Large scale structure power spectrum and constraining model parameters in <i>f</i> ( <i>Q</i> ,ℒ _<i>m</i> ) gravity by MCMC method

In this work, we present a comprehensive cosmological viability analysis of two representative phenomenological models in the framework of f(Q,ℒ m ) gravity, where the gravitational action depends on both the nonmetricity scalar Q and the matter Lagrangian ℒ m . Unlike previous studies that were mainly restricted to homogeneous background evolution, the present analysis extends these models to the linear perturbation regime in order to examine whether their late-time accelerated behaviour remains compatible with the observed formation of large-scale structures. To this end, we first employ the dynamical systems approach to construct the corresponding autonomous equations and investigate the cosmological phase-space evolution of the models. We then use the 1 + 3 covariant formalism to derive the full gauge-invariant matter perturbation equations and solve the resulting energy density contrast evolution numerically. The obtained density contrast exhibits a decaying behaviour with increasing redshift for suitable parameter choices, from which the associated matter power spectra are computed and compared with the standard ΛCDM predictions. In addition, Markov Chain Monte Carlo simulations are performed using several combinations of background and growth-sensitive observational datasets in order to statistically constrain the free model parameters and test the observational consistency of each scenario. Our analysis shows that the non-minimal matter-nonmetricity coupling leaves measurable imprints on structure growth, and among the two considered models only Model A remains compatible with the combined observational datasets, while Model B is observationally disfavoured. These results demonstrate that perturbative and growth-based diagnostics provide a stringent and necessary test for assessing the cosmological viability of f(Q,ℒ m ) gravity beyond background expansion alone.