Velocity-dependent interacting dark energy and dark matter with a
  Lagrangian description of perfect fluids
  We consider a cosmological scenario where the dark sector is described by two
perfect fluids that interact through a velocity-dependent coupling. This
coupling gives rise to an interaction in the dark sector driven by the relative
velocity of the components, thus making the background evolution oblivious to
the interaction and only the perturbed Euler equations are affected at first
order. We obtain the equations governing this system with the Schutz-Sorkin
Lagrangian formulation for perfect fluids and derive the corresponding
stability conditions to avoid ghosts and Laplacian instabilities. As a
particular example, we study a model where dark energy behaves as a radiation
fluid at high redshift while it effectively becomes a cosmological constant in
the late Universe. Within this scenario, we show that the interaction of both
dark components leads to a suppression of the dark matter clustering at late
times. We also argue the possibility that this suppression of clustering
together with the additional dark radiation at early times can simultaneously
alleviate the $\sigma_8$ and $H_0$ tensions.
