Hybrid-star asteroseismology across sharp phase interfaces: Slow and rapid conversion in the $f$- and $p_1$-mode spectra

Abstract

First-order phase transitions in neutron-star interiors can leave signatures in the polar quasi-normal-mode spectrum. We compute the complex quadrupolar $f$- and $p_1$-mode spectra in full general relativity for cold, nonrotating hybrid stars with a sharp hadron--quark interface. We model the equation of state by matching a hadronic SLy4 phase to a constant-sound-speed quark-phase. We compare slow phase conversion, in which fluid elements retain their phase and the interface moves with the fluid, with rapid phase conversion, in which matter changes phase as it crosses the boundary and the interface follows phase equilibrium rather than the fluid, together with an algebraic interpolation between these limits. On identical equilibrium backgrounds, the two physical limits produce a sizable separation in the real part of the $p_1$-mode frequency. This $p_1$ frequency separation is largest when the local tangential-displacement fraction of the eigenfunction changes rapidly across the interface. Across the surveyed EOS parameter space, rapid conversion generally lowers the $p_1$-mode frequency relative to slow conversion, with compactness-averaged separations reaching $\sim3.5\,{\rm kHz}$ on disconnected twin-star branches. The largest separations occur at relatively low transition pressures and large density discontinuities. By contrast, the $f$-mode is only weakly affected by the interface condition for individual stellar models. Across the sampled constant-sound-speed parameter space, departures exceeding the hadronic SLy4 reference residual occur mainly at low transition pressure, large density discontinuity, or both, with different boundaries for the real- and imaginary-part fitting variables. Thus, the $p_1$-mode probes the local interface response, whereas the tracked $f$-mode reflects the global departure of the hybrid sequence from the hadronic calibration.