Modification of the universal relation between mass, radius and nonradial f-mode oscillation in proto-neutron stars

Neutron stars are usually assumed to be cold; however, in certain dynamical astrophysical scenarios such as newly born neutron stars or binary star mergers, the temperature effects play a non-negligible role. We systematically derive the equation of state at finite-temperature within a relativistic mean-field hadronic model applicable to such proto-neutron stars. The equation of state so derived considerably affects the mass-radius curve, thereby affecting the nonradial quadruple $f$-mode oscillation frequencies.} Temperature effectively makes the equation of state stiffer at relatively low and intermediate densities, thereby making the star less compact and flattening the mass-radius curve. The $f$-mode frequency for low and intermediate-mass neutron stars decreases with temperature and thus should be easier to detect. The universal relation (connecting $f$-mode frequency, mass, and radius) changes nonlinearly with temperature. The parameters defining the universal relation [$\omega M = a(T) \left(\frac{M}{R}\right) + b(T)$] becomes temperature dependent with the coefficients following a parabolic relation with temperature.