Unveiling $f(R)$ Gravity with Void-Galaxy Cross-Correlation Multipoles

Cosmic voids provide low-density environments where the scalar fifth force predicted by $f(R)$ modified gravity can be weakly screened. We present a semi-analytical calculation of the monopole, dipole, and quadrupole of the void-galaxy cross-correlation function $\xi^{s}(s,\mu)$ in redshift space for the Hu-Sawicki $f(R)$ model ($n=1$), combining scale-dependent growth induced by the scalaron with nonlinear spherical shell dynamics. The same framework can be generalized to metric $f(R)$ theories for which $G_{\rm eff}(k,a)/G$ is specified in the quasi-static limit. Our key results are: (1)~the monopole deviation from $\Lambda{\rm CDM}$ grows from $+2.8\%$ for large voids ($r_v=30 h^{-1}{\rm Mpc}$) to $+29.7\%$ for small voids ($r_v=11.7 h^{-1}{\rm Mpc}$) at $|f_{R0}|=10^{-5}$, a distinctive size-dependent signature of the Compton-scale scalaron response, with $\lambda_C\approx 8 h^{-1}{\rm Mpc}$; (2)~nonlinear evolution amplifies the modified-gravity signal by $\mathcal{A}_0\approx 4$, bringing it within reach of ongoing and upcoming spectroscopic surveys such as DESI, Subaru PFS, Euclid, and Roman; (3) the gravitational potential contains a finite-range Yukawa component, producing a radially dependent dipole signature complementary to the density and velocity multipoles; (4) for the fiducial Hu-Sawicki evolution, the signal generally decreases toward higher redshift as the scalaron Compton wavelength becomes shorter, but remains potentially detectable at Stage-IV spectroscopic void samples. We show that the void-scale transition in the modified-gravity response, the joint sensitivity to density, velocity, and fifth-force contributions, and the nonlinear amplification around void shells make redshift-space void-galaxy multipoles a powerful semi-analytical probe of $f(R)$ gravity and effective dark-energy inhomogeneities in modified gravity.