Gamma-ray burst spectral-luminosity correlations in the synchrotron scenario

For over two decades, gamma-ray burst (GRB) prompt emission spectra were modelled with smoothly-broken power laws (Band function), and a positive and tight correlation between the spectral rest-frame peak energy $E_p$ and the total isotropic-equivalent luminosity $L_{iso}$ was found, constituting the so-called Yonetoku relation. However, more recent studies show that many prompt emission spectra are well described by the synchrotron radiation model, hence significantly deviating from the Band function. In this work, we test the impact of a more suited spectral model such as an idealized synchrotron spectrum from non-thermal electrons on the Yonetoku relation and its connection with physical parameters. We select GRBs with measured redshift observed by Fermi/GBM together with high energy observations (>30 MeV), and perform spectral analysis dividing them in two samples: the single-bin sample, using the light curve peak spectrum of each GRB, and the multiple-bins sample, where we explore the whole duration of 13 bright bursts with time-resolved spectral analysis. We observed that the $E_p$ of synchrotron spectra in fast-cooling regime ($\nu_m/\nu_c\gg1$) is generally larger than the one provided by the Band function. For this reason, we do not find any $E_p-L_{iso}$ correlation in our samples except for the GRBs in an intermediate-cooling regime ($1<\nu_m/\nu_c<3$), namely where peak and break energies are very close. We instead find in both our samples a new tight correlation between the rest-frame cooling frequency $\nu_{c,z}$ and $L_{iso}$: $\nu_{c,z} \propto L_{iso}^{(0.53 \pm 0.06)}$. These results suggest that, assuming that prompt emission spectra are produced by synchrotron radiation, the physical relation is between $\nu_{c,z}$ and $L_{iso}$. The fit of the Band function to an intrinsic synchrotron spectrum returns peak energy values $E_{p,z}^{Band} \sim \nu_{c,z}$.