Impact of minimal geometric deformation and anisotropy on compact objects in modified rainbow gravity

In the current research article, we delve into the theoretical implications of Rastall–Rainbow Gravity within the framework of modified gravitational theories in which the curvature of spacetime is taken into account during the variation of the gravitational constant. We investigate gravitational decoupling phenomena through the Minimal Deformation Method (MGD), which consists of independent modifications of different matter and energy components under the action of the revised gravitational field equations. We thoroughly outline the roles of these decouplings in physical cosmological measurements, which include spectral properties of distant astrophysical sources and the redshift – distance relation, with particular emphasis on their implications on redshifts. Further, we study the behavior of the four energy conditions to ensure the physical viability of the model, with particular emphasis on the Dominant Energy Condition (DEC). We demonstrate that Rastall–Rainbow Gravity is in agreement with the underlying relativistic energy bounds, especially given the DEC even if this allows for large deformations of the gravitational field. This is a robust framework for studying cosmic dynamics and structure formation.