Effective field theory of coupled dark energy and dark matter

Abstract

We formulate an effective field theory (EFT) of coupled dark energy (DE) and dark matter (DM) interacting through energy and momentum transfers. In the DE sector, we exploit the EFT of vector-tensor theories with the presence of a preferred time direction on the cosmological background. This prescription allows one to accommodate shift-symmetric and non-shift-symmetric scalar-tensor theories by taking a particular weak coupling limit, with and without consistency conditions respectively. We deal with the DM sector as a non-relativistic perfect fluid, which can be described by a system of three scalar fields. By choosing a unitary gauge in which the perturbations in the DE and DM sectors are eaten by the metric, we incorporate the leading-order operators that characterize the energy and momentum transfers besides those present in the conventional EFT of vector-tensor and scalar-tensor theories and the non-relativistic perfect fluid. We express the second-order action of scalar perturbations in real space in terms of time- and scale-dependent dimensionless EFT parameters and derive the linear perturbation equations of motion by taking into account additional matter (baryons, radiation). In the small-scale limit, we obtain conditions for the absence of both ghosts and Laplacian instabilities and discuss how they are affected by the DE-DM interactions. We also compute the effective DM gravitational coupling $G_{\rm eff}$ by using a quasi-static approximation for perturbations deep inside the DE sound horizon and show that the existence of momentum and energy transfers allow a possibility to realize $G_{\rm eff}$ smaller than in the uncoupled case at low redshift.

Figures (2)

• Figure 1: Evolution of the DM density contrast for different Fourier modes (normalized to the present Hubble horizon $k_0=a_0H_0$) and EFT parameters. We have set initial conditions with $\delta_{\text{c}}(t_{\text{ini}})=a(t_{\text{ini}})$, $\dot{\delta}_{\rm c}(t_{\text{ini}})=a(t_{\text{ini}})H(t_{\text{ini}})$, and $\zeta(t_{\text{ini}})=\dot{\zeta}(t_{\text{ini}})=0$. We have checked that $\zeta$ quickly reaches the attractor solution, so our results are not sensitive to the initial condition on $\zeta$ for the considered range of parameters. In all cases we have chosen the parameters $\alpha_{m_1}$ and $\alpha_{m_2}$, so that the sound horizon during matter domination is fixed and varied only $r$. The upper panels show the evolution of three Fourier modes that enter the sound horizon at different times. In the upper left panel, we can see how the sub-horizon evolution corresponds to a slow mode, while in the upper right panel the sub-horizon evolution oscillates, thus signalling the invalidity of the quasi-static approximation. We corroborate this in the lower panels, where we have compared the exact numerical solution with the solution that matches the super-horizon solution and the quasi-static solution at horizon crossing time $t_s$ defined as $c_s k=a(t_s) H(t_s)$. In all cases, however, we can observe how the clustering is reduced with respect to $\Lambda$CDM.
• Figure 2: In this Figure, we show the transfer function normalized to $\Lambda$CDM for different combinations of parameters. We normalize $k$ to the present Hubble horizon $k_0=a_0H_0$ and the vertical dashed lines correspond to the sound horizon scale. In the left panel, we keep $r$ fixed and plot today's CDM density contrast $\delta_{\rm c}$ for several different values of $c_{\text{s}}^2$, while in the right panel we vary $r$ and keep the propagation speed fixed. We observe that $c_{\text{s}}$ determines the scales that undergo the suppression on small scales while $r$ drives the amount of suppression on those scales, in agreement with our analytical estimates in the main text. We also see the presence of acoustic oscillations that signal the failure of the quasi-static approximation. However, we can corroborate the suppressed growth on small scales in all cases. We also observe the small suppression on large scales due to the slower growth of super-Hubble modes.