The imprint of $f(R)$ gravity on weak gravitational lensing I:
  Connection between observables and large scale structure
  We study the effect of $f(R)$ gravity on the statistical properties of
various large-scale structures which can be probed in weak gravitational
lensing measurements. A set of ray-tracing simulations of gravitational lensing
in $f(R)$ gravity enables us to explore cosmological information on (i)
stacking analyses of weak lensing observables and (ii) peak statistics in
reconstructed lensing mass maps. For the $f(R)$ model proposed by Hu \&
Sawicki, the measured lensing signals of dark matter haloes in the stacking
analysis would show a $\simlt10\%$ difference between the standard $\Lambda$CDM
and the $f(R)$ model when the additional degree of freedom in $f(R)$ model
would be $|f_{\rm R0}|\sim10^{-5}$. Among various large-scale structures to be
studied in stacking analysis, troughs, i.e, underdensity regions in projected
plane of foreground massive haloes, could be promising to constrain the model
with $|f_{\rm R0}|\sim10^{-5}$, while stacking analysis around voids is found
to be difficult to improve the constraint of $|f_{\rm R0}|$ even in future
lensing surveys with a sky coverage of $\sim1000$ square degrees. On the peak
statistics, we confirm the correspondence between local maxima and dark matter
haloes along the line of sight, regardless of the modification of gravity in
our simulation. Thus, the number count of high significance local maxima would
be useful to probe the mass function of dark matter haloes even in the $f(R)$
model with $|f_{\rm R0}|\simlt10^{-5}$. We also find that including local
minima in lensing mass maps would be helpful to improve the constant on $f(R)$
gravity down to $|f_{\rm R0}|=10^{-5}$ in ongoing weak lensing surveys.
