Universal relation for supernova gravitational waves

Using the numerical simulation data for two-dimensional core-collapse supernova, we examine the protoneutron star (PNS) asteroseismology with the relativistic Cowling approximation. As shown in the previous study, e.g., [H. Sotani and T. Takiwaki, Mon. Not. R. Astron. Soc. 498, 3503 (2020).], the gravitational wave signals appearing in the numerical simulation can be well identified with the gravity (fundamental) oscillation in the early (later) phase before (after) the avoided crossing between the gravity and fundamental oscillations. On the other hand, the time evolution of supernova gravitational waves strongly depends on the PNS models, such as the progenitor mass and the equation of state for dense matter. Nevertheless, we find that the fundamental and gravity mode frequencies according to the gravitational wave signals appearing in the numerical simulations can be expressed as a function of the protoneutron star average density independently of the PNS models. Using the average density, we derive the empirical formula for supernova gravitational wave frequency. In addition, we confirm that the dependence of the PNS surface density on the PNS average density is almost independent of the PNS models, and we also discuss how the different treatment of the nonuniform matter in the equation of state affects the observables.