Large-scale curvature and entropy perturbations for multiple interacting fluids

TL;DR

The paper develops a gauge-invariant formalism for the coupled evolution of curvature ($ζ$) and isocurvature ($S_{αβ}$) perturbations in a multi-fluid FRW universe with energy transfer. It shows that on superhorizon scales adiabatic perturbations remain adiabatic despite inter-fluid energy exchange, and provides explicit evolution equations for $ζ$, $ζ_α$, and $S_{αβ}$. The curvaton decay scenario is analyzed as a concrete application, revealing how an initial curvaton isocurvature perturbation can generate an adiabatic curvature perturbation after decay, with a transfer coefficient $r$ that matches well with the sudden-decay approximation. The framework generalizes to other cosmological fluids and clarifies the role of non-adiabatic energy transfer in shaping large-scale perturbations.

Abstract

We present a gauge-invariant formalism to study the evolution of curvature perturbations in a Friedmann-Robertson-Walker universe filled by multiple interacting fluids. We resolve arbitrary perturbations into adiabatic and entropy components and derive their coupled evolution equations. We demonstrate that perturbations obeying a generalised adiabatic condition remain adiabatic in the large-scale limit, even when one includes energy transfer between fluids. As a specific application we study the recently proposed curvaton model, in which the curvaton decays into radiation. We use the coupled evolution equations to show how an initial isocurvature perturbation in the curvaton gives rise to an adiabatic curvature perturbation after the curvaton decays.

Figures (5)

• Figure 1: Phase-plane showing trajectories for the background solutions in the curvaton model in Eqs.(\ref{['din3']}--\ref{['con1']}).
• Figure 2: Evolution of the normalised curvature perturbation on uniform curvaton density hypersurfaces, $\zeta_\sigma/\zeta_{\sigma,{\rm in}}$, and of the normalised total curvature perturbation, $\zeta/\zeta_{\sigma,{\rm in}}$, as a function of the number of e-foldingss, starting with $\zeta_\sigma/\zeta_{\sigma,{\rm in}}=1$ and initial density and decay rate $\Omega_\sigma=10^{-2}$ and $\Gamma/H=10^{-3}$.
• Figure 3: Evolution of the normalised curvature perturbation on uniform curvaton density hypersurfaces, $\zeta_\sigma/\zeta_{\sigma,{\rm in}}$, and of the normalised total curvature perturbation, $\zeta/\zeta_{\sigma,{\rm in}}$, as a function of the number of e-foldingss, starting with $\zeta_\sigma/\zeta_{\sigma,{\rm in}}=1$ and initial density and decay rate $\Omega_\sigma=10^{-2}$ and $\Gamma/H=10^{-6}$.
• Figure 4: Transfer parameter $r$ defined in Eq. (\ref{['defr']}) obtained from numerical solutions as a function of the initial value of $\Omega_\sigma/(\Gamma/H)^{1/2}$.
• Figure 5: Comparison of full numerical solution for $r$ in Eq. (\ref{['transfer']}) with sudden-decay approximation, $f_{\rm dec}$ given in Eq. (\ref{['fdec']}).