Bispectrum BAO and the baryon-dark matter relative velocity

We evaluate the Baryon Acoustic Oscillation (BAO) signal in the bispectrum as a tool to detect and characterize the relative velocity effect. We extend the existing framework by presenting an updated model for the redshift-space tree-level bispectrum that comprehensively incorporates all relative velocity terms. We introduce a novel, unbiased technique to extract the isotropic BAO dilation parameter ($\alpha_{\rm iso}$) solely from the bispectrum monopole. Validated against N-body simulations, this template-based extraction successfully recovers the acoustic scale and enhances the statistical constraining power by $\sim30\%$, when analyzed in tandem with the pre-reconstruction power spectrum, offering a powerful complement to standard post-reconstruction pipelines. We quantify how individual relative velocity components distort both two- and three-point statistics. We find that these effects induce distinct systematic shifts in the extracted $\alpha_{\rm iso}$ between the two probes. We find systematic discrepancies of up to $2\%$ for $b_{v^2}=\pm0.05$ and up to $20\%$ for $b_{\delta^{bc}}\le-2$. This differences demonstrate that a direct comparison of independent power spectrum and bispectrum BAO measurements can break parameter degeneracies and isolate the amplitude of these biases. Finally, we provide a concrete prescription to detect and constrain the three associated relative velocity bias parameters, showing that a joint analysis is highly sensitive to the $b_{v^2}$ and $b_{\delta^{bc}}$ amplitudes. This establishes the bispectrum BAO as a robust cosmological probe for current and next-generation galaxy surveys, serving both as a cross-check for standard analyses and a crucial diagnostic tool against systematic biases.