What can the information paradox tell us about the early Universe?

In recent years we have come to understand how the information paradox is resolved in string theory. The huge entropy $S_{bek}={A\over 4G}$ of black holes is realized by an explicit set of horizon sized `fuzzball' wavefunctions. The wavefunction of a collapsing shell spreads relatively quickly over this large phase space of states, invalidating the classical black hole geometry the shell would have created. We argue that a related effect may occur in the early Universe. When matter is crushed to high densities we can access a similarly large phase space of gravitational `fuzzball' solutions. While we cannot estimate specific quantities at this point, a qualitative analysis suggests that spreading over phase space creates an extra `push' expanding the Universe to larger volumes.