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.
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