Modified Einstein versus Modified Euler for Dark Matter

Modifications of General Relativity generically contain additional degrees of freedom that can mediate forces between matter particles. One of the common manifestations of a fifth force in alternative gravity theories is a difference between the gravitational potentials felt by relativistic and non-relativistic particles, also known as "the gravitational slip". In contrast, a fifth force between dark matter particles, due to dark sector interaction, does not cause a gravitational slip, making the latter a possible smoking gun of modified gravity. In this article, we point out that a force acting on dark matter particles, as in models of coupled quintessence, would also manifest itself as a measurement of an effective gravitational slip by cosmological surveys of large-scale structure. This is linked to the fact that redshift-space distortions due to peculiar motion of galaxies do not provide a measurement of the true gravitational potential if dark matter is affected by a fifth force. Hence, it is extremely challenging to distinguish a dark sector interaction from a modification of gravity with cosmological data alone. Future observations of gravitational redshift from galaxy surveys can help to break the degeneracy between these possibilities, by providing a direct measurement of the distortion of time. We discuss this and other possible ways to resolve this important question.