Exploring Physically Motivated Models to Fit Gamma-Ray Burst Spectra

We explore fitting gamma-ray burst (GRB) spectra with three physically motivated models, and thus revisit the viability of synchrotron radiation as the primary source of GRB prompt emission. We pick a sample of 100 bright GRBs observed by the Fermi Gamma-ray Burst Monitor (GBM), based on their energy flux values. In addition to the standard empirical spectral models used in previous GBM spectroscopy catalogs, we also consider three physically motivated models; (a) a thermal synchrotron model, (b) a Band model with a high-energy cutoff, and (c) a smoothly broken power-law (SBPL) model with a multiplicative broken power law (MBPL). We then adopt the Bayesian information criterion to compare the fits obtained and choose the best model. We find that 42% of the GRBs from the fluence spectra and 23% of GRBs from the peak-flux spectra have one of the three physically motivated models as their preferred one. From the peak-flux spectral fits, we find that the low-energy index distributions from the empirical model fits for long GRBs peak around the synchrotron value of −2/3, while the two low-energy indices from the SBPL+MBPL fits of long GRBs peak close to the −2/3 and −3/2 values expected for a synchrotron spectrum from marginally fast-cooling electrons.


INTRODUCTION
The detailed nature of the radiative process responsible for gamma-ray burst (GRB) prompt emission has not yet been identified.The radiative process expected to dominate the emission is synchrotron radiation, due to the nonthermal appearance of the observed spectra and the likely presence of accelerated electrons and intense magnetic fields (Katz 1994;Rees & Meszaros 1994;Tavani 1996), but the inconsistency between the observed spectral shape at low energies and predictions from the synchrotron theory represents a challenge for this interpretation.In case of fast-cooling synchrotron radiation, the part of the spectrum immediately below the νF ν peak energy should display a power-law behavior with a slope ∼ −3/2, which breaks to a harder spectral shape (slope ∼ −2/3) at lower energies (Sari et al. 1998).The prompt emission spectra of GRBs are usually fit with empirical functions, such as the Band function (Band et al. 1993), which consists of two smoothly connected power laws N(E) ∝ E α and N(E) ∝ E β , describing the photon spectrum at low and high energies, respectively.The typical slope of the low-energy power law is α ∼ −1 (Gruber et al. 2014;Goldstein et al. 2012).This is higher than the value expected in the case of fast-cooling synchrotron radiation.
Recent works have shown that a number of GRBs have an additional spectral break between ∼1 and a few hundred keV (Oganesyan et al. 2017;Ravasio et al. 2019;Toffano et al. 2021) and the slopes of the power-law below and above this break are consistent with the values expected from synchrotron emission.It has been suggested that the value of α is an average value between the two power-law segments below and above the break energy.The low-energy power-law index is of particular interest for determining the emission mechanism that converts the kinetic or/and magnetic energy of the bulk relativistic outflow into radiation.Under the assumption that the emission is dominated by synchrotron radiation, the low-energy photon index should be no harder than −3/2 in the case of non-adiabatic cooling, and no harder than −2/3 in the case of adiabatic cooling (Rybicki & Lightman 1979;Katz 1994).
In this work we explore three physically-motivated models to fit GRB Spectra, and thus revisit the viability of synchrotron radiation as the source of GRB prompt emission.We pick a sample of 100 bright GRBs observed by GBM, based on their energy flux values.In addition to the empirical models used in the latest Spectral Catalog We then adopt the Bayesian information criterion (Neath & Cavanaugh 2012) to compare the fits obtained and choose the best one.

Instrument and Sample Selection
Fermi GBM consists of 14 detector modules: 12 Sodium Iodide (NaI) detectors, covering the energies 8 -1000 keV, and two Bismuth Germanate (BGO) detectors, covering 200 keV to 40 MeV (Meegan et al. 2009).Data selection is identical to that as described in previous spectroscopy catalogs (Gruber et al. 2014;Poolakkil et al. 2021).In brief, up to three NaI detectors with observing angles to the source less than 60 • are selected, along with the BGO detector that has the smallest observing angle of the burst.For each of these, standard energy ranges that avoid unmodeled effects, such as an electronic roll-off at low energies and high-energy overflow bins are selected.Each data set is binned according to whether the burst is long (1.024 s binning) or short (0.064 s binning), with a dividing line at T 90 = 2 s (Kouveliotou et al. 1993), where T 90 is the time between the 5% and 95% values of the total fluence.Next, a background model (polynomial in time) is chosen to fit regions of the light curve that bracket the emission interval.
The analyses presented herein are comprised of two spectra for each burst: a 'fluence' (F ) spectrum that represents the entire duration of emission and a 'peak flux' (P ) spectrum that depicts the brightest portion of each burst, on a fixed timescale of 1.024 s for long GRBs and 64 ms for short GRBs.The selection of fluence time bins for each of these two classes is made by including every (energy-integrated) time bin that has flux that is at least 3.5 sigma in excess of the background model for that bin.The data is then joint fit with RMfit (version 4.5.3 1 , available at the Fermi Science Support Center), using a set of standard model functions (Section 2.2).For a fit statistic, we have chosen the pgstat statistic (Arnaud et al. 2011), which properly accounts for the Gaussian uncertainties in the background, arising from the temporal fit.

Models
We chose seven spectral models to fit the spectra of GRBs in our sample.In addition to the empirical spectral functions used in the latest spectroscopy catalog, we consider three additional models that are physically motivated (Bauke 2007;Arnold 2015).We also consider the COMP and BAND models with a Black Body component added to them.All models are formulated in units of photon flux with energy (E ), temperature (kT ) in keV and multiplied by a normalization constant A (photon s −1 cm −2 keV −1 ).Note that for the SBPL+MBPL model, the two low-energy indices before and after the MBPL break energy will be referred to as α 1 and α 2 respectively, as highlighted in Figure 1(c).The pivot energy (E piv ) normalizes the model to the energy range under consideration and helps reduce cross-correlation of other parameters.In all cases, E piv is held fixed at 100 keV.
where: a = m∆ ln e q + e −q 2 , a piv = m∆ ln e q piv +e −q piv 2 , • MBPL: A Multiplicative Broken Power Law, which can be used as a low-energy or high-energy cutoff by fixing λ h or λ l at zero, respectively • Black Body: A Black Body spectrum with normalization (A) and temperature (kT ) • High-energy Cutoff: A multiplicative component with a cutoff energy (E cut ) and folding energy (E F ) • Thermal Synchrotron: For this model, we assume an electron distribution that consists of electrons in a thermal pool, given by Here n 0 normalizes the distribution to total number or energy, γ is the electron Lorentz factor in the fluid frame and γ th is the thermal electron Lorentz factor.We convolve this simplified distribution with the standard isotropic synchrotron kernel (Rybicki & Lightman 1979) where expresses the single-particle synchrotron emissivity (i.e., energy per unit time per unit volume) in dimensionless functional form.

PGSTAT and BIC
In previous works we have used a variant of the Cash-statistic likelihood (Cash 1979), called C-Stat in RMfit and pstat in Xspec (Arnaud et al. 2011), which assumes the background model uncertainty to be negligible.For this work, we have chosen to use pgstat, which correctly accounts for the Gaussian uncertainties in the temporally-interpolated background model, while retaining the Poisson statistics of the source counts.
We then adopt the Bayesian information criterion (BIC) to compare the fits obtained and choose the best one.The BIC is a well-known general approach to model selection that favors more parsimonious models over more complex models, i.e. it adds a penalty based on the number of parameters being estimated in the model.One form for calculating the BIC is given by: where L is the maximized value of the likelihood function of the model M k , k is the number of free parameters in the model and n is the number of data points.The model with the lowest BIC is considered the best.We can also calculate the ∆ BIC; the difference between a particular model and the 'best' model with the lowest BIC, and use it as an argument against the other model.Applying these criteria, the number of bursts that classify as 'best' for each model can be seen in Table 2.We find that 73% of Short GRBs in the fluence spectra and 84% of Short GRBs in the peak-flux spectra have COMP as the preferred model.The low-energy index distributions from the empirical model fits for long GRBs are shown in Figure 2. Table 1 shows that out of the 100 GRBs used in this sample, 42 in the fluence spectra and 23 in the peak-flux spectra have one of the three physically motivated models as their preferred one.A detailed list of the BIC values for all bursts, for both the fluence and peak-flux spectra can be found in Table 4 and Table 5 respectively.The distribution of spectral parameters help us place each burst in relation to the ensemble of all bursts.The time-integrated spectral distributions depict the overall emission properties and any spectral evolution is averaged out.For peak spectra however, the time intervals are reasonably short for any significant change of the spectral parameters.
A single electron emitting synchrotron radiation will have a spectra with slope -2/3 (photon index) and an exponential cutoff at high energies.The spectrum of a population of radiating electrons in the optically thin regime is just a superposition of individual spectra.For this reason the -2/3 value is also a hard upper limit on the overall spectral slope, also referred to as the synchrotron "line of death" (Preece et al. 1998).For a population of shock accelerated electrons, the part of the spectrum immediately below the νF ν peak energy will display a power-law behavior with a slope ∼ −3/2, which breaks to a harder spectral shape (slope ∼ −2/3, following the single electron spectrum) at lower energies.This is in the so called fast-cooling regime, where random Lorentz factor of an electron that cools on the dynamic timescale, is lower than the typical injected electron's energy (Sari et al. 1998).
Figure 2 shows the distribution of low-energy indices of long GRBs (Peak-flux spectra) from the four empirical models used in this work.They have a combined median value of ∼ −0.69 +0.20  −0.22 , aligning with the −2/3 expectation.The distribution of the two low-energy indices from the SBPL+MBPL fits for peak-flux spectra of long GRBs are shown  in Figure 3; the two distributions have median values of −0.72 +0.20 −0.21 (α 1 ) and −1.32 +0.33 −0.24 (α 2 ) respectively (Table 3).These values are in line with the -2/3 and -3/2 expected indices for synchrotron spectrum from fast-cooling electrons.This result lends strong support for the case of synchrotron radiation as the primary emission mechanism for GRB prompt emission.
It is evident from Figure 2, 3 that a non-negligible fraction of GRBs have a low-energy index value steeper than -2/3, which presents a challenge for the synchrotron theory.One possibility in such cases would be the appearance of a quasi-thermal component of photospheric origin (Mészáros & Rees 2000;Daigne & Mochkovitch 2002).The emission from both the photosphere and internal shocks have a similar duration, the latter having only a very short lag behind the first.The intensity of the photospheric emission depends strongly on the unknown mechanism responsible for the acceleration of the relativistic outflow (Daigne et al. 2011).Another possibility to consider is the effect of electron pitch angle scattering (Lloyd & Petrosian 2000); where a small pitch angle leads to a harder low-energy index value (−2/3 ≲ α ≲ 0) and a large pitch angle can accommodate bursts below the "line of death" (α ≲ −2/3).Spectral indices harder than -3/2 can be obtained by more elaborate models.For example, Derishev (2007); Zhao et al. (2014) showed that fast cooling electrons in decaying magnetic fields can lead to such hard spectra.
Figure 4 shows the distribution of the two break energies from the SBPL+MBPL model.Interestingly, we find a subgroup of GRBs that have a low-energy break between 30-40 keV in both the fluence and peak-flux spectral fits.A similar low-energy break was found in GRB 221009A (Lesage et al. 2023) and will be investigated further in a future The goal of this work is to explore the the use of physically-motivated models to fit GRB spectra and directly compare them to the empirical models used in previous GBM spectroscopy catalogs (Goldstein et al. 2012;Gruber et al. 2014).The spectral properties presented here are from time-integrated and peak flux analysis, produced using seven photon models; four empirical and three physically-motivated.We use a sample of 100 GRBs, that were selected based on their energy flux values (Poolakkil et al. 2021).We have used pgstat as our fitting statistic; which accounts for the Gaussian uncertainties in the background model, while retaining the Poisson statistics of the source counts, and then use the Bayesian information criterion (BIC) to compare fits and choose the best model.
We found that 42% of the GRBs from the fluence spectra and 23% of GRBs from the peak-flux spectra had one of the three physically-motivated models as their preferred one.The low-energy index distributions from the empirical model fits for long GRBs (peak-flux spectra), highlighted in Figure 2, peak around the synchrotron value of -2/3, while the two low-energy indices from the SBPL+MBPL fits of long GRBs (peak-flux spectra, Figure 3) peak close to the -2/3 and -3/2 values expected for synchrotron spectrum below and above the cooling frequency.These results present a strong case for synchrotron radiation as a leading mechanism for the origin of prompt emission from GRBs and further encourage the transition from empirical models towards physically-motivated models to fit GRB spectra.
The UAH co-authors gratefully acknowledge NASA funding from co-operative agreement 80MSFC22M0004.

Figure 1 .
Figure 1.νFν (photons keV cm −2 s −1 ) spectrum for TS (left), Band+H.E.Cutoff (middle) and SBPL+MBPL (right) fits in RMfit for GRB 110921912.The pink/blue data points represent data from the NaI/BGO detectors respectively and the red dashed line represents the spectral shape predicted by the model.

Figure 2 .
Figure 2. Distribution of the low-energy indices from COMP, BAND, COMP+BB and BAND+BB fits for long GRBs (Peakflux Spectra).Gaussian functions showing the central value and standard deviation of the distributions are overlapped to the histograms (colour-coded dashed curves).The black dashed-line represents the synchrotron "line-of-death" value of -2/3.

Figure 3 .
Figure 3. Distribution of the two low-energy indices from the SBPL+MBPL fits for long GRBs (Peak-flux Spectra).Gaussian functions showing the central value and standard deviation of the distributions are overlapped to the histograms (dashed curves).

Figure 4 .
Figure 4. Distribution of the two break energies from the fluence (a) and peak-flux (b) spectral fits for SBPL+MBPL.

Table 1 .
Best GRB Models based on BIC

Table 2 .
BIC Comparison (Number of GRBs for which the given model has the lowest value of BIC)

Table 3 .
The median parameter values and the 68% CL for all long and short GRBs

Table 4 .
BIC Comparison (Fluence Spectra).The best model is indicated with an *

Table 4
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Table 4
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Table 5 .
BIC Comparison (Peak-flux Spectra).The best model is indicated with an *

Table 5
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