$f(R,T_{\mu\nu}T^{\mu\nu})$ gravity and Cardassian-like expansion as one of its consequences

Abstract.

We propose a new model of gravity where the Ricci scalar $(R)$ in the Einstein-Hilbert action is replaced by an arbitrary function of $R$ and of the norm of energy-momentum tensor, i.e., $f(R,T_{\mu\nu}T^{\mu\nu})$ . Field equations are derived in the metric formalism. We find that the equation of motion of massive test particles is non-geodesic and these test particles are acted upon by a force which is orthogonal to the four-velocity of the particles. We also find the Newtonian limit of the model to calculate the extra acceleration which can affect the perihelion of Mercury. There is a deviation from the general relativistic (GR) result unless the energy density of fluid is constant. Arranging the $\alpha$ parameter gives an opportunity to cure the inconsistency between the observational values for the abundance of light elements and the standard Big Bang nucleosynthesis results. Even the dust-dominated universe undergoes an accelerated expansion without using a cosmological constant in Model II. With this specific choice of $f(R,T_{\mu\nu}T^{\mu\nu})$, we get Cardassian-like expansion.