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

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 $V(ϕ)=m^{2}ϕ^{2}/2$ with $m=2\times10^{13}$ GeV and cutoff at $M_{\rm Pl}=2.4\times10^{18}$ GeV, we find an expectation value of $2\times10^{10}$ $e$-folds on the set of Friedmann-Robertson-Walker trajectories. For cosine inflation $V(ϕ)=Λ^{4}[1-\cos(ϕ/f)]$ with $f=1.5\times10^{19}$ 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.

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