Measuring the properties of $f-$mode oscillations of a protoneutron star by third generation gravitational-wave detectors

Core-collapse supernovae are among the astrophysical sources of gravitational waves that could be detected by third-generation gravitational-wave detectors. Here, we analyze the gravitational-wave strain signals from two- and three-dimensional simulations of core-collapse supernovae generated using the code F{\sc{ornax}}. A subset of the two-dimensional simulations has non-zero core rotation at the core bounce. A dominant source of time changing quadrupole moment is the $l=2$ fundamental mode ($f-$ mode) oscillation of the proto-neutron star. From the time-frequency spectrogram of the gravitational-wave strain we see that, starting $\sim 400$ ms after the core bounce, most of the power lies within a narrow track that represents the frequency evolution of the $f-$mode oscillations. The $f-$mode frequencies obtained from linear perturbation analysis of the angle-averaged profile of the protoneutron star corroborate what we observe in the spectrograms of the gravitational-wave signal. We explore the measurability of the $f-$mode frequency evolution of protoneutron star for a supernova signal observed in the third-generation gravitational-wave detectors. Measurement of the frequency evolution can reveal information about the masses, radii, and densities of the proto-neutron stars. We find that if the third generation detectors observe a supernova within 10 kpc, we can measure these frequencies to within $\sim$90\% accuracy. We can also measure the energy emitted in the fundamental $f-$mode using the spectrogram data of the strain signal. We find that the energy in the $f-$mode can be measured to within 20\% error for signals observed by Cosmic Explorer using simulations with successful explosion, assuming source distances within 10 kpc.