Horizon Ratio Bound for Inflationary Fluctuations

We demonstrate that the gravity wave background amplitude implies a robust upper bound on the wavelength-to-horizon ratio at the end of inflation: $\lambda /H^{-1}\lesssim e^{60}$, as long as the cosmic energy density does not drop faster than radiation subsequent to inflation. This limit implies that $N$, the number of $e$-folds between horizon exit and the end of inflation for wave modes of interest, is $\lesssim 60$ plus a model-dependent factor—for vast classes of slow-roll models, $N\lesssim 67$. As an example, this bound solidifies the tension between observations of the cosmic microwave background anisotropies and chaotic inflation with a ${\phi }^4$ potential by closing the escape hatch of large $N$ ($<62$).