Trans-Planckian censorship and single-field inflaton potential

Abstract

It was recently proposed that a field theory cannot be consistent with quantum gravity if it allows a mode shorter than the Planck length to exit the Hubble horizon. This is called the Trans-Planckian Censorship Conjecture (TCC). We discuss the implications of the TCC on the possible shape of the inflaton potential in single-field slow-roll inflation. We point out that (1) there is generically an initial condition in which the total e-folding number $N_{\text{total}}$ is doubled or more compared to the e-folds necessary for the cosmic microwave background fluctuations, and (2) a sizable negative running of spectral index is generically expected to make $N_{\text{total}}$ small. In concrete setups, we find a stringent constraint on the inflationary energy scale, $V_{\text{inf}}^{1/4}<\mathcal{O}(10)\text{TeV}$ with $r<\mathcal{O}(10^{-50})$, and the running parameter is bounded above as ${\alpha }_{\text{s}}\lesssim -4\times 10^{-3}$.

Comments: 30 pages, 4 figures; published version (minor revision)