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Discontinuity gravity modes may arise in perturbed quark-hadron hybrid starswhen a sharp density jump exists in the stellar interior and are a potentialfingerprint to infer the existence of quark matter cores in compact objects.When a hybrid star is perturbed, conversion reactions may occur at thequark-hadron interface and may have a key role in global stellar propertiessuch as the dynamic stability and the quasi-normal mode spectrum. In this workwe study the role of the conversion rate at the interface. To this end, wefirst derive the junction conditions that hold at the sharp interface of anon-radially perturbed hybrid star in the case of slow and rapid conversions.Then, we analyse the discontinuity $g$-mode in both cases. For rapidconversions, the discontinuity $g$-mode has zero frequency because a displacedfluid element near the phase splitting surface adjusts almost immediately itscomposition to its surroundings and gravity cannot provide a buoyancy force.For slow conversions, a $g$-mode exists and its properties are analysed hereusing modern hadronic and quark equations of state. Moreover, it has been shownrecently that in the case of slow conversions an extended branch of stablehybrid configurations arises for which $\partial M/ \partial \epsilon_c <0$. Weshow 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 withprevious results in the literature. However, $g$-modes of the extended branchhave significantly larger frequencies (in the range $1-2 \, \mathrm{kHz}$) andmuch shorter damping times (few seconds in some cases). We discuss thedetectability of $g$-mode GWs with present and planned GW observatories.

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