We investigate nonradial $f$-mode oscillations of protoneutron stars in full general relativity, employing equations of state described by the Brueckner-Hartree-Fock theory or the relativistic mean field model, while assuming isentropy and fixed lepton fractions for the internal structure. The validity of various universal relations for cold neutron stars involving $f$-mode characteristics and macroscopic properties of the star is confirmed for those isentropic protoneutron stars (PNSs). Prospects of observations are also discussed. According to simulation results, we then model details of the thermal and trapping profiles in a PNS with the canonical mass. The corresponding $f$-mode frequencies and gravitational-wave strain amplitudes are presented. The validity of the universal relations during the evolution to the formation of a cold neutron star is confirmed.

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• Nuclear matter in neutron stars