Dirac fields, torsion and Barbero-Immirzi parameter in Cosmology

We consider cosmological solution for Einstein gravity with massive fermions with a four-fermion coupling, which emerges from the Holst action and is related to the Barbero-Immirzi (BI) parameter. This gravitational action is an important object of investigation in a non-perturbative formalism of quantum gravity. We study the equation of motion for for the Dirac field within the standard Friedman-Robertson-Walker (FRW) metric. Finally, we show the theory with 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 the equation of state \ $p = w\rho$, with $w = 1$ which is independent of the BI parameter. The existence of mass imposes a variable $w$, which creates either an 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 $m\to\infty$).