  We consider the de Sitter cosmology deformed by the presence of a thermal
bath of radiation and/or time-dependent moduli fields. Depending on the
parameters, either a first or second order phase transition can occur.
  In the first case, an instanton allows a double analytic continuation. It
induces a probability to enter the inflationary evolution by tunnel effect from
another cosmological solution. The latter starts with a big bang and, in the
case the transition does not occur, ends with a big crunch. A temperature
duality exchanges the two cosmological branches. In the limit where the pure de
Sitter universe is recovered, the tunnel effect reduces to a "creation from
nothing", due to the vanishing of the big bang branch. However, the latter may
be viable in some range of the deformation parameter. In the second case, there
is a smooth evolution from a big bang to the inflationary phase.
