Stability of the protoneutron stars toward black hole formation

We examine the protoneutron star (PNS) stability in this study by solving the radial oscillation equations. For this purpose, we adopt the numerical results of massive PNS toward the black hole formation obtained by spherically symmetric numerical simulations for core-collapse supernova with general relativistic neutrino-radiation hydrodynamics. We find that the PNSs are basically stable in their evolution against the radial perturbations, while the PNS finally becomes unstable before the apparent horizon appears inside the PNS. We also examine the gravitational wave frequencies from PNS with the relativistic Cowling approximation. Then, we derive the empirical formula for the $f$-mode frequency, which weakly depends on the PNS models. This kind of universality tells us the PNS property, which is a combination of the PNS mass and radius in this study, once one would observe the $f$-mode gravitational waves.