Rotational effects on the oscillation frequencies of newly born proto-neutron stars

In this paper we study the effects of rotation on the frequencies of the quasi-normal modes of a proto-neutron star (PNS) born in a gravitational collapse during the first minute of life. Our analysis starts a few tenths of a second after the PNS formation, when the stellar evolution can be described by a sequence of equilibrium configurations. We use the evolutionary models which describe how a non-rotating star cools down and contracts while neutrino diffusion and thermalization processes dominate the stellar dynamics. For assigned values of the evolution time, we set the star into slow rotation and integrate the equations of stellar perturbations in the Cowling approximation, both in the time domain and in the frequency domain, to find the quasi-normal mode frequencies. We study the secular instability of the g modes, which are present in the oscillation spectrum due to the intense entropy and composition gradients that develop in the stellar interior, and we provide an estimate of the growth time of the unstable modes based on a post-Newtonian formula.