spectral lag and peak luminosity for lGRB, but this relationship is different for sGRB and lGRB.Gehrels et al. (2006) found that the lag and luminosity of GRB 060614 with EE belong to the sGRB plane.Recently, the lGRB 211211A was characterized by a main emission (ME) phase (13 s) and an EE phase lasting 55 s (Rastinejad et al. 2022;Yang et al. 2022;Chang et al. 2023).
The existence of EE makes it confusing to distinguish between lGRB and sGRB by relying only on the criterion of  90 .There is a general method involving the peak energy in the rest frame  , and isotropic energy   , named as Amati correlation (Amati et al. 2002;Amati 2005), which is used to classify different types of GRBs (van Putten et al. 2014;Zhang et al. 2018;Minaev & Pozanenko 2020;Li et al. 2023;Zhu et al. 2023).Qin & Chen (2013) investigated the distribution of the logarithmic deviation of the peak energy in rest frame ( , ).They proposed a statistical classification of GRBs in the  , versus   plane (Amati GRBs and non-Amati GRBs), in which the Amati type bursts well follow the Amati relation, non-Amati type bursts do not.Zhang et al. (2020a) divided long/short GRB with EE into two subclasses (EE-I and EE-II) again based on their positions in the  , −   plane, and suggested that these two subclasses have different origins by comparing the empirical relationship (e.g., Yonetoku correlation and peak energy distribution, etc..).According to their results, it is more reasonable to classify GRB with EE into types I and II, which motivates us to investigate their progenitors further.
It is acceptable that lGRBs are associated with the deaths of massive stars.Therefore, it is reasonable to use lGRBs to investigate the star formation rate (SFR) (Yonetoku et al. 2004;Wang & Dai 2009;Butler et al. 2010;Yu et al. 2015).The sGRB is thought to be produced from a coalescence of compact objects.Yonetoku et al. (2014) pointed out that the key to confirming this idea is the formation rate (FR) of sGRB.Theoretically, the sGRB FR will track the SFR with some delay time.The methods to estimate the coalescence rates of binary compact object systems have large difficulties and uncertainties (Lipunov et al. 1995;Fryer et al. 1999;Belczynski et al. 2002).If sGRBs are expected to be accompanied by gravitational-wave emission (Abramovici et al. 1992;Narayan et al. 1992) (e.g., GW 170817 and sGRB 170817A; Rossi et al. 2018), then the local sGRB FR is directly related to the expected number of GW events in the future.The FR of sGRB has been extensively explored in previous research (Ando 2004;Zhang & Wang 2018;Dainotti et al. 2021).Two of the critical properties characterizing the population of GRBs are their formation rate (FR) and luminosity function (LF), which are helpful to profoundly understand the nature of GRBs (Dermer 2007;Pescalli et al. 2016).FR and LF respectively represent the number of bursts per unit comoving volume and the relative fraction of bursts with a certain luminosity.The construction of these two distributions, however, requires measuring the redshift.Only some GRBs EE had well redshift measurements a few years ago, such as GRB 060614 (Gehrels et al. 2006).The luminosity function and formation rate must be urgently studied with the number of GRBs EE with known redshifts increasing.
The previous studies on GRB LF and FR usually used the log  − log  distribution (Fenimore & Ramirez-Ruiz 2000;Cao et al. 2011;Sun et al. 2015).However, the distribution is produced by the luminosity and redshift convolved (Yonetoku et al. 2004).Coward (2007) pointed out that several selection effects for the observed redshift distribution of GRBs, such as Malmquist bias, observational limit of the satellite, which is the most important selection effect.The Swift have a flux limit 2 × 10 −8 / 2 /.This means that we can't observe a GRB below the flux limit.To correct this selection effect and obtain the intrinsic LF and FR, Lynden-Bell (1971) put forward a non-parametric approach named as Lynden-Bell's  − method to calculate the FR.This method has been applied to many transient phenomena, such as GRB (Yonetoku et al. 2004;Yu et al. 2015;Deng et al. 2016;Zhang & Wang 2018;Liu et al. 2021;Dong et al. 2022Dong et al. , 2023)), Active galactic nucleus (Singal et al. 2011;Zeng et al. 2021) and FRB (Deng et al. 2019).The premise of this method is that the luminosity and redshift are independent of each other.Therefore, we should first test for independence between them by using Kendall  test method (Efron & Petrosian 1992).Importantly, Dong et al. (2023) also used the non-parameters method to investigate the progenitors of low-and high-luminosity GRB samples.
In this paper, Our main purpose is to study the LF function and FR of the subclass of GRB EE using Lynden-Bell's  − method to distinguish their origin.Section 2 introduces the sample source and K-correction method.In Section 3 and Section 4, we describe the Amati relation, Lynden-Bell's  − and Kendall  test method in detail, respectively.In Section 5, we present the result of the luminosity function and formation rate of GRB EE (EE-I and EE-II).Finally, Section 6 presents the conclusion.
The spectra of GRB are generally fitted by two spectral models, including Band model (Band et al. 1993) and a single/cut-off power law model (Sakamoto et al. 2008).The form of the Band function is as follows: and the cut off power law can be expressed as Since the peak flux is observed in different energy ranges, we will use the same K-correction method to convert the flux into the 1−10 4 keV band to get bolometric luminosities (e.g., Yu et al. 2015;Zhang et al. 2018).The bolometric luminosity can be calculated by  = 4 2  ()  or  = 4 2  (). and  are the K-correction factor and the peak flux observed in the energy range, respectively.if P is in units of / 2 /, The K can be expressed as  else if P is in unit of ℎ/ 2 /, the   can be expressed as The value of the flux limit is  lim = 2.0 × 10 −8 / 2 / (Yu et al. 2015).Then, luminosity limit is given as  lim = 4 2  () lim .

AMATI RELATION
The duration of a GRB is a key indicator of its physical origin, with lGRBs perhaps associated with the collapse of massive stars and sGRBs with mergers of neutron stars.However, there is a substantial overlap in the properties of both lGRB and sGRB.To date, no other parameter fully distinguishes the origins of these two groups, such as Li et al. (2023) who verified that both GRBs associated with supernova/kilonova comply with the Amati relations that match those of long/short GRBs, but two kinds of GRB also have obviously overlapped.Classifying GRBs based on their prompt emission is a useful task, as it has the potential to quickly identify the possible properties of their progenitor and plan the most comprehensive follow-up actions within a few minutes of detecting GRB.However, the situation in this field is puzzling due to the complexity of the GRB light curves and the diversity of possible progenitors (the combination of different types of compact stars, collapsar with or without short-lived active neutron stars, etc.).Fortunately, the sample size of GRB is large enough to establish correlation relationships for classification.The  , −   relation proposed by Amati et al. (2002) is a universal method to classify the GRB into lGRB and sGRB.Amati (2006) implied the sGRB is an outlier of this correlation, so sGRB could have its own Amati relation.They found that the slope of lGRBs is 0.5 (see also Zhang et al. 2009), showing that it can be a powerful tool for discriminating different classes of GRBs and understanding their nature and differences.Zhang et al. (2018) established the  , −   relationship for lGRB and sGRB, where the lGRB and sGRB distributions are in different locations, although the slopes are consistent.This claim has been challenged by some authors (Band & Preece 2005;Krimm et al. 2009;Heussaff et al. 2013), suggesting that the relationship is the result of selection and instrumental effects, but some authors have argued that these effects are relatively small (Amati & Della Valle 2013;Demianski et al. 2017a,b).Zhang et al. (2020a) found that reclassifying the GRB with EE into EE-I and EE-II types can result in a tighter correlation.This classification is similar to that of Qin & Chen (2013), who proposed that the GRB in the Amati plane could be divided into two groups: Amati GRB and no-Amati GRB based on the logarithmic deviation of the   .Therefore, according to their empirical relationship, we divide our samples into two categories (see Fig. 2 right panel): Amati GRB (renamed as type EE-I) and no-Amati GRB (renamed as type EE-II), defined as GRBs that are located below and above the empirical relationship ( , ,   = 493 0.57  ), and the   is in units of 10 52 .The isotropic bolometric energy can be expressed as  = 4 2  ()  /(1 + ), and the peak energy in the rest frame can be calculated by  , =   (1 + ).Fig. 2a shows the  , −   locations of lGRB and sGRB with EE.The best fits the Amati relations of short and long bursts are taken from Zhang et al. (2018).We redivide these EE GRB into type EE-I and EE-II as Fig. 2b, according to the empirical relationship proposed by Qin & Chen (2013).Out of 56 EE-I types, 54 belong to lGRB (94%), and Out of 24 EE-II types, 18 belong to sGRB (75%), which indicates that most EE-I type bursts are lGRB and most non EE-II type bursts are of sGRBs.

LYNDEN-BELL'S 𝐶 − METHOD AND NON-PARAMETERIC TEST METHOD
The Lynden-Bell  − method is an effective non-parametric method to analyze the distribution of the bolometric luminosity/energy and redshift of the astronomical objects with the truncated data sample (eg., Yonetoku et al. 2004;Yu et al. 2015;Deng et al. 2016;Liu et al. 2021;Dong et al. 2022), Active galactic nucleus (Singal et al. 2011;Zeng et al. 2021) and FRB (Deng et al. 2019).This work also uses this method to study LF and FR.
We use the non-parametric test method raised by Efron & Petrosian (1992) to derive the evolution function ().In the (, ) plane as shown in Fig. 3, for the ith point (  ,   ), we can define   as where   is the luminosity of the th GRB EE and  max  is the maximum redshift at which a GRB EE (EE-I and EE-II) with the luminosity   can be detected by Swift detector.This range is shown as a black rectangle in Fig. 3.The number included in this range is   , and the   is defined as   − 1. which means take th out.and the  1  also can be defined as where  min  is the limit luminosity at the redshift   .This range is shown as the red rectangle in Fig. 3.The number included in this region is   .In the black rectangle,   is defined as the events number that have redshift  less than   .  should be uniformly distributed between 1 and   based on the fact that  and  are independent.The Kendall  test statistic is (Efron & Petrosian 1992) where   = 1+  2 and   =  2  −1 12 are respectively the expected mean and variance of   . will be zero if the size of the sample of   ≤   is equal to the size of the sample with   ≥   .After we find the function form of (), the effection of luminosity evolution can be removed by transforming  into  0 .
and  are independent on each other until the test statistic  is zero by changing the value of k.We show how  changes with varying .The  value is 2.64 and 6.66 for EE-I and EE-II in Fig. 4. Therefore, the non-evolving luminosity can be written as  0 = /(1 + )  in Fig. 5.We can use a non-parametric method to derive the local cumulative LF distribution from the following equation (Lynden-Bell 1971;Efron & Petrosian 1992) and the cumulative number distribution can be obtained from Next, the FR can be calculated by is the differential comoving volume, which can be expressed as and the expected number of GRBs can be estimated by (Lan et al. 2019) The Swift instrument has been running for approximately T=19 years.the field of view of this telescope is Ω = 1.33 (Sun et al. 2015).

RESULT
In this part, We present the LF and FR of subclasses of GRBs EE, respectively.

Luminosity function
Fig. 6 shows the distribution of normalized LF ( 0 ).Using the broken power law, we can fit this curves to obtain the forms of LF for the dim segment and bright segment for EE-I GRB as and for EE-II as where break point   0 = 3.03 × 10 49 / for EE-II GRBs, which is small two order than the break luminosity   0 = 1.00 × 10 51 / of EE-I GRBs.Yu et al. (2015) described the cumulatively luminosity distribution by a broken power law function with  = −0.14 ± 0.02,  = −0.7 ± 0.03,   0 = 1.43 × 10 51 / for 127 lGRBs.Pescalli et al. (2016) estimated the luminosity distribution of complete 99 lGRBs with  = −1.32 ± 0.21,  = −1.84± 0.24 and   0 = 2.82 × 10 51 /.Wanderman & Piran (2015) used sGRB, which originated from non-corecollapsars to estimate the LF and FR, and they acquire the breaking point as   = 2.0 × 10 52 / with power law indices of 0.95 and 2.0 for the dim and bright segments, respectively.Liu et al. (2021) used 324 Fermi sGRB to derive the break point as  = −0.45± 0.01,  = −1.11± 0.01, with the break luminosity   0 = 4.92 × 10 49 /.It is worth noting that this result is roughly consistent with our work.The break luminosity of EE-I GRBs is larger by two orders than EE-II GRBs, similar with lGRB to sGRB after removing the luminosity evolution with redshift.We must emphasize that the luminosity function only presents the local distribution at  = 0. Therefore, the LF at redshift z will be rewritten as

Formation rate
Fig. 7 shows the normalized cumulative distribution of redshift ().
From Eq.10, we can calculate the FR of GRB EE.Firstly, the differential cumulative redshift distribution form should be derived.In Fig. 8, the blue and purple stepwise line is the FR of GRBs EE.It is obvious that EE-II GRBs has kept a decreasing trend.The error bars is calculate by the GRBs number of black or red rectangle,   and   .The final error of the FR obtained through the error transfer formula.The error bar gives a 1  poisson error (Gehrels 1986).We also fit the different segments using the broken power law function.
Fig. 9 shows the SFR compared with the FR of EE-I and EE-II GRBs.From the qualitative perspective, The FR of EE-I GRB is The formation rate for 56 EE-I and 24 EE-II GRBs.The datas of SFR is taken from Hopkins & Beacom (2006).
roughly consistent with SFR at  > 1.There is still fierce debate regarding the excess of  < 1. Possible reasons include completeness, unclear definition of  90 , different origins of high-and lowluminosity bursts, etc (Pescalli et al. 2016;Dong et al. 2022Dong et al. , 2023)).
Table 1 lists samples associated with supernovae and kilonovae (reference Li et al. 2023).The GRB associated with supernovae or kilonovae is believed to originate from the death of massive stars and the merger of binary compact objects, respectively.According to the statistical results, there are a total of 6 supernova GRBs and 5 kilonovae GRB with EE.It is worth emphasizing that EE not only exists in Type I GRB (Compact Star) but also in Type II GRB (Massive Star) (Li et al. 2020), and it is not a unique process for a particular type of GRB.It is a generic process that commonly exists in these two kinds of GRBs.

CONCLUSION
Gamma-ray bursts are brief and violent gamma-ray explosions in the universe, lasting from a few milliseconds to a few thousand seconds.GRBs are essential tools for tracing star formation history and studying the merger of compact objects.Using non-parametric methods to investigate the LF and FR can better understand the intrinsic properties of GRB with EE, because the co-evolution between redshift and luminosity can be removed.
In this work, we used for the first time 80 GRBs with EE (56 EE-I and 24 EE-II) that they have known redshifts and well-measured spectra parameters to drive the bolometric luminosity.Then, we used a non-parametric method to derive the isotropic luminosity and formation rate based on the constructed  −  plane.Before that, we had addressed the flux-truncation effect by the Kendall  method.In our analysis, the evolution function () = (1 + )  can transform  into  0 .The normalized luminosity distribution can be fitted by a broken power law after removing the redshift dependence.This fitted form can be expressed as ( 0 ) ∝  −0.34±0.010 for the dim segment and ( 0 ) ∝  −0.67±0.020 for bright segment with EE-I GRB, the broken point is   0 = 1.00 × 10 51 /.The form can be expressed as ( 0 ) ∝  −0.43±0.020 for dim segment and ( 0 ) ∝  −0.93±0.010 for bright segment with EE-II GRB, the broken point is   0 = 3.03 × 10 49 /.We also found that the FR of the GRB EE subclass keeps decreasing, and that a broken power law can fit it.The fitting function of EE-I GRBs can be written as () ∝ (1 + ) −0.34±0.04 for  < 2.39 and () ∝ (1 + ) −2.34±0.24for  > 2.39.The FR of EE-II can describe as () ∝ (1 + ) −1.05±0.03for  < 0.43 and () ∝ (1 + ) −8.44±1.10 for  > 0.43.Using Eq.10,The local formation rate are (0) = 0.03 Gpc −3 yr −1 for EE-I GRBs and (0) = 0.32 Gpc −3 yr −1 for EE-II GRBs.It can be found that the formation rates of EE-I and EE-II GRBs are significantly different (see Fig. 8), which further suggests that these two kinds of GRB may have different origins.It is worth noting that it is difficult to search for EE-II GRBs at high redshifts because they are weaker in luminosity than EE-I GRBs, and current instruments are not sensitive to EE-II GRBs above  > 1.This leads to a relatively small number of EE-II GRBs at high redshift.Therefore, our result have speculative to derive a reliable FR for EE-II GRBs at  > 1.When the sensitivity of instruments is further improved, The results would be more reliable if a sample of high redshift E-II bursts is adopted in the future.
Since EE-I GRBs have a similar position in the Amati relation to lGRBs, which are thought to originate from core collapse, we further compare the formation rate of EE-I GRBs with the SFR.The results of the comparison show that the evolution of the formation rate of EE-I GRBs is similar to that of the SFR (see Fig. 9), suggesting that EE-I GRBs may arise from the death of massive stars, whereas EE-II GRBs, which are unrelated to the SFR, may come from other processes unrelated to the SFR.Therefore, we suggest the GRB with EE could have multiple production channels from the perspective of their formation rates.

Figure 1 .
Figure 1.The light curve of prompt emission of GRB 080413A and GRB 140430A in 15-350 keV energy band.The blue dotted line shows the fitted line of the Bayesian block.The red line represents the 2  line.

Figure 2 .
Figure 2. The distribution of 60 lGRB (orange rectangles) and 20 sGRB (green dots) in the  , −   plane (left panel).The best solid orange and green fitting lines is derived from Zhang et al. (2018) for normal lGRB and sGRB.Right: The distribution of 56 EE-I (blue rectangle) and 24 EE-II GRB (purple dots) in the  , −   plane (right panel).The brown solid line is taken from Qin & Chen (2013).

Figure 3 .Figure 4 .
Figure 3.The distribution of luminosity and redshift in the  −  plane.The blue squares and purple dots represent the EE-I and EE-II GRBs, respectively.The flux limit is 2 × 10 −8  / 2 /.

Figure 5 .
Figure 5. Non-evolving luminosity L 0 = L/(1 + z)  of 60 GRBs EE (56 EE-I GRBs and 24 EE-II GRBs) above the truncation line.The k value is 2.64 and 6.66 for EE-I GRB and EE-II GRB, respectively.

Figure 7 .
Figure 7.The normalized cumulative distribution of redshift for 56 EE-I and 24 EE-II GRBs).

Table 1 :
spectral Parameters of GRBs with EE