Inflationary perturbations near horizon crossing

TL;DR

The paper investigates the evolution of inflationary perturbations near horizon crossing in a single-field setup using mode-by-mode numerical integration of the Mukhanov-Sasaki equation. It identifies two regimes where standard analytic formulas fail: a temporary interruption of inflation that drives entropy perturbations to amplify curvature perturbations on super-horizon scales, and the true end of inflation where perturbations do not reach the asymptotic regime and re-entry amplitudes can be enhanced, with potential implications for primordial black hole formation. The results show that entropy perturbations can source super-horizon growth and produce broad features in the scalar power spectrum, producing large deviations from slow-roll and Stewart-Lyth predictions. Altogether, the work highlights the limitations of conventional approximations and the necessity of mode-by-mode calculations to accurately predict the power spectrum and PBH-related constraints in these non-standard inflationary histories.

Abstract

We study the behaviour of inflationary density perturbations in the vicinity of horizon crossing, using numerical evolution of the relevant mode equations. We explore two specific scenarios. In one, inflation is temporarily ended because a portion of the potential is too steep to support inflation. We find that perturbations on super-horizon scales can be modified, usually leading to a large amplification, because of entropy perturbations in the scalar field. This leads to a broad feature in the power spectrum, and the slow-roll and Stewart--Lyth approximations, which assume the perturbations reach an asymptotic regime well outside the horizon, can fail by many orders of magnitude in this regime. In the second scenario we consider perturbations generated right at the end of inflation, which re-enter shortly after inflation ends --- such perturbations can be relevant for primordial black hole formation.

Figures (5)

• Figure 1: The comoving Hubble wavenumber, $aH$, increases with the number of $e$-folds of expansion, $N$, during inflation. We set $N=0$ when inflation first ends, and inflation is suspended for around 1 $e$-fold. The quantity $1+\epsilon-\eta$ remains negative for around 5 $e$-folds. We took $B=0.55$.
• Figure 2: The evolution of scalar and tensor perturbations, $|{\mathcal{R}}_{{\rm k}}|$ and $|V_{{\rm k}}|$, for a single mode $k=10^{-2}$ which is outside the horizon by a factor of 100 when inflation is suspended. The solid line is the quantity $aH/k$, and $z'/z<0$ between the two vertical dotted lines. The arrow indicates when $k=aH$, the instant of horizon crossing. The absolute normalization of the perturbations is arbitrary, though the relative one is correct.
• Figure 3: The evolution of two scalar modes, $|u_{{\rm k}}|$, as in Fig. \ref{['fig:mode']}. The arrows again indicate horizon crossing for each mode. For the first mode, $k_1$, horizon crossing occurs while $z'/z<0$, while for the second it occurs after $z'/z$ becomes positive again. Both modes asymptote to approximately the same amplitude.
• Figure 4: The scalar power spectrum as determined from mode-by-mode integration. The upper panel shows $B=0.55$, while the lower is for $B=0.3$ (note that for inflation to be interrupted requires $B > 0.19$RLL). The scales between the two dotted lines correspond to the epoch when $z'/z<0$. The first discontinuity in the analytic expressions occurs in the region where modes make multiple horizon crossings, while the second sharp feature in the Stewart--Lyth case occurs due to an accidental cancellation of the entire Stewart--Lyth coefficient and has no physical significance.
• Figure 5: The scalar power spectrum plotted at various stages of its evolution. The numerical spectra are evaluated (from top to bottom) at horizon exit, at the end of inflation, and at horizon re-entry for each mode. The right-hand edge of the plot corresponds to the wavenumber equalling the Hubble radius at the end of inflation. The Stewart--Lyth and slow-roll spectra are evaluated, as usual, at horizon exit.