  We study the growth of subhorizon perturbations in brane-induced gravity
using perturbation theory. We solve for the linear evolution of perturbations
taking advantage of the symmetry under gauge transformations along the
extra-dimension to decouple the bulk equations in the quasistatic
approximation, which we argue may be a better approximation at large scales
than thought before. We then study the nonlinearities in the bulk and brane
equations, concentrating on the workings of the Vainshtein mechanism by which
the theory becomes general relativity (GR) at small scales. We show that at the
level of the power spectrum, to a good approximation, the effect of
nonlinearities in the modified gravity sector may be absorbed into a
renormalization of the gravitational constant. Since the relation between the
lensing potential and density perturbations is entirely unaffected by the extra
physics in these theories, the modified gravity can be described in this
approximation by a single function, an effective gravitational constant for
nonrelativistic motion that depends on space and time. We develop a resummation
scheme to calculate it, and provide predictions for the nonlinear power
spectrum. At the level of the large-scale bispectrum, the leading order
corrections are obtained by standard perturbation theory techniques, and show
that the suppression of the brane-bending mode leads to characteristic
signatures in the non-Gaussianity generated by gravity, generic to models that
become GR at small scales through second-derivative interactions. We compare
the predictions in this work to numerical simulations in a companion paper.
