Void–Galaxy Cross-Correlation Overview

• Void–galaxy cross-correlation is defined as the statistical link between galaxy positions and underdense cosmic voids, forming a basis for precise cosmological inference.
• It uses methods such as Landy–Szalay estimators and multipole expansions to model redshift-space distortions and account for the Alcock–Paczynski effect.
• Advanced emulators and reconstruction techniques applied in surveys like DESI and Euclid yield tight constraints on parameters including Ωₘ, σ₈, and fσ₈.

Void–galaxy cross-correlation quantifies the statistical relationship between the distribution of galaxies and the distribution of cosmic voids—large underdense regions in the matter density field. The void–galaxy cross-correlation function (CCF), denoted $\xi_{vg}$, measures the excess probability of finding a galaxy at a separation $r$ from a void center compared to a random distribution, and forms the basis for precision cosmological inference, tests of gravity, and studies of the cosmic web's structure and evolution. Modern analyses operate in both real space and the observed redshift space, modeling the impact of peculiar velocities (redshift-space distortions, RSD), cosmological geometry (Alcock–Paczynski, or AP, effect), galaxy bias, and void identification methodology. The void–galaxy CCF is a core observable for next-generation surveys such as DESI and Euclid, providing highly competitive constraints on parameters including $\Omega_m$, $\sigma_8$, $f\sigma_8$, and the expansion history.

1. Definition and Theoretical Framework

The 3D void–galaxy CCF in real space is defined as

$$\xi_{vg}(r) = \langle \delta_v(\mathbf{x})\,\delta_g(\mathbf{x} + \mathbf{r}) \rangle,$$

where $\delta_v$ and $\delta_g$ are the fractional overdensity fields of void centers and galaxies, respectively. The corresponding cross-power spectrum in Fourier space is $$P_{vg}(k) = \langle \delta_v(\mathbf{k})\,\delta_g(-\mathbf{k}) \rangle$$.

In redshift space, peculiar velocities shift galaxy and void positions along the line of sight, so separations are decomposed into transverse ($s_\perp$) and line-of-sight ($s_\parallel$) components. The CCF $\xi_{vg}(s,\mu)$, with $\mu = s_\parallel / s$, is typically expanded into Legendre multipoles: $$\xi_{vg}(s, \mu) = \sum_{\ell=0,2,4} \xi_{vg}^{(\ell)}(s) P_\ell(\mu),$$ where $P_\ell$ are Legendre polynomials and $\ell=0$ (monopole), 2 (quadrupole), 4 (hexadecapole) capture the isotropic and anisotropic components (Hamaus et al., 2013, Radinović et al., 2023, Degni et al., 10 Sep 2025).

The cross-correlation is commonly estimated using Landy–Szalay or Davis–Peebles estimators. For projected statistics, the line-of-sight and plane-of-sky projections are evaluated by integrating over finite ranges in $s_\parallel$ or $s_\perp$ to compress information and mitigate small-scale systematics (Correa et al., 2018, Correa et al., 2021). The galaxy density profile around voids is related directly to the CCF via $1 + \xi_{vg}(r) = \rho_g(r)/\bar \rho_g$.

Theoretical modeling in the linear regime relates $\xi_{vg}$ to the underlying matter correlation, with bias parameters (galaxy bias $b_g$, void bias $b_v$) entering multiplicatively: $$P_{vg}(k) \approx b_v\,b_g\,u_v(k)\,P_{mm}(k) + \frac{1}{\bar n_v} u_v(k),$$ where $u_v(k)$ is the Fourier transform of the normalized void density profile, and $P_{mm}(k)$ is the matter power spectrum (Hamaus et al., 2013).

2. Redshift-Space Distortions and Alcock–Paczynski Effect

Peculiar velocities induce coherent outflows around voids, stretching the cross-correlation along the line of sight ("Kaiser effect"). The Gaussian streaming model (GSM) is widely adopted: $$1 + \xi^s_{vg}(\vec{s}) = \int \left[1 + \xi^r_{vg}(\vec{r})\right] P(v_\parallel \mid \vec{r}) dv_\parallel,$$ where $s_\parallel = r_\parallel + v_\parallel/(aH)$ and $P(v_\parallel \mid \vec{r})$ is the probability distribution of pairwise velocities, often taken as Gaussian with scale-dependent dispersion (Radinović et al., 2023, Correa et al., 2021).

The AP effect arises from using a fiducial cosmology to convert angles and redshifts to distances, leading to physical separations

$$s_\parallel = \alpha_\parallel s_\parallel^{\rm fid}, \qquad s_\perp = \alpha_\perp s_\perp^{\rm fid},$$

where $\alpha_\parallel = D_H(z)/D_H^{\rm fid}(z)$ and $\alpha_\perp = D_M(z)/D_M^{\rm fid}(z)$. The parameter $\epsilon = \alpha_\parallel / \alpha_\perp$ quantifies the geometric anisotropy, and is primarily constrained by the quadrupole of $\xi_{vg}$ (Fraser et al., 2024, Nadathur et al., 2019).

Recent simulation analyses demonstrate that, contrary to previous assumptions, void centers identified after AP transformation do not simply scale by $(\alpha_\perp, \alpha_\parallel)$; the void-finder's nonlinear response masks part of the AP-induced anisotropy, reducing the signal by a finder-specific, nontrivial correction term (Radinović et al., 2024). Proper modeling requires either direct measurement of the real-space CCF or an iterative template-fitting approach, both yielding unbiased recovery of $\epsilon$ (Radinović et al., 2024).

3. Void Finding and Galaxy Catalog Construction

Void identification is performed on galaxy (or halo) catalogs using either watershed-based algorithms (e.g., ZOBOV/VIDE/REVOLVER/voxel), spherical underdensity criteria, or grid-based methods (Degni et al., 10 Sep 2025, Fraser et al., 2024, Radinović et al., 2023). Watershed-based finders segment the density field into basins around minima, recovering complex, non-spherical void geometries and enabling accurate sampling of deep underdensities ($\delta \to -1$). Key void properties are the central position (e.g., volume-weighted barycenter or cell minimum) and effective radius $R_{\rm eff}$.

Galaxy catalogs for forward modeling are constructed with halo occupation distribution (HOD) prescriptions, populating $N$-body halo catalogs with central and satellite galaxies, including assembly bias when necessary (Salcedo et al., 11 Apr 2025, Fraser et al., 2024). In emulator-based approaches, cosmological and HOD parameters are varied jointly in network training.

Surveys utilize pre-reconstruction or reconstructed galaxy density fields for void finding. Reconstruction is typically implemented via Zeldovich approximation, shifting observed galaxy positions to minimize RSD, thereby reducing selection biases and enhancing void statistics at small scales (Degni et al., 10 Sep 2025, Nadathur et al., 2019, Radinović et al., 2023).

4. Cosmological Parameter Inference and Forecasts

The void–galaxy CCF encodes information about the growth of structure ($f\sigma_8$) via the amplitude and shape of RSD multipoles, and the expansion history via the AP dilation parameter $\epsilon$. Likelihood analyses use the full shape of the CCF multipoles, typically over $\sim$30–60 radial bins and several multipole orders. Covariances are estimated from large ensembles of mock catalogs, jackknife resampling, or combinations thereof (Fraser et al., 2024, Radinović et al., 2023).

Joint analysis of void–galaxy CCF with BAO and galaxy RSD significantly improves precision. In BOSS CMASS, inclusion of voids reduces errors on $D_A H$ by a factor of $\sim$2 compared to BAO+RSD alone, achieving $F_{\rm AP}=D_A H/c=0.4367\pm0.0045$ (1%) and $f\sigma_8=0.501\pm0.051$ (10%) at $z=0.57$ (Nadathur et al., 2019). Emulator-based approaches further tighten these, with recent neural network models yielding $\Omega_m=0.330\pm0.020$ and $\sigma_8=0.777^{+0.047}_{-0.062}$, outperforming classic templates (by 28% on $\Omega_m$) (Fraser et al., 2024).

Forecasts for DESI and Euclid generically find that void–galaxy CCF alone can constrain $\Omega_m$ or $D_M/D_H$ to $\sim$0.3–0.9% per redshift bin, and $f\sigma_8$ to 5–8%—competitive with, or surpassing, galaxy clustering and weak lensing for the same survey volume (Radinović et al., 2023, Salcedo et al., 11 Apr 2025). Inclusion of all void sizes, enabled by RSD reconstruction, boosts statistical power and tightens AP constraints by $\sim$23% (Degni et al., 10 Sep 2025).

5. Sensitivity to Modified Gravity, Massive Neutrinos, and Systematics

Void environments are theoretically sensitive to modifications of gravity and the suppression of clustering by massive neutrinos. In $f(R)$ models, the Compton scale sets a fifth-force range, altering void expansion and profiles. Massive neutrinos reduce $\sigma_8$, modify cumulative underdensities $\Delta(r)$, and hence change outflow velocities and the CCF shape. Where both are present, their effects can partially offset. However, limitations in linear velocity modeling and simulation fidelity currently impede the robust separation of these signatures via CCF analysis alone (Mauland et al., 2023).

Systematic effects arise from methodology, including the void finder used, density thresholding, RSD modeling, and choice of reconstruction parameters. The response of the void sample to AP-induced distortions depends strongly on the finder; not accounting for this can bias $\epsilon$ estimates by several percent—far exceeding statistical errors for DESI or Euclid-like samples. Model likelihoods must include emulator or template uncertainties, with robust validation on mocks (Radinović et al., 2024, Fraser et al., 2024).

6. Advanced Modeling: Emulators and Small-Scale Expansion

Simulation-trained emulators are now the state of the art for void–galaxy CCF modeling, fully capturing the impact of cosmology, HOD, RSD, AP effect, and void-finder selection on $\xi_{vg}$ (Fraser et al., 2024). These emulators enable full-shape fits and explicit marginalization over galaxy bias, assembly bias, and small-scale nonlinearities.

Reconstruction (via Zeldovich or similar) allows inclusion of small voids into the analysis, greatly increasing the void sample and hence the statistical power. Direct void finding and correlation measurement in reconstructed space both removes large-scale RSD and delivers constraints on geometric parameters at the $\lesssim$1% level (Degni et al., 10 Sep 2025). Workflows now routinely incorporate consistency checks for smoothing scales, growth-rate assumptions, void-radius cuts, and covariance estimation.

Survey/Data Set	$\sigma_\epsilon$ (AP error)	$f\sigma_8$ Precision	Notes
BOSS CMASS	0.0045 (1%)	10%	Voids+RSD+BAO (Nadathur et al., 2019)
Euclid forecast	0.0027–0.0040 (0.3–0.4%)	5–8%	Per $z$ bin (Radinović et al., 2023)
DESI Y5 forecast	1.5% ($\Omega_m$)	0.8% ($\sigma_8$)	Multi-probe (Salcedo et al., 11 Apr 2025)
Reconstruction gain	23% tighter	—	All void sizes included (Degni et al., 10 Sep 2025)

7. Specialized Probes: Gravitational Redshift and Environmental Consistency

Beyond monopole and quadrupole, the dipole of the void–galaxy CCF is sensitive to gravitational redshift effects, reflecting the gravitational potential profile of voids. The magnitude of the gravitational redshift dominates the dipole, exceeding second-order velocity effects by an order of magnitude, making dipole measurements a direct test of gravity on tens of Mpc scales (Nan et al., 2018).

Measurements of $f\sigma_8$ from void and galaxy environments show consistency within current uncertainties, disfavouring models with strong environment-dependent modifications to gravity or growth (Achitouv et al., 2016). Voids also act as static rulers when exactly mass-compensated, providing an additional geometric anchor akin to BAO but on smaller scales (Hamaus et al., 2013).

---

The void–galaxy cross-correlation serves as a powerful and robust cosmological probe, sensitive to growth, expansion, gravity, and new physics, provided observational systematics and modeling complexities—especially those linked to redshift-space distortions, AP response, and void-finder selection—are rigorously accounted for. The field continues to advance rapidly, with simulation-based full-shape emulation, improved reconstruction techniques, and multi-probe analyses establishing the void–galaxy CCF as a cornerstone observable for upcoming survey cosmology.