Tracing signatures of modified gravity in redshift-space galaxy bispectrum multipoles: Prospects for E u c l i d

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 fR0, 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 Euclid-like survey. The monopole (B₀⁰) and quadrupole (B₂⁰) show the strongest signatures, with relative deviations of $2%$--$8%$ at $z=0.7$ and k₁0.3\,h\, Mpc⁻¹ (largest side of the triangle) for fR0=10⁻⁵. Higher multipoles provide weaker but complementary signals. For Euclid, we forecast signal-to-noise ratios up to ~30 for the monopole and ~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 $Λ$CDM.