Non-minimal Energy-momentum squared gravity

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