The kinematic Sunyaev-Zel'dovich effect of the large-scale structure
  (II): the effect of modified gravity
  The key to understand the nature of dark energy relies in our ability to
probe the distant Universe. In this framework, the recent detection of the
kinematic Sunyaev-Zel'dovich (kSZ) effect signature in the cosmic microwave
background obtained with the South Pole Telescope (SPT) is extremely useful
since this observable is sensitive to the high-redshift diffuse plasma. We
analyse a set of cosmological hydrodynamical simulation with 4 different
realisations of a Hu & Sawicki $f(R)$ gravity model, parametrised by the values
of $\overline{f}_{\rm R,0}=(0,-10^{-6},-10^{-5},-10^{-4})$, to compute the
properties of the kSZ effect due to the ionized Universe and how they depend on
$\overline{f}_{\rm R,0}$ and on the redshift of reionization, $z_{\rm re}$. In
the standard General Relativity limit ($\overline{f}_{\rm R,0}$=0) we obtain an
amplitude of the kSZ power spectrum of $\mathcal{D}^{\rm
kSZ}_{3000}=4.1\,$$\mu$K$^2$ ($z_{\rm re}$=8.8), close to the $+1\sigma$ limit
of the $\mathcal{D}^{\rm kSZ}_{3000}=(2.9\pm1.3)\,$$\mu$K$^2$ measurement by
SPT. This corresponds to an upper limit on the kSZ contribute from patchy
reionization of $\mathcal{D}^{\rm kSZ,patchy}_{3000}<0.9\,$$\mu$K$^2$ (95 per
cent confidence level). Modified gravity boosts the kSZ signal by about 3, 12
and 50 per cent for $\overline{f}_{\rm R,0}=(-10^{-6},-10^{-5},-10^{-4})$,
respectively, with almost no dependence on the angular scale. This means that
with modified gravity the limits on patchy reionization shrink significantly:
for $\overline{f}_{\rm R,0}=-10^{-5}$ we obtain $\mathcal{D}^{\rm
kSZ,patchy}_{3000}<0.4\,$$\mu$K$^2$. Finally, we provide an analytical formula
for the scaling of the kSZ power spectrum with $z_{\rm re}$ and
$\overline{f}_{\rm R,0}$ at different multipoles: at $\ell=3000$ we obtain
$\mathcal{D}^{\rm kSZ}_{3000}\propto z_{\rm
re}^{0.24}\left(1+\sqrt{\left|\overline{f}_{\rm R,0}\right|}\right)^{41}$.
