Large-scale instability in interacting dark energy and dark matter fluids

Abstract

If dark energy interacts with dark matter, this gives a new approach to the coincidence problem. But interacting dark energy models can suffer from pathologies. We consider the case where the dark energy is modelled as a fluid with constant equation of state parameter w. Non-interacting constant-w models are well behaved in the background and in the perturbed universe. But the combination of constant w and a simple interaction with dark matter leads to an instability in the dark sector perturbations at early times: the curvature perturbation blows up on super-Hubble scales. Our results underline how important it is to carefully analyze the relativistic perturbations when considering models of coupled dark energy. The instability that we find has been missed in some previous work where the perturbations were not consistently treated. The unstable mode dominates even if adiabatic initial conditions are used. The instability also arises regardless of how weak the coupling is. This non-adiabatic instability is different from previously discovered adiabatic instabilities on small scales in the strong-coupling regime.

Figures (3)

• Figure 1: The background evolution of dark energy density $\rho_x$ (top panels) and the evolution of the gauge-invariant curvature perturbation $\zeta$ for a super-Hubble scale $k=7\times10^{-5}\,$Mpc$^{-1}$ (bottom panels), as functions of scale factor $a$, for the model with coupling given by Eq. (\ref{['C']}). In panels on the left, we vary $w_x$ with $\Gamma$ fixed, while the right-hand panels have fixed $w_x$ and varying $\Gamma$. For each case, the vertical lines (in matching style) indicate the moment when $|3\mathcal{H}(1+w_{x})\rho_{x}|$ and $|a\Gamma\rho_c|$ are equal in Eq. (\ref{['kg1']}). To the right of these lines the background evolves as in the uncoupled case, i.e, $\rho_x \propto a^{-3(1+w_x)}$. To the left, the coupling modifies the background evolution to $\rho_x \propto a^{-1}$, Eq. (\ref{['radrx']}). (Note that in the left panels for $w_x=-0.80$ this happens very far in the past, not shown in the figure.) All the curves show the full numerical solution obtained with our modified version of CAMB, with the initial amplitude of $\psi$ set to $10^{-25}$. The analytical solution for the blow-up of $\zeta$, Eq. (\ref{['cper']}), is practically indistinguishable from the numerical solution at early times.
• Figure 2: The evolution of the gauge-invariant curvature perturbation $\zeta$ for three different scales as a function of scale factor $a$, for the model with coupling given by Eq. (\ref{['C']}). In the panel on the left, the coupling $|\Gamma|$ is very small while in the right-hand panel, $|\Gamma|$ is larger. Vertical lines indicate the moment when each mode enters the horizon ($k\tau \sim 1$). The largest scale ($k = 7\times10^{-5}\,$Mpc$^{-1}$) stays super-Hubble all the way up to today. The intermediate scale ($k = 1.5\times10^{-3}\,$Mpc$^{-1}$) enters the horizon during matter domination, and the smallest scale ($k = 5\,$Mpc$^{-1}$) enters deep in the radiation era.
• Figure 3: The evolution of the background $\rho_x/\rho_c$ (top panel), and gauge-invariant curvature perturbation $\zeta$ for a super-Hubble scale $k=7\times10^{-5}\,$Mpc$^{-1}$ (bottom panel) for the model with coupling given by Eq. (\ref{['QC']}). The figure shows a full numerical solution for $w_x=-0.87$ and $\beta=-0.003$, starting with the initial value $\zeta = 10^{-80}$ and ending up with oscillations of amplitude $|\zeta|>10^{+300}$.