Covariantizing the interaction between dark energy and dark matter

TL;DR

The paper develops a covariant framework for interactions between dark energy and dark matter by modeling each component as an imperfect fluid with a timelike energy-transfer current $q^c=\alpha(t)u^c$, linking the covariant exchange to a phenomenological $Q$ via $\dot{\alpha}+3H\alpha+\frac{Q}{2}=0$ and $\alpha(t)=-\frac{1}{2a^3}\int dt\,a^3 Q(t)$. It shows that a consistent Lagrangian description is challenging for multi-fluid systems, but proposes a formal single-fluid Lagrangian obtained by a Kerr–Schild transformation to an effective metric $\bar{g}_{ab}$, where the fluid becomes perfect and the Lagrangian is ${\cal L}=\sqrt{-\bar{g}}\,P$. The work extends the covariant construction to scalar-field couplings, with $Q=\Gamma\dot{\phi}^2$ and an associated $\alpha$, and further to two scalar-field fluids, where velocity-dependent friction terms arise. Together these results offer a covariant foundation for interacting dark sectors, clarify the role of imperfectness in energy exchange, and highlight the limitations and future directions for Lagrangian formulations in multi-fluid cosmologies.

Abstract

Coupling dark energy and dark matter through an effective fluid description is a very common procedure in cosmology, however it always remains in comoving coordinates in the special FLRW space. We construct a consistent, general, and covariant formulation, where the interaction is a natural implication of the imperfectness of the fluids. This imperfectness makes difficult the final step towards a robust formulation of interacting fluids, namely the construction of a Lagrangian whose variation would give rise to the interacting equations. Nevertheless, we present a formal solution to this problem for a single fluid, through the introduction of an effective metric.