Probing baryonic feedback and cosmology with 3$\times$2-point statistic of FRBs and galaxies

The impact of galaxy formation processes on the matter power spectrum is uncertain and may bias cosmological parameters inferred by large-scale structure surveys. Fast Radio Bursts (FRBs), through their dispersion measures (DMs) encoding the integrated column density of baryons, offer a unique window into the distribution of gas. In this work, we investigate the constraining power of a 3x2-point correlation statistic of FRB DMs and galaxies. We present the correlation formalism, derive covariance matrices and forecast signal-to-noise ratios and Fisher parameter constraints. Assuming host galaxy DM variance of 90 pc cm$^{-3}$, for $10^4$ ($10^5$) FRBs across 35% of the sky, the angular DM power spectrum is noise dominated at multipoles $\ell \gtrsim 20$ ($\ell \gtrsim 100$), which implies that the analysis can be conducted using arcmin-scale localizations, where the redshift distribution of the FRB population can be modeled through the FRB luminosity function or FRB position cross-correlations with galaxies. We show that while $10^4$ ($10^5$) FRB DM correlations can constrain cosmological parameters at 40-70% (30-40%) level, this is a factor of 2-3 (1.5-2) weaker than the precision attainable with galaxy clustering alone due to shot noise from the limited FRB number density, variance of the field and host DMs. On the contrary, feedback-sensitive scales are not accessible in galaxy surveys. We demonstrate that combining FRB DMs and galaxies auto- and cross-correlations in a 3x2-point analysis breaks feedback-cosmology degeneracies, yielding 10-18% (7-13%) precision on cosmological parameters and 3% (2%) constraints on feedback using $10^4$ ($10^5$) FRBs. This work positions the 3x2-point statistic of FRB DMs and galaxies as a promising multi-probe strategy, bridging the gap between constraining astrophysical feedback models and precise measurement of cosmological parameters.