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.