Classical Inflationary and Ekpyrotic Universes in the No-Boundary Wavefunction

This paper investigates the manner in which classical universes are obtained in the no-boundary quantum state. In this context, universes can be characterised as classical (in a WKB sense) when the wavefunction is highly oscillatory, i.e. when the ratio of the change in the amplitude of the wavefunction becomes small compared to the change in the phase. In the presence of a positive or negative exponential potential, the WKB condition is satisfied in proportion to a factor $e^{-(\epsilon - 3)N/(\epsilon -1)},$ where $\epsilon$ is the (constant) slow-roll/fast-roll parameter and $N$ designates the number of e-folds. Thus classicality is reached exponentially fast in $N$, but only when $\epsilon < 1$ (inflation) or $\epsilon > 3$ (ekpyrosis). Furthermore, when the potential switches off and the ekpyrotic phase goes over into a phase of kinetic domination, the level of classicality obtained up to that point is preserved. Similar results are obtained in a cyclic potential, where a dark energy plateau is added. Finally, for a potential of the form $-\phi^n$ (with $n=4,6,8$), where the classical solution becomes increasingly kinetic-dominated, there is an initial burst of classicalisation which then quickly levels off. These results demonstrate that inflation and ekpyrosis, which are the only dynamical mechanisms known for smoothing the universe, share the perhaps even more fundamental property of rendering space and time classical in the first place.