  We study a cosmological scenario in which the DBI action governing the motion
of a D3-brane in a higher-dimensional spacetime is supplemented with an induced
gravity term. The latter reduces to the quartic Galileon Lagrangian when the
motion of the brane is non-relativistic and we show that it tends to violate
the null energy condition and to render cosmological fluctuations ghosts. There
nonetheless exists an interesting parameter space in which a stable phase of
quasi-exponential expansion can be achieved while the induced gravity leaves
non trivial imprints. We derive the exact second-order action governing the
dynamics of linear perturbations and we show that it can be simply understood
through a bimetric perspective. In the relativistic regime, we also calculate
the dominant contribution to the primordial bispectrum and demonstrate that
large non-Gaussianities of orthogonal shape can be generated, for the first
time in a concrete model. More generally, we find that the sign and the shape
of the bispectrum offer powerful diagnostics of the precise strength of the
induced gravity.
