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We consider a cosmological scenario where the dark sector is described by twoperfect fluids that interact through a velocity-dependent coupling. Thiscoupling gives rise to an interaction in the dark sector driven by the relativevelocity of the components, thus making the background evolution oblivious tothe interaction and only the perturbed Euler equations are affected at firstorder. We obtain the equations governing this system with the Schutz-SorkinLagrangian formulation for perfect fluids and derive the correspondingstability conditions to avoid ghosts and Laplacian instabilities. As aparticular example, we study a model where dark energy behaves as a radiationfluid at high redshift while it effectively becomes a cosmological constant inthe late Universe. Within this scenario, we show that the interaction of bothdark components leads to a suppression of the dark matter clustering at latetimes. We also argue the possibility that this suppression of clusteringtogether with the additional dark radiation at early times can simultaneouslyalleviate the $\sigma_8$ and $H_0$ tensions.

Velocity-dependent interacting dark energy and dark matter with a Lagrangian description of perfect fluids