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Enhancement of superhorizon scale inflationary curvature perturbations

Samuel M Leach, Misao Sasaki, David Wands, Andrew R Liddle

TL;DR

The paper shows that the curvature perturbation on superhorizon scales, $R_c$, can be amplified during single-field inflation if slow-roll is briefly violated and the background quantity $z = a\dot\phi/H$ becomes small. A long-wavelength analysis decomposes $R_c$ into growing and decaying modes, with two integrals $D_k$ and $F_k$ controlling the degree of enhancement; when either integral becomes large, the late-time amplitude far exceeds the horizon-crossing value. The authors illustrate the mechanism with Starobinsky-type potentials, where a rapid change in the slope drives a transient fast-roll and yields sizable enhancement across a broad range of scales, in agreement with numerical results. They also discuss a duality invariance of spectra and show that the conventional slow-roll amplitude formula remains a useful guide even when horizon crossing occurs outside slow-roll.

Abstract

We show that there exists a simple mechanism which can enhance the amplitude of curvature perturbations on superhorizon scales during inflation, relative to their amplitude at horizon crossing. The enhancement may occur even in a single-field inflaton model, and occurs if the quantity $a\dotφ/H$ becomes sufficiently small, as compared to its value at horizon crossing, for some time interval during inflation. We give a criterion for this enhancement in general single-field inflation models.

Enhancement of superhorizon scale inflationary curvature perturbations

TL;DR

The paper shows that the curvature perturbation on superhorizon scales, , can be amplified during single-field inflation if slow-roll is briefly violated and the background quantity becomes small. A long-wavelength analysis decomposes into growing and decaying modes, with two integrals and controlling the degree of enhancement; when either integral becomes large, the late-time amplitude far exceeds the horizon-crossing value. The authors illustrate the mechanism with Starobinsky-type potentials, where a rapid change in the slope drives a transient fast-roll and yields sizable enhancement across a broad range of scales, in agreement with numerical results. They also discuss a duality invariance of spectra and show that the conventional slow-roll amplitude formula remains a useful guide even when horizon crossing occurs outside slow-roll.

Abstract

We show that there exists a simple mechanism which can enhance the amplitude of curvature perturbations on superhorizon scales during inflation, relative to their amplitude at horizon crossing. The enhancement may occur even in a single-field inflaton model, and occurs if the quantity becomes sufficiently small, as compared to its value at horizon crossing, for some time interval during inflation. We give a criterion for this enhancement in general single-field inflation models.

Paper Structure

This paper contains 7 sections, 37 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Enhancement of the curvature perturbation
  3. Long-wavelength approximation
  4. Starobinsky's model
  5. Invariant spectra
  6. Summary
  7. If $\dot\phi$ crosses zero

Figures (2)

  • Figure 1: The power spectrum for the Starobinsky model Starobinsky92 with $A_{+}/A_{-}=10^4$. Plotted are the exact asymptotic value of the curvature perturbation ${\cal R}_c^2(\eta_*)$, the horizon-crossing value ${\cal R}_c^2(\eta_k)$, and the enhanced horizon-crossing amplitude $\alpha^2{\cal R}_c^2(\eta_k)$ using the long-wavelength approximation. The range of scales between the dotted lines corresponds to modes leaving the horizon during the transient epoch, defined as the region where $z'/z<0$. Also plotted is the slow-roll amplitude ${\cal R}^2_s$ given by Eq. (\ref{['eqn:sr']}).
  • Figure 2: Power-spectrum for the false-vacuum quartic model as in Fig. \ref{['fig:staro_spec1']}.