Eternal inflation in light of Wheeler-DeWitt equation

The Wheeler-DeWitt equation provides the probability distribution for the curvature perturbation, the gauge invariant quantum fluctuation of the inflaton. From this, we can find a tower of power spectra which is not found in a perturbative approach. Since the power spectrum for the modes that cross the horizon contributes to the uncertainty in the classical inflaton displacement, we obtain new conditions for eternal inflation. In the presence of the patch in the higher excitations, the bound on the slow-roll parameter allowing eternal inflation is given by at most $\epsilon \lesssim (2n+1)(H/m_{\rm Pl})^2$ with $n$ integer indicating the quantum number labelling the excitation. For large $n$, the bound on $\epsilon$ is relaxed such that eternal inflation can take place with even larger value of $\epsilon$. While the second law of thermodynamics implies that $n=0$ state is preferred, we cannot ignore such large $n$ effect since the nonlinear interaction inducing transitions to the $n=0$ state is suppressed.