Year: 2014
Type: article
Abstract: We address the issue of how many $e$-folds we would naturally expect if inflation occurred at an energy scale of order ${10}^{16}\text{ }\text{ }\mathrm{GeV}$. 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 $V(\ensuremath{\phi})={m}^{2}{\ensuremath{\phi}}^{2}/2$ with $m=2\ifmmode\times\else\texttimes\fi{}{10}^{13}\text{ }\text{ }\mathrm{GeV}$ and cutoff at ${M}_{\text{Pl}}=2.4\ifmmode\times\else\texttimes\fi{}{10}^{18}\text{ }\text{ }\mathrm{GeV}$, we find an expectation value of $2\ifmmode\times\else\texttimes\fi{}{10}^{10}$ $e$-folds on the set of Friedmann-Robertson-Walker trajectories. For cosine inflation $V(\ensuremath{\phi})={\mathrm{\ensuremath{\Lambda}}}^{4}[1\ensuremath{-}\mathrm{cos}(\ensuremath{\phi}/f)]$ with $f=1.5\ifmmode\times\else\texttimes\fi{}{10}^{19}\text{ }\text{ }\mathrm{GeV}$, 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.
Authors: Grant N. Remmen, Sean M. Carroll
Institution: California Institute of Technology
Language: English
Subfield: Astronomy and Astrophysics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W1966537924