Study in Bianchi-IX space time with linear dependencies on  R & T with gravity

This communication involves the exploration of Bianchi-IX spacetime with linear dependencies on the Ricci scale, denoted as , and the trace of the energy-momentum tensor, referred to as . This implies a modified gravity theory described by the function . The detailed investigation is structured into seven comprehensive sections. We present solutions to the field equations governing the cosmological model by introducing the concept of a fractional linear varying deceleration parameter (FLVDP). This innovative approach enables us to unveil the complex dynamics of the universe. We analyze our findings through graphical representation, providing insights into various key parameters. The visual depictions of important parameters, such as the Hubble parameter, scale factor, shear scalar, expansion scalar, and cosmological constant, serve to validate the results obtained by us.