Planckian corrections to the Friedmann flat equations from
  thermodynamics at the apparent horizon
  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.
