Compact stars with dark matter induced anisotropy in complexity-free background and effect of dark matter on GW echoes

In this work, we consider the vanishing complexity factor scenario which has opened up a whole new way of generating solutions to the Einstein field equations for the spherically symmetric structure of celestial bodies. By using this very rare condition on the system of two metric potentials, viz. gtt and grr, we make reduce it to a uni-metric potential system satisfying all physical conditions. Along with this, we further have considered that the space-time is deformed by dark matter (DM) content in DM haloes resulting into perturbations in the gtt and grr metric potentials. This DM deformation is mathematically done by the complete geometric decoupling method where the decoupling parameter beta decides the amount of DM content. In connection to the claimed post-merger object in the GW170817 event we have argued that if these compact stars were in galactic DM haloes with the assumption that the radius remains the same, the compactness factor can grow within the range 1/3 to 4/9 and therefore can generate gravitational waves (GW) echoes. Additionally, we have presented effect of beta on the generation of GW echoes in accordance with the observational constraints related to the compact stars GW190814, PSR J0740+6620, PSR J1614-2230, Cen X-3, and LMC X-4.