Prospect of unraveling the first-order phase transition in neutron stars with f and p₁ modes

Quasi-normal modes of NSs are an exciting prospect for analyzing the internal composition of NSs and studying matter at high densities. In this work, we focus on studying the f- and p- quadrupolar oscillation modes, which couple with gravitational waves. We construct two different EOS ensembles, one without and one with a first-order phase transition, and examine how f- and p-modes might help us differentiate them. We find ensemble specific exclusion regions in the $65%$ and $95%$ confidence contours of the frequency-damping time relations. The exclusion regions become more prominent for the higher-order oscillation modes. However, these modes have higher frequencies, which are beyond the detection capabilities of present gravitational wave detectors. The quasi-universal relations of dimensionless quantities prove to be ineffective in differentiating the EOS ensembles, as they obscure the details of the EOS.