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Although it is conjectured that a phase transition from hadronic todeconfined quark matter is possible in the ultrahigh density environment inNeutron Stars, the nature of such a transition is still unknown. Depending onwhether there is a sharp or slow phase transition, one may expect a thirdfamily of stable compact stars or ``twin stars" to appear, with the same massbut different radii compared to Neutron stars. The possibility of identifyingtwin stars using astrophysical observations has been a subject of interest,which has gained further momentum with the recent detection of gravitationalwaves from binary neutron stars. In this work, we investigate for the firsttime the prospect of probing the nature of hadron-quark phase transition withfuture detection of gravitational waves from unstable fundamental (f-) modeoscillations in Neutron Stars. By employing a recently developed model thatparametrizes the nature of the hadron-quark phase transition via ``pastaphases", we calculate f-mode characteristics within a full general relativisticformalism. We then recover the stellar properties from the detected modeparameters using Universal Relations in GW asteroseismology. Our investigationssuggest that the detection of gravitational waves emanating from the f-modeswith the third-generation gravitational wave detectors offers a promisingscenario for confirming the existence of the twin stars. We also estimate thevarious uncertainties associated with the determination of the mode parametersand conclude that these uncertainties make the situation more challenging toidentify the nature of the hadron-quark phase transition.

Probing hadron-quark phase transition in twin stars using $f$-modes