Creation of an inflationary epoch from an Emergent Universe through quantum tunneling
We consider that the very early Universe was well described by an Emergent Universe, whose geometry was that of a static and classically stable Einstein Universe, which is possible in certain Two Measures Theories (TMT). These solutions do not have a big bang, but rather they extend to arbitrarily early times. The TMT we consider are theories with spontaneously broken scale invariance and contain a dilaton field, crucial for the implementation of the scale invariance, that after the symmetry breaking acquires a nontrivial effective potential, which we study in the Einstein frame. Using the conserved quantity associated with the dilaton, we recast the cosmological evolution in terms of an effective potential $V(\dot{\phi})$, which exhibits a divergent barrier separating the static solution from the subsequent expanding evolution. Quantizing the corresponding minisuperspace Hamiltonian, we evaluate the tunneling probability in the WKB approximation. The tunneling action is well defined and finite once the branch of the square root is fixed by requiring a consistent semiclassical interpretation of the tunneling probability. We find that the barrier gives rise to a logarithmic contribution to the tunneling exponent, yielding a power-law rather than the usual exponential dependence of the tunneling probability, which for the parameter values considered is close to unity. The Universe thus emerges with a finite scale factor into a superinflationary phase that develops into slow-roll inflation. This quantum creation of an inflationary Universe does not represent a quantum creation of spacetime, which exists before the tunneling.