Non-minimal energy–momentum squared gravity

We consider a gravitational theory with an additional non-minimal coupling between baryonic matter fields and geometry. The is second order in the energy momentum tensor can be seen as generalization of energy-momentum squared gravity model. will add constraint through Lagrange multiplier to ensure conservation tensor. Background cosmological implications together its dynamical system analysis investigated details. Also we growth perturbation at first order, estimate model parameter from observations on $H$ also $f\sigma_8$. show that should small positive 2$\sigma$ confidence interval. shown good agreement observational data.