The imprint of $f(R)$ gravity on weak gravitational lensing II :
  Information content in cosmic shear statistics
  We investigate the information content of various cosmic shear statistics on
the theory of gravity. Focusing on the Hu-Sawicki-type $f(R)$ model, we perform
a set of ray-tracing simulations and measure the convergence bispectrum, peak
counts and Minkowski functionals. We first show that while the convergence
power spectrum does have sensitivity to the current value of extra scalar
degree of freedom $|f_{\rm R0}|$, it is largely compensated by a change in the
present density amplitude parameter $\sigma_{8}$ and the matter density
parameter $\Omega_{\rm m0}$. With accurate covariance matrices obtained from
1000 lensing simulations, we then examine the constraining power of the three
additional statistics. We find that these probes are indeed helpful to break
the parameter degeneracy, which can not be resolved from the power spectrum
alone. We show that especially the peak counts and Minkowski functionals have
the potential to rigorously (marginally) detect the signature of modified
gravity with the parameter $|f_{\rm R0}|$ as small as $10^{-5}$ ($10^{-6}$) if
we can properly model them on small ($\sim 1\, \mathrm{arcmin}$) scale in a
future survey with a sky coverage of 1,500 squared degrees. We also show that
the signal level is similar among the additional three statistics and all of
them provide complementary information to the power spectrum. These findings
indicate the importance of combining multiple probes beyond the standard power
spectrum analysis to detect possible modifications to General Relativity.
