Gravitational wave echoes from compact stars in $f(\mathcal{R},T)$ gravity

We have calculated the static and spherically symmetric solutions for compact stars in the $f(\mathcal{R},T)$ gravity metric formalism. To describe the matter of compact stars, we have used the MIT Bag model equation of state (EoS) and the color-flavor-locked (CFL) EoS. Solving the hydrostatic equilibrium equations i.e., the modified TOV equations in $f(\mathcal{R},T)$ gravity, we have obtained different stellar models. The mass-radius profiles for such stars are eventually discussed. The stability of these configurations are then analysed using different parameters. From the obtained solutions of TOV equations for mass and radius, we have checked the compactness of such objects. It is found that similar to the unrealistic EoS, like the stiffer form of the MIT Bag model, under some considerations the realistic interacting quark matter CFL EoS can give stellar structures which are compact enough to possess a photon sphere outside the stellar boundary and hence can echo GWs. The obtained echo frequencies are found to lie in the range of 39-55 kHz. Also we have shown that for different parametrizations of the gravity theory, the structure of stars and also the echo frequencies differ significantly. Moreover, we have constrained the pairing constant value $\beta$ from the perspective of emission of echo frequencies. For the stiffer MIT Bag model $\beta\geq-2.474$ and for the CFL phase with massless quark condition $\beta\geq-0.873$, whereas for the massive case $\beta\geq-0.813$.