We consider cosmological solution for Einstein gravity with massivefermions with a four-fermion coupling, which emerges from the Holstaction and is related to the Barbero-Immirzi (BI) parameter. Thisgravitational action is an important object of investigationin a non-perturbative formalism of quantum gravity. We study theequation of motion for the Dirac field within the standardFriedman-Robertson-Walker (FRW) metric. Finally, we show the theorywith BI parameter and minimally coupling Dirac field, in the zero masslimit, is equivalent to an additional term which looks like a perfectfluid with the equation of state p = wρ, with w = 1 which isindependent of the BI parameter. The existence of mass imposes avariable w, which creates either an inflationary phase with w = −1,or assumes an ultra hard equation of states w = 1 for very earlyuniverse. Both phases relax to a pressure less fluid w = 0 forlate universe (corresponding to the limit m → ∞).