A statistical representation of the cosmological constant from finite size effects at the apparent horizon

In this paper we present a statistical description of the cosmological constant in terms of massless bosons (gravitons). To this purpose, we use our recent results implying a non vanishing temperature ${T_{\Lambda}}$ for the cosmological constant. In particular, we found that a non vanishing $T_{\Lambda}$ allows us to depict the cosmological constant $\Lambda$ as composed of elementary oscillations of massless bosons of energy $\hbar\omega$ by means of the Bose-Einstein distribution. In this context, as happens for photons in a medium, the effective phase velocity $v_g$ of these massless excitations is not given by the speed of light $c$ but it is suppressed by a factor depending on the number of quanta present in the universe at the apparent horizon. We found interesting formulas relating the cosmological constant, the number of quanta $N$ and the mean value $\overline{\lambda}$ of the wavelength of the gravitons. In this context, we study the possibility to look to the gravitons system so obtained as being very near to be a Bose-Einstein condensate. Finally, an attempt is done to write down the Friedmann flat equations in terms of $N$ and $\overline{\lambda}$.