Spacetime curvature as a probe of exotic core phases in neutron stars within modified gravity

In this study, we investigate the effect of Energy-Momentum Squared Gravity (EMSG) on the curvature of neutron stars (NSs) by using three relativistic mean-field (RMF) equations of state (EOSs) and three hadron-quark phase transition (HQPT) EOSs. Neutron stars, with their extreme densities and strong gravitational fields, provide an ideal laboratory for testing General Relativity (GR) in the high-curvature regime and for exploring possible deviations via modified gravity. EMSG extends GR by including nonlinear terms involving the energy-momentum tensor, characterized by a coupling parameter $α$. We focus on the Kretschmann, Ricci, and Weyl curvature scalars, analyzing their dependence on baryon density and radial coordinate for varying values of $α$. Our results indicate that EMSG can significantly alter the curvature profiles of neutron stars. In particular, the magnitude of both Weyl and Kretschmann scalars increases (decreases) for a positive (negative) EMSG parameter, with the former exhibiting a larger dependence. Similarly, the surface curvature (SC) is notably affected by $α$. Interestingly, we further observe distinct discontinuities in the curvature profiles at hadron-quark phase transitions, especially in the soft and intermediate HQPT models. These signatures may provide observable imprints of exotic core phases in neutron stars.