High-energy spectral cutoffs in the prompt emission of Fermi gamma-ray bursts: bulk Lorentz factors, emission radii, and a cutoff-peak energy relation

The high-energy end of the gamma-ray burst (GRB) prompt spectrum carries information on the physical conditions of the relativistic outflow, but the number of well-characterized spectral cutoffs is small. We present a systematic search for high-energy spectral cutoffs in the time-integrated prompt spectra of 139 GRBs observed by Fermi between 2008 July and 2025 December, selected to have broadband coverage either through a LAT detection or through bright BGO emission. Joint GBM, LLE, and LAT spectra over $T_{90}$ are fitted with four empirical models and compared using the Bayesian information criterion, yielding 106 bursts with well-constrained cutoff energies $E_c$. An empirical two-component Gaussian mixture model applied to ${\log }_{10}(E_c\mathrm{/}\mathrm{M}\mathrm{e}\mathrm{V})$ separates a low-$E_c$ group from the main distribution at $E_c=2.12$ MeV; the 77 bursts lying firmly above this boundary are interpreted within the internal $\gamma \gamma$ pair-opacity framework. Combining $E_c$ with wavelet-based minimum variability timescales (median $0.119$ s), we infer bulk Lorentz factors of $11.7$ to $2.78\times {10}^3$ (median 145) and emission radii of $\sim {10}^{13}$ to ${10}^{15}\mathrm{c}\mathrm{m}$. Neither the $\mathrm{\Gamma }$-$L_{\mathrm{i}\mathrm{s}\mathrm{o}}$ nor the $\mathrm{\Gamma }$-$E_{\mathrm{i}\mathrm{s}\mathrm{o}}$ relation is statistically significant for the full sample, whereas the 27 bursts with measured redshifts show a moderate $\mathrm{\Gamma }$-$L_{\mathrm{i}\mathrm{s}\mathrm{o}}$ correlation with a slope of $0.36_{-0.23}^{+0.22}$. We further find a strong rest-frame correlation between the cutoff and peak energies, $E_{c,z}\propto E_{p,z}^{2.15_{-0.97}^{+0.96}}$, for the 26 bursts with measured redshifts, which persists after controlling for redshift.