The FRB--Galaxy Overdensity Cross-Correlation Statistic in Dispersion Space
Cross-correlating the dispersion of fast radio bursts (FRBs) with galaxies provides a means to study the distribution of the baryons in the Universe, even in the absence of FRB redshifts. To this end, two variants of angular cross-power spectrum statistics have been proposed: one between DM and galaxy density binned by redshift $C^{Dg}_l(z_g)$ (abbreviated $D \times g$), and one between FRB counts binned by dispersion measure (DM) and galaxy density binned by redshift $C^{fg}_l(\textrm{DM}, z_g)$ (abbreviated $f \times g$). Here we show the $D \times g$ statistic can be recovered as a DM-moment of $f \times g$, implying the latter is strictly more informative. By slicing in both DM space and galaxy redshift space, the $f\times g$ statistic separates contributions from the clustering of free electrons and from the clustering of FRB sources. We perform Fisher forecasts for FRB samples consistent with CHIME (1,600 FRBs) and the upcoming CHORD (20,000 FRBs) survey cross-correlated against the DESI Legacy Survey BGS sample and Euclid galaxy surveys, respectively. We show that, compared to the $D \times g$ statistic, the $f \times g$ statistic results in $S/N\approx 12$ for CHIME$\times$DESI(LS) and SNR $\approx 54$ for CHORD$\times$Euclid. The $f \times g$ statistics is more sensitive to the redshift distribution of FRBs with forecasted errors on a simple parameterization of order $10 \%$. It measures the logarithmic cutoff scale for clustering of baryons due to feedback $k_{cut}$ to 26\% precision with CHIME and 14\% precision with CHORD. Since most FRBs currently lack host identifications, and scaling optical followup to large samples will remain challenging even with precise localizations, reliable redshifts will be unavailable for most FRBs for the foreseeable future. The $f \times g$ statistic provides a means to extract maximum cosmological information in their absence.