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We consider cosmological solution for Einstein gravity with massive fermionswith a four-fermion coupling, which emerges from the Holst action and isrelated to the Barbero-Immirzi (BI) parameter. This gravitational action is animportant object of investigation in a non-perturbative formalism of quantumgravity. We study the equation of motion for for the Dirac field within thestandard Friedman-Robertson-Walker (FRW) metric. Finally, we show the theorywith BI parameter and minimally coupling Dirac field, in the zero mass limit,is equivalent to an additional term which looks like a perfect fluid with theequation of state \ $p = w\rho$, with $w = 1$ which is independent of the BIparameter. The existence of mass imposes a variable $w$, which creates eitheran inflationary phase with $w=-1$, or assumes an ultra hard equation of states$w = 1$ for very early universe. Both phases relax to a pressure less fluid $w= 0$ for late universe (corresponding to the limit $m\to\infty$).

Dirac fields, torsion and Barbero-Immirzi parameter in Cosmology