How Many e-Folds Should We Expect from High-Scale Inflation?

Abstract

We address the issue of how many e-folds we would naturally expect if inflation occurred at an energy scale of order 10^(16) GeV. We use the canonical measure on trajectories in classical phase space, specialized to the case of flat universes with a single scalar field. While there is no exact analytic expression for the measure, we are able to derive conditions that determine its behavior. For a quadratic potential V(ϕ)=m^2ϕ^2/2 with m=2×10^(13) GeV and cutoff at M_(Pl)=2.4×10^(18) GeV, we find an expectation value of 2×1010 e-folds on the set of FRW trajectories. For cosine inflation V(ϕ)=Λ^4[1−cos(ϕ/f)] with f=1.5×10^(19) GeV, we find that the expected total number of e-folds is 50, which would just satisfy the observed requirements of our own Universe; if f is larger, more than 50 e-folds are generically attained. We conclude that one should expect a large amount of inflation in large-field models and more limited inflation in small-field (hilltop) scenarios.