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 $V(\phi)=m^{ 2}\phi^{ 2}/2$ 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 $V(\phi)=\Lambda^{ 4}[1-\cos(\phi/f)]$ 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.
Categories: hep-th astro-ph.CO gr-qc