Evolutionary sequences of rotating protoneutron stars

We investigate the evolution of rigidly and differentially rotating protoneutron stars during the first twenty seconds of their life. We solve the equations describing stationary axisymmetric configurations in general relativity coupled to a finite temperature, relativistic equation of state, to obtain a sequence of quasi-equilibrium configurations describing the evolution of newly born neutron stars. The initial rotation profiles have been taken to mimic the situation found immediately following the gravitational collapse of rotating stellar cores. By analyzing the output of several models, we estimate that the scale of variation of the angular velocity in a newly born neutron star is of the order of 7-10 km. We obtain the maximum rotation frequency that can be reached as the protoneutron stars deleptonizes and cools down, as well as other relevant parameters such as total angular momentum or the instability parameter |T/W|. Our study shows that imposing physical constraints (conservation of baryonic mass and angular momentum) and choosing reasonable thermodynamical profiles as the star evolves gives results consistent with the energetics of more complex simulations of non-rotating protoneutron stars. It appears to be unlikely that newly born protoneutron stars formed in nearly axisymmetric core collapses reach the critical angular velocity to undergo the bar mode instability. They could, however, undergo secular or low |T/W| rotational instabilities a few seconds after birth, resulting in a strong emission of gravitational waves retarded with respect to the neutrino luminosity peak. We also found that the geometry of strongly differentially rotating protoneutron stars can become toroidal-like for large values of the angular velocity, before reaching the mass shedding limit.