Quintessence with quadratic coupling to dark matter

TL;DR

The paper studies a phenomenological quadratic coupling between dark energy (quintessence) and dark matter in a background cosmology with an exponential potential $V(\varphi) = V_0 e^{-\kappa\lambda\varphi}$. It analyzes three specific couplings, $Q = (\alpha/H_0)\rho_{\varphi}^2$, $Q = (\beta/H_0)\rho_c^2$, and $Q = (\gamma/H_0)\rho_c\rho_{\varphi}$, plus a superposition $Q = (\alpha/H_0)\rho_{\varphi}^2 + (\gamma/H_0)\rho_c\rho_{\varphi}$, within a compact phase-space formulation using $x, y, z$ where $x^2 = \frac{\kappa^2\dot{\varphi}^2}{6H^2}$, $y^2 = \frac{\kappa^2 V}{3H^2}$ and $z = \frac{H_0}{H+H_0}$. The main findings are that the $\mathcal{B}$-type coupling cannot realize a standard matter era, while $\mathcal{A}$ and $\mathcal{C}$ permit a matter-dominated epoch transitioning to a late-time accelerated, dark-energy-dominated attractor when the potential is sufficiently flat ($\lambda^2<2$); the composite $\mathcal{A}+\mathcal{C}$ model preserves this behavior. In all viable cases the late-time state is not a scaling solution, i.e., $\Omega_{c*}=0$, $\Omega_{\varphi*}=1$, analogous to $\Lambda$CDM, which means the coincidence problem is not addressed at the background level. These results set the stage for perturbation analyses to tighten observational constraints on the couplings $\alpha$ and $\gamma$.

Abstract

We introduce a new form of coupling between dark energy and dark matter that is quadratic in their energy densities. Then we investigate the background dynamics when dark energy is in the form of exponential quintessence. The three types of quadratic coupling all admit late-time accelerating critical points, but these are not scaling solutions. We also show that two types of coupling allow for a suitable matter era at early times and acceleration at late times, while the third type of coupling does not admit a suitable matter era.

Figures (4)

• Figure 1: Phase-space trajectories for model $\mathcal{A}$. The left plot shows the stable node G, with $\lambda=1.2$ and $\alpha=10^{-3}$. The right plot shows the stable focus F with $\lambda=2.3$ and $\alpha=10^{-3}$
• Figure 2: Phase-space trajectories for model $\mathcal{B}$. The left plot shows the stable node G, with $\lambda=1.2$ and $\beta=10^{-3}$. The right plot shows the stable focus F with $\lambda=2.3$ and $\beta=10^{-3}$.
• Figure 3: Phase-space trajectories for model $\mathcal{C}$. The left plot shows the stable node G, with $\lambda=1.2$ and $\gamma=10^{-3}$. The right plot shows the stable focus F with $\lambda=2.3$ and $\gamma=10^{-3}$.
• Figure 4: Phase-space trajectories for the superposition of couplings. The left plot shows the stable node G, with $\lambda=1.2$ and $\alpha= 2 \gamma=2 \times 10^{-3}$. The right plot shows the stable focus F with $\lambda=2.3$ and $\alpha= 2 \gamma=2 \times 10^{-3}$.