Modification of the Universal Relation between Mass, Radius, and Non-Radial f-Mode Oscillation in Proto-Neutron Stars

Proto-neutron stars exhibit both high density and relatively high temperatures. This study investigates the impact of temperature on the equation of state (EoS) for proto-neutron stars, which is critical to understanding their structure and behavior. We derive an EoS using a relativistic mean field model with density-dependent coupling parameters. This temperature-dependent EoS notably influences the mass-radius relationship and, consequently, the f-mode non-radial oscillation frequency. Increased temperature stiffens the EoS at low and intermediate densities, leading to less compact stars and a flatter mass-radius curve. For neutron stars with lower and intermediate mass, the f-mode frequency of the non-radial oscillation decreases with rising temperature, which could facilitate detection. Furthermore, the universal relation that connects f-mode non-radial oscillation frequency, mass, and radius becomes non-linearly dependent on temperature. The parameters of this relation, expressed as ωM = a(T) (M/R) + b(T), vary with temperature, with the coefficients following a parabolic dependence on temperature.