Abstract

Spectral lag, which is defined as the difference in time of arrival of high- and low-energy photons, is a common feature in gamma-ray bursts (GRBs). Previous investigations have shown a correlation between this lag and the isotropic peak luminosity for long duration bursts. However, most of the previous investigations used lags extracted in the observer frame only. In this work (based on a sample of 43 Swift long GRBs with known redshifts), we present an analysis of the lag–luminosity relation in the GRB source frame. Our analysis indicates a higher degree of correlation −0.82 ± 0.05 (chance probability of ∼5.5 × 10−5) between the spectral lag and the isotropic peak luminosity, Liso, with a best-fitting power-law index of −1.2 ± 0.2, such that Liso ∝ lag−1.2. In addition, there is an anticorrelation between the source-frame spectral lag and the source-frame peak energy of the burst spectrum, Epk(1 + z).

1 INTRODUCTION

Gamma-ray bursts (GRBs) are extremely energetic events and produce highly diverse light curves. A number of empirical correlations between various properties of the light curves and GRB energetics have been discovered. However, the underlying physics of these correlations is far from being understood.

One such correlation is the relation between isotropic peak luminosity of long bursts and their spectral lags (Norris, Marani & Bonnell 2000). Various authors have studied this relation using arbitrary observer-frame energy bands of various instruments (Norris 2002; Gehrels et al. 2006; Schaefer 2007; Hakkila et al. 2008; Ukwatta et al. 2010c, hereafter U10). These investigations support the existence of the relation, however with considerable scatter in the extracted results. Recently, Margutti et al. (2010) investigated spectral lags of X-ray flares and found that X-ray flares of long GRBs also exhibit the lag–luminosity correlation observed in the prompt emission.

The spectral lag is defined as the difference in time of arrival of high- and low-energy photons and is considered to be positive when the high-energy photons arrive earlier than the low-energy ones. Typically, the spectral lag is extracted between two arbitrary energy bands in the observer frame. However, because of the redshift dependence of GRBs, these two energy bands can correspond to a different pair of energy bands in the GRB source frame, thus potentially introducing an arbitrary energy dependence to the extracted spectral lag.

In order to explore whether the lag–luminosity relation is intrinsic to the GRB, it is preferable to extract spectral lags in the source frame as opposed to the observer frame. At least two corrections are needed to accomplish this: (1) correct for the time-dilation effect (z-correction) and (2) take into account the fact that for GRBs with various redshifts, observed energy bands correspond to different energy bands at the GRB source frame (K-correction; Gehrels et al. 2006).

The first correction is straightforward and is achieved by multiplying the extracted lag value (in the observer frame) by (1 + z)−1. The second correction, on the other hand, is not so straightforward. Gehrels et al. (2006) attempted to approximately correct the spectral lag by multiplying the lag value (in the observer frame) by (1 + z)0.33. We note here that this correction is based on the assumption that the spectral lag is proportional to the pulse width and that the pulse width itself is proportional to the energy (Fenimore et al. 1995; Zhang et al. 2009). These approximations depend on clearly identifying corresponding pulses in the light curves of each energy band, and may be of limited validity for a large fraction of GRBs in which the light curves are dominated by overlapping multipulse structures.

Using a sample of 31 Swift GRBs, U10 found that the correlation coefficient improves significantly after the z-correction is applied. However, this correlation does not improve further after the application of the K-correction as defined by Gehrels et al. (2006).

An alternative is to make the K-correction by choosing two appropriate energy bands fixed in the GRB source frame and projecting these bands into the observer frame using the relation Eobserver= Esource/(1 + z). Ukwatta et al. (2010b) used this method for the first time to investigate the lag–luminosity relation in the source frame of the GRB. They selected two source-frame energy bands (100–200 and 300–400 keV) and used background-subtracted as well as non-background-subtracted Swift data to extract lags. Non-background-subtracted data were used to improve the signal-to-noise ratio for weak bursts. They found that the source-frame relation seems a bit tighter, but with a slope consistent with previous studies. Arimoto et al. (2010) also looked at a limited sample of High Energy Transient Explorer (HETE)-II bursts (eight GRBs) both in the observer frame and the source frame, and concluded that there is no significant effect from the redshift. However, the redshift distribution of their burst sample is very narrow and peaks around 1. In contrast to Ukwatta et al. (2010b), in this study we used only background-subtracted data and measured the lag between source-frame energy bands 100–150 and 200–250 keV (the reason for selecting these particular energy bands is described in Section 2) for a sample of 43 Swift bursts with spectroscopic redshifts.

In this work, we have investigated only long GRBs, i.e. bursts with duration greater than ∼2 s. It is rather difficult to test the lag–Liso relation effectively for short GRBs due to a lack of spectroscopically measured redshifts. None of the short bursts detected so far has any redshift measurements obtained from a spectroscopic analysis of their optical afterglow. Moreover, it has been shown that short GRBs have either small or negligible lags (Norris & Bonnell 2006; Zhang et al. 2006). According to the lag–Liso relation, these small lag values imply short bursts to be highly luminous. However, based on the redshift measurements of their host galaxies, we can show that short GRBs are generally less luminous than long bursts. Hence, short bursts seem to not follow the lag–luminosity relation (Gehrels et al. 2006).

The structure of this paper is the following. In Section 2, we discuss briefly our methodology for extracting spectral lags. In Section 3, we present our results for a sample of 43 Swift GRBs. We discuss our results with two candidate models in Section 4. Finally, in the last section (Section 5), we summarize our results and conclusions. Throughout this paper, the quoted uncertainties are at the 68 per cent confidence level.

2 METHODOLOGY

The Swift Burst Alert Telescope (BAT) is a highly sensitive instrument using a coded-mask aperture (Barthelmy et al. 2005). BAT uses the shadow pattern resulting from the coded mask to facilitate localization of the source. When a gamma-ray source illuminates the coded mask, it casts a shadow on to a position-sensitive detector. The shadow cast depends on the position of the gamma-ray source on the sky. If one knows the tile pattern in the coded mask and the geometry of the detector, it is possible to calculate the shadow patterns created by all possible points in the sky using a ray-tracing algorithm. Hence, by correlating the observed shadow with the pre-calculated shadow, one can find the location of the source. However, each detector can be illuminated by many sources and a given source can illuminate many detectors. Hence, in order to disentangle each sky position, special algorithms have been developed and integrated into the data analysis software by the Swift BAT team.

To generate background-subtracted light curves, we used a process called mask weighting. The mask weighting assigns a ray-traced shadow value for each individual event, which then enables the user to calculate light curves or spectra. We used the batmaskwtevt and batbinevt tasks in ftools to generate mask-weighted, background-subtracted light curves, for various observer-frame energy bands, as shown in Table 1. These are the energy bands that correspond to fixed energy bands in the source frame, i.e. 100–150 and 200–250 keV. These particular energy bands were selected so that after transforming to the observer frame they lie in the detectable energy range of the Swift BAT instrument (see Fig. 1). Even though the BAT can detect photons up to 350 keV, we limited the upper boundary to 200 keV in the observer frame. This is because the mask-weighted effective area of the detector falls rapidly after 200 keV, and as a result the contribution to the light curve from energies greater than ∼200 keV (in observer frame) is negligible (Sakamoto et al. 2011).

Table 1

The observer-frame energy bands and energy gaps (the energy difference between the mid-points of energy bands) for bursts in the sample.

GRB Redshift Low-energy band (keV) High-energy band (keV) Energy gap (keV) 
GRB 050401 2.8991 26–38 51–64 26 
GRB 050603 2.8212 26–39 52–65 26 
GRB 050922C 2.1993 31–47 63–78 32 
GRB 051111 1.5494 39–59 78–98 39 
GRB 060206 4.0565 20–30 40–49 20 
GRB 060210 3.9136 20–31 41–51 21 
GRB 060418 1.4907 40–60 80–100 40 
GRB 060904B 0.7038 59–88 117–147 59 
GRB 060908 1.8849 35–52 69–87 35 
GRB 060927 5.46410 15–23 31–39 16 
GRB 061007 1.26211 44–66 88–111 45 
GRB 061021 0.34612 74–111 149–186 75 
GRB 061121 1.31513 43–65 86–108 43 
GRB 070306 1.49614 40–60 80–100 40 
GRB 071010B 0.94715 51–77 103–128 52 
GRB 071020 2.14516 32–48 64–79 32 
GRB 080319B 0.93717 52–77 103–129 52 
GRB 080319C 1.94918 34–51 68–85 34 
GRB 080411 1.03019 49–74 99–123 50 
GRB 080413A 2.43320 29–44 58–73 29 
GRB 080413B 1.10121 48–71 95–119 48 
GRB 080430 0.76722 57–85 113–141 56 
GRB 080603B 2.68923 27–41 54–68 27 
GRB 080605 1.64024 38–57 76–95 38 
GRB 080607 3.03625 25–37 50–62 25 
GRB 080721 2.59126 28–42 56–70 28 
GRB 080916A 0.68927 59–89 118-148 59 
GRB 081222 2.77028 27–40 53–66 26 
GRB 090424 0.54429 65–97 130–162 65 
GRB 090618 0.54030 65–97 130–162 65 
GRB 090715B 3.00031 25–38 50–63 25 
GRB 090812 2.45232 29–43 58–72 29 
GRB 090926B 1.24033 45–67 89–112 45 
GRB 091018 0.97134 51–76 101–127 51 
GRB 091020 1.71035 37–55 74–92 37 
GRB 091024 1.09136 48–72 96–120 48 
GRB 091029 2.75237 27–40 53–67 27 
GRB 091208B 1.06338 48–73 97–121 49 
GRB 100621A 0.54239 65–97 130–162 65 
GRB 100814A 1.44040 41–61 82–102 41 
GRB 100906A 1.72741 37–55 73–92 37 
GRB 110205A 2.22042 31–47 62–78 31 
GRB 110213A 1.46043 41–61 81–102 41 
GRB Redshift Low-energy band (keV) High-energy band (keV) Energy gap (keV) 
GRB 050401 2.8991 26–38 51–64 26 
GRB 050603 2.8212 26–39 52–65 26 
GRB 050922C 2.1993 31–47 63–78 32 
GRB 051111 1.5494 39–59 78–98 39 
GRB 060206 4.0565 20–30 40–49 20 
GRB 060210 3.9136 20–31 41–51 21 
GRB 060418 1.4907 40–60 80–100 40 
GRB 060904B 0.7038 59–88 117–147 59 
GRB 060908 1.8849 35–52 69–87 35 
GRB 060927 5.46410 15–23 31–39 16 
GRB 061007 1.26211 44–66 88–111 45 
GRB 061021 0.34612 74–111 149–186 75 
GRB 061121 1.31513 43–65 86–108 43 
GRB 070306 1.49614 40–60 80–100 40 
GRB 071010B 0.94715 51–77 103–128 52 
GRB 071020 2.14516 32–48 64–79 32 
GRB 080319B 0.93717 52–77 103–129 52 
GRB 080319C 1.94918 34–51 68–85 34 
GRB 080411 1.03019 49–74 99–123 50 
GRB 080413A 2.43320 29–44 58–73 29 
GRB 080413B 1.10121 48–71 95–119 48 
GRB 080430 0.76722 57–85 113–141 56 
GRB 080603B 2.68923 27–41 54–68 27 
GRB 080605 1.64024 38–57 76–95 38 
GRB 080607 3.03625 25–37 50–62 25 
GRB 080721 2.59126 28–42 56–70 28 
GRB 080916A 0.68927 59–89 118-148 59 
GRB 081222 2.77028 27–40 53–66 26 
GRB 090424 0.54429 65–97 130–162 65 
GRB 090618 0.54030 65–97 130–162 65 
GRB 090715B 3.00031 25–38 50–63 25 
GRB 090812 2.45232 29–43 58–72 29 
GRB 090926B 1.24033 45–67 89–112 45 
GRB 091018 0.97134 51–76 101–127 51 
GRB 091020 1.71035 37–55 74–92 37 
GRB 091024 1.09136 48–72 96–120 48 
GRB 091029 2.75237 27–40 53–67 27 
GRB 091208B 1.06338 48–73 97–121 49 
GRB 100621A 0.54239 65–97 130–162 65 
GRB 100814A 1.44040 41–61 82–102 41 
GRB 100906A 1.72741 37–55 73–92 37 
GRB 110205A 2.22042 31–47 62–78 31 
GRB 110213A 1.46043 41–61 81–102 41 
Figure 1

Fixed energy bands at the GRB source frame are projected to various energy bands at the observer frame, depending on the redshift.

Figure 1

Fixed energy bands at the GRB source frame are projected to various energy bands at the observer frame, depending on the redshift.

The spectral lags were extracted using the improved cross-correlation function (CCF) analysis method described in U10. In this method, the spectral lag is defined as the time delay corresponding to the global maximum of the CCF. The CCF with a delay index d is defined as  

1
formula

where xi and yi are two sets of time-sequenced data spread over N bins. The time delay is obtained by multiplying d by the time bin size of the light curves. A Gaussian curve was fitted to the CCF (plotted as a function of time delay) to extract the spectral lag. The uncertainty in the spectral lag is obtained by simulating 1000 light curves using the Monte Carlo technique (see U10 for more details).

The isotropic peak luminosity (Liso) and its uncertainty for each GRB are obtained using the method described in U10. In essence, a typical GRB spectrum can be described by the Band function (Band et al. 1993), for the photon flux per unit photon energy using  

2
formula

which has four model parameters: the amplitude (A), the low-energy spectral index (α), the high-energy spectral index (β) and the peak (Epk) of E2N(E) spectrum (also called the νFν spectrum, apart from a factor of Planck’s constant). Using these spectral parameters, the observed peak flux can be calculated for the source-frame energy range E1= 1.0 keV to E2= 10 000 keV using  

3
formula

The isotropic peak luminosity is defined by  

4
formula

where dL is the luminosity distance:  

5
formula

For the current universe, we take ΩM= 0.27, ΩL= 0.73 and the Hubble constant H0 to be 70 km s−1 Mpc−1 (Komatsu et al. 2009). For more details of the Liso calculation, see U10.

3 RESULTS

We employed an additional 12 long bursts to the GRB sample (31 GRBs) that was used in U10, which increased the total sample to 43. This sample has redshifts ranging from 0.346 (GRB 061021) to 5.464 (GRB 060927), with an average redshift of ∼2.0. The spectral information for the additional 12 bursts used in this paper is given in Table 2. The calculated peak isotropic luminosities, spanning three orders of magnitude, are given in U10 and Table 2.

Table 2

GRB redshift and spectral information. Note that uncertainties of parameters that are reported with 90 per cent confidence level have been reduced to 1σ level for consistency.

GRB Peak fluxa Epkb α β Liso (erg s−1Reference 
GRB 090812 3.60 ± 0.13 572+99− 156 −1.03+0.04− 0.04 −2.50+0.16− 0.16 (7.86+1.95− 0.87) × 1052 Baumgartner et al. (2009b); Pal’Shin et al. (2009) 
GRB 090926B 3.20 ± 0.19 91+1− 1 −0.13+0.04− 0.04 −2.36+0.31− 0.31 (5.22+3.88− 0.82) × 1051 Baumgartner et al. (2009c); Briggs (2009) 
GRB 091018 10.30 ± 0.25 28+10− 6 −1.53+0.24− 0.37 −2.44+0.15− 0.15 (6.96+1.76− 0.58) × 1051 Golenetskii et al. (2009a); Markwardt et al. (2009) 
GRB 091020 4.20 ± 0.19 47+4− 4 −0.20+0.25− 0.25 −1.70+0.01− 0.01 (2.81+0.19− 0.16) × 1052 Chaplin (2009); Palmer et al. (2009) 
GRB 091024 2.00 ± 0.19 500+100− 100 −1.10+0.13− 0.13 −2.36+0.31− 0.31 (5.56+2.43− 0.89) × 1051 Golenetskii et al. (2009b); Sakamoto et al. (2009) 
GRB 091029 1.80 ± 0.06 61+10− 10 −1.46+0.17− 0.17 −2.36+0.31− 0.31 (1.67+0.60− 0.15) × 1052 Barthelmy et al. (2009) 
GRB 091208B 15.20 ± 0.63 124+12− 12 −1.44+0.04− 0.04 −2.32+0.29− 0.12 (1.68+0.65− 0.09) × 1052 Baumgartner et al. (2009a);McBreen (2009) 
GRB 100621A 12.80 ± 0.19 95+8− 11 −1.70+0.08− 0.08 −2.45+1.44− 1.44 (2.55+0.83− 0.34) × 1051 Golenetskii et al. (2010a); Ukwatta et al. (2010b) 
GRB 100814A 2.50 ± 0.13 106+7− 8 −0.64+0.08− 0.09 −2.02+0.08− 0.06 (8.27+1.13− 0.69) × 1051 Krimm et al. (2010); von Kienlin (2010) 
GRB 100906A 10.10 ± 0.25 180+25− 28 −1.10+0.06− 0.06 −2.20+0.19− 0.13 (4.90+1.23− 0.43) × 1052 Barthelmy et al. (2010); Golenetskii et al. (2010) 
GRB 110205A 3.60 ± 0.13 222+46− 46 −1.52+0.09− 0.09 −2.36+0.31− 0.31 (2.78+0.57− 0.20) × 1052 Golenetskii et al. (2011) 
GRB 110213A 1.60 ± 0.38 98+4− 5 −1.44+0.03− 0.03 −2.36+0.31− 0.31 (3.53+1.97− 0.53) × 1051 Barthelmy et al. (2011); Foley (2011) 
GRB Peak fluxa Epkb α β Liso (erg s−1Reference 
GRB 090812 3.60 ± 0.13 572+99− 156 −1.03+0.04− 0.04 −2.50+0.16− 0.16 (7.86+1.95− 0.87) × 1052 Baumgartner et al. (2009b); Pal’Shin et al. (2009) 
GRB 090926B 3.20 ± 0.19 91+1− 1 −0.13+0.04− 0.04 −2.36+0.31− 0.31 (5.22+3.88− 0.82) × 1051 Baumgartner et al. (2009c); Briggs (2009) 
GRB 091018 10.30 ± 0.25 28+10− 6 −1.53+0.24− 0.37 −2.44+0.15− 0.15 (6.96+1.76− 0.58) × 1051 Golenetskii et al. (2009a); Markwardt et al. (2009) 
GRB 091020 4.20 ± 0.19 47+4− 4 −0.20+0.25− 0.25 −1.70+0.01− 0.01 (2.81+0.19− 0.16) × 1052 Chaplin (2009); Palmer et al. (2009) 
GRB 091024 2.00 ± 0.19 500+100− 100 −1.10+0.13− 0.13 −2.36+0.31− 0.31 (5.56+2.43− 0.89) × 1051 Golenetskii et al. (2009b); Sakamoto et al. (2009) 
GRB 091029 1.80 ± 0.06 61+10− 10 −1.46+0.17− 0.17 −2.36+0.31− 0.31 (1.67+0.60− 0.15) × 1052 Barthelmy et al. (2009) 
GRB 091208B 15.20 ± 0.63 124+12− 12 −1.44+0.04− 0.04 −2.32+0.29− 0.12 (1.68+0.65− 0.09) × 1052 Baumgartner et al. (2009a);McBreen (2009) 
GRB 100621A 12.80 ± 0.19 95+8− 11 −1.70+0.08− 0.08 −2.45+1.44− 1.44 (2.55+0.83− 0.34) × 1051 Golenetskii et al. (2010a); Ukwatta et al. (2010b) 
GRB 100814A 2.50 ± 0.13 106+7− 8 −0.64+0.08− 0.09 −2.02+0.08− 0.06 (8.27+1.13− 0.69) × 1051 Krimm et al. (2010); von Kienlin (2010) 
GRB 100906A 10.10 ± 0.25 180+25− 28 −1.10+0.06− 0.06 −2.20+0.19− 0.13 (4.90+1.23− 0.43) × 1052 Barthelmy et al. (2010); Golenetskii et al. (2010) 
GRB 110205A 3.60 ± 0.13 222+46− 46 −1.52+0.09− 0.09 −2.36+0.31− 0.31 (2.78+0.57− 0.20) × 1052 Golenetskii et al. (2011) 
GRB 110213A 1.60 ± 0.38 98+4− 5 −1.44+0.03− 0.03 −2.36+0.31− 0.31 (3.53+1.97− 0.53) × 1051 Barthelmy et al. (2011); Foley (2011) 

a1-s peak photon flux measured in photons  cm−2 s−1 in the energy range 15–150 keV.

bPeak energy, Epk, is given in keV.

By choosing appropriate energy bands in the observer frame (according to the redshift of each burst), we extracted mask-weighted background-subtracted light curves for the selected source-frame energy bands 100–150 and 200–250 keV. The observer-frame energy bands used for each burst are shown in Table 1. Note that the energy gap between the mid-points of the two source-frame energy bands is fixed at 100 keV, whereas in the observer frame, as expected, this gap varies depending on the redshift of each burst (see Table 1). For example, in GRB 060927, this gap is 16 keV and in GRB 061021 it is 75 keV. This is in contrast to the spectral lag extractions performed in the observer frame where this gap is treated as a constant.

The extracted spectral lags for the source-frame energy bands 100–150 and 200–250 keV are listed in Table 3. The Swift BAT trigger ID, the segment of the light curve used for the lag extraction (T + XS and T + XE, T is the trigger time), the time binning of the light curve and the Gaussian curve fitting range of the CCF versus time delay plot (with start time and end time denoted as LS and LE, respectively) are also given in Table 3. Of 43 bursts in the sample, there are 24 bursts which have lags greater than zero. The remaining 19 bursts have lags either consistent with zero (16 bursts) or negative values (three bursts).

Table 3

Source-frame spectral lag values of long duration Swift BAT GRBs.

GRB Trigger ID T + XS (s) T + XE (s) Bin size (ms) LS (s) LE (s) Lag value (ms) Significance 
GRB 050401 113120 23.03 29.43 64 −2.00 2.00 310 ± 145 2.14 
GRB 050603 131560 −3.83 3.08 16 −0.40 0.40 −16 ± 21 −0.76 
GRB 050922C 156467 −2.70 2.94 16 −1.00 1.00 136 ± 68 2.00 
GRB 051111 163438 −6.96 28.62 64 −4.00 4.00 333 ± 251 1.33 
GRB 060206 180455 −1.29 8.18 16 −2.00 2.00 86 ± 111 0.77 
GRB 060210 180977 −3.37 5.08 128 −4.00 4.00 658 ± 259 2.54 
GRB 060418 205851 −7.66 33.04 64 −2.00 2.00 −110 ± 106 −1.04 
GRB 060904B 228006 −1.97 10.32 512 −6.00 6.00 124 ± 436 0.28 
GRB 060908 228581 −10.91 3.68 32 −2.00 2.00 78 ± 124 0.63 
GRB 060927 231362 −1.69 8.04 32 −1.00 1.00 18 ± 75 0.24 
GRB 061007 232683 23.86 65.08 −0.20 0.20 52 ± 22 2.36 
GRB 061021 234905 −0.46 14.64 512 −4.00 4.00 −430 ± 975 −0.44 
GRB 061121 239899 60.44 80.66 −0.20 0.20 22 ± 10 2.20 
GRB 070306 263361 90.00 118.42 32 −4.00 2.00 −362 ± 247 −1.47 
GRB 071010B 293795 −1.70 17.24 64 −2.00 2.00 404 ± 159 2.54 
GRB 071020 294835 −3.22 1.14 −0.20 0.40 35 ± 13 2.69 
GRB 080319B 306757 −2.85 57.57 −0.10 0.14 23 ± 6 3.83 
GRB 080319C 306778 −0.77 13.31 32 −1.00 1.00 174 ± 91 1.91 
GRB 080411 309010 38.46 48.45 −0.50 0.50 116 ± 25 4.64 
GRB 080413A 309096 −0.42 9.05 −1.00 1.00 107 ± 59 1.81 
GRB 080413B 309111 −1.44 4.96 32 −1.00 1.00 115 ± 50 2.30 
GRB 080430 310613 −1.24 12.84 256 −4.00 4.00 91 ± 431 0.21 
GRB 080603B 313087 −0.54 5.10 16 −1.00 1.00 5 ± 59 0.08 
GRB 080605 313299 −5.46 15.53 −0.20 0.20 35 ± 18 1.94 
GRB 080607 313417 −6.13 12.05 −0.50 0.50 26 ± 30 0.87 
GRB 080721 317508 −3.39 8.64 64 −2.00 2.00 −86 ± 110 −0.78 
GRB 080916A 324895 −2.66 39.58 128 −2.00 4.00 585 ± 214 2.73 
GRB 081222 337914 −0.80 15.58 −1.00 1.00 227 ± 51 4.45 
GRB 090424 350311 −0.94 4.95 16 −0.20 0.20 14 ± 14 1.00 
GRB 090618 355083 46.01 135.35 −2.00 2.00 267 ± 72 3.71 
GRB 090715B 357512 −4.80 21.06 16 −2.00 3.00 275 ± 155 1.77 
GRB 090812 359711 −6.93 41.20 256 −6.00 6.00 −22 ± 202 −0.11 
GRB 090926B 370791 −22.00 36.00 512 −10.00 8.00 746 ± 627 1.19 
GRB 091018 373172 −0.28 2.92 64 −2.00 1.00 143 ± 297 0.48 
GRB 091020 373458 −2.54 13.84 128 −3.00 2.00 −187 ± 177 −1.06 
GRB 091024 373674 −9.58 27.29 512 −10.00 10.00 912 ± 604 1.51 
GRB 091029 374210 −4.03 38.98 256 −10.00 10.00 −112 ± 395 −0.28 
GRB 091208B 378559 7.66 10.61 64 −1.00 1.00 105 ± 66 1.59 
GRB 100621A 425151 −6.79 40.31 256 −3.00 3.00 1199 ± 311 3.86 
GRB 100814A 431605 −4.40 29.39 256 −4.00 4.00 862 ± 147 5.86 
GRB 100906A 433509 −1.49 26.16 128 −2.00 2.00 105 ± 79 1.33 
GRB 110205A 444643 118.89 293.99 64 −1.00 1.00 −29 ± 52 −0.56 
GRB 110213A 445414 −3.42 5.29 512 −3.00 3.50 602 ± 746 0.81 
GRB Trigger ID T + XS (s) T + XE (s) Bin size (ms) LS (s) LE (s) Lag value (ms) Significance 
GRB 050401 113120 23.03 29.43 64 −2.00 2.00 310 ± 145 2.14 
GRB 050603 131560 −3.83 3.08 16 −0.40 0.40 −16 ± 21 −0.76 
GRB 050922C 156467 −2.70 2.94 16 −1.00 1.00 136 ± 68 2.00 
GRB 051111 163438 −6.96 28.62 64 −4.00 4.00 333 ± 251 1.33 
GRB 060206 180455 −1.29 8.18 16 −2.00 2.00 86 ± 111 0.77 
GRB 060210 180977 −3.37 5.08 128 −4.00 4.00 658 ± 259 2.54 
GRB 060418 205851 −7.66 33.04 64 −2.00 2.00 −110 ± 106 −1.04 
GRB 060904B 228006 −1.97 10.32 512 −6.00 6.00 124 ± 436 0.28 
GRB 060908 228581 −10.91 3.68 32 −2.00 2.00 78 ± 124 0.63 
GRB 060927 231362 −1.69 8.04 32 −1.00 1.00 18 ± 75 0.24 
GRB 061007 232683 23.86 65.08 −0.20 0.20 52 ± 22 2.36 
GRB 061021 234905 −0.46 14.64 512 −4.00 4.00 −430 ± 975 −0.44 
GRB 061121 239899 60.44 80.66 −0.20 0.20 22 ± 10 2.20 
GRB 070306 263361 90.00 118.42 32 −4.00 2.00 −362 ± 247 −1.47 
GRB 071010B 293795 −1.70 17.24 64 −2.00 2.00 404 ± 159 2.54 
GRB 071020 294835 −3.22 1.14 −0.20 0.40 35 ± 13 2.69 
GRB 080319B 306757 −2.85 57.57 −0.10 0.14 23 ± 6 3.83 
GRB 080319C 306778 −0.77 13.31 32 −1.00 1.00 174 ± 91 1.91 
GRB 080411 309010 38.46 48.45 −0.50 0.50 116 ± 25 4.64 
GRB 080413A 309096 −0.42 9.05 −1.00 1.00 107 ± 59 1.81 
GRB 080413B 309111 −1.44 4.96 32 −1.00 1.00 115 ± 50 2.30 
GRB 080430 310613 −1.24 12.84 256 −4.00 4.00 91 ± 431 0.21 
GRB 080603B 313087 −0.54 5.10 16 −1.00 1.00 5 ± 59 0.08 
GRB 080605 313299 −5.46 15.53 −0.20 0.20 35 ± 18 1.94 
GRB 080607 313417 −6.13 12.05 −0.50 0.50 26 ± 30 0.87 
GRB 080721 317508 −3.39 8.64 64 −2.00 2.00 −86 ± 110 −0.78 
GRB 080916A 324895 −2.66 39.58 128 −2.00 4.00 585 ± 214 2.73 
GRB 081222 337914 −0.80 15.58 −1.00 1.00 227 ± 51 4.45 
GRB 090424 350311 −0.94 4.95 16 −0.20 0.20 14 ± 14 1.00 
GRB 090618 355083 46.01 135.35 −2.00 2.00 267 ± 72 3.71 
GRB 090715B 357512 −4.80 21.06 16 −2.00 3.00 275 ± 155 1.77 
GRB 090812 359711 −6.93 41.20 256 −6.00 6.00 −22 ± 202 −0.11 
GRB 090926B 370791 −22.00 36.00 512 −10.00 8.00 746 ± 627 1.19 
GRB 091018 373172 −0.28 2.92 64 −2.00 1.00 143 ± 297 0.48 
GRB 091020 373458 −2.54 13.84 128 −3.00 2.00 −187 ± 177 −1.06 
GRB 091024 373674 −9.58 27.29 512 −10.00 10.00 912 ± 604 1.51 
GRB 091029 374210 −4.03 38.98 256 −10.00 10.00 −112 ± 395 −0.28 
GRB 091208B 378559 7.66 10.61 64 −1.00 1.00 105 ± 66 1.59 
GRB 100621A 425151 −6.79 40.31 256 −3.00 3.00 1199 ± 311 3.86 
GRB 100814A 431605 −4.40 29.39 256 −4.00 4.00 862 ± 147 5.86 
GRB 100906A 433509 −1.49 26.16 128 −2.00 2.00 105 ± 79 1.33 
GRB 110205A 444643 118.89 293.99 64 −1.00 1.00 −29 ± 52 −0.56 
GRB 110213A 445414 −3.42 5.29 512 −3.00 3.50 602 ± 746 0.81 

For the 24 bursts which have positive lags with significance 1σ or greater (see Table 3), we find that the redshift-corrected lag is anticorrelated with Liso. The correlation coefficient for this relation is −0.82 ± 0.05 with a chance probability of ∼5.54 × 10−5. The extracted correlation coefficient is significantly higher than the correlation coefficient (averaged over the six combinations of standard BAT energy channels) of ∼−0.68 reported in U10. Various correlation coefficients of the relation are shown in Table 4, where uncertainties in the correlation coefficients were obtained through a Monte Carlo simulation utilizing uncertainties in Liso and the lag values. The null probability that the correlation occurs due to random chance is also given for each coefficient type.

Table 4

Correlation coefficients of the lag–luminosity relation.

Coefficient type Correlation coefficient Null probability 
Pearson’s r −0.82 ± 0.05 5.54 × 10−5 
Spearman’s rs −0.70 ± 0.06 1.49 × 10−4 
Kendall’s τ −0.50 ± 0.05 6.63 × 10−4 
Coefficient type Correlation coefficient Null probability 
Pearson’s r −0.82 ± 0.05 5.54 × 10−5 
Spearman’s rs −0.70 ± 0.06 1.49 × 10−4 
Kendall’s τ −0.50 ± 0.05 6.63 × 10−4 

Fig. 2 shows a log–log plot of isotropic peak luminosity versus redshift-corrected spectral lag. The solid line shows the following best-fitting power-law curve:  

6
formula

Figure 2

The spectral lags between the source-frame energy range bands 100–150 and 200–250 keV and the isotropic peak luminosity are plotted in a log–log plot.

Figure 2

The spectral lags between the source-frame energy range bands 100–150 and 200–250 keV and the isotropic peak luminosity are plotted in a log–log plot.

Since there is considerable scatter, the uncertainties of the fit parameters are multiplied by a factor of forumla. The dash lines indicate the estimated 1σ confidence level, which is obtained from the cumulative fraction of the residual distribution taken from 16 to 84 per cent.

The best-fitting power-law index (−1.2 ± 0.2) is consistent with observer-frame results obtained by Norris et al. (2000) (∼−1.14) and the average power-law index of −1.4 ± 0.3 reported in U10.

4 DISCUSSION

4.1 Spectral lags: observer frame versus source frame

U10 extracted spectral lags in fixed energy bands in the observer frame, and in this work for the same sample of 31 bursts we extracted lags in fixed energy bands in the source frame. In the observer-frame case, there are four energy channels [canonical BAT energy bands: channel 1 (15–25 keV), 2 (25–50 keV), 3 (50–100 keV) and 4 (100–200 keV)], thus six lag extractions per burst. It is interesting to study to what degree these different lags correlate with source-frame lags (between fixed source-frame energy channels 100–150 and 200–250 keV). In Fig. 3, we show all combinations of observer-frame lags as a function of source-frame lags. The red data points show lags with the time-dilation correction due to cosmological redshift, and black data points show lags without the time-dilation correction. From Fig. 3, it is clear that all plots show some correlation both in the time-dilation-corrected (shown in red) and time-dilation-uncorrected (shown in black) cases. We note that the correlation coefficients are greater than 0.5 in time-dilation-uncorrected cases where BAT channel 1 is involved in the lag extraction. In the time-dilation-corrected case, all plots show correlation coefficients greater than 0.5 except for the lag 43 plot. Despite these moderate correlation coefficients, the large scatter seen in these plots indicates that the observer-frame lag does not directly represent the source-frame lag.

Figure 3

All combinations of fixed observer-frame energy channel [canonical BAT energy bands: channel 1 (15–25 keV), 2 (25–50 keV), 3 (50–100 keV) and 4 (100–200 keV)] spectral lag values as a function of fixed source-frame energy channel (between 100–150 and 200–250 keV) lag values. Black and red data points and labels correspond to redshift-uncorrected and redshift-corrected cases, respectively. The blue dashed line corresponds to the equality line of the two parameters in each panel.

Figure 3

All combinations of fixed observer-frame energy channel [canonical BAT energy bands: channel 1 (15–25 keV), 2 (25–50 keV), 3 (50–100 keV) and 4 (100–200 keV)] spectral lag values as a function of fixed source-frame energy channel (between 100–150 and 200–250 keV) lag values. Black and red data points and labels correspond to redshift-uncorrected and redshift-corrected cases, respectively. The blue dashed line corresponds to the equality line of the two parameters in each panel.

4.2 Lag–forumla relation: observer frame versus source frame

There are two important changes in the lag–luminosity relation which may occur when going from fixed observer-frame energy bands to fixed source-frame energy bands: a change in the power-law index, and a change in the dispersion of the data measured by the correlation coefficient. Table 5 summarizes these two parameters for various energy bands both in the observer frame and in the source frame.

Table 5

Observer-frame and source-frame slopes and correlation coefficients of the lag–Liso relation. Conservative 10 per cent uncertainty is assumed for cases without uncertainties.

Energy bands Frame Slope Correlation coefficient Number of GRBs Reference 
(0.3–1), (3–10) keV Observer 0.95 ± 0.23 – Margutti et al. (2010) 
(6–25), (50–400) keV Observer 1.16 ± 0.07 −0.79+0.16− 0.05 Arimoto et al. (2010) 
(15–25), (25–50) keV Observer 1.4 ± 0.1 −0.63 ± 0.06 21 U10 
(15–25), (50–100) keV Observer 1.5 ± 0.1 −0.60 ± 0.06 28 U10 
(15–25), (100–200) keV Observer 1.8 ± 0.1 −0.67 ± 0.07 27 U10 
(25–50), (50–100) keV Observer 1.2 ± 0.1 −0.66 ± 0.07 27 U10 
(25–50), (100–200) keV Observer 1.4 ± 0.1 −0.75 ± 0.07 25 U10 
(25–50), (100–300) keV Observer 1.14 ± 0.1 – Norris et al. (2000) 
(25–50), (100–300) keV Observer 0.62 ± 0.04 −0.72 ± 0.07 Hakkila et al. (2008) 
(50–100), (100–200) keV Observer 1.4 ± 0.1 −0.77 ± 0.08 22 U10 
(20–100), (100–500) keV Source 1.23 ± 0.07 −0.90+0.12− 0.02 Arimoto et al. (2010) 
(100–200), (300–400) keV Source 0.9 ± 0.1 −0.76 ± 0.06 22 Ukwatta et al. (2010b) 
(100–150), (200–250) keV Source 1.2 ± 0.2 −0.82 ± 0.05 24 This work 
Energy bands Frame Slope Correlation coefficient Number of GRBs Reference 
(0.3–1), (3–10) keV Observer 0.95 ± 0.23 – Margutti et al. (2010) 
(6–25), (50–400) keV Observer 1.16 ± 0.07 −0.79+0.16− 0.05 Arimoto et al. (2010) 
(15–25), (25–50) keV Observer 1.4 ± 0.1 −0.63 ± 0.06 21 U10 
(15–25), (50–100) keV Observer 1.5 ± 0.1 −0.60 ± 0.06 28 U10 
(15–25), (100–200) keV Observer 1.8 ± 0.1 −0.67 ± 0.07 27 U10 
(25–50), (50–100) keV Observer 1.2 ± 0.1 −0.66 ± 0.07 27 U10 
(25–50), (100–200) keV Observer 1.4 ± 0.1 −0.75 ± 0.07 25 U10 
(25–50), (100–300) keV Observer 1.14 ± 0.1 – Norris et al. (2000) 
(25–50), (100–300) keV Observer 0.62 ± 0.04 −0.72 ± 0.07 Hakkila et al. (2008) 
(50–100), (100–200) keV Observer 1.4 ± 0.1 −0.77 ± 0.08 22 U10 
(20–100), (100–500) keV Source 1.23 ± 0.07 −0.90+0.12− 0.02 Arimoto et al. (2010) 
(100–200), (300–400) keV Source 0.9 ± 0.1 −0.76 ± 0.06 22 Ukwatta et al. (2010b) 
(100–150), (200–250) keV Source 1.2 ± 0.2 −0.82 ± 0.05 24 This work 

In the observer frame, the power-law index varies from ∼0.6 to ∼1.8, with mean around 1.3. In the source frame, the index changes from 0.9 to 1.23 with a mean of ∼1.1. Meanwhile, the correlation coefficient varies from 0.60 to 0.79 in the observer frame, and in the source frame it changes from 0.76 to 0.90. Hence, according to Table 5, the source-frame lag–Liso relation seems to be tighter than the observer-frame case with a slope closer to 1.

4.3 Spectral lag–Epk relation

Now we investigate the relation between source-frame spectral lag and source-frame average peak energy [Epk(1 + z)] of the burst spectrum. In Fig. 4, we plotted Epk(1 + z) as a function of source-frame lags. There is a correlation between these two parameters with a correlation coefficient of −0.57 ± 0.14. Various correlation coefficients of the relation are shown in Table 6, with uncertainties and null probabilities.

Figure 4

The source-frame peak energy [Epk(1 + z)] versus source-frame spectral lags. The energy bands, 100–150 and 200–250 keV, corresponding to the lag extractions are shown in hashed red bands on the plot.

Figure 4

The source-frame peak energy [Epk(1 + z)] versus source-frame spectral lags. The energy bands, 100–150 and 200–250 keV, corresponding to the lag extractions are shown in hashed red bands on the plot.

Table 6

Correlation coefficients of the lag–Epk relation.

Coefficient type Correlation coefficient Null probability 
Pearson’s r −0.57 ± 0.14 4.83 × 10−3 
Spearman’s rs −0.50 ± 0.12 1.36 × 10−2 
Kendall’s τ −0.37 ± 0.14 1.18 × 10−2 
Coefficient type Correlation coefficient Null probability 
Pearson’s r −0.57 ± 0.14 4.83 × 10−3 
Spearman’s rs −0.50 ± 0.12 1.36 × 10−2 
Kendall’s τ −0.37 ± 0.14 1.18 × 10−2 

The best fit is shown as a dashed line in Fig. 4, yielding the following relation between Epk(1 + z) and lag/(1 + z):  

7
formula

The uncertainties in the fitted parameters are expressed with the factor of forumla.

According to equation (6), Liso ∝ [lag/(1 + z)]−1.2. From the Yonetoku relation, we know that Liso ∝ [Epk(1 + z)]2.0 (Yonetoku et al. 2004). Hence, from these two relations, we expect to see a correlation between Epk(1 + z) and forumla such as forumla.

The best-fitting slope of 0.56 ± 0.06 is consistent with the expected slope of ∼0.6 based on the source-frame lag–luminosity and the Yonetoku relation. However, note that the correlation coefficient is significantly smaller than the coefficient for the lag–luminosity relation. This lower degree of correlation may be suggestive of brightness and detector-related selection effects that have been noted in the literature (Butler et al. 2007) for the Yonetoku relation.

4.4 Some models for spectral lags

U10 and this work have provided more evidence for the existence of the lag–luminosity relation based on a sample of Swift BAT GRBs with measured spectroscopic redshifts. This analysis calls for a physical interpretation for spectral lag and a lag–luminosity relation. In the literature, several possible interpretations have been discussed (Dermer 1998; Salmonson 2000; Ioka & Nakamura 2001; Kocevski & Liang 2003; Qin et al. 2004; Schaefer 2004; Ryde 2005; Shen, Song & Li 2005; Lu et al. 2006; Peng et al. 2011).

One proposed explanation for the observed spectral lag is the spectral evolution during the prompt phase of the GRB (Dermer 1998; Kocevski & Liang 2003; Ryde 2005). Due to cooling effects, Epk moves to a lower energy channel after some characteristic time. When the peak energy (Epk) moves from a higher energy band to a lower energy band, the temporal peak of the light curve also moves from a higher energy band to a lower one, which results in the observed spectral lag. In a recent study, Peng et al. (2011) suggest that spectral evolution can be invoked to explain both positive and negative spectral lags. Hard-to-soft evolution of the spectrum produces positive spectral lags, while soft-to-hard evolution would lead to negative lags. In addition, these authors also suggest that soft-to-hard-to-soft evolution may produce negative lags.

A schematic diagram showing a hard-to-soft scenario is depicted in Fig. 5. Initially, Epk of the spectrum is in the high-energy band, which results in a pulse in the light curve of the high-energy band. Then Epk moves to the lower energy band resulting in a pulse in the low-energy light curve. The temporal difference between the two pulses in the light curves would then be a measure of the cooling time-scale of the spectrum.

Figure 5

The time evolution of the Epk across energy bands may cause the observed spectral lags in GRBs.

Figure 5

The time evolution of the Epk across energy bands may cause the observed spectral lags in GRBs.

If this were the only process that caused the lag, then in a simple picture one would expect the source-frame average Epk to lie within the two energy bands in question. According to Fig. 4, for the majority of bursts the source-frame Epk lies outside the energy band 100–250 keV, indicating that the simple spectral evolution scenario described above may not be the dominant process responsible for the observed lags. However, it is worth noting that a pulse in a specific energy band may not always mean that the Epk is also within that energy band. There are other issues associated with this model: (1) the calculated cooling times based on simple synchrotron models are, in general, relatively small compared to the extracted lags and (2) short bursts which exhibit considerable spectral evolution do not show significant lags.

Another model that purports to explain spectral lags is based on the curvature effect, i.e. a kinematics effect due to the observer looking at increasingly off-axis annulus areas relative to the line of sight (Salmonson 2000; Ioka & Nakamura 2001; Dermer 2004; Shen et al. 2005; Lu et al. 2006). Fig. 6 illustrates how the spectral lag could arise due to the curvature effect of the shocked shell. Due to a smaller Doppler factor and a path difference, the radiation from shell areas which are further off-axis will be softer and therefore lead to a lag. As with spectral evolution models, there are difficulties associated with the curvature models too. These kinematic models generally predict only positive lags. As can be seen from Table 3, some of the measured lags are negative, and therefore these lags present a real challenge for the simple curvature models.

Figure 6

Spectral lags could arise due to the curvature effect of the shocked shell. At the source, the relativistically expanding shell emits identical pulses from all latitudes. However, when the photons reach the detector, on-axis photons get boosted to higher energy (hard). Meanwhile, off-axis photons get relatively smaller boost and travel longer to reach the detector. Thus, these photons are softer and arrive later than the on-axis photons.

Figure 6

Spectral lags could arise due to the curvature effect of the shocked shell. At the source, the relativistically expanding shell emits identical pulses from all latitudes. However, when the photons reach the detector, on-axis photons get boosted to higher energy (hard). Meanwhile, off-axis photons get relatively smaller boost and travel longer to reach the detector. Thus, these photons are softer and arrive later than the on-axis photons.

It is possible that spectral lags are caused by multiple mechanisms. Peng et al. (2011) investigated spectral lags caused by intrinsic spectral evolution and the curvature effect combined. They showed that the curvature effect always tends to increase the observed spectral lag in the positive direction. Even for cases with soft-to-hard spectral evolution, when the curvature effect is introduced lags become positive. Hence, they predict that the majority of measured spectral lags should be positive, which is consistent with the findings of this work and U10.

5 SUMMARY AND CONCLUSION

We have investigated the spectral lag between 100–150 and 200–250 keV energy bands at the GRB source frame by projecting these bands to the observer frame. This is a step forward in the investigation of lag–luminosity relations since most of the previous investigations used arbitrary observer-frame energy bands.

Our analysis has produced an improved correlation between spectral lag (τ) and isotropic luminosity over those previously reported with the following relation:  

8
formula

We also find a modest correlation between the source-frame spectral lag and the peak energy of the burst, which is given by the relation  

9
formula

Finally, we mentioned two simple models and noted their limitations in explaining the observed spectral lags.

We thank the anonymous referee for comments that significantly improved the paper. The NSF grant 1002432 provided partial support for the work of TNU and is gratefully acknowledged. The work of CDD is supported by the Office of Naval Research and Fermi Guest Investigator grants. We acknowledge that this work has been performed via th eauspices of the GRB Temporal Analysis Consortium (GTAC), which represents a comprehensive effort dedicated towards the systematic study of spectral variation in Gamma-ray Bursts.

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