Cosmic baryon census with fast radio bursts and gravitational waves

The cosmic baryon density fraction ($\Omega_{\rm b}$) is intrinsically correlated with the Hubble constant ($H_0$) through the critical density of the Universe. In the context of the decade-long $H_0$ tension, the significant discrepancy between early- and late-Universe measurements of $H_0$ implies that fixing its value or imposing an external prior could bias the baryon census. To address this concern, we construct a late-Universe probe framework that unifies fast radio bursts (FRBs) and gravitational-wave (GW) standard sirens, which can respectively resolve the ''missing baryon'' problem and the $H_0$ tension through their dispersion measures (DMs) and absolute luminosity distances. By combining $104$ localized FRBs with $47$ GW events, we obtain an $H_0$-free measurement of $\Omega_{\rm b}=0.0488\pm0.0064$ ($1\sigma$), in concordance with early-Universe observations of CMB + BBN. The result is tightly anchored by GW-inferred $H_0$ through the strong $\Omega_{\rm b}$-$H_0$ degeneracy. Although the current precision ($\sim 13\%$) is limited by sample size, the growing detections of both FRBs and GWs will make their synergy a powerful probe of low-redshift cosmology.