Cosmological long-wavelength solutions in non-adiabatic multi-fluid systems

TL;DR

This paper develops fully nonlinear long-wavelength solutions for cosmological perturbations in non-adiabatic multi-fluid systems using the ADM formalism combined with a spatial gradient expansion, with ε ≡ k/(aH). It defines nonlinear adiabatic and entropy perturbations and analyzes how curvature and density perturbations evolve on superhorizon scales, highlighting the physical degrees of freedom associated with entropy in multi-fluid cosmologies. The authors derive zeroth- and leading-order solutions for general N-fluid and specifically two-fluid (e.g., matter–radiation) systems, illustrating how the curvature perturbation can acquire time dependence at leading order and how slicing choices (geodesic, N-slice, CMC) impact gauge freedom. They demonstrate the framework with a matter–radiation example, showing how pure entropy initial conditions drive evolution through the equality epoch and yield nontrivial density-perturbation behavior, offering nonlinear initial data for PBH formation simulations and broader early-universe phenomenology.

Abstract

We develop a formulation of nonlinear cosmological perturbations on superhorizon scales in multi-fluid systems. It is based on the Arnowitt-Deser-Misner formalism combined with a spatial gradient expansion characterized by a small expansion parameter defined as the ratio of the comoving wavenumber to the Hubble scale. The background spacetime is assumed to be a flat Friedmann-Lemaitre-Robertson-Walker universe. Within this framework, we explicitly construct nonlinear long-wavelength solutions for cosmological perturbations. Since multi-fluid systems are inherently non-adiabatic, these solutions admit both adiabatic and entropy modes already at leading nonlinear order. We define adiabatic and entropy perturbations and discuss the non-uniqueness in defining pure entropy perturbations. Using different choices of pure entropy initial conditions, we analyze the time evolution of physical quantities such as the curvature perturbation and density perturbations in the geodesic slicing for two-fluid systems.

Figures (5)

• Figure 1: The evolution of scale factor $a(t)$ for matter and radiation fluid with $a_\text{eq}=D_{(\mathrm{r})}/D_{(\mathrm{m})}=1,~H_\text{eq}t_\text{eq}=2\sqrt{2}\qty(2-\sqrt{2})/3$, radiation fluid with $\Gamma_{(\mathrm{r})}=4/3$, and matter fluid with $\Gamma_{(\mathrm{m})}=1$.
• Figure 2: The spatial dependence of the integral functions $\bar{C}_{(\mathrm{r})}$ and $\bar{C}_{(\mathrm{m})}$ (a), and the time evolution of the spherically symmetric curvature perturbation (b) satisfying the primeval isocurvature condition
• Figure 3: Same as Fig. \ref{['en']} but for the zero-average condition.
• Figure 4: Same as Fig. \ref{['en']} but for the initially isocurvature condition.
• Figure 5: The snapshots of the time evolution of the spherically symmetric density perturbation for the radiation (r) (a), matter (m) (b), and total fluid (c) under the conditions of Eqs. (\ref{['dimlessHubble']}) and (\ref{['seed2']}) with the values $\beta=3,~\sigma_2=3,~{}^{(b)}H_\text{eq}B=15$.