Trans-Planckian censorship and single-field inflaton potential

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 $α_\text{s} \lesssim - 4 \times 10^{-3}$.

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