I-Love-Q for proto-neutron stars

We study some universal relations (or relations that are independent of the equation of state) for proto-neutron stars in the post-bounce phase of core-collapse supernovae. Using results from three-dimensional core-collapse supernova simulations, we extract the equation of state of the system and calculate the structure of spherically symmetric proto-neutron stars. We then solve the equations that describe the gravitational field of perturbed proto-neutron stars: in the slow rotation approximation, we calculate the moment of inertia ($I$) and the spin-induced quadrupole moment ($Q$); in the small tidal deformation approximation, we calculate the tidal deformability ($\Lambda$, which is related to the tidal Love number). There are known universal relations among the I-Love-Q trio for cold neutron stars. We verify that the relations remain universal for proto-neutron stars in their late evolutionary stage and report how they connect to those for cold neutron stars. We also solve the axial gravitational perturbation equation in order to calculate the oscillation frequency and the damping time (or the real and imaginary parts of the complex frequency) of the fundamental axial $w$-mode for proto-neutron stars. We discuss the universality between such oscillation frequencies and I-Love-Q for proto-neutron stars.