In trace-free Einstein gravity, the energy-momentum tensor of matter is not necessarily conserved and so the theory offers a natural framework for interacting dark energy models with a constant equation of state w=−1. From the point of view of quantum gravity phenomenology, it has been argued that such violations of energy-momentum conservation might originate from discreteness at the Planck scale. We show that within this framework it is possible to build models that are free from perturbative instabilities, which are known to affect a large class of interacting dark energy models. It has been suggested that models within this class may also help alleviate the Hubble tension.
We analyze in detail a simple such model where the energy-momentum transfer potential is proportional to the energy density of cold dark matter, which is also equivalent to a generalized dark matter model with a constant equation of state. Interestingly, requiring that there are no gradient instabilities implies that energy is transferred from dark matter to dark energy. We study the evolution of cosmological perturbations in this model and discuss observational constraints.