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

We argue that the $\Lambda$CDM tensions of the Hubble-Lemaitre expansion rate $H_0$ and the clustering normalization $\sigma_8$ 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 $\phi$ 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 $\phi$-matter-dominated-era ($\phi$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 $\phi$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 $\phi$MDE.