Binary neutron-star mergers offer crucial insights into the matter properties of neutron stars. We present the possible imprints in the gravitational wave signal from the nature of phase transition from such events. Our study employs a one-parameter family of equation of states built using a polytropic approach with a control parameter $\mathrm{\Delta }p$ surveying the features of hadron-quark phase transition, from Maxwell construction to the Gibbs construction. It allows us to explore the extent of mixed phases and analyze their direct impact on merger dynamics. Post-merger gravitational wave emissions reveal the expression of specific signatures in the spectrogram and power spectral density, serving as a distinct signature of equations of state with mixed phases. We found additional peaks in power spectral density that are exclusively generated from the post-merger remnant experiencing a phase transition. Additionally, the nature of phase transition leaves specific imprints on the spectrogram, leading to a twofolded signature from gravitational wave analysis. Furthermore, we establish the first correlation between $\mathrm{\Delta }p$ and the threshold mass for prompt collapse. Our analysis shows that $\mathrm{\Delta }p\lesssim 0.04$ is required if GW170817 formed a long-lived remnant or has experienced a delayed collapse into a black hole.

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• Gravitational wave sources
• Gravitational waves
• Nuclear matter in neutron stars
• Numerical relativity
• Relativistic hydrodynamics