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 ${10}^{16}\mathrm{GeV}$. 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 $V(\varphi )=m^2{\varphi }^2/2$ with $m=2\times {10}^{13}\mathrm{GeV}$ and cutoff at $M_{\text{Pl}}=2.4\times {10}^{18}\mathrm{GeV}$, we find an expectation value of $2\times {10}^{10}$ $e$-folds on the set of Friedmann-Robertson-Walker trajectories. For cosine inflation $V(\varphi )={\mathrm{\Lambda }}^4[1-\cos (\varphi /f)]$ with $f=1.5\times {10}^{19}\mathrm{GeV}$, 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.

Figure 1

Trajectories in effective phase space for quadratic inflation. The field value and velocity are parametrized by the variables $x$ and $y$ defined in Eq. (34). The dark nearly horizontal lines indicate the apparent attractors, where the conserved measure grows large. For clarity we used the (unrealistic) value of $m=0.2M_{\text{Pl}}$ to make this plot.

Figure 2

Trajectories in effective phase space for cosine inflation. The field value and velocity are parametrized by the variables $x$ and $y$ defined in Eq. (57). The dark spirals indicate the apparent attractors, where the conserved measure grows large. For this plot we used $f=3M_{\text{Pl}}$, $\mathrm{\Lambda }=0.1M_{\text{Pl}}$.

Figure 3

Expected number of $e$-folds $⟨N_{\text{tot}}⟩$, as computed in Eq. (72) using the canonical measure on the space of trajectories, for cosine inflation with potential $V(\varphi )={\mathrm{\Lambda }}^4[1-\cos (\varphi /f)]$.

Figure 4

The probability of obtaining 50 or more $e$-folds of inflation as a function of $f$ for cosine inflation with potential $V(\varphi )={\mathrm{\Lambda }}^4[1-\cos (\varphi /f)]$, as computed using the canonical measure on the space of trajectories and starting on the $H=M_{\text{Pl}}$ surface.