A Horizon Ratio Bound for Inflationary Fluctuations

Abstract

We demonstrate that the gravity wave background amplitude implies a robust upper bound on the ratio: \lambda / H^{-1} < e^60, where \lambda is the proper wavelength of fluctuations of interest and H^{-1} is the horizon at the end of inflation. The bound holds as long as the energy density of the universe does not drop faster than radiation subsequent to inflation. This limit implies that the amount of expansion between the time the scales of interest leave the horizon and the end of inflation, denoted by e^N, is also bounded from above, by about e^60 times a factor that involves an integral over the first slow-roll parameter. In other words, the bound on N is model dependent -- we show that for vast classes of slow-roll models, N < 67. The quantities, \lambda / H^{-1} or N, play an important role in determining the nature of inflationary scalar and tensor fluctuations. We suggest ways to incorporate the above bounds when confronting inflation models with observations. As an example, this bound solidifies the tension between observations of cosmic microwave background (CMB) anisotropies and chaotic inflation with a \phi^4 potential by closing the escape hatch of large N (< 62).