Parametrized post-Newtonian expansion and FRW scalar perturbations in $n$-DBI gravity

$n$-DBI gravity explicitly breaks Lorentz invariance by the introduction of a unit timelike vector field, thereby giving rise to an extra (scalar) degree of freedom. We look for observational consequences of this mode in two setups. First, we compute the parametrized post-Newtonian (PPN) expansion of the metric to first post-Newtonian order. Surprisingly, we find that the PPN parameters are exactly the same as in general relativity (GR), and no preferred-frame effects are produced. In particular this means that $n$-DBI gravity is consistent with all GR solar system experimental tests. We discuss the origin of such degeneracy between $n$-DBI gravity and GR and suggest it may also hold in higher post-Newtonian order. Second, we study gravitational scalar perturbations of a Friedmann-Robertson-Walker space-time with a cosmological constant $\mathrm{\Lambda }\ge 0$. In the case of de Sitter space, we show that the scalar mode grows as the Universe expands and, in contrast with a canonical scalar field coupled to GR, it does not freeze on superhorizon scales.