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.
