  In a quantum theory of cosmology spacetime behaves classically only in
limited patches of the configuration space on which the wave function of the
universe is defined. Quantum transitions can connect classical evolution in
different patches. Working in the saddle point approximation and in
minisuperspace we compute quantum transitions connecting inflationary histories
across a de Sitter like throat or a singularity. This supplies probabilities
for how an inflating universe, when evolved backwards, transitions and branches
into an ensemble of histories on the opposite side of a quantum bounce.
Generalising our analysis to scalar potentials with negative regions we
identify saddle points describing a quantum transition between a classically
contracting, crunching ekpyrotic phase and an inflationary universe.
