I-Love-Q and quasinormal modes for proto-neutron stars

We investigate the time evolution of some universal relations that are well established for cold neutron stars but not for proto-neutron stars. We use an effective time-dependent equation of state extracted from three-dimensional core-collapse supernova simulations for different progenitors. We then obtain the structure of spherically symmetric proto-neutron stars and study their radial stability as the post-bounce time increases. Next, we solve the equations of non-radial oscillations for proto-neutron stars and calculate their quasinormal modes, more specifically, the oscillation frequency and the damping time of the polar f-mode and the fundamental axial w-mode. Finally, we solve the perturbation equations for slowly-rotating and tidally-deformed proto-neutron stars and obtain their moment of inertia (I), quadrupole moment (Q), and tidal deformability (related to the Love number). There are well-known universal relations between the aforementioned parameters for cold neutron stars, such as I-Love-Q or f-Love and w-Love. We verify that these relations, for different progenitors, approach those for cold neutron stars to within a post-bounce time of about 1 s, and we discuss the reliability of these relations in the context of future observations of, e.g., gravitational waves from core-collapse supernovae.