The FRB--Galaxy Overdensity Cross-Correlation Statistic in Dispersion Space

Technical Summary: The FRB–Galaxy Overdensity Cross-Correlation Statistic in Dispersion Space

Problem Statement
Fast Radio Bursts (FRBs) serve as unique tracers for the distribution of baryons in the Universe, particularly the diffuse intergalactic medium (IGM) which is difficult to observe via other means. The dispersion measure (DM) of an FRB integrates the free electron density along the line of sight. While previous work (e.g., Wang et al. [30]) established a method to cross-correlate the DM overdensity field with galaxy overdensity fields (the $D \times g$ statistic) to study baryon clustering, this approach faces a fundamental limitation: most FRBs lack host galaxy identifications and thus precise redshifts. Consequently, the $D \times g$ statistic collapses all FRBs into a single DM field, making it difficult to disentangle the clustering of free electrons from the inherent clustering of FRB sources (progenitors) with galaxies. This degeneracy limits the ability to constrain feedback models and the redshift distribution of FRB sources.

Methodology
This paper introduces a generalized cross-correlation statistic, denoted as $f \times g$, which cross-correlates FRB spatial overdensity binned by Dispersion Measure (DM) against galaxy overdensity binned by redshift ($z_g$).

1. Derivation of the $f \times g$ Statistic:
   The authors derive the angular cross-power spectrum $C^{\text{fg}}_{\ell}(D, z_g)$ by treating the FRB field in DM space. The derivation accounts for the "DM budget," separating the contribution from the host galaxy ($D_h$) and the cosmic line-of-sight integral. The resulting signal is decomposed into two distinct physical terms:

   • The "Background" Term: Represents FRBs passing through the redshift of the galaxy bin, interacting with electrons correlated to those galaxies. This term is non-negligible only when the DM bin corresponds to an effective redshift greater than the galaxy redshift.
   • The "Contact" Term: Represents the spatial clustering of FRB progenitors with the galaxies themselves. This term is significant only when the DM bin aligns with the effective redshift of the galaxies.

2. Theoretical Connection:
   The paper demonstrates that the traditional $D \times g$ statistic can be mathematically recovered as a specific linear combination of the first two moments (weighted by $D^0$ and $D^1$) of the $f \times g$ statistic. This proves that $f \times g$ contains a strict superset of the information available in $D \times g$.

3. Fisher Forecasting Framework:
   The authors perform Fisher matrix forecasts to compare the constraining power of $f \times g$ versus $D \times g$.

   • Surveys: Forecasts are conducted for the CHIME survey (1,600 FRBs) cross-correlated with the DESI Legacy Survey BGS sample, and the upcoming CHORD survey (20,000 FRBs) cross-correlated with Euclid galaxy surveys.
   • Models: The analysis utilizes a Schechter luminosity function for FRB redshift distribution, parameterized by $\alpha$ and $z^*$. Clustering is modeled using electron bias ($b_e$), FRB bias ($b_f$), and an exponential scale cutoff ($k_{\text{cut}}$) to account for feedback. Host DM contributions are modeled as a log-normal distribution with mean $\mu_{\text{host}}$ and variance $\sigma_{\text{host}}$.
   • Noise: Noise spectra are calculated considering shot noise, line-of-sight variance, and localization uncertainties (modeled via a suppression factor $\ell_{\text{loc}}$).

Key Contributions and Results

• Increased Signal-to-Noise Ratio (SNR): The $f \times g$ statistic increases the total detection SNR by a factor of 2.5 to 4 compared to $D \times g$. The authors forecast a total SNR of $\approx 12$ for CHIME $\times$ DESI and $\approx 54$ for CHORD $\times$ Euclid using the $f \times g$ statistic.
• Parameter Constraints:
  • Redshift Distribution: The $f \times g$ statistic is significantly more sensitive to the FRB redshift distribution parameters ($\alpha$ and $z^*$), with forecasted errors on the order of 10%.
  • Clustering and Feedback: The statistic measures the logarithmic cutoff scale for baryon clustering due to feedback ($k_{\text{cut}}$) with 26% precision for CHIME and 14% for CHORD. This represents a factor-of-two improvement for CHORD over the $D \times g$ approach.
  • Host DM Properties: Unlike $D \times g$, the $f \times g$ statistic allows for the direct constraint of the host DM variance ($\sigma_{\text{host}}$), as it measures the scattering of FRBs into different DM bins. It also provides tighter constraints on the host DM mean ($\mu_{\text{host}}$).
• Bias Degeneracy Breaking: By resolving the "background" and "contact" terms individually, the $f \times g$ statistic breaks degeneracies between the FRB bias ($b_f$) and other parameters that persist in the $D \times g$ analysis.

Significance and Claims
The paper claims that the $f \times g$ statistic provides a means to extract maximum cosmological information from FRB samples even in the absence of host redshifts, a condition expected to persist for the foreseeable future given the prohibitive cost of optical follow-up for large samples.

The primary significance lies in the ability to:

1. Disentangle Clustering Sources: Separate the clustering of free electrons from the clustering of FRB progenitors, which is impossible with the collapsed $D \times g$ statistic.
2. Constrain Feedback Physics: Provide tighter constraints on feedback models (via $k_{\text{cut}}$) that govern the distribution of baryons around galaxies.
3. Characterize FRB Progenitors: Offer insights into the nature of FRB progenitors by constraining their redshift distribution and their tendency to reside in specific galaxy types (via $b_f$).
4. Serve as Priors: The results from DM field analysis can serve as priors for other probes (such as Sunyaev–Zel'dovich or X-ray analyses) to break degeneracies in weighting fields.

The authors note that while the simple parameterizations used (e.g., ad-hoc $k_{\text{cut}}$) may not capture the full complexity of the underlying physics, the $f \times g$ framework establishes a necessary step toward more sophisticated modeling using effective field theories or perturbation theory as FRB samples grow.