  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.
