The Integrated Sachs-Wolfe effect in $f(R)$ gravity
  We study the late-time Integrated Sachs-Wolfe (ISW) effect in $f(R)$ gravity
using N-body simulations. In the $f(R)$ model under study, the linear growth
rate is larger than that in general relativity (GR). This slows down the decay
of the cosmic potential and induces a smaller ISW effect on large scales.
Therefore, the $\dot\Phi$ (time derivative of the potential) power spectrum at
$k<0.1h$/Mpc is suppressed relative to that in GR. In the non-linear regime,
relatively rapid structure formation in $f(R)$ gravity boosts the non-linear
ISW effect relative to GR, and the $\dot\Phi$ power spectrum at $k>0.1h$/Mpc is
increased (100$\%$ greater on small scales at $z=0$). We explore the
detectability of the ISW signal via stacking supercluster and supervoids. The
differences in the corresponding ISW cold or hot spots are $\sim 20\%$ for
structures of $\sim 100$Mpc/$h$. Such differences are greater for smaller
structures, but the amplitude of the signal is lower. The high amplitude of ISW
signal detected by Granett et al. can not explained in the $f(R)$ model. We
find relatively big differences between $f(R)$ and GR in the transverse bulk
motion of matter, and discuss its detectability via the relative frequency
shifts of photons from multiple lensed images.
