Discontinuity gravity modes in hybrid stars: assessing the role of rapid and slow phase conversions

Discontinuity gravity modes may arise in perturbed quark-hadron hybrid stars when a sharp density jump exists in the stellar interior and are a potential fingerprint to infer the existence of quark matter cores in compact objects. When a hybrid star is perturbed, conversion reactions may occur at the quark-hadron interface and may have a key role in global stellar properties such as the dynamic stability and the quasi-normal mode spectrum. In this work we study the role of the conversion rate at the interface. To this end, we first derive the junction conditions that hold at the sharp interface of a non-radially perturbed hybrid star in the case of slow and rapid conversions. Then, we analyse the discontinuity $g$-mode in both cases. For rapid conversions, the discontinuity $g$-mode has zero frequency because a displaced fluid element near the phase splitting surface adjusts almost immediately its composition to its surroundings and gravity cannot provide a buoyancy force. For slow conversions, a $g$-mode exists and its properties are analysed here using modern hadronic and quark equations of state. Moreover, it has been shown recently that in the case of slow conversions an extended branch of stable hybrid configurations arises for which $\partial M/ \partial \epsilon_c <0$. We show that $g$-modes of the standard branch (that is, the one with $\partial M/ \partial \epsilon_c > 0$) have frequencies and damping times in agreement with previous results in the literature. However, $g$-modes of the extended branch have significantly larger frequencies (in the range $1-2 \, \mathrm{kHz}$) and much shorter damping times (few seconds in some cases). We discuss the detectability of $g$-mode GWs with present and planned GW observatories.