Large-scale structure power spectrum and constraining model parameters in f(Q,ℒm) gravity by MCMC method

In this work, we present a comprehensive cosmological viability analysis of two representative phenomenological models in the framework of [Formula: see text] gravity, where the gravitational action depends on both the nonmetricity scalar [Formula: see text] and the matter Lagrangian [Formula: see text]. 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 behavior 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 [Formula: see text] 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 behavior with increasing redshift for suitable parameter choices, from which the associated matter power spectra are computed and compared with the standard [Formula: see text]CDM 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 disfavored. These results demonstrate that perturbative and growth-based diagnostics provide a stringent and necessary test for assessing the cosmological viability of [Formula: see text] gravity beyond background expansion alone.