  We analyze the early stage of evolution of a universe with two scale factors
proposed in [1] when matter is present. The scale factors describe two causally
disconnected patches of the universe interacting trough a non-trivial Poisson
bracket structure in the momentum sector characterized by one parameter \kappa.
We studied two scenarios in which one of the patches is always filled with
relativistic matter while the other contains relativistic matter in one case,
and non-relativistic matter in the second case. By solving numerically the set
of equations governing the dynamics, we found that the energy content of one
sector drains to the other and from here it is possible to constraint the
deformation parameter \kappa by imposing that the decay of the energy density
happens, at most, at the Big Bang Nucleosynthesis temperature in order to
return to the usual behavior of radiation. The relation with Non Standard
Cosmologies is also addressed.
