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.