In this paper we use our recently generalized black hole entropy formula to propose a quantum version of the Friedmann equations .
In particular , starting from the differential version of the first law of thermodynamics , we are able to find planckian ( non commutative ) corrections to the Friedmann flat equations .
The so modified equations are formally similar to the ones present in Gauss-Bonnet gravity , but in the ordinary 3+1 dimensions .
As a consequence of these corrections , by considering negative fluctuations in the internal energy that are allowed by quantum field theory , our equations imply a maximum value both for the energy density \rho and for the Hubble flow H , i.e .
the big bang is ruled out .
Conversely , by considering positive quantum fluctuations , we found no maximum for \rho and H .
Nevertheless , by starting with an early time energy density \rho \sim 1 / t ^ { 2 } , we obtain a value for the scale factor a ( t ) \sim e ^ { \sqrt { t } } , implying a finite planckian universe at t = 0 , i.e .
the point-like big bang singularity is substituted by a universe of planckian size at t = 0 .
Finally , we found possible higher order planckian terms to our equations together with the related corrections of our generalized Bekenstein-Hawking entropy .