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We examine the frequencies of the fundamental ($f$-) and the gravity ($g_i$-)mode in gravitational waves from accreting protoneutron stars (PNSs) towardblack hole formation. For this purpose, we analyze numerical results ofgravitational collapse of massive stars for two different progenitors withthree equations of state. We adopt profiles of central objects obtained fromthe numerical simulations by solving general relativistic neutrino-radiationhydrodynamics under the spherical symmetry. Using a series of snapshots as astatic configuration at each time step, we solve the eigenvalue problem todetermine the specific frequencies of gravitational waves from the evolvingPNSs with accretion from core bounce to black hole formation by therelativistic Cowling approximation. We find that the frequency of the $f$-modegravitational waves can be expressed as a function of the average density ofthe PNS at each time step almost independently of the progenitor models inaccord with the previous studies on ordinary PNSs. We also find that the ratioof the $g_1$-mode frequency to the $f$-mode frequency is insensitive to theprogenitor, but strongly depends on the compactness of the PNSs. Having asimultaneous observation of the $f$- and $g_1$-mode gravitational waves, itwould provide the information of the average density and compactness of PNS,which can reconstruct the evolution of mass and radius of the PNSs as aconstraint on the equation of state at high densities. Furthermore, we showthat the highest average density of the PNS with the maximum mass can beestimated with the detection of the $f$-mode gravitational waves together withthe help of the neutrino observations.

Determination of properties of protoneutron stars toward black hole formation via gravitational wave observations