Cosmological Perturbations in Models of Coupled Dark Energy

TL;DR

This work develops a phase-space, dynamical framework to calculate cosmological perturbations in models where dark energy interacts with dark matter, for a general quintessence potential. By formulating perturbations along background attractors and using dimensionless variables $x,y,z,u,v$ and $\gamma$, the authors derive coupled second-order equations for the dark-matter density contrast $\delta_c$ and the quintessence perturbation $\kappa\delta\phi$, including the perturbation $\delta\gamma$ of the interaction. The analysis reveals that dark-energy perturbations can cluster on small scales in the presence of coupling, and that adiabatic initial conditions are not preserved once $\gamma$ is nonzero, potentially generating isocurvature modes. The paper examines three couplings, showing that $Q=\beta H\rho_c$ can induce an instability near $x=0$ and make some initial conditions finely tuned, while $Q=\sqrt{2/3}\,b\kappa\dot\phi\rho_c$ avoids this instability and allows dark-energy perturbations to remain constant or grow, with significant implications for structure formation and cosmological observables.

Abstract

Models in which dark energy interacts with dark matter have been proposed in the literature to help explain why dark energy should only come to dominate in recent times. In this paper, we present a dynamical framework to calculate cosmological perturbations for a general quintessence potential and interaction term. Our formalism is built upon the powerful phase-space approach often used to analyse the dynamical attractors in the background. We obtain a set of coupled differential equations purely in terms of dimensionless, bounded variables and apply these equations to calculate perturbations in a number of scenarios. Interestingly, in the presence of dark-sector interactions, we find that dark energy perturbations do not redshift away at late times, but can cluster even on small scales. We also clarify the initial conditions for the perturbations in the dark sector, showing that adiabaticity is no longer conserved in the presence of dark-sector interactions, even on large scales. Some issues of instability in the perturbations are also discussed.

Figures (7)

• Figure 1: Trajectories in the $x-y$ plane for the exponential potential $V=V_0 e^{-\kappa\lambda\phi}$ with $\lambda=\sqrt5$ and interaction term of the form $Q=\beta H\rho_c$. From left to right, the trajectories correspond to $\beta=0.04, 0.035, 0.02, 0, -0.1$ and $-0.2$. The behaviour near the $y$ axis seen in the left-most trajectory is discussed in the text. The dotted hyperbola is the locus of scaling solutions [Equation (\ref{['locus']})].
• Figure 2: Evolution of large-scale perturbations over $\Delta N=10$ during the scaling regime for the exponential potential with $\lambda=\sqrt5$, and interaction term of the form $Q=\beta H\rho_c$ (with $\beta = 0,\pm0.1$). The panel on the left shows the evolution of the dark-matter density contrast $\delta_c$, while the panel on the right shows the quintessence density contrast $|\delta_\phi|$, showing similar trends. Dashed lines indicate negative values of the perturbations.
• Figure 3: Evolution of large-scale perturbations over $\Delta N=10$ during the quintessence-dominated regime for the exponential potential with $\lambda=0.01$, and interaction term of the form $Q=\beta H\rho_c$ (with $\beta = 0,\pm0.1$). The panel on the left shows the evolution of the dark-matter density contrast $\delta_c$, while the panel on the right shows the quintessence density contrast $|\delta_\phi|$. Dashed lines indicate negative values of the perturbations. Asymptotic forms of these curves are given in the text. Note especially that $\delta_\phi$ is no longer constant if $\beta\neq0$.
• Figure 4: Trajectories in the $x-y$ plane for the exponential potential with $\lambda=\sqrt5$ and interaction term $Q=\sqrt{2/3}b\kappa\dot\phi\rho_c$. From left to right, the trajectories within the semicircle correspond to $b=1, 0.5, 0, -0.5$ and $-1$. The dotted hyperbola is the locus of scaling solutions (Equation (\ref{['locus']})). In contrast with Figure (\ref{['figbeta']}), the region $x<0$ is now accessible to the trajectories since there is no singularity at $x=0$.
• Figure 5: Evolution of large-scale perturbations over $\Delta N=10$ during the scaling regime for the exponential potential with $\lambda=\sqrt5$ and interaction term $Q=\sqrt{2/3}b\kappa\dot\phi\rho_c$ (with $b=0,\pm0.1$). The panel on the left shows the evolution of density contrast, $\delta_c$, while the panel on the right shows the quintessence energy contrast, $\delta_\phi$. The dashed lines indicate negative values.