We aim to obtain a measure of the curvature of time-resolved spectra that can be compared directly to theory. This tests the ability of models such as synchrotron emission to explain the peaks or breaks of GBM prompt emission spectra. We take the bur
st sample from the official Fermi GBM GRB time-resolved spectral catalog. We re-fit all spectra with a measured peak or break energy in the catalog best-fit models in various energy ranges, which cover the curvature around the spectral peak or break, resulting in a total of 1,113 spectra being analysed. We compute the sharpness angles under the peak or break of the triangle constructed under the model fit curves and compare to the values obtained from various representative emission models: blackbody, single-electron synchrotron, synchrotron emission from a Maxwellian or power-law electron distribution. We find that 35% of the time-resolved spectra are inconsistent with the single-electron synchrotron function, and 91% are inconsistent with the Maxwellian synchrotron function. The single temperature, single emission time and location blackbody function is found to be sharper than all the spectra. No general evolutionary trend of the sharpness angle is observed, neither per burst nor for the whole population. It is found that the limiting case, a single temperature Maxwellian synchrotron function, can only contribute up to $58^{+23}_{-18}$% of the peak flux. Our results show that even the sharpest but non-realistic case, the single-electron synchrotron function, cannot explain a large fraction of the observed GRB prompt spectra. Because of the fact that any combination of physically possible synchrotron spectra added together will always further broaden the spectrum, emission mechanisms other than optically thin synchrotron radiation are likely required in a full explanation of the spectral peaks or breaks of the GRB prompt emission phase.
We study the time-resolved spectra of eight GRBs observed by Fermi GBM in its first five years of mission, with 1 keV - 1 MeV fluence $f>1.0times10^{-4}$ erg cm$^{-2}$ and signal-to-noise level $text{S/N}geq10.0$ above 900 keV. We aim to constrain in
detail the spectral properties of GRB prompt emission on a time-resolved basis and to discuss the theoretical implications of the fitting results in the context of various prompt emission models. We perform time-resolved spectral analysis using a variable temporal binning technique according to optimal S/N criteria, resulting in a total of 299 time-resolved spectra. We fit the Band function to all spectra and obtain the distributions for the low-energy power-law index $alpha$, the high-energy power-law index $beta$, the peak energy in the observed $ u F_ u$ spectrum $E_text{p}$, and the difference between the low- and high-energy power-law indices $Delta s=alpha-beta$. Using the distributions of $Delta s$ and $beta$, the electron population index $p$ is found to be consistent with the moderately fast scenario which fast- and slow-cooling scenarios cannot be distinguished. We also apply a physically motivated synchrotron model, which is a triple power-law with constrained power-law indices and a blackbody component, to test for consistency with a synchrotron origin for the prompt emission and obtain the distributions for the two break energies $E_text{b,1}$ and $E_text{b,2}$, the middle segment power-law index $beta$, and the Planck function temperature $kT$. A synchrotron model is found consistent with the majority of time-resolved spectra for these eight energetic Fermi GBM bursts with good high-energy photon statistics, as long as both the cooling and injection break are included and the leftmost spectral slope is lifted either by inclusion of a thermal component or when an evolving magnetic field is accounted for.
