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

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