  We present the cosmic shear signal predicted by two viable cosmological
models in the framework of modified-action f(R) theories. We use f(R) models
where the current accelerated expansion of the Universe is a direct consequence
of the modified gravitational Lagrangian rather than Dark Energy (DE), either
in the form of vacuum energy/cosmological constant or of a dynamical scalar
field (e.g. quintessence). We choose Starobinsky's (St) and Hu & Sawicki's (HS)
f(R) models, which are carefully designed to pass the Solar System gravity
tests. In order to further support - or rule out - f(R) theories as alternative
candidates to the DE hypothesis, we exploit the power of weak gravitational
lensing, specifically of cosmic shear. We calculate the tomographic shear
matrix as it would be measured by the upcoming ESA Cosmic Vision Euclid
satellite. We find that in the St model the cosmic shear signal is almost
completely degenerate with LCDM, but it is easily distinguishable in the HS
model. Moreover, we compute the corresponding Fisher matrix for both the St and
HS models, thus obtaining forecasts for their cosmological parameters. Finally,
we show that the Bayes factor for cosmic shear will definitely favour the HS
model over LCDM if Euclid measures a value larger than ~0.02 for the extra HS
parameter n_HS.
