The cosmology in f(R, τ) gravity without dark energy

We examine the evolution of universe in f(R, tau) gravity where R is the Ricci curvature and tau is the trace of energy-momentum tensor. We focus on the specific form f(R, tau) = R + f(tau) and explicitly find out the relation between f(tau) and state of equation w by adiabatic condition. All possible modes of evolution of the scale factor for the five kinds of state of equation, w = +/-1, +/-1/3, 0 are exhibited. We simulate numerically the plots of redshift versus distance modulus for three interesting modes and then obtain the good fitting curve compared with the astronomical observation data.