A New Measurement of the Spectral Lag of Gamma-Ray Bursts and its Implications for Spectral Evolution Behaviors

Gamma-ray bursts (GRBs) are prolific emitters of the nonthermal radiation that is found at cosmological distances that can span many decades in frequency. The prompt emission phase of GRBs is still not well understood given their complex temporal structures and spectral properties (see Kumar & Zhang 2015; Pe'er 2015, for recent reviews). Study of the prompt GRB spectra is essential for understanding the radiation mechanisms and their physical origins. In general, the integrated spectrum could be well described by the so-called Band function, which is a smoothly joined broken power law peaking at ${E}_{{\rm{p}}}$ in the $\nu {f}_{\nu }$ spectrum (Band et al. 1993). This peak energy ${E}_{{\rm{p}}}$ is usually well correlated with the estimated isotropic total GRB energy (${E}_{\gamma ,\mathrm{iso}}$) and the peak luminosity (${L}_{{\rm{p}}}$) (Amati et al. 2002; Wei & Gao 2003; Yonetoku et al. 2004; Amati 2006; Nava et al. 2012; Zhang et al. 2012c).

As GRBs are typically highly variable in intensity and significant spectral evolution apparently prevails, time-resolved spectral analyses have been considered more crucial to determine the radiation mechanism of GRBs. The correlation between the time-resolved peak energy and the corresponding isotropic peak luminosity has been well-established within individual bursts (e.g., Ghirlanda et al. 2010; Frontera et al. 2012; Guiriec et al. 2015a). Multiple spectral components have been proposed by time-resolved analyses of data from different satellites and detectors (e.g., Zhang et al. 2011; Frontera et al. 2013; Tierney et al. 2013; Basak & Rao 2014; Guiriec et al. 2015b; Iyyani et al. 2015; Yu et al. 2016). Different radiation mechanisms have been tried to interpret the spectral shape (e.g., Ryde et al. 2011; Zhang et al. 2012a, 2016; Guiriec et al. 2013, 2016; Lazzati et al. 2013; Burgess et al. 2014; Uhm & Zhang 2014; Chhotray & Lazzati 2015; Pe'er et al. 2015; Yu et al.2015).

As abundant information about the spectral properties has been revealed by time-resolved analyses, complexity and unexpected confusion have also been introduced. Different analysis methods that include different time-binning schemes, samples, and energy ranges can lead to different results. The shapes of the spectra and/or the patterns of the spectral evolution have been found to be affected by the blend of different pulses. Two different correlations between the spectral peak energy and the corresponding luminosity have been proposed: "hard-to-soft" (HTS) and "hardness-intensity tracking" (HIT), which are seemingly incompatible with each other (Hakkila & Preece 2011; Lu et al. 2012; Burgess & Ryde 2015; Hakkila et al. 2015). The binning method is also found to play an important role in analyzing the pattern of the spectral evolution (Burgess 2014).

Since the temporal profiles of GRBs are very complex and the spectral property is difficult to identify in blended pulses, efforts have been made to analyze the bursts with a single or well-separated pulse. The pulse width and the arrival time have both been found to depend on energy in most GRBs, revealing an important connection between the temporal profile and the spectral evolution (Cheng et al. 1995; Fenimore et al. 1995; Norris et al. 1996; Band 1997). The spectral lag can be measured between two given energy ranges by cross-correlating the light curves of two corresponding energy channels. The spectral lag has been found well correlated with the peak luminosity (Norris et al. 2000; Norris 2002; Ukwatta et al. 2010). The spectral lag has also been considered as an important tool to classify long and short GRBs (Gehrels et al. 2006; Norris & Bonnell 2006; Zhang et al. 2006; Bernardini et al. 2015). Detailed analyses of the spectral lag would use both the temporal and spectral information and help to reveal the radiation mechanisms of GRBs. The spectral lag can be considered as a consequence of the spectral evolution in the radiation processes (e.g., Dermer 1998; Daigne & Mochkovitch 2003; Kocevski & Liang 2003; Ryde 2005; Mochkovitch et al. 2016; Uhm & Zhang 2016). For synchrotron and synchrotron self-Compton (SSC) processes in relativistic plasma outflows, the pulse width should be correlated with the photon energy E as ${E}^{-1/2}$ and ${E}^{-1/4}$ , respectively (Chiang 1998; Dermer 1998; Kazanas et al. 1998). On the other hand, spectral lags can also be explained by some geometric effect including the curvature effect (e.g., Qin et al. 2004; Shen et al. 2005; Lu et al. 2006) or the pulse confusion of different spectral components (Eichler & Manis 2008; Guiriec et al. 2013).

In the previous works, the spectral lag is calculated using the cross-correlation function (CCF) between two given energy channels. The value of the spectral lag is highly sensitive to the energy channels, which are chosen arbitrarily, and a GRB will have a different lag when different energy channels are selected. To take this effect into account and further reveal the observational details on the spectral evolution, we carry out a systematic research of 50 bright GRB pulses detected by the Fermi Gamma-Ray Burst Monitor (GBM) and perform a new analysis of the spectral lag using the light curves in nine consecutive energy channels for each GRB. By investigating the energy dependencies of both the arrival time and the pulse width in universal forms, we are able to provide a new description of the spectral lag effect over all energy channels for each burst, independent of energy channel selections. Details of the sample selection and data reduction are presented in Section 2, followed by the description of our analyses in Section 3. We discuss the implications of the spectral hardness evolution behaviors in Section 4. Brief discussion and conclusion are summarized in Section 5.

This work made extensive use of the data from the GBM on board the Fermi Gamma-ray Space Telescope (Meegan et al. 2009). For the first step, we searched in the official GBM online burst catalog (Gruber et al. 2014; von Kienlin et al. 2014) for bright bursts with a total fluence $F\gt 5\times {10}^{-6}\,\mathrm{erg}\,{\mathrm{cm}}^{-2}$ in 10–1000 keV. For simplicity, we selected the bright bursts based on their total fluence instead of their peak flux, since the total fluence is generally a well-determined property. On the other hand, the peak flux is significantly dependent on the selection of time resolution and therefore has several different values in different time intervals for each burst, as provided in the burst catalog. As shown in Figure 1, the total fluence is generally well correlated with the peak fluxes in different time resolutions. Our sample (which we introduce below) is a bright sample based on both measures. For the second step, we then carefully selected the bursts with only one well-defined pulse by manually examining their light curves recorded in their brightest sodium iodide [NaI(Tl)] detector. For each burst, we only used the time-tagged event (TTE) data of the brightest NaI detector. The TTE data consist of individual photon arrival times with 2 $\mu s$ temporal resolution and 128-channel spectral resolution, recorded from about 30 s before to 300 s after the burst trigger.

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By selecting the brightest bursts based on their total fluence, we have included in our sample only one short ($\lt 2$ s) burst—GRB 140209313 with a duration of ∼1.4 s and a fluence of $\sim 9\times {10}^{-6}\,\mathrm{erg}\,{\mathrm{cm}}^{-2}$, which turns out to be the brightest short burst (without considering the distance) detected in the GBM as of 2016 May 30. Six short bursts in the online burst catalog meet the fluence criterion. Of these, GRBs 090227772 (${T}_{90}\sim 1.3\,{\rm{s}}$), 090228204 (${T}_{90}\sim 0.4\,{\rm{s}}$), 120624309 (${T}_{90}\sim 0.6\,{\rm{s}}$), 140901821 (${T}_{90}\sim 0.2\,{\rm{s}}$), and 150819440 (${T}_{90}\sim 1.0\,{\rm{s}}$) all have multiple pulses. We have excluded these bursts without carrying out further analyses. The fluence of the previously well-studied short burst 090510016 is only $\sim 3\times {10}^{-6}\,\mathrm{erg}\,{\mathrm{cm}}^{-2}$ and also has multiple overlapping pulses, therefore it does not meet our the criterion. Most other long bursts have also been excluded for the same reason.

As the third and the most important step for our sample selection, the TTE data for each burst were rebinned into nine energy channels evenly separated in the logarithmic scale. The light curve in each channel was then fit independently by a first-order polynomial (to subtract the background) together with a Gaussian function f(t) (to pinpoint the pulse),

$$\begin{eqnarray}&&f(t)=a\,{e}^{-\displaystyle \frac{{(t-{t}_{\mathrm{peak}})}^{2}}{2{\delta }^{2}}},\end{eqnarray} \tag{ 1 }$$

where a is a constant and ${t}_{\mathrm{peak}}$ and δ represent the arrival time of the peak of the pulse and the rms width of the Gaussian shape, respectively.

In general, the typical shape of a single GRB pulse was proposed to be asymmetric, e.g., with a fast rise and an exponential decay, as in the BATSE GRBs (Fishman et al. 1994). A two-sided exponential or Gaussian profile with a significantly longer decay time than the rise time (with a ratio ranging from about 2 to 3) was also proposed to be suitable for BATSE GRBs (e.g., Norris et al. 1996, 2005; Kocevski et al. 2003). During our fitting processes, we have tried several asymmetric functions to fit the pulse profiles, but none of them worked perfectly for most the light curves in our sample. Indeed, we found that a Gaussian profile was adequate to pinpoint the bulk photons of the pulses using the standard IDL fitting function, GAUSSFIT, and a more complex function was not necessary. Unfortunately, many very bright pulses that previously appeared to be single as manually selected from the whole TTE data with a very raw time resolution in the second step would still fail to be pinpointed by a Gaussian function in one of the nine energy channels as the time bin size was lowered in the third step. Pulses would also become very noisy and indistinguishable from the background when the bin size was very small. As a standard, the bin size in our work has been automatically set as $5 \%$ of the duration T₉₀ for each burst. T₉₀ is the duration of the time interval during which the detector accumulates from the 5% to the 95% of the photons in a given energy range (50–300 keV for GBM), which has been provided by the official GBM online burst catalogs (Gruber et al. 2014; von Kienlin et al. 2014). Since ${T}_{90}$ is not always representative of the width of the single pulse, some modifications to the bin sizes have been made manually to keep them close to $5 \%$ of the genuine pulse width. Given a determined time bin, any burst that has one or more pulses that could not be well pinpointed by the Gaussian profile was excluded as a final step of the sample selection process.

Since each burst has a different range of the spectral distribution, the uppermost and lowermost edges of the energy channels were selected manually to ensure enough signal above background (S/N > 5) in each channel. Table 1 lists our final sample of 50 bursts (between 2008 August and 2016 May) that are qualified for our selection criteria above. The final bin size of each burst is listed in the third column in Table 1. The nine-channel light curves of each burst are shown in the left panels in Figure Set 2, where the total light curve summed in the nine channels is shown at the top. The Gaussian profile fitting is shown at the top of each light curve as a red curve. For a visual aid to manually check the effect of spectral lag, the peaks of each Gaussian profile are connected and plotted as green dashed lines.

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Table 1.  Our Sample of 50 GRBs Detected by GBM and Results on the Spectral Lag

Burst	T90 (s)a,b	Bin (s)	t0 (s)	τ (s)	β	ω (s)	γ	NaIc	${\rm{\Delta }}{t}_{\mathrm{HR}}$ (s)
081125496	9.28 ± 0.61	0.46	$-{15.770}_{-0.142}^{+4.420}$	${23.560}_{-4.066}^{+0.225}$	${0.047}_{-0.003}^{+0.035}$	${8.456}_{-0.746}^{+0.932}$	${0.336}_{-0.022}^{+0.026}$	na	2.521 ± 0.126
081224887	16.45 ± 1.16	0.82	$-{11.103}_{-0.116}^{+3.040}$	${18.545}_{-2.713}^{+0.131}$	${0.058}_{-0.003}^{+0.033}$	${4.135}_{-0.408}^{+0.559}$	${0.171}_{-0.022}^{+0.028}$	n6	1.174 ± 0.125
090719063	11.39 ± 0.47	0.57	${2.926}_{-0.632}^{+0.609}$	${18.732}_{-2.311}^{+5.054}$	${0.488}_{-0.085}^{+0.127}$	${8.557}_{-0.621}^{+0.755}$	${0.246}_{-0.019}^{+0.022}$	n8	1.678 ± 0.131
090804940	5.57 ± 0.36	0.28	$-{16.344}_{-0.258}^{+5.194}$	${20.837}_{-4.702}^{+0.340}$	${0.022}_{-0.003}^{+0.090}$	${4.735}_{-0.393}^{+0.489}$	${0.238}_{-0.021}^{+0.025}$	n5	0.828 ± 0.086
090809978	11.01 ± 0.32	0.55	$-{4.747}_{-6.670}^{+4.876}$	${14.166}_{-3.057}^{+6.186}$	${0.110}_{-0.047}^{+0.130}$	${6.128}_{-0.527}^{+0.656}$	${0.241}_{-0.021}^{+0.025}$	n4	1.908 ± 0.128
090820027	12.42 ± 0.18	0.62	${34.182}_{-0.014}^{+0.177}$	${12.640}_{-0.016}^{+58.098}$	${0.786}_{-0.291}^{+0.291}$	${4.985}_{-0.387}^{+0.507}$	${0.142}_{-0.016}^{+0.020}$	n2	1.157 ± 0.151
090922539	87.04 ± 0.81	0.60	${2.522}_{-0.031}^{+0.311}$	${15.469}_{-1.196}^{+58.945}$	${0.882}_{-0.013}^{+0.672}$	${5.710}_{-0.941}^{+1.592}$	${0.228}_{-0.044}^{+0.062}$	n9	0.941 ± 0.159
091010113	5.95 ± 0.14	0.05	${2.219}_{-0.060}^{+0.060}$	${1.224}_{-0.300}^{+0.559}$	${0.374}_{-0.088}^{+0.158}$	${0.760}_{-0.087}^{+0.124}$	${0.339}_{-0.029}^{+0.037}$	n3	0.102 ± 0.009
100324172	17.92 ± 2.06	0.90	$-{19.702}_{-0.295}^{+6.729}$	${27.893}_{-6.158}^{+1.088}$	${0.037}_{-0.008}^{+0.050}$	${2.596}_{-0.295}^{+0.472}$	${0.032}_{-0.027}^{+0.039}$	n2	1.416 ± 0.150
100515467	10.62 ± 1.43	0.53	${0.922}_{-0.054}^{+0.054}$	${1.764}_{-1.151}^{+32.415}$	${0.359}_{-0.283}^{+0.477}$	${2.085}_{-0.268}^{+0.415}$	${0.212}_{-0.033}^{+0.045}$	n7	0.256 ± 0.068
100528075	22.46 ± 0.75	1.12	${7.425}_{-0.139}^{+0.728}$	${7.593}_{-0.118}^{+44.195}$	${0.477}_{-0.052}^{+0.831}$	${8.132}_{-0.662}^{+0.881}$	${0.135}_{-0.020}^{+0.024}$	n7	1.374 ± 0.215
100612726	8.58 ± 3.21	0.43	$-{19.597}_{-0.402}^{+6.497}$	${26.430}_{-5.708}^{+0.683}$	${0.029}_{-0.004}^{+0.096}$	${2.607}_{-0.203}^{+0.274}$	${0.088}_{-0.021}^{+0.026}$	n4	2.656 ± 0.117
100707032	81.79 ± 1.22	1.00	${0.726}_{-0.190}^{+0.032}$	${97.539}_{-8.153}^{+2.460}$	${0.804}_{-0.060}^{+0.000}$	${49.183}_{-5.219}^{+6.636}$	${0.665}_{-0.022}^{+0.025}$	n8	1.305 ± 0.165
101126198	43.84 ± 1.75	0.50	${9.244}_{-0.416}^{+0.416}$	${4.837}_{-1.232}^{+64.333}$	${0.085}_{-0.068}^{+2.734}$	${6.185}_{-0.476}^{+0.615}$	${0.104}_{-0.020}^{+0.024}$	n7	1.749 ± 0.159
110301214	5.69 ± 0.36	0.50	${2.031}_{-2.158}^{+0.520}$	${3.559}_{-1.317}^{+2.614}$	${0.350}_{-0.242}^{+0.394}$	${2.371}_{-0.173}^{+0.221}$	${0.088}_{-0.020}^{+0.024}$	n8	2.029 ± 0.107
110605183	82.69 ± 3.08	1.00	$-{7.452}_{-0.140}^{+9.839}$	${25.367}_{-1.900}^{+19.457}$	${0.149}_{-0.003}^{+0.452}$	${16.274}_{-2.665}^{+5.620}$	${0.310}_{-0.039}^{+0.066}$	n2	2.459 ± 0.237
110721200	21.82 ± 0.57	0.50	$-{3.272}_{-0.143}^{+1.258}$	${8.470}_{-0.848}^{+0.848}$	${0.098}_{-0.009}^{+0.065}$	${2.497}_{-0.242}^{+0.329}$	${0.109}_{-0.024}^{+0.031}$	n7	0.964 ± 0.088
110817191	5.95 ± 0.57	0.30	$-{3.319}_{-0.528}^{+1.153}$	${8.290}_{-0.623}^{+0.623}$	${0.114}_{-0.017}^{+0.070}$	${3.971}_{-0.423}^{+0.591}$	${0.299}_{-0.025}^{+0.032}$	n9	0.717 ± 0.059
111009282	20.74 ± 4.22	1.04	$-{17.923}_{-1.835}^{+6.918}$	${27.799}_{-6.463}^{+6.463}$	${0.024}_{-0.007}^{+0.215}$	${5.177}_{-0.496}^{+0.686}$	${0.105}_{-0.027}^{+0.034}$	n1	3.317 ± 0.207
111017657	11.07 ± 0.41	0.55	$-{0.226}_{-1.110}^{+1.447}$	${7.894}_{-1.361}^{+1.361}$	${0.078}_{-0.024}^{+0.171}$	${4.466}_{-0.474}^{+0.676}$	${0.184}_{-0.024}^{+0.031}$	n6	0.991 ± 0.101
120119170	55.30 ± 6.23	2.77	${1.771}_{-3.838}^{+3.838}$	${22.122}_{-5.626}^{+38.296}$	${0.100}_{-0.078}^{+0.438}$	${16.617}_{-2.092}^{+3.023}$	${0.193}_{-0.034}^{+0.045}$	nb	3.775 ± 0.439
120426090	2.88 ± 0.18	0.14	$-{0.155}_{-3.346}^{+0.656}$	${2.679}_{-0.303}^{+3.113}$	${0.143}_{-0.099}^{+0.111}$	${1.464}_{-0.086}^{+0.105}$	${0.151}_{-0.015}^{+0.017}$	n2	0.684 ± 0.039
120427054	5.63 ± 0.57	0.50	$-{1.847}_{-0.088}^{+0.845}$	${8.166}_{-1.091}^{+0.229}$	${0.153}_{-0.014}^{+0.080}$	${2.731}_{-0.362}^{+0.564}$	${0.162}_{-0.032}^{+0.043}$	na	0.953 ± 0.079
120625119	7.42 ± 0.57	0.37	${2.649}_{-0.145}^{+0.314}$	${4.745}_{-0.743}^{+14.566}$	${0.456}_{-0.093}^{+0.514}$	${7.166}_{-0.788}^{+1.243}$	${0.427}_{-0.026}^{+0.038}$	n5	0.665 ± 0.086
120727681	10.50 ± 1.64	0.53	${1.764}_{-0.094}^{+0.094}$	${2.965}_{-2.960}^{+36.079}$	${0.200}_{-0.146}^{+3.338}$	${6.622}_{-0.815}^{+1.352}$	${0.247}_{-0.032}^{+0.046}$	n2	1.198 ± 0.203
120919309	21.25 ± 1.81	0.60	${2.818}_{-0.125}^{+0.228}$	${11.280}_{-2.432}^{+16.023}$	${0.708}_{-0.104}^{+0.360}$	${6.275}_{-0.605}^{+0.836}$	${0.293}_{-0.023}^{+0.029}$	n1	0.830 ± 0.095
121122885	7.94 ± 0.57	0.40	$-{19.883}_{-0.116}^{+5.070}$	${34.420}_{-4.068}^{+1.431}$	${0.081}_{-0.009}^{+0.038}$	${4.030}_{-0.434}^{+0.655}$	${0.106}_{-0.026}^{+0.035}$	na	1.607 ± 0.101
121223300	11.01 ± 0.72	0.55	${2.334}_{-0.641}^{+0.487}$	${45.081}_{-17.034}^{+39.613}$	${0.755}_{-0.182}^{+0.224}$	${7.409}_{-1.193}^{+2.277}$	${0.237}_{-0.040}^{+0.063}$	n7	2.266 ± 0.127
130206482	11.26 ± 1.95	0.50	${2.064}_{-0.217}^{+0.217}$	${8.709}_{-5.110}^{+30.579}$	${0.617}_{-0.393}^{+0.393}$	${3.412}_{-0.454}^{+0.788}$	${0.199}_{-0.035}^{+0.053}$	n1	1.345 ± 0.123
130325203	6.91 ± 0.72	0.35	${1.042}_{-0.443}^{+0.914}$	${5.117}_{-1.736}^{+5.929}$	${0.303}_{-0.072}^{+0.556}$	${3.418}_{-0.442}^{+0.706}$	${0.248}_{-0.031}^{+0.043}$	n7	0.738 ± 0.065
130509078	24.32 ± 3.59	1.22	${2.433}_{-0.033}^{+0.422}$	${14.821}_{-14.715}^{+33.794}$	${0.827}_{-0.361}^{+0.361}$	${7.527}_{-1.189}^{+2.358}$	${0.259}_{-0.043}^{+0.070}$	n9	0.604 ± 0.282
130518580	48.58 ± 0.92	1.00	${18.033}_{-0.538}^{+2.341}$	${10.493}_{-2.069}^{+2.069}$	${0.036}_{-0.014}^{+0.180}$	${3.276}_{-0.253}^{+0.340}$	${0.041}_{-0.018}^{+0.023}$	n3	0.943 ± 0.164
130630272	17.15 ± 0.57	0.86	${5.245}_{-1.479}^{+0.345}$	${73.604}_{-46.599}^{+12.555}$	${0.893}_{-0.386}^{+0.071}$	${17.024}_{-2.551}^{+5.267}$	${0.264}_{-0.038}^{+0.065}$	n4	0.886 ± 0.209
130701060	20.22 ± 1.73	0.50	${2.861}_{-0.069}^{+0.069}$	${2.853}_{-2.820}^{+34.833}$	${0.295}_{-0.233}^{+3.402}$	${3.994}_{-0.724}^{+1.549}$	${0.153}_{-0.045}^{+0.073}$	na	0.275 ± 0.104
130704560	6.40 ± 0.57	0.50	${1.229}_{-0.384}^{+0.349}$	${10.478}_{-1.005}^{+2.006}$	${0.448}_{-0.078}^{+0.103}$	${4.876}_{-0.301}^{+0.358}$	${0.247}_{-0.017}^{+0.019}$	n4	5.186 ± 4.424
131014215	3.20 ± 0.09	0.16	$-{6.485}_{-0.126}^{+2.125}$	${10.698}_{-2.065}^{+0.155}$	${0.040}_{-0.006}^{+0.025}$	${2.186}_{-0.130}^{+0.155}$	${0.153}_{-0.017}^{+0.020}$	na	0.572 ± 0.039
131028076	17.15 ± 0.57	0.86	${5.962}_{-1.970}^{+0.942}$	${13.255}_{-0.623}^{+2.759}$	${0.305}_{-0.111}^{+0.111}$	${9.094}_{-0.633}^{+0.775}$	${0.211}_{-0.016}^{+0.018}$	n2	1.625 ± 0.151
131216081	19.26 ± 3.60	0.50	$-{14.778}_{-1.151}^{+1.151}$	${17.919}_{-12.889}^{+20.736}$	${0.023}_{-0.010}^{+0.519}$	${2.498}_{-0.497}^{+1.094}$	${0.179}_{-0.047}^{+0.079}$	n9	0.251 ± 0.072
131231198	31.23 ± 0.57	1.56	${20.476}_{-0.695}^{+0.815}$	${35.391}_{-2.863}^{+5.885}$	${0.460}_{-0.051}^{+0.084}$	${20.614}_{-1.129}^{+1.374}$	${0.331}_{-0.015}^{+0.017}$	n3	8.519 ± 0.534
140209313	1.41 ± 0.26	0.05	${1.477}_{-0.082}^{+0.068}$	${0.922}_{-0.067}^{+0.300}$	${0.353}_{-0.099}^{+0.165}$	${0.571}_{-0.056}^{+0.079}$	${0.269}_{-0.023}^{+0.029}$	na	0.122 ± 0.009
140821997	32.51 ± 1.64	1.00	$-{19.073}_{-0.075}^{+13.702}$	${56.024}_{-12.322}^{+0.034}$	${0.032}_{-0.004}^{+0.059}$	${11.621}_{-1.303}^{+1.823}$	${0.201}_{-0.027}^{+0.034}$	n5	3.551 ± 0.296
141028455	31.49 ± 2.43	1.00	${6.716}_{-8.459}^{+3.587}$	${14.993}_{-0.696}^{+7.844}$	${0.185}_{-0.103}^{+0.194}$	${8.961}_{-0.806}^{+1.045}$	${0.184}_{-0.022}^{+0.026}$	n6	2.393 ± 0.197
150306993	18.94 ± 1.15	0.95	$-{0.424}_{-2.952}^{+2.193}$	${27.668}_{-2.744}^{+20.100}$	${0.358}_{-0.126}^{+0.238}$	${23.230}_{-3.543}^{+5.790}$	${0.413}_{-0.036}^{+0.049}$	n4	1.716 ± 0.166
150314205	10.69 ± 0.14	0.53	$-{0.004}_{-2.038}^{+1.155}$	${9.060}_{-0.149}^{+3.836}$	${0.269}_{-0.115}^{+0.199}$	${6.024}_{-0.611}^{+0.901}$	${0.264}_{-0.024}^{+0.032}$	n9	1.065 ± 0.083
150721242	18.43 ± 0.57	0.92	$-{4.904}_{-2.286}^{+1.778}$	${51.275}_{-2.343}^{+3.957}$	${0.368}_{-0.057}^{+0.060}$	${35.815}_{-2.679}^{+3.347}$	${0.544}_{-0.020}^{+0.023}$	n7	11.371 ± 1.728
151021791	7.23 ± 0.60	0.36	${0.943}_{-0.033}^{+0.161}$	${7.772}_{-0.849}^{+28.501}$	${0.713}_{-0.042}^{+0.551}$	${1.397}_{-0.175}^{+0.255}$	${0.111}_{-0.029}^{+0.036}$	na	0.422 ± 0.051
151107851	139.01 ± 6.45	1.00	${8.162}_{-0.048}^{+0.885}$	${11.509}_{-0.722}^{+47.452}$	${0.474}_{-0.016}^{+0.650}$	${23.531}_{-2.965}^{+4.388}$	${0.427}_{-0.030}^{+0.039}$	n9	1.568 ± 0.239
160101030	4.67 ± 0.60	0.40	$-{1.767}_{-0.096}^{+1.029}$	${4.882}_{-0.828}^{+0.828}$	${0.067}_{-0.018}^{+0.182}$	${1.423}_{-0.132}^{+0.179}$	${0.019}_{-0.027}^{+0.033}$	n2	1.059 ± 0.080
160113398	24.58 ± 0.26	1.23	$-{14.304}_{-1.444}^{+13.188}$	${54.794}_{-12.636}^{+1.614}$	${0.030}_{-0.002}^{+0.041}$	${8.164}_{-0.585}^{+0.703}$	${0.180}_{-0.019}^{+0.021}$	nb	5.217 ± 0.377
160530667	9.02 ± 0.18	0.45	${1.513}_{-0.158}^{+1.207}$	${6.642}_{-0.950}^{+0.223}$	${0.070}_{-0.007}^{+0.060}$	${3.393}_{-0.197}^{+0.226}$	${0.112}_{-0.014}^{+0.015}$	n2	0.913 ± 0.084

Notes.

^aData provided by the official GBM online burst catalogs (Gruber et al. 2014; von Kienlin et al. 2014). ^bAll of the errors in this work indicate a confidence interval of 1 σ uncertainty. ^cOnly the data from the brightest NaI detector are used for the analysis in this work.

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As the pulses in each energy channel have been well fit by Gaussian profiles for each burst, the peak arrival time ${t}_{\mathrm{peak}}$ and the rms Gaussian width δ could be well determined. Both ${t}_{\mathrm{peak}}$ and δ, each as a function of the photon energy E (the middle value of each energy channel), are shown in the right panels of Figure Set 2 for each burst. The uncertainties in ${t}_{\mathrm{peak}}$ and δ are the 1σ errors evaluated for the returned parameters from the Guassian fitting using the standard IDL fitting function GAUSSFIT. An anticorrelation between ${t}_{\mathrm{peak}}$ and E apparently exists for almost all the bursts. We fit this anticorrelation with a three-parameter power law,

$$\begin{eqnarray}&&{t}_{\mathrm{peak}}(E)={t}_{0}+\tau \times {\left(\displaystyle \frac{E}{1\mathrm{keV}}\right)}^{-\beta },\end{eqnarray} \tag{ 2 }$$

where t₀ and τ are constants measured in seconds and β is the power-law index. We note that there are some practical meanings of t₀ and τ. If $\beta \gt 0$, then we have

$$\begin{eqnarray}&&{t}_{\mathrm{peak}}(1\ \mathrm{keV})={t}_{0}+\tau ,\end{eqnarray} \tag{ 3 }$$

and

$$\begin{eqnarray}&&{t}_{\mathrm{peak}}(\infty )={t}_{0},\end{eqnarray} \tag{ 4 }$$

in two limiting cases with the photon energy of 1 keV and $\infty$, respectively. So ${t}_{0}\,\equiv \,{t}_{\mathrm{peak}}(E=\infty )$ can be regarded as the limiting value of the earliest arrival time of the most energetic photons. τ is the difference between ${t}_{\mathrm{peak}}(1\,\mathrm{keV})$ and ${t}_{\mathrm{peak}}(\infty )$, thus it is the "limiting" spectral lag at 1 keV. We use τ as a fundamental timing measurement of the spectral lags independent of the energy channel selections.

There is also an apparent anticorrelation between δ and E for almost all the bursts in our sample. We fit this anticorrelation with a two-parameter power law,

$$\begin{eqnarray}&&\delta (E)=\omega \times {\left(\displaystyle \frac{E}{1\mathrm{keV}}\right)}^{-\gamma },\end{eqnarray} \tag{ 5 }$$

where ω is a constant in the unit of second and γ is the power-law index. ω could be considered as the limiting half-pulse width at 1 keV.

The effects of spectral lag have been manifested as both the delay of the peak arrival time and the broadening of the pulse width in a lower energy channel. These two phenomena appear to be closely connected. The best-fitting functions given by Equations (2) and (5) are shown by the red curves in the right panels in Figure Set 2. To constrain the values and uncertainties of each parameter in the power laws well, we have adopted the efficient Nested Sampling Monte Carlo algorithm in the framework of Bayesian analysis, which is used in the generic package PyMultiNest, as first applied to X-ray spectral analysis (Buchner et al. 2014). The best-fitting parameters of t₀, τ , β, ω, and γ are provided in Table 1. The distributions for these parameters together with the T₉₀ in our sample are shown in Figure 3.

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The median value of t₀ is −1.03 s, which can be understood as the mean value of the limiting initial occurrence time of the GRB pulse. The median value of T₉₀, τ, and ω is 12.4 s, 12.6 s, and 5.2 s, respectively. There is a clear trend for the limiting spectral lag τ to be correlated with the duration T₉₀, as shown in Figure 4(a). ω is also correlated with the duration T₉₀, however, as shown in Figure 4(c), and τ is correlated with twice ω, as shown in Figure 4(e). These correlations suggest that ${T}_{90}$ is a modestly good indicator of the intrinsic duration of the GRB pulses in single-pulse events. The really interesting correlation is between the pulse width and the spectral lag. The correlation between the spectral lag and T₉₀ is the consequence. A similar correlation between the pulse width and the spectral lag (defined slightly differently, however) was also found in bright X-ray flares, suggesting a common origin of prompt emission and the X-ray flares (e.g., Margutti et al. 2010). In contrast, neither β nor γ appears to be correlated with the duration T₉₀, as shown in Figures 4(b) and (d).

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The connection between the delay of the peak arrival time and the broadening of the pulse width can also be revealed by the similar median values of $\beta =0.27$ and $\gamma =0.21$. While γ tends to have a unimodal distribution that peaks at ∼0.2, β tends to have a bimodal distribution that peaks toward ∼0.1. This bimodal distribution of β is also indicated by the heart-shaped correlation between τ and β, as shown in Figure 4(g), where two distinction components, i.e., an anticorrelation between τ and β at $\beta \lt 0.3$ and a correlation at $\beta \gt 0.3$, are apparent. These two components form the two peaks at $\beta \sim 0.1$ and $\beta \sim 0.4$ shown in Figure 3(e). Figure 4(g) also indicates an intriguing trend that no cases exist with both $\tau \sim 0$ and $\beta \sim 0$. For comparison, an apparent correlation between ω and γ is shown in Figure 4(h). This indicates that the width of narrower pulses is less dependent on the photon frequencies.

It is worth mentioning that a mean value of the power-law index $\gamma =0.21$ appears to be inconsistent with the previously proposed values of $\gamma \sim 0.4$ for BATSE and Swift GRBs, which were all based on analyses of the average pulse width as determined either by the CCF or by specific fits of the light curves of individual pulses in given broad energy channels (Fenimore et al. 1995; Norris et al. 1996, 2005). Recently, the energy dependence of minimum variability timescales has also been studied for Fermi/GRM bursts in four energy channels, and the power-law index has been found to be between 0.53 and 0.97 (Golkhou et al. 2015). A value of $\gamma \sim 0.4$ has also been found for X-ray flares (Chincarini et al. 2010; Margutti et al. 2010). Alternatively, the recent work on the extremely bright GRB 130427A detected by Fermi/GBM estimated a value of $\gamma =0.27\pm 0.03$ (Preece et al. 2014), which is consistent with our results here. The energy dependence of the pulse width with a power-law index of $\gamma \sim 0.5$ might indicate the signature of synchrotron cooling (e.g., Chiang 1998; Kazanas et al. 1998). In contrast, a power-law index of $\gamma \sim 0.25$ might favor the dominance of the SSC processes (e.g., Dermer 1998), the relativistic curvature effect (e.g., Shen et al. 2005), or the Doppler effect (e.g., Qin et al. 2004). However, as discussed in recent works, these models still have difficulties in fully accounting for the observational features, which instead indicate a combination of several specific constraints on the radiation mechanism (e.g., Uhm & Zhang 2016). It was proposed that the spectral lag might be produced by a combination of spectral evolution and the curvature effect (Peng et al. 2011).

Even though short GRBs generally show negligible spectral lags (Norris & Bonnell 2006) or even "negative" lags (Yi et al. 2006), the only short burst in our sample, i.e., 140209313 (${T}_{90}=1.41\,{\rm{s}}$), does show an apparent spectral lag of $\tau \,=1.28\,{\rm{s}}$ that is comparable to its duration, as shown by the blue data points in Figure 4. The correlations between the spectral lag or the pulse width and the duration apply nicely for both short and long bursts in our sample, suggesting some similarity or connection between the two classes, as suggested by previous works (e.g., Ghirlanda et al. 2009; Shao et al. 2011; Zhang et al. 2012c). While the values of β and γ for GRB 140209313 are similar to those of the long bursts, it is, however, an outlier in the ω–γ diagram, as shown in Figure 4(h). Another special burst is GRB 091010113, as shown by the purple data point. Although GRB 091010113 has a long duration of ${T}_{90}=5.95\,{\rm{s}}$ as listed in the online catalog, its main pulse (as shown in our Figure Set 2) lasts obviously shorter than 2 s. Two special bursts are also located at the bottom of heart shape in the τ-β diagram in panel (g) of Figure 4 or in the bottom right corner in the ω–γ diagram in panel (h) of Figure 4, suggesting a potential subgroup.

The peak energy ${{\rm{E}}}_{{\rm{p}}}$ has been considered as an important spectral parameter of GRB pulses and found to be in correlation with many measured quantities, such as the isotropic total energy ${E}_{\gamma ,\mathrm{iso}}$ (Amati et al. 2002). However, none of the four newly measured parameters related with the spectral lag shows any clear correlation with ${E}_{{\rm{p}}}$, as shown by Figure 5, where the values of ${E}_{{\rm{p}}}$ have been adopted as the peak energy of a Band function fit to the single spectrum over the duration of the burst provided by the official GBM online burst catalogs (Gruber et al. 2014; von Kienlin et al. 2014). Given the significant spectral evolution in GRB pulses, time-resolved spectral analysis is a key to determine the underlying radiation mechanism. A recent hot topic is to distinguish the HTS and HIT patterns in the evolution of the hardness of the spectra.

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Instead of using the peak energy ${E}_{{\rm{p}}}$, in this work we directly use the hardness ratio as an indicator of the spectral properties of the GRBs. The hardness ratio is defined as the ratio between the two light curves in a harder and a softer energy channel, which are determined here for a preliminary result by evenly cutting the full energy channel in half in the logarithmic scale. This hardness ratio is easily available and is not subject to the uncertainties in determining ${E}_{{\rm{p}}}$ for a Band function fitting. The latter might have some statistical issues, as suggested by previous works (e.g., Preece et al. 2016). Our results of the hardness ratio are shown as the thick blue histogram in Figure 2. For almost all the bursts in our sample, the hardness ratio shows a uniform pattern that includes both soft-to-hard and HTS phases, and it has a very similar temporal profile as the total light curve. In a few of them, GRB 130630272, for instance, the hardness ratio is nicely consistent with the total light curve, clearly showing an HIT pattern. Most of the GRBs that cannot be considered as strictly HIT instead show a delay between the temporal profile of the hardness ratio and the total light curve. Strictly speaking, the majority of them seem to differ with either the HTS or HIT patterns. For example, GRBs 081224887, 090809978, 100612726, and 110817191 were categorized as HTS pattern based on an analysis of the temporal evolution of the peak energy ${E}_{{\rm{p}}}$ (e.g., Lu et al. 2012). However, an initial "soft-to-hard" phase seems to be indispensable in these four cases, as shown here. A very similar pattern between the hardness ratio and the total light curve can be perceived for almost all the bursts in our sample. A pure HTS pattern seems to be disfavored for most of the bursts in our sample.

To further illustrate the discrepancy in the pattern of the spectral evolution between the previous analyses using the peak energy ${E}_{{\rm{p}}}$ and our analysis using the hardness ratio, we made a direct comparison of the results based on these two spectral quantities for the four previously proposed HTS bursts mentioned above (i.e., 081224887, 090809978, 100612726, and 110817191). First, we reproduced the spectral analyses for the evolution of the peak energy ${E}_{{\rm{p}}}$ using the software package GBM RMFIT tool (version 4.3pr2),⁸ and the results are shown in the top panels in Figure 6 for each burst. We adopted the Bayesian blocks (BBs) as the binning method (Scargle et al. 2013), which is considered more accurate in reproducing the intrinsic spectral evolution pattern in highly variable GRB light curves, as suggested by Burgess (2014). The adopted time bins are listed in the second column of Table 2. We can reproduce the HTS pattern in the measure of ${E}_{{\rm{p}}}$, which is consistent with the previous result. For a straight comparison, we rebinned the light curve in the same time bins and calculated the hardness ratio as introduced above. The results are shown in the bottom panels in Figure 6 for each burst. Dramatically, the four putative HTS bursts determined by ${E}_{{\rm{p}}}$ appear to have a semi-HIT pattern, although this is not highly statistically significant in GRBs 100612726 and 110817191. This suggests that an obvious ambiguity may exist in distinguishing HTS and HIT behaviors. The greatest difference is in the first time bin, which is understandable since the pulse is weak and the time bin is narrow, therefore the uncertainty in determining the value of ${E}_{{\rm{p}}}$ is the largest.

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Table 2.  Four Putative HTS Bursts

Burst Name	Time Bin (s)	${E}_{{\rm{p}}}$ (keV)	Hardness Ratio
081224887	0.001–0.383	1232.640 ± 413.737	2.403 ± 0.523
	0.383–1.791	645.221 ± 33.499	3.783 ± 0.300
	1.791–4.287	408.177 ± 12.684	2.176 ± 0.079
	4.287–6.015	286.788 ± 16.786	1.273 ± 0.066
	6.015–8.511	214.702 ± 14.967	1.071 ± 0.068
	8.511–11.263	192.329 ± 22.508	0.873 ± 0.075
	11.263–18.495	178.013 ± 35.499	0.780 ± 0.093
090809978	−1.727–1.023	382.599 ± 206.356	1.709 ± 0.424
	1.023–1.983	255.765 ± 50.279	2.427 ± 0.282
	1.983–4.863	209.860 ± 15.778	1.502 ± 0.049
	4.863–8.127	131.351 ± 13.241	0.831 ± 0.034
	8.127–11.263	67.924 ± 13.042	0.512 ± 0.045
	11.263–20.159	40.761 ± 29.170	0.383 ± 0.105
100612726	−0.063–1.087	181.979 ± 56.287	1.221 ± 0.153
	1.087–2.943	132.842 ± 9.858	1.309 ± 0.070
	2.943–5.247	105.570 ± 4.407	0.857 ± 0.029
	5.247–6.143	85.145 ± 8.302	0.568 ± 0.044
	6.143–7.487	54.283 ± 8.528	0.381 ± 0.038
	7.487–9.791	61.054 ± 39.131	0.341 ± 0.050
110817191	−0.063–0.447	395.760 ± 118.835	2.528 ± 0.644
	0.447–0.831	316.608 ± 48.255	3.065 ± 0.459
	0.831–1.919	193.306 ± 12.979	1.872 ± 0.105
	1.919–2.943	124.718 ± 11.275	1.235 ± 0.080
	2.943–4.415	57.360 ± 9.035	0.622 ± 0.052
	4.415–7.551	29.179 ± 11.825	0.430 ± 0.081

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More importantly, the evolution behavior of the spectral hardness for most GRBs in our sample is neither HTS nor HIT. An obvious time delay between the hardness evolution pattern and the light curve exists. To study the property of this hardness-intensity time delay ${\rm{\Delta }}{t}_{\mathrm{HR}}$, we calculated the CCF between the evolution curve of hardness ratio (the thick blue curve in each left panel of Figure Set 2) and the total light curve (the thick black curve in each left panel of Figure Set 2) using the same method as described in Zhang et al. (2012b). The measured values of ${\rm{\Delta }}{t}_{\mathrm{HR}}$ are listed in the last column of Table 1. The origin of this time delay can be revealed by the correlation between ${\rm{\Delta }}{t}_{\mathrm{HR}}$ and the spectral lag τ, as shown in Figure 7. The best fit yields

$$\begin{eqnarray}&&{\rm{\Delta }}{t}_{\mathrm{HR}}=0.29\times {\tau }^{0.64\pm 0.01},\end{eqnarray} \tag{ 6 }$$

with a Spearman rank correlation coefficient r = 0.67 and a chance probability $P=1.0\times {10}^{-7}$. The suggests that the spectral evolution behavior (hardness versus light curve) is closely related with the spectral lag phenomenon. Given an intrinsic time delay between the evolution of hardness and light curve, neither HTS or HIT is an appropriate categorization of the spectral evolution behavior for most GRBs. This time delay is an indication of the spectral lag. As a consequence, only the bursts with negligible spectral lag would exhibit a pure HIT behavior. The bursts with a long spectral lag would more likely exhibit an HTS behavior. Most GRBs have neither HIT nor HTS behaviors, but form a major and continuous transition between HIT and HTS categories. Since the spectral lag is well correlated with the pulse width, as found above, short pulses tend to show an HIT behavior and long pulses tend to show an HTS behavior. As shown in Figure 7, the only two short GRBs 091010113 and 140209313 that have negligible ${\rm{\Delta }}{t}_{\mathrm{HR}}$ are located well in the lower left HIT corner, and the four previously proposed HTS GRBs with longer pulse widths all have a moderate ${\rm{\Delta }}{t}_{\mathrm{HR}}$.

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In this work, we revisited the measurement of GRB spectral lag between a group of consecutive energy channels instead of two given energy channels. The idea of splitting the light curves into more consecutive energy channels was triggered by the studies of pulsars in the radio frequencies, even though the physics behind them are totally different. For pulsars there is a quadratic frequency dependence on the arrival time (${\rm{\Delta }}{t}_{\mathrm{peak}}\propto {\nu }^{-2}$), as is typical for the propagation effect in cold plasma along the line of sight, known as the dispersion (e.g., Lorimer et al. 2007). Meanwhile, the significant pulse width broadening with a quadratic frequency dependence ($\propto {\nu }^{-4}$) is consistent with the Kolmogorov-like spectrum due to interstellar scattering (Lee & Jokipii 1976). On the other hand, the spectral lag for GRBs in the MeV energies is still an open issue, which more likely has an internal origin that is due to the radiation mechanisms. Based on a systematical analysis of 50 single-pulsed GRBs, we find that the arrival time of the GRB pulse is universally anticorrelated with the photon energy, showing an intrinsic soft spectral lag (i.e., softer photons come later) below ∼800 keV, as observed by the NaI detectors of Fermi/GBM. By investigating the pulse profiles in multiple consecutive energy channels, we can determine an intrinsic spectral lag τ and a pulse width ω for each burst, independent of the energy channel selection.

The spectral lag τ and pulse width ω are found to be well correlated with each other, which may favor the relativistic geometric effects previously proposed, e.g., the spectral lag arises because the observer is looking at the increasing latitudes with respect to the line of the sight with time (see Zhang et al. 2009 for a discussion and the references therein). For the first time, we can evaluate the energy dependency for the pulse arrival time by the power-law index β, which is widely distributed between ∼0.02 and 0.9 with a mean value of ∼0.27. The pulse width is also found universally to be anticorrelated with the photon energy (i.e., softer pulses have a wider width), which is consistent with previous studies. The power-law index γ for the energy dependency of the pulse width is also widely distributed between ∼0.02 and 0.7, but with a mean value of ∼0.21 that is distinct from previous studies (e.g., Fenimore et al. 1995; Norris et al. 1996, 2005). Our new result on the power-law index γ would also favor the relativistic geometric effects. We also caution that for some cases, e.g., GRB 100324172, the anticorrelations of the energy dependence may not be well constrained, potential systematic uncertainties may exist, and they probably translate into a very low value of γ (or β in some other case).

It is worth mentioning that our work reveals some interesting mutual correlations between the burst duration T₉₀, the spectral lag τ, and the pulse width ω. There might also exist a correlation between the power-law indices β and γ as shown in Figure 4(f), although with a mild Spearman rank correlation coefficient r of 0.4086 and a corresponding chance probability P of 0.0032. Since a physical model that is fully consistent with the data is still lacking, we may provide some indications about the connection between the energy dependencies of both the arrival time and the pulse width. As a simple demonstration, we can show that given the energy dependency of the arrival time that is fully described in Equations (2), the energy dependency of the pulse width described in Equation (5) might be a natural consequence. Let us assume that the energy dependence of the peak arrival time (i.e., Equation (2)) is a realization of the intrinsic radiation mechanism that applies to each emitted photon. Now, we consider a very simple pulse that consists of only two photons with different energies E₁ and E₂ (${E}_{1}\lt {E}_{2}$), respectively. According to Equation (2), they each arrive at the time t₁ and t₂ by

$$\begin{eqnarray}&&{t}_{1}={t}_{0}+\tau \times {\left(\displaystyle \frac{{E}_{1}}{1\mathrm{keV}}\right)}^{-\beta },\end{eqnarray} \tag{ 7 }$$

and

$$\begin{eqnarray}&&{t}_{2}={t}_{0}+\tau \times {\left(\displaystyle \frac{{E}_{2}}{1\mathrm{keV}}\right)}^{-\beta }.\end{eqnarray} \tag{ 8 }$$

Now we collect these two photons in the energy channel [E₁, E₂] and measure the width W of this simple pulse formed by only the two photon, which is the time difference between t₁ and t₂ by

$$\begin{eqnarray}&&W={t}_{1}-{t}_{2}=\tau \times \left[{\left(\displaystyle \frac{{E}_{1}}{1\mathrm{keV}}\right)}^{-\beta }-{\left(\displaystyle \frac{{E}_{2}}{1\mathrm{keV}}\right)}^{-\beta }\right],\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&\approx \tau \times {\left(\displaystyle \frac{{E}_{1}}{1\mathrm{keV}}\right)}^{-\beta }.\end{eqnarray} \tag{ 10 }$$

Comparing Equations (5) and (10) suggests that $\tau \approx 2\times \omega$ and $\beta \approx \gamma$, which is consistent with the results, as shown in Figures 4(e) and (f). This means that the energy dependence of the characteristic arrival time of a group of photons should inevitably be reflected in a similar energy dependence of the pulse width of the same group of photons. We doubt that a group of photons can be distributed that has significant energy-dependent arrival times but can also have an energy-independent pulse width. Based on our analysis above, we therefore propose that the energy dependency of the arrival time of the emitted photons (Equation (2)) is more likely the internal and intrinsic radiation mechanism and that the energy dependency of the pulse width (Equation (5)) is its external and a natural or inevitable consequence. A complete theoretical framework that can fully account for the energy dependencies of both arrival time and pulse width is still lacking.

We used a symmetric Guassian profile to describe the single-pulse profiles of the bursts in our sample. Given the potential asymmetry of the pulses, there might be possible systematical uncertainties of the pulse peak times. Since we had some difficulties in fitting all the light curves in our sample with an exact function provided in the literature, we also adopted the CCF method to check the delay in the peak time between two consecutive energy channels and compared it with our results based on Guassian fitting. Our CCF method has been applied and explained in Zhang et al. (2012b). The comparison for each burst is shown in Figure 8. The same time ranges for each burst as shown in the left panels of Figure Set 2 were used to calculate the CCF lags. All the uncertainties of CCF lags in this work were estimated by Monte Carlo simulation, as described in Zhang et al. (2012b), and the uncertainties of Gaussian peak-time lags were estimated by error propagation of the peak times. Most of the results from different methods are consistent with each other, although a few cases, such as GRB 100707032 and 150314205, do reveal a systematical overestimation because of the pulse confusion, as can be perceived by the fittings for the light curves in the lower energy channels shown in Figure Set 2. Given that their numbers are few, the general results for the sample in this work have not been significantly affected by our current choice of the fitting function. The intrinsic shape of GRB pulses is still an open issue. A more complex fitting scheme (e.g., Norris et al. 2005) for the pulse profile should be explored for the work in the future.

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As an important result in this work, we found a new clue to solve the controversy over the spectral evolution of GRB pulses, i.e., the so-called HTS and HIT behaviors. We used the hardness ratio instead of the peak energy ${E}_{{\rm{p}}}$ as the characteristic spectral parameter since the latter is subject to uncertainties in the very early stage of the pulse. We find that most of the GRB pulses exhibit neither HTS nor HIT behaviors. Instead, they show a behavior close to HIT, but with a noticeable time delay. This time delay ${\rm{\Delta }}{t}_{\mathrm{HR}}$ is well correlated with the spectral lag τ that we have measured in this work. A GRB pulse with negligible spectral lag would appear to have an HIT behavior. On the other hand, a GRB pulse with significant spectral lag would appear to have an HTS behavior, which, in fact, is not a genuine behavior given that there is always a very short soft-to-hard phase at the very beginning. Even though most GRBs are highly variable with multiple pulses, single pulses seem to be fundamental elements of GRB prompt emission (e.g., Hakkila et al. 2008). Our results on single-pulse GRBs can provide important insights into the nature of GRB pulses.

This work was supported in part by the the National Basic Research Program ("973" Program) of China (Grant No. 2014CB845800), the National Natural Science Foundation of China (Grants No. 11103083, 11433009, 11673068, and 11533003) and the Strategic Priority Research Program Multi-wavelength Gravitational Wave Universe of the Chinese Academy of Sciences (No. XDB23000000). L.S. acknowledges the supported by the Joint NSFC-ISF Research Program (No. 11361140349), jointly funded by the National Natural Science Foundation of China and the Israel Science Foundation. B.B.Z. acknowledges the support by the Spanish Ministry Projects AYA 2012-39727-C03-01 and AYA2015-71718-R. X.F.W. acknowledges the support by the Youth Innovation Promotion Association (2011231) and the Key Research Program of Frontier Sciences (QYZDB-SSW-SYS005) of the Chinese Academy of Sciences. D.X. acknowledges the support by the One-Hundred-Talent Program of the Chinese Academy of Sciences. Part of this work used BBZ's personal IDL code library ZBBIDL and personal Python library ZBBPY. The computation resources used in this work are owned by Scientist Support LLC. This research has made use of NASA's Astrophysics Data System Bibliographic Services.

Software: GAUSSFIT, GBM RMFIT (http://fermi.gsfc.nasa.gov/ssc/data/analysis/rmfit/), PyMultiNest (Buchner et al.2014).