Tracing Signatures of Modified Gravity in Redshift-Space Galaxy Bispectrum Multipoles: Prospects for Euclid

We study the galaxy bispectrum multipoles in the Hu-Sawicki $f(R)$ gravity model, where a scalar degree of freedom mediates a fifth force that is screened in high-density environments. The model is specified by $f_{R0}$, the present-day background value of the scalar field, which controls the strength of deviations from General Relativity (GR). Using perturbation theory, we compute the redshift-space galaxy bispectrum with the full scale- and time-dependent second-order kernels, incorporating corrections from the scale-dependent growth rate and nonlinear screening. Expanding the bispectrum in spherical harmonics, we analyze the sensitivity of the multipoles to modified gravity and forecast their detectability in a $\mathit{\text{Euclid}}$-like survey. The monopole ($B_0^0$) and quadrupole ($B_2^0$) show the strongest signatures, with relative deviations of $2\mathrm{\%}$–$8\mathrm{\%}$ at $z=0.7$ and $k_1\simeq 0.3h{\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ (largest side of the triangle) for $f_{R0}={10}^{-5}$. Higher multipoles provide weaker but complementary signals. For $\mathit{\text{Euclid}}$, we forecast signal-to-noise ratios up to $\sim 30$ for the monopole and $\sim 15$ for the quadrupole including the Finger-of-God damping and shot noise effect. These results demonstrate that bispectrum multipoles are a powerful probe of gravity, capable of breaking degeneracies with bias and velocity effects and strengthening constraints on deviations from $\mathrm{\Lambda }$CDM.