  We extend the theory of weak gravitational lensing to cosmologies with
generalized gravity, described in the Lagrangian by a generic function
depending on the Ricci scalar and a non-minimal coupled scalar field. We work
out the generalized Poisson equations relating the dynamics of the fluctuating
components to the two gauge invariant scalar gravitational potentials, fixing
the new contributions from the modified background expansion and fluctuations.
We show how the lensing equation gets modified by the cosmic expansion as well
as by the presence of the anisotropic stress, which is non-null at the linear
level both in scalar-tensor gravity and in theories where the gravitational
Lagrangian term features a non-minimal dependence on the Ricci scalar. Starting
from the geodesic deviation, we derive the generalized expressions for the
shear tensor and projected lensing potential, encoding the spacetime variation
of the effective gravitational constant and isolating the contribution of the
anisotropic stress, which introduces a correction due to the spatial
correlation between the gravitational potentials. Finally, we work out the
expressions of the lensing convergence power spectrum as well as the
correlation between the lensing potential and the Integrated Sachs-Wolfe effect
affecting Cosmic Microwave Background total intensity and polarization
anisotropies. To illustrate phenomenologically the new effects, we work out
approximate expressions for the quantities above in Extended Quintessence
scenarios where the scalar field coupled to gravity plays the role of the dark
energy.
