Scaling solutions and weak gravity in dark energy with energy and momentum couplings

TL;DR

The paper develops a coupled dark-energy–dark-matter framework with both energy and momentum exchange to address the $H_0$ and $\sigma_8$ tensions. It constructs a scaling Lagrangian of the form $L=-\left[e^{Q\phi/M_{\rm pl}} g_1(Y_1,Y_2)-1\right]\rho_m(n)+X g_2(Y_1,Y_2)$ with $Y_1=X e^{\lambda\phi/M_{\rm pl}}$, $Y_2=Z e^{\lambda\phi/(2M_{\rm pl})}$, enabling scaling solutions and a φMDE. The analysis identifies fixed points (scaling (a), scalar-field dominated (b), and φMDE (c)) and demonstrates that momentum transfer via $Z=u^{\mu}\nabla_{\mu}\phi$ can yield $G_{\rm eff}<G$ at low redshifts, potentially suppressing structure growth. A concrete background model shows a sequence from radiation to φMDE to acceleration, with $G_{cc}$ and $G_{cb}$ adjustable by $Q$ and $\beta$ to reduce $f\sigma_8$ while preserving stability, suggesting a viable path to reconcile data, pending detailed observational constraints.

Abstract

We argue that the $Λ$CDM tensions of the Hubble-Lemaitre expansion rate $H_0$ and the clustering normalization $σ_8$ can be eased, at least in principle, by considering an interaction between dark energy and dark matter in such a way to induce a small and positive early effective equation of state and a weaker gravity. For a dark energy scalar field $φ$ interacting with dark matter through an exchange of both energy and momentum, we derive a general form of the Lagrangian allowing for the presence of scaling solutions. In a subclass of such interacting theories, we show the existence of a scaling $φ$-matter-dominated-era ($φ$MDE) which can potentially alleviate the $H_0$ tension by generating an effective high-redshift equation of state. We also study the evolution of perturbations for a model with $φ$MDE followed by cosmic acceleration and find that the effective gravitational coupling relevant to the linear growth of large-scale structures can be smaller than the Newton gravitational constant $G$ at low redshifts. The momentum exchange between dark energy and dark matter plays a crucial role for realizing weak gravity, while the energy transfer is also required for the existence of $φ$MDE.

Figures (3)

• Figure 1: (Left) Evolution of $\Omega_{\phi}$, $\Omega_c$, $\Omega_b$, and $\Omega_r$ versus $z+1$ for $\lambda=1$, $Q=0.07$, and $\beta=0.5$, where $z=a_0/a-1$ and $a_0$ is today's value of $a$. The initial conditions are chosen to be $x=1.0 \times 10^{-13}$, $\tilde{y}=1.0 \times 10^{-14}$, $\Omega_b=5.8 \times 10^{-6}$, and $\Omega_r=0.999962$ at redshift $z=8.3 \times 10^7$. (Right) Evolution of $x$, $\tilde{y}$, $w_{\phi}$, and $w_{\rm eff}$ for the same model parameters and initial conditions as those used in the left.
• Figure 2: (Left) Evolution of $G_{cc}/G$ versus the redshift $z$ for $\lambda=1$ in three different cases: (i) $Q=0.04$, $\beta=0$, $m=2$, (ii) $Q=0.04$, $\beta=1$, $m=2$, and (iii) $Q=0.02$, $\beta=0.5$, $m=3$. The background initial conditions are chosen to realize $\Omega_{\phi} \simeq 0.68$, $\Omega_b \simeq 0.05$, and $\Omega_r \simeq 10^{-4}$ today. (Right) Evolution of $f \sigma_8$ versus the redshift $z$ for the three cases shown in the left panel. Today's values of $\sigma_8$ and ${\cal K}$ are chosen to be $\sigma_8 (z=0)=0.811$ and ${\cal K}(z=0)=300$, respectively.
• Figure 3: Phase-space analysis for the same model parameters as those used in Fig. \ref{['fig1']}, but with neither baryons nor radiation ($\Omega_{b}=0$ and $\Omega_{r}=0$). The $\phi$MDE saddle point and the final accelerated attractor are denoted as c and b, respectively.