ν − ν interactions in supernovae environments as a function of density and temperature

Neutrinos in astrophysical environments, such as Supernovae (SNe), may interact through pair interactions. These interactions are density and temperature-dependent. Under certain conditions, a neutrino-condensate may form. In this work, we have adapted the separable nonlocal pairing interactions to the SN conditions and studied the temperature and density dependence of the pairing gap as well as the collective response with respect to these variables. We have applied the Bogoliubov transformations and solved the Bardeen, Cooper and Schrieffer (BCS) equations to construct the free quasi-particle sector of the Hamiltonian. The residual two-quasi-particle terms have been treated by applying the Tamm–Dancoff approximation (TDA) and random-phase approximation (RPA), suited to the extended neutrino media. With this, we have constructed neutrino energy distributions both in the superfluid and normal regimes and extracted critical values of the density and temperature.