Probing Curvature Effects in the Fermi GRB 110920
A. Shenoy, E. Sonbas, C. Dermer, L. C. Maximon, K. S. Dhuga, P. N. Bhat, J. Hakkila, W. C. Parke, G. A. Maclachlan, T. N. Ukwatta
aa r X i v : . [ a s t r o - ph . H E ] S e p Draft version October 18, 2018
Preprint typeset using L A TEX style emulateapj v. 5/2/11
PROBING CURVATURE EFFECTS IN THE FERMI GRB 110920
A. Shenoy , E. Sonbas , C. Dermer, , L. C. Maximon , K. S. Dhuga , P. N. Bhat , J. Hakkila , W. C. Parke , G.A. Maclachlan , T. N. Ukwatta Department of Physics, The George Washington University, Washington, DC 20052, USA University of Adiyaman, Department of Physics, 02040 Adiyaman, Turkey NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA Space Science Division, Code 7653, Naval Research Laboratory, Washington, D.C. 20375, USA CSPAR, University of Alabama in Huntsville, Huntsville, AL 35805, USA Department of Physics and Astronomy, College of Charleston, Charleston, S.C. 29424, USA and Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA
Draft version October 18, 2018
ABSTRACTCurvature effects in Gamma-ray bursts (GRBs) have long been a source of considerable interest. Ina collimated relativistic GRB jet, photons that are off-axis relative to the observer arrive at later timesthan on-axis photons and are also expected to be spectrally softer. In this work, we invoke a relativelysimple kinematic two-shell collision model for a uniform jet profile and compare its predictions toGRB prompt-emission data for observations that have been attributed to curvature effects such asthe peak-flux–peak-frequency relation, i.e., the relation between the ν F ν flux and the spectral peak,E pk in the decay phase of a GRB pulse, and spectral lags. In addition, we explore the behavior ofpulse widths with energy. We present the case of the single-pulse Fermi GRB 110920, as a test forthe predictions of the model against observations. Subject headings:
Gamma-ray bursts: general INTRODUCTION
Pulses in GRB light curves are thought to be pro-duced by collisions between relativistic shells ejectedfrom an active central engine (Rees and Meszaros 1994).The interception of a more slowly moving shell by asecond shell that is ejected at a later time, but withgreater speed, produces a shock that dissipates internalenergy and accelerates the particles that emit the GRBradiation. This scenario is widely adopted in orderto model pulses in GRB light curves (e.g., Daigneand Mochkovitch 1998; Zhang et al. 2009). Studiesof pulses are important to determine if GRB sourcesrequire engines that are long-lasting or impulsive, andto determine the likely radiation mechanism(s), withimportant implications for the nature of the centralengine.Spectral lags, where low energy photons reach theobserver at later times than high-energy photons,are seen in a significant fraction of GRBs. Cheng etal. (1995) were the first to analyze the spectral lag ofGRBs, which they determined as the time delay betweenthe peaks in the Burst and Transient Source Experiment(BATSE) Large Area Detector (LAD) channel 1 (25- 50 keV) and channel 3 (100 - 300 keV) light curves.Since then, several authors have analyzed spectral lagsin GRBs, while also extending these observations tothe Swift and Fermi GRB samples (e.g., Norris et al.1996; Norris, Marani & Bonnell 2000, Wu & Fenimore2000; Chen et al. 2005; Ukwatta et al. 2010; Sonbaset. al. 2012) The leading model to explain the spectrallag is the curvature effect, i.e. the kinematic effectdue to the observer looking at an increasingly off-axisannulus area relative to the line-of-sight (Fenimore et [email protected] al. 1996; Salmonson 2000; Kumar & Panaitescu 2000;Ioka and Nakumura 2001; Qin 2002; Qin et al. 2004;Dermer 2004; Shen et al. 2005; Lu et al. 2006). Softerlow-energy radiation comes from the off-axis annulusarea due to smaller Doppler factors. This radiation isalso delayed at the observer end with respect to on-axisobservation due to the geometric curvature of the shell.The existing models as well as observations suggestthat a connection exists between the observed hard-to-soft spectral evolution of GRB pulses and spectral lags.It is therefore important to understand the mechanismthat produces this evolution. Tavani (1996) proposedthat this hard-to-soft spectral evolution is caused by thevariation of the average Lorentz factor of pre-acceleratedparticles and the strength of the local magnetic field atthe GRB site as the synchrotron emission evolves withinthe burst. Liang (1997) proposed a physical modelof hard-to-soft spectral evolution in which impulsivelyaccelerated non-thermal leptons cool by saturatedCompton up-scattering of soft photons. Kocevski &Liang (2003) have analyzed a sample of 19 GRBs andfound a positive correlation between the decay rateof the peak energy and the spectral lag. Ukwatta etal. (2010) have analyzed a sample of 31 Swift GRBswith known red-shifts and determined the spectrallags between fixed source frame bands, 100 – 150 keVand 200 – 250 keV. They also determined that thesource-frame E pk lies beyond the higher-energy band,100 – 250 keV for a majority of these bursts. Based onthis result, they suggest that spectral evolution may notbe the dominant process causing the observed spectrallag.Borgonovo and Ryde (2001) studied the spectral evo-lution in the prompt-emission in GRBs by performing a Shenoy et al.time-resolved spectral analysis of BATSE single pulsesand showed that in many cases the νF ν flux at the E pk for each time segment was ∝ E ηpk (hereafter referredto as the peak-flux–peak-frequency relation), with η ranging from ≈ . η was found tostay roughly constant for pulses within the same GRB.Dermer (2004) modeled and analyzed GRB pulses basedon curvature effects with a Broken-Power-Law (BPL)rest-frame spectrum and showed that in the curvaturelimit, η was equal to 3 for pulses with a wide range oftemporal properties.While several studies (Qin 2002; Qin et al. 2004;Shen et al. 2005; Lu et al. 2006) have been performedto determine the role played by curvature effects, bothin the spectral evolution of GRB prompt-emissionas well as in producing spectral lags, a number ofquestions remain unanswered. Specifically, we note thatthe dependence of the lag on the radius at which theemission takes place is quite unclear at this stage withseemingly contradictory results being reported in theliterature (Shen et al. 2005; Lu et al. 2006). While thatis not the focus of our current study, it is a matter ofconsiderable interest and will be the central topic of aforthcoming work. In this work we focus on the roleplayed by the thickness of colliding shells on observablessuch spectral lags, the νF ν flux vs. E pk relation, andthe behavior of the corresponding pulse widths as afunction of energy. Other studies have also attemptedto determine the role played by the evolution of therest-frame spectrum on the observables such as thespectral lags and the evolution of the pulse-widths withenergy within the context of a curvature model(Qinet al.2005; Lu et al. 2006; Qin et al. 2009; Penget al. 2011). Again, such considerations are veryimportant but we do not specifically consider the effectsof the evolution of the rest-frame spectrum in this paper.The paper is organized as follows: the basic featuresof the model are presented in section 2, followed bya description of the sample selection criteria, analysismethodology, and a case study in section 3. The discus-sion of our main results is presented in section 4, followedby a summary of our conclusions in section 5. THE MODEL
We have used a particular representation of theinternal shock model for our purposes (Dermer 2004).This model consists of a single two-shell collision eventoccurring at a radius r from the source, generatinga uniform spherical shell, with Lorentz factor Γ, thatradiates for a co-moving time between t ′ and t ′ + ∆ t ′ with ∆ t ′ = η t Γ t var / (1 + z ), where t var is the observedvariability time scale. The rest-frame–emission profile isassumed to be rectangular with instantaneous rise anddecay phases. The opening angle of the jet is assumedto be 4 / Γ. Emission from angles greater than 4 / Γare ignored. This is a suitable compromise betweenconsidering emissions from an entire fireball surface(0 < θ < π/
2) and a collimated jet ( θ ∼ / Γ). As notedby Qin et al. 2004, limiting the radiation to θ < / Γleads to a cutoff-tail problem whereas the contributionsfrom areas at θ > / Γ fall off very rapidly. We findsuch a choice to be suitable for producing pulse profiles
Fig. 1.—
Normalized light curves obtained using selected param-eters except: Thin-shell pulse, η ∆ = 0 .
1; and the Curvature pulse, η ∆ = η t = 0 .
1. The case of η t = η ∆ = η r = 1 is referred to asa Causal pulse (see Dermer 2004 for a detailed discussion of thesethree generic types of pulses). that may be directly compared with observations. Theco-moving width of the shell ∆ r ′ is assumed to remainconstant during the period of illumination and givenby ∆ r ′ = η ∆ Γ c t var / (1 + z ). It is in this respectthat the chosen model differs from previous studies.Previous studies (see for instance Qin et al. 2004, Shenet al. 2005) have studied the effects of curvature from aspherical surface in great detail. As shown subsequently,the effect of a finite shell where ∆ r ′ is comparable to c ∆ t ′ , has a significant effect on the predicted observablessuch as spectral lags and pulse widths as a functionof energy when compared to the models that studycurvature effects from a surface. As ∆ r ′ << c ∆ t ′ ,we approach the infinitesimal shell or emission-surfacesituation and are able to recover many of the predictionsof the aforementioned models. The emission spectrum inthe co-moving frame may be described by any suitablespectral function such as a Broken Power-Law (BPL),Band, or Comptonized–E pk , peaking at a co-movingphoton energy E ′ pk, = (1 + z) E pk, /2Γ where E pk, is the observer frame E pk at the start of the pulse.The spectral indices at E < E pk, and at E > E pk, incounts space are denoted α and β , respectively. Thecurvature constraint requires that r . c t var / (1 + z )(Fenimore et al. 1996). The radius is thus writtenusing the expression r = 2 η r Γ c t var / (1 + z ), with0 . η r .
1. The parameters η t , η ∆ and η r thus controlthe blast-wave duration, shell-thickness and radius ofemission respectively.Unless otherwise stated, the selected parameters usedfor the numerical calculations of the light curves, spectraand spectral lags are shown in Table. 1. Fig. 1 showsthe normalized, generic pulse shapes obtained using theselected parameters. As can be seen, the cases of thethin-shell pulse ( η ∆ << η t ) and the curvature pulse( η ∆ = η t << η r ) produce the sharp featured light curvesnoted by Qin et. al. 2004 for emission from a rectan-gular pulse profile in the co-moving frame, and whichare attributed to the effect of a suddenly-dimming emis-sion profile. These two cases most closely correspond torobing Curvature Effects in The Fermi GRB 110920 3 TABLE 1Selected parameters for the generation of the model light curves unless otherwise stated. η r η t η ∆ t var ( s ) Γ z θ jet E pk, (keV) d L (cm) u (ergs cm − )1.0 1.0 1.0 1.0 300 1.0 4.0/Γ 250.0 2 . × Fig. 2.—
Evolution of the spectral energy distribution due tocurvature effects for the case of the causal pulse in Fig. 1. In thedeclining phase of the pulse, the value of f E pk ∝ E pk is shown bythe red line. Fig. 3.—
Lag vs. Energy for selected parameters except shellthickness η ∆ . Red: η ∆ = 1 .
0; Green: η ∆ = 0 .
5; Blue: η ∆ = 0 . a long duration and a short duration pulse in the co-moving frame respectively, and where the effects of afinite shell are suppressed. The case of the causal pulse( η ∆ = η t = η r = 1) however, shows that emission froma finite shell can produce smooth light curves withoutthe need for a slowly dimming co-moving emission pro-file. Fig. 2 shows the evolution of the spectral energydistribution for the case of the curvature pulse shown inFig. 1. The ν F ν peak flux f E pk ∝ E pk equality line isshown in the decay portion of the pulse. Dermer (2004)shows that this equality holds in the declining phase forall pulses (a similar result has been derived by Qin etal. 2009). We also note that the presence of a finite shellaffects the low energy and high energy fluxes equally andtherefore does not effect the shape of the spectrum as afunction of time as first noted by Qin et. al. 2002 inthe context of emission from a fireball surface. Fig. 3 Fig. 4.—
Lag vs. Energy due to different rest-frame spectralfunctions with selected parameters except E pk, = 200 keV. Here,Red: Broken-power-law with α = − / β = − .
5, Green: Bandfunction with α = − . β = − .
25 and Blue: Comptonized-E pk with α = − . Fig. 5.—
Pulse FWHM vs. Energy for different shell-collisionparameters η ∆ and η t for selected parameters except E pk = 200keV, a Band spectrum with α = − . β = − .
25 and t var =1 . η ∆ = 0 .
001 and η t = 0 .
001 Red: Curvature pulsewith η ∆ = 0 . η t = 0 .
1; Green: Thin-shell pulse with η ∆ = 0 . η t = 1; and Blue: Causal pulse with η ∆ = 1 and η t = 1. Alsoshown is the exponent of the power-law that best fits the modelpoints for the thin-shell and Causal pulses. shows the spectral lag as a function of energy for a casewith selected parameters but with varying shell thick-ness parameter, η ∆ . Fig. 4 shows the lag as a functionof energy for a case with selected parameters but withdifferent rest-frame spectral functions (BPL, Band, andComptonized–E pk ). Fig. 5 shows the pulse Full-width at Shenoy et al. Fig. 6.—
KRL pulse-fit for the light curve for GRB 110920 withresiduals.
Fig. 7.—
Light curve segments for 110920 with equal fluences. half-maximum (FWHM) as a function of energy for thevarious profiles shown in Fig. 1. For the purpose of com-parison we have used a Band function with α = − . β = − .
25, identical to Qin et. al. 2005. These authorshave shown that the Doppler effect of a relativisticallyexpanding fireball could lead to a power-law trend forthe pulse width as a function of energy within a cer-tain energy range. By taking a sizable sample of BATSEGRBs, they demonstrated that the pulse widths exhibita plateau/power-law/plateau feature as a function of en-ergy. They also note that the power-law index dependsstrongly on E pk and the rest-frame radiation spectrum.The plateau/power-law/plateau feature reported by Qinet al. 2005 is well reproduced here. Furthermore, we notethat the power-law exponent is sensitive to the assumedthickness of the shell. SAMPLE SELECTION AND METHODOLOGY.
As a first step we analyze either single-pulse GRBs,or GRBs with relatively simple light curves where theindividual pulses within a multi-pulse structure in thelight curve can be distinguished. In addition, we requirethat the GRB pulses be bright enough and of sufficientduration (the duration of the pulse is particularlyrelevant to tests of the peak-flux – peak-frequencyrelation) so that we may obtain reliable results from our
Fig. 8.—
Comparable pulses generated using the BPL (E peak =300 keV, α = − , β = − . peak = 334 keV, α = − . β = − . pk (E peak = 280 . α = − . Fig. 9.— νF ν
Flux vs. E pk for the data from GRB 110920. Thedata were fit with the best-fit Comptonized–E pk function in therange 100-985 keV. analyses.After identifying a potential candidate GRB, wepulse-fit the GRB light curves using a suitable pulsefunction (such as the Kocevski-Ryde-Liang (KRL) pulsefunction: see Kocevski Ryde & Liang 2003 or the Norrispulse function: see Norris et al. 2005) in multiple energybands. In order to support the supposition that a givenstrong pulse (obtained from a suitable pulse-fit) is notmade from overlapping multiple pulses (within statis-tics), we perform a wavelet based minimum-variabilityrobing Curvature Effects in The Fermi GRB 110920 5 Fig. 10.— νF ν
Flux vs. E pk from the model for the three rest-frame spectral functions described in the text with Blue: BPL;slope = 3.11 +/- 0.04, Pink: Band; slope = 3.39 +/- 0.10 andBlack: Comptonized–E pk ; slope = 2.57 +/- 0.01, for time segmentsidentical to those used for the data. The flux scale has been offsetfor better viewing. Fig. 11.—
Lag vs. Energy from the model for the pulses shownin Fig. 7 with Red: Data, Solid blue squares: Comptonized–E pk ,Hollow black squares: Band function and Pink crosses: BPL. TheBand 2 energies are the mid-point of the energy in the second band. time-scale (MTS) extraction. In essence, the MTSis a measure of the smallest temporal structure ina lightcurve. The full details of its extraction, andthe technique in general, are given in MacLachlan etal. (2013). The correlation between MTS and pulseproperties such as rise times and widths is discussed inMacLachlan et al. (2012). The best-fit pulse profile isthen used as a representation of the light curve from thedata. Fig. 12.—
Pulse FWHM vs. Energy for the data and the modelfor GRB 110920 for identical energy bands. Also shown is theexponent of the power-law that best fits the model points.
The time-integrated spectrum of the GRB is fit witha suitable function (Band, Comptonized–E pk etc.). Thebest-fit spectral function is used as the rest-frame emis-sion spectrum in the model. The model parameters arethen varied to generate a light curve that best matchesthe best-fit pulse profile. The light curve is subdividedinto time segments with equal, and sufficiently highbackground-subtracted fluence in order to minimize theeffects of varying signal-to-noise and an E pk is extractedvia a spectral fit for each time segment. The ν F ν flux isextracted at E pk in a range spanned by the E pk -error.The model light curve is treated in an identical fashionas the data with regard to segments. Model fluxes andE pk ’s are extracted and the peak-flux – peak-frequencyrelation is tested. In addition, we extract spectral lagsin suitable, identical energy bands from the data andthe model and compare the predicted and the observedspectral-lag-energy evolution. The spectral lags areextracted using the cross-correlation-function analysismethod as described in Ukwatta et. al (2010). GRB 110920 - A Test Case
The Fermi GRB 110920 is a single-pulse burst witha relatively long fast-rise, exponential-decay structurewith a T of 170 ±
17 seconds. The best fit Bandparameters (see McGlynn et al. 2012 for a detaileddiscussion on the properties of this GRB) for the timeinterval [ T + 0 . , T + 52 . T is the triggertime) were α = − . ± . β = − . +0 . − . and E peak = 334 ± E peak = 978 +154 − keV and the temperature of theblackbody was found to be kT = 61 . +0 . − . keV. Thelow energy index α became ( − . ± . ∼ pk function (in the range 100-985keV) and these, along with a theoretical BPL function(originally used by Dermer 2004) were used as rest-framespectra for the model. We varied the shell-collisionparameters and extracted best-fit pulses using thesethree spectral functions (see Fig. 8 for the pulses aswell as details of the corresponding spectral parame-ters). The resulting best-fit shell-collision parameters,together with a best-fit value of t var = 44 seconds andthe chosen values of Γ and z, yielded a radius, r of1.5 × cm, a shell thickness, ∆ r ′ of 8.8 × cmand a co-moving frame pulse-duration, ∆ t ′ of 2.0 × seconds for the Comptonized–E pk spectral function. Thecorresponding values for the Band and BPL functionswere very similar. Figs. 9 and 10 show our results forthe peak-flux–peak-frequency relation for the data withthe best-fit Comptonized-E pk function and the modelusing the three different rest-frame spectra. Fig. 11shows the spectral lags for the light curves of Fig. 8.In order to explore the evolution of the pulse widthwith energy, we extracted the FWHM and plotted thisas a function of energy. The plot is shown in Fig. 12.The energy bands chosen (in keV) were 8–25, 25–50,50–100, 100–150, 150–200, 200–250, 250–350, 350–985.These energy bands were so chosen as to ensure a suffi-cient number of counts in each energy band for a reliablemeasurement of pulse FWHM while also providing a suf-ficient number and spacing of bands to show the trendcurve. As the KRL function does not fit the pulses below150 keV accurately, we employed a Monte-Carlo simula-tion where 1000 light curves were simulated in each en-ergy band using the square-root of the counts as theirerrors (assuming independent Poisson distributions forthe counts), the pulse FWHM was extracted for eachlight curve, and the mean and standard deviation of the1000 pulse FWHMs were used as the pulse FWHM andits error respectively for each band. We fit a power-lawof the form C E a to the model points and extracted anexponent of -0.31 +/- 0.03. DISCUSSION
Before we turn to the test-case GRB, we note that inthe model calculation for a broken-power-law rest-framespectrum, the lag shows a relatively well-defined trendwith no lags at energies below ∼ pk, and a constantlag for all energies above E pk, (Fig. 4). Increasing ordecreasing the value of E pk, , while keeping all othermodel parameters fixed, only shifts the entire curve inthe direction of increase or decrease. Shen et al. (2005)studied the lags due to curvature effects using differentrest-frame emission profiles. They found that for aninfinitesimal shell, a rectangular profile produces nolags. We find the situation to be quite different for thecase of a finite shell thickness. In addition, as depictedin Fig 4, a change in the rest-frame spectrum also hasa significant effect in the evolution of the spectral lagwith energy. In the case of a Band or Comptonized–E pk function, the lags are small for energies small comparedto E pk, , and the lags for a Comptonized–E pk functiondo not show a saturation energy. This is primarilybecause the Comptonized–E pk function varies monoton-ically at all energies and does not have a well-definedbreak energy. The value of r obtained for the testcase is consistent with estimates for an internal shockmodel (see for e.g. Hascoet et al. 2012). While thevalue of ∆ r ′ ( ∼ cm) seems reasonable, it is difficultto infer the significance of its absolute magnitude inthe context of the current analysis. The inclusion of afinite-shell-thickness component in our curvature modelalso produces relatively large lags ( ∼ a few seconds)without a need for extreme physical parameters suchas Γ <
50 (as concluded by Shen et al. 2005), or alarge local pulse width ( ∼ seconds as concluded byLu et al. 2006). We find that a rest-frame pulse dura-tion, ∆ t ′ ∼ seconds is sufficient to produce such lags.Our results for the test case show that an internalshock model with a rest-frame spectrum identical tothat used to fit the data (the Comptonized–E pk andBand functions), reproduces the observed pulse profileas well as the observed spectral lags. It was difficult todetermine if there was a saturation energy present in thelag-energy-evolution (Fig. 11) as there were insufficientcounts to extract lags in higher energy bands. As notedabove, a finite shell thickness can account for observedlags even for the case of a rectangular rest-frame emis-sion profile. A second Band function component with apeak shifted to 1 MeV (when a blackbody component isincluded in the fit) would imply a 1 MeV break energyin the lag-energy plot, and would also predict no lags(or small lags in the case of the Comptonized–E pk )below ∼
300 keV. This does not match the observations.It can be seen from Fig. 10 that the exponent in thepeak-flux–peak-frequency relation is close to 3 in allcases for the model pulses shown in Fig. 8. This confirmsthe observations of Dermer (2004) that the exponent attimes after the peak of the light curve is close to 3 evenwith different choices of rest-frame spectra. However,this does not match the exponent obtained from thedata (Fig. 9). While a connection may exist betweenthe observed spectral evolution and the spectral lags, itappears that curvature effects alone cannot describe thisconnection and additional emission mechanisms may berobing Curvature Effects in The Fermi GRB 110920 7needed. We note in passing that Guirec et al. (2012)have included a blackbody component to describe thetemporal and spectral properties of a number of GRBs.We have also explored the behavior of the pulse widthwith energy. As shown in Fig. 12, the pulse-widthdecreases with energy approximately as a power law.Both the data and the model predictions exhibit similartrends although the low-energy agreement is marginal.The exponent of the power law (-0.31 +/- 0.03) matcheswell the exponent extracted by Fenimore et. al. (1995),who analyzed a large sample of bright BATSE burstsand obtained an average power-law exponent of about-0.4. A similar result was also obtained by Peng et al.2006 who analyzed a sizeable sample of bright singlepulses in BATSE GRBs. In addition, in a recent workbased on an analysis of 51 long-duration FRED-likesingle-pulses from the BATSE data, Peng et. al. 2012showed that the curvature effect combined with a Bandrest-frame spectrum can explain the energy dependenceof the pulse widths. Cohen et al. (1997) have suggestedthat such an exponent is consistent with a populationof electrons losing energy via synchrotron radiation, aprocess for which the exponent is predicted to be -0.5. CONCLUSIONS
We have used a simple two-shell collision model toinvestigate curvature effects in the prompt emission ofGRBs. We have examined the effects of emission spectrasuch as the Band, the Comptonized–E pk , and the BPLfunctions. We have focused primarily on the peak-flux –peak-frequency relation and the evolution of the spectrallags and the pulse widths with energy. We compare ourmodel results with the results of similar models in theliterature and also present a test case study of GRB110920.We summarize our main findings as follows: • We find that introduction of a finite shell thicknessin the curvature formulation can produce smoothlight curves, i.e., without a rapid transition fromthe rise to the decay portions even for the case ofa rectangular rest-frame emission profile. As weapproach the infinitesimal shell (surface) approxi-mation, we recover the sharp featured light-curveprofile of Qin et. al. 2004 for a rapidly dimmingintrinsic emission profile. • While we agree with Shen et al. (2005) that aninfinitesimal shell produces no discernible spectrallag using a rectangular emission-pulse-profile, wefind the situation to be different for a shell of finitethickness i.e., a finite spectral lag can be producedeven with a rectangular pulse profile; • The spectral lag evolution as a function of energy isquite sensitive to the type of rest-frame spectrum.For example, the Comptonized–E pk model does notappear to exhibit a saturation energy at which thespectral lags reach a plateau phase as in the case ofthe Band and the broken-power-law functions. Weagree with Shen at al (2005) that the spectral lagsseem to approach a maximum when E pk is near the high-energy channel used in extracting the lag.Most likely this simply reflects the break energypresent in the assumed rest-frame spectrum (i.e.,Band and BPL); • All rest-frame spectral models tested exhibit thepeak-flux – peak-frequency relation although withexponents that differ from the predicted exponentof 3. The significance of this discrepancy is notclear at this stage and warrants further investiga-tion; • The peak-flux – peak-frequency test for GRB110920 yields an exponent of 1.64 +/- 0.012 com-pared to the theoretical one of 2.57 +/- 0.01 (withComptonized–E pk as the rest-frame spectrum). Weconsider this discrepancy to be significant andthe result to be in disagreement with the predic-tion based purely on effects of curvature. Similarconclusions were reached by Dermer (2004), Qin(2009) and Borgonovo and Ryde (2001); • Both the data (test GRB) and the model exhibita very similar power-law trend for the pulse widthwith energy. The plateau/power-law/plateau fea-ture noted by Qin et al. 2005 is well reproducedwith a given choice of key model parameters. Inaddition, we note that the power-law exponent issensitive to the assumed shell thickness. For thetest-case GRB, the power-law exponent matcheswell with exponents extracted from a larger sampleof GRBs from earlier studies (Fenimore et al. 1995;Peng et. al. 2006, 2012); and • Relatively good agreement is obtained with all rest-frame spectral models for the spectral lag versus en-ergy for the test GRB. This is somewhat surprisinggiven the result of the peak-flux – peak-frequencytest noted above. It would seem that some complexinterplay is at work between various model param-eters such as shell thickness, variability time scale,the energy evolution of E pk and the Lorentz fac-tor. The investigation of the dependencies of thesevarious parameters is ongoing.The role of the reported soft component of the lightcurve for GRB 110920 has not been fully investigatedin this study and is worth pursuing, particularly withregard to the behavior of the peak-flux–peak-frequencyrelation. Finally, we note that these studies are beingextended to a larger sample of GRBs. ACKNOWLEDGEMENTS