Adiabatic and entropy perturbations from inflation

TL;DR

The paper develops a general formalism for adiabatic and entropy perturbations in multi-field inflation by rotating the field basis to an adiabatic field $\sigma$ and an orthogonal entropy field $s$ and deriving gauge-invariant evolution equations. It shows that on super-Hubble scales an entropy perturbation can source the curvature perturbation $\mathcal{R}$ when the background trajectory is curved in field space, while $\mathcal{R}$ does not source entropy in the long-wavelength limit. The authors apply the framework to two cases: a two-field double inflation model to compute cross-correlations between $\mathcal{R}$ and $\mathcal{S}$, and a preheating scenario with a nontrivial potential to study how entropy perturbations can alter large-scale curvature, highlighting the importance of the entropy mass $\mu_s^2=V_{ss}+3\dot{\theta}^2$. They also demonstrate numerical stability advantages of evolving the rotated variables and discuss observational implications of correlated adiabatic/isocurvature perturbations.

Abstract

We study adiabatic (curvature) and entropy (isocurvature) perturbations produced during a period of cosmological inflation that is driven by multiple scalar fields with an arbitrary interaction potential. A local rotation in field space is performed to separate out the adiabatic and entropy modes. The resulting field equations show explicitly how on large scales entropy perturbations can source adiabatic perturbations if the background solution follows a curved trajectory in field space, and how adiabatic perturbations cannot source entropy perturbations in the long-wavelength limit. It is the effective mass of the entropy field that determines the amplitude of entropy perturbations during inflation. We present two applications of the equations. First, we show why one in general expects the adiabatic and entropy perturbations to be correlated at the end of inflation, and calculate the cross-correlation in the context of a double inflation model with two non-interacting fields. Second, we consider two-field preheating after inflation, examining conditions under which entropy perturbations can alter the large-scale curvature perturbation and showing how our new formalism has advantages in numerical stability when the background solution follows a non-trivial trajectory in field space.

Figures (2)

• Figure 1: An illustration of the decomposition of an arbitrary perturbation into an adiabatic ($\delta \sigma$) and entropy ($\delta s$) component. The angle of the tangent to the background trajectory is denoted by $\theta$. The usual perturbation decomposition, along the $\phi$ and $\chi$ axes, is also shown.
• Figure 2: Numerical simulations of the entropy and comoving curvature perturbations during inflation and preheating, with $\lambda=0$, $g=2\times10^{-3}$, $\tilde{g}=8\times10^{-3}g$ and $m=10^{-6}M_{\rm pl}$. The 'new' prefix indicates that the field perturbations were evaluated by numerically integrating Eqs. (\ref{['eq:entropy']}), and (\ref{['eq:adiabatic']}), while the 'old' prefix indicates that the perturbations were evaluated by integrating the original field equations (\ref{['eq:perturbation']}). We have not included any higher-order corrections such as backreaction from small-scale perturbations which would shut down the resonant amplification of $\delta s$ at some point.