Cosmological Perturbations with Multiple Fluids and Fields

TL;DR

This work tackles scalar cosmological perturbations in a universe with multiple interacting fluids and scalar fields within Einstein gravity. It develops a gauge-ready perturbation formalism and formulates evolution equations using curvature ($\Phi$ or $\varphi_\chi$) and isocurvature ($S_{(ij)}$ or $\delta\phi_{(ij)}$) variables for general backgrounds with curvature $K$ and cosmological constant $\Lambda$. Key contributions include the gauge-ready fluid–field framework, explicit component and multi-component equations, Newtonian correspondence, and slow-roll linear-order results for multi-field systems. The results provide a robust toolkit for analyzing structure formation in complex multi-component cosmologies, with applications to quintessence and warm inflation and implications for both early- and late-time cosmic evolution.

Abstract

We consider the evolution of perturbed cosmological spacetime with multiple fluids and fields in Einstein gravity. Equations are presented in gauge-ready forms, and are presented in various forms using the curvature (Φor φ_χ) and isocurvature (S_{(ij)} or δφ_{(ij)}) perturbation variables in the general background with K and Λ. We clarify the conditions for conserved curvature and isocurvature perturbations in the large-scale limit. Evolutions of curvature perturbations in many different gauge conditions are analysed extensively. In the multi-field system we present a general solution to the linear order in slow-roll parameters.

Figures (3)

• Figure 1: In the first figure, evolutions of $\varphi_\chi$ (black, solid line), ${1 \over 3(1 + w)} \delta_{v}$ (red, solid line), $v_{\chi}$ (blue, dot and long-dash line) are compared with $\varphi_\kappa$ (black, dotted line), ${1 \over 3(1 + w)} \delta_{\kappa}$ (red, dotted line), $v_{\kappa}$ (blue, dot and short-dash line). In the second figure, evolutions of $\varphi_\chi$ (black, solid line), ${1 \over 3} \delta_{(c)v}$ (red, solid line), $v_{(c)\chi}$ (blue, dot and long-dash line) are compared with $\varphi_\kappa$ (black, dotted line), ${1 \over 3} \delta_{(c)\kappa}$ (red, dotted line), $v_{(c)\kappa}$ (blue, dot and short-dash line). The vertical scale indicates the relative amplitude; we normalized $\varphi_\kappa = 1$ in the early radiation era. We took $K = 0 = \Lambda$, $\Omega_{b0} = 0.06$, $H = 65 km/sec Mpc$, and ignored the direct coupling between the baryon and the photon before recombination. Radiation-matter equality ($t_{eq}$) occurs around $\log(a_{eq}/a_0) \simeq -4.2$. We took the adiabatic initial condition in the radiation dominated era. As the scale we considered $\lambda_0 \equiv 2 \pi a_0/k = 4 \pi Mpc$. We indicate the horizon-crossing epochs of the given scale as $t_1$ where $k/aH = 1$, and $t_2$ where $\lambda/\lambda_H = 2 \pi aH/k = 1$.
• Figure 2: The first figure shows evolutions of $\varphi_\chi$ (black, solid line), $\varphi_\kappa$ (red, dotted line), $\varphi_v$ (blue, dot and short-dash line), $\varphi_{v(c)}$ (blue, dot and long-dash line). The same evolutions are reproduced in the second figure which also shows evolutions of $\varphi_\delta$ (red, solid line), $\varphi_{\delta_{(c)}}$ (red, dotted line), ${1 \over 3 (1 + w)} \delta_v$ (blue, dot and short-dash line), ${1 \over 3} \delta_{(c)v_{(c)}}$ (blue, dot and long-dash line). The conditions used are the same as in Fig. \ref{['Fig-Newtonian']}.
• Figure 3: The same as first figure of Fig. \ref{['Fig-varphi']} for scales $\lambda_0 = 40 \pi Mpc$ and $400 \pi Mpc$, respectively.