Cavendish experiment with fast radio bursts on cosmological scales

A key measure of gravity is the relation between the Weyl potential $\Psi+\Phi$ and the matter overdensity $\delta_m$, encapsulated as an effective gravitational constant $G_{\rm light}$ for light motion. Its value, along with possible spatial and temporal variations, is essential for probing physics beyond Einstein gravity. However, the absence of an unbiased proxy for $\delta_m$ prevents the direct measurement of $G_{\rm light}$. In this work, we show that within a theoretical framework respecting the weak equivalence principle, the dispersion measure (DM) of localized fast radio bursts (FRBs) serve as a good proxy for $\delta_m$. We further propose an FRB-based estimator $F_G$ to directly measure $G_{\rm light}$, combining galaxy-DM of localized FRBs and galaxy-weak lensing cross-correlations. With a conservative cut $k\leq 0.1\, h/{\rm Mpc}$, the measurement can achieve a precision of $\lesssim 10\% \sqrt{10^5/N_{\rm FRB}}$ over 10 equal-width redshift bins at $z\lesssim 1$. The major systematic error, arising from the clustering bias of electrons traced by the FRB DM, remains subdominant at the $5\%$ level. It can be further mitigated to the $\lesssim 1\%$ level, based on the gastrophysics-agnostic behavior that the clustering bias of total baryons (ionized diffuse gas, stars, neutral hydrogen, etc) approaches unity at sufficiently large scales. Therefore, FRBs shed light on gravitational physics across spatial and temporal scales spanning 20 orders of magnitude.