Growth Rate Analysis in $f(R,L_m)$ Gravity: A Comparative Study with \boldmath$\Lambda$CDM Cosmology

We investigate the evolution of cosmic structures within the framework of modified gravity, specifically focusing on theories described by the function $f(R, L_m)$, where $R$ is the Ricci scalar and $L_m$ is the matter Lagrangian. This class of models introduces a non-minimal coupling between geometry and matter, leading to modifications in the dynamics of density perturbations. We derive the linear growth equation and compute the observable growth rate $f\sigma_8(z)$, which is directly accessible from redshift-space distortion (RSD) data. Using recent observational constraints from galaxy surveys such as eBOSS and DESI, we perform a comparative analysis between predictions from $f(R, L_m)$ gravity and the standard $\Lambda$CDM model. Our results indicate that while $\Lambda$CDM remains broadly consistent with current data, the $f(R, L_m)$ framework can accommodate subtle deviations in structure growth, offering a possible resolution to existing tensions in large-scale structure observations. We also outline the implications of our findings for future surveys, including Euclid and LSST.