Instability of the cosmological DBI-Galileon in the non-relativistic limit

The DBI-Galileon model is a tensor-scalar theory of gravity which finds its foundation as the most general theory of the dynamics of a 4D brane embedded in a 5D bulk. It is of particular interest as it provides a few free parameters with a physical meaning, such as the cosmological constant which is there related to the brane tension. Most studies of this model have been performed assuming a maximally symmetric geometry for the 5D bulk, in which it has been shown that the theory reduces to various types of Galileon. In contrast, the general case for the geometry of the bulk provides a different covariantization of the Galileon model than the covariant Galileon: the DBI-Galileon. From the tight constraints on the gravitational waves speed, we are naturally led to consider the non-relativistic limit of the model where the kinetic energy of the brane is small compared to its tension, that we study in the context of late-time cosmology. The DBI-Galileon in the non-relativistic limit is simply an expansion around General Relativity (GR) which can be expressed as a shift-symmetric Horndeski theory. We developed the description of this theory at the background and perturbation level. However, by studying the scalar and tensor perturbations around a flat FLRW background, we found that they contain a ghost degree of freedom leading to fatal instability of the vacuum for every combination of the free parameters. As a lesson, we emphasized which of the Horndeski terms competes to avoid this instability in more general cases.