Testing modified gravity with 21 cm intensity mapping, HI galaxy, cosmic
  microwave background, optical galaxy, weak lensing, galaxy clustering, type
  Ia supernovae and gravitational wave surveys
  In modern cosmology, an important task is investigating whether there exists
a signal of modified gravity in the universe. Due to the limited resolutions
and sensitivities of facilities, current observations can not detect any signal
of modified gravity. As a consequence, it is urgent to predict the constraining
power of future cosmological surveys on modified gravity. We constrain the
Hu-Sawicki $f(R)$ gravity with eight future mainstream probes encompassing 21
cm intensity mapping, HI galaxy, cosmic microwave background, optical galaxy,
weak lensing, galaxy clustering, type Ia supernovae and gravitational wave. We
find that the HI galaxy survey SKA2 gives the strongest constraint
$\sigma_{f_{R0}}=1.36\times10^{-8}$ among eight probes. The promising 21 cm
intensity mapping survey SKA1-MID-B1 and optical galaxy survey Euclid also
reach the order $\mathcal{O}$(-8). The fourth-generation CMB experiments SO and
CORE produces the order $\mathcal{O}$(-6), while large scale structure surveys
Euclid weak lensing, Euclid galaxy clustering and CSST weak lensing obtain the
order $\mathcal{O}$(-5). Interestingly, CSST galaxy clustering gives the same
order $\mathcal{O}$(-6) as SO and CORE, and the gravitational wave survey ET
also obtain the order $\mathcal{O}$(-5). The combination of eight probes gives
the tightest constraint $1.14\times10^{-8}$, which is just a little stronger
than $1.34\times10^{-8}$ from the combination of SKA2 and SK1-MID-B1. This
indicates that, to a large extent, future 21 cm intensity mapping and HI galaxy
surveys can improve our understanding of modified gravity and energy budget in
the cosmic pie.
