  In this letter, we argue that the past of the Universe, extrapolated from
standard physics and measured cosmological parameters, might be a non-singular
bounce. We also show that, in this framework, quite stringent constraints can
be put on the reheating temperature and number of inflationary e-folds,
basically fixing $T_{RH}\sim T_{GUT}$ and $N\sim 70$. We draw some conclusions
about the shape of the inflaton potential and raise the "naturalness" issue in
this context. Finally, we argue that this could open a very specific window on
the "pre big bounce" universe.
