Classical and Stochastic Trans-Planckian Censorship

The trans-Planckian censorship conjecture (TCC) limits inflation to a deadline of $N_{\mathrm{T}\mathrm{C}\mathrm{C}}=\ln (M_{\mathrm{P}\mathrm{l}}\mathrm{/}{H}_{\mathrm{e}\mathrm{n}\mathrm{d}})$ e-folds. We formulate it in stochastic inflation as a bound at confidence $1-\delta$ on the probability that a history outlasts the deadline, with three readings: one worldline, the region grown from one Hubble patch, and the reheated volume. Each reading becomes a closed-form condition, rigorous for a linear roll in leading-order stochastic inflation (for the region, in a branching model) and sharp at leading order in $N_{\mathrm{T}\mathrm{C}\mathrm{C}}$: a linear roll whose classical history lasts $N_{\mathrm{c}\mathrm{l}}$ e-folds respects the TCC if $1-N_{\mathrm{c}\mathrm{l}}\mathrm{/}{N}_{\mathrm{T}\mathrm{C}\mathrm{C}}\ge \{2{\mathcal{P}}_{\zeta }[w+\ln (1\mathrm{/}\delta )\mathrm{/}{N}_{\mathrm{T}\mathrm{C}\mathrm{C}}]{\}}^{1\mathrm{/}2}$, with $w=0$ for a worldline and $w=3$ for a region. On a linear slope without vacuum decay, the worldline and region readings are never weaker than the classical TCC, and on the CMB-normalized attractor all readings agree closely with it, so the bounds on the scale of inflation stand. A classical phase that ends in a flat quantum well, which the overdamped classical TCC excludes, is allowed with a margin of a few e-folds per worldline. Near the onset of eternal inflation, the readings differ. On a linear slope, a region must leave the fraction $\sqrt{6{\mathcal{P}}_{\zeta }}$ of the deadline unused at leading order: eternal inflation is excluded for every classical history that uses a finite fraction of the deadline, as is a band of non-eternal slopes, and the slope bound becomes $\mathrm{\mid }{V}^{\mathrm{\prime }}\mathrm{\mid }>V^{3\mathrm{/}2}\mathrm{/}[\sqrt{2}\pi (1-N_{\mathrm{c}\mathrm{l}}\mathrm{/}{N}_{\mathrm{T}\mathrm{C}\mathrm{C}})]$. Per worldline, every slope is allowed, given a margin of a few standard deviations. A false vacuum must decay at about the eternal threshold $\mathrm{\Gamma }=9H^4\mathrm{/}(4\pi )$ or faster for a region or the volume, but only faster than $3H^4\ln (1\mathrm{/}\delta )\mathrm{/}(4\pi N_{\mathrm{T}\mathrm{C}\mathrm{C}})$ per worldline. Whether the TCC forbids eternal inflation depends on the reading.