Void–Galaxy Cross-Correlation Function

• Void–galaxy cross-correlation function is a two-point statistic that quantifies the excess or deficit of galaxies around void centres compared to a random distribution.
• It employs multiple formulations—3D isotropic, anisotropic, projected, and angular—to extract cosmological parameters, including growth rate and Alcock–Paczynski distortions.
• Advanced reconstruction and modeling techniques reduce redshift-space distortions, enabling precise measurements in surveys such as SDSS, BOSS, Euclid, and DESI.

The void–galaxy cross-correlation function is the two-point statistic between cosmic-void centres and galaxies, defined as the excess or deficit of galaxies at a given void-centric separation relative to a random distribution. In configuration space it is equivalent to the stacked radial density profile of tracers around void centres; in redshift space it becomes anisotropic and encodes coherent outflows, the Alcock–Paczynski effect, and higher-order relativistic contributions. It is used as a diagnostic of void environments in SDSS, as a growth-rate test in 6dF, as a high-precision AP/RSD observable in BOSS, and as a forecasted Euclid and DESI probe in recent survey analyses (Achitouv et al., 2016, Nadathur et al., 2019, Radinović et al., 2023, Salcedo et al., 11 Apr 2025).

1. Definitions and mathematical representations

In its most direct form, the void–galaxy cross-correlation function is the tracer density contrast around void centres,

$$\xi_{vg}(r) \equiv \frac{\rho_g(r)}{\bar{\rho}_g}-1 \equiv \delta_g(r),$$

or, more generally for a tracer population $T$,

$$1+\xi_{vT}(r)=\frac{\langle n_T(r)\rangle}{\bar n_T}.$$

This identifies the cross-correlation with the stacked density profile around voids. The generic profile recalled in void analyses is a strong underdensity near the centre, a rise toward the void edge, an overdense ridge or wall at the boundary, and an asymptotic approach to the mean density at larger radii (Panchal et al., 2020).

Representation	Expression	Typical use
3D isotropic CCF	$\xi_{vg}(r)$	Stacked tracer profile around void centres
Anisotropic CCF	$\xi_{vg}(r_p,\pi)$ or $\xi^s(s,\mu)$	RSD and AP analyses
Projected CCF	$$w_{p,vg}(r_p)=2\int_0^{\Pi_{\max}}\xi_{vg}(r_p,\pi)\,d\pi$$	Real-space or weak-RSD compressed observable
Angular tomographic form	$C^{vg}_{ij}(\ell)$	Photometric and harmonic-space analyses

The projected statistic used in forward-model DESI forecasts is

$$w_{p,vg}(r_p)=2\int_0^{\Pi_{\max}}\xi_{vg}(r_p,\pi)\,d\pi,$$

with $\Pi_{\max}=200\,h^{-1}\,\mathrm{Mpc}$ in that analysis, and with forecast fits restricted to $T$0 to reduce sensitivity to noisy derivatives and modelling challenges (Salcedo et al., 11 Apr 2025). On large scales, the Fourier-space cross-spectrum is commonly written as

$T$1

with $T$2 and $T$3 the void and galaxy biases (Bonici et al., 2022).

A tomographic angular counterpart follows the same projection structure used for void–lensing. Replacing the lensing kernel by a galaxy tracer kernel yields

$T$4

with $T$5. This is mathematically parallel to the void–lensing cross-spectrum and makes explicit that the void–galaxy CCF is one instance of a broader void–tracer cross-correlation framework (Bonici et al., 2022).

2. Redshift-space formulation and multipole structure

The central observable in spectroscopic analyses is the anisotropic redshift-space cross-correlation. A standard streaming description writes

$T$6

where $T$7 is the real-space CCF, $T$8 is the line-of-sight velocity PDF, and the mapping between $T$9 and $$1+\xi_{vT}(r)=\frac{\langle n_T(r)\rangle}{\bar n_T}.$$0 is supplied by the galaxy peculiar velocity field. The coherent radial outflow is linked to the enclosed density contrast through the linear continuity equation,

$$1+\xi_{vT}(r)=\frac{\langle n_T(r)\rangle}{\bar n_T}.$$1

so the anisotropy of $$1+\xi_{vT}(r)=\frac{\langle n_T(r)\rangle}{\bar n_T}.$$2 directly probes $$1+\xi_{vT}(r)=\frac{\langle n_T(r)\rangle}{\bar n_T}.$$3 once the profile amplitude is fixed (Radinović et al., 2023).

The observed anisotropy is usually compressed into Legendre multipoles,

$$1+\xi_{vT}(r)=\frac{\langle n_T(r)\rangle}{\bar n_T}.$$4

with the monopole and quadrupole carrying most of the cosmological information in BOSS analyses, and the hexadecapole often added in Euclid forecasts (Nadathur et al., 2019, Radinović et al., 2023). In the BOSS configuration-space treatment where void centres are approximately in real space and galaxy velocities are radial and spherically symmetric about the void centre, the base redshift-space mapping can be written as

$$1+\xi_{vT}(r)=\frac{\langle n_T(r)\rangle}{\bar n_T}.$$5

which isolates the anisotropic RSD contribution in the term proportional to $$1+\xi_{vT}(r)=\frac{\langle n_T(r)\rangle}{\bar n_T}.$$6 (Nadathur et al., 2019).

Beyond the even multipoles, the dipole $$1+\xi_{vT}(r)=\frac{\langle n_T(r)\rangle}{\bar n_T}.$$7 is a distinct higher-order observable. When the real-to-redshift mapping is extended to second order in peculiar velocity and gravitational potential, the dipole is dominated by gravitational redshift inside voids, whereas the monopole is almost unaffected and the quadrupole receives smaller higher-order corrections. In that formulation the dipole becomes a probe of the gravitational potential profile of voids rather than merely of the density or coherent outflow field (Nan et al., 2018).

3. Void definition and measurement practice

The measured CCF depends on how voids are defined. In VIDE/ZOBOV-like methods, the tracer field is tessellated with Voronoi cells, local densities are taken as inversely proportional to cell volumes, and a watershed transform merges low-density basins into voids; the resulting centre may be the macrocenter or volume-weighted barycentre, and the effective radius is usually

$$1+\xi_{vT}(r)=\frac{\langle n_T(r)\rangle}{\bar n_T}.$$8

Alternative choices include the centre of the largest empty sphere in a watershed basin, the minimum-density voxel in a gridded watershed, the area-weighted centre of a 2D photometric void, and the centre of a spherical underdensity in excursion-set-like finders. These centre definitions are not equivalent, and later systematics analyses show that the choice matters for AP and dipole measurements (Panchal et al., 2020, Nadathur et al., 2019, Fraser et al., 2024, Xiong et al., 6 Nov 2025).

The CCF is estimated from pair counts. Survey analyses often use a Landy–Szalay form,

$$1+\xi_{vT}(r)=\frac{\langle n_T(r)\rangle}{\bar n_T}.$$9

with voids and galaxies as the two species, while periodic-box analyses can use natural estimators such as $\xi_{vg}(r)$0 because no survey mask is present (Radinović et al., 2023, Correa et al., 2021). Earlier SDSS void work also used a Davis–Peebles estimator,

$\xi_{vg}(r)$1

and verified that a symmetric Landy–Szalay-type estimator gives essentially the same result for the data considered (Paz et al., 2013).

Binning conventions are analysis-specific but conceptually stable. BOSS measured $\xi_{vg}(r)$2 in 30 bins over $\xi_{vg}(r)$3 and 80 $\xi_{vg}(r)$4-bins, then compressed to monopole and quadrupole (Nadathur et al., 2019). The DESI forward-model study measured $\xi_{vg}(r)$5 in 20 logarithmic transverse bins over $\xi_{vg}(r)$6 and 200 linear bins over $\xi_{vg}(r)$7, then projected to $\xi_{vg}(r)$8 (Salcedo et al., 11 Apr 2025). Euclid forecasts use 30 radial bins up to $\xi_{vg}(r)$9, 200 angular bins in $\xi_{vg}(r_p,\pi)$0, and the multipoles $\xi_{vg}(r_p,\pi)$1 as the redshift-space data vector (Radinović et al., 2023).

4. Alcock–Paczynski distortions, reconstruction, and systematic effects

A central assumption in many AP analyses has been that void centres transform under a change of fiducial cosmology in the same way as galaxies. Stretched-box tests show that this is not generally correct: applying AP distortions to the tracer field and then running the void finder is not equivalent to finding voids first and then rescaling their centres. The response of the void finder reduces the amplitude of the AP signal in the CCF, the size of the effect depends on the void-finding package, and incorrect treatment biases recovered parameters for $\xi_{vg}(r_p,\pi)$2, $\xi_{vg}(r_p,\pi)$3, $\xi_{vg}(r_p,\pi)$4, and the spherical finder in $\xi_{vg}(r_p,\pi)$5 (Radinović et al., 2024).

A more complete redshift-space picture identifies several distinct effects on voids and on the void–galaxy CCF: a systematic expansion induced by galaxy dynamics, the AP volume effect, a systematic off-centring along the line of sight caused by void dynamics, and distortions associated with void ellipticity. Off-centring and ellipticity are detectable in projected versions of the CCF and generate additional anisotropies beyond the standard Gaussian streaming description unless they are explicitly included. In the simplified tests where all these effects are accounted for, the Gaussian streaming model remains robust (Correa et al., 2021).

Velocity-field reconstruction is therefore used to mitigate anisotropic selection bias before void finding. In BOSS, reconstruction was introduced specifically to remove the complicating effects of RSD in the void centre positions themselves, enabling a joint AP and RSD fit to the anisotropic CCF (Nadathur et al., 2019). Euclid forecasts adopt the same principle: remove large-scale redshift-space distortions from the galaxy field, find voids in the reconstructed catalogue, and interpolate the void catalogue and the CCF over a grid of reconstruction efficiencies parametrized by $\xi_{vg}(r_p,\pi)$6. This yields nearly isotropic post-reconstruction galaxy clustering on large scales and stabilizes the modelling of both $\xi_{vg}(r_p,\pi)$7 and $\xi_{vg}(r_p,\pi)$8 (Radinović et al., 2023).

5. Cosmological information content

The CCF already provided a low-redshift consistency test in 6dF. Fitting a self-consistent RSD model to the 2D galaxy–galaxy and void–galaxy correlation functions recovered

$\xi_{vg}(r_p,\pi)$9

from galaxy clustering and

$\xi^s(s,\mu)$0

from the void–galaxy CCF, indicating consistency of the growth rate measured in overdense and underdense environments within the same dataset (Achitouv et al., 2016).

In BOSS CMASS, the anisotropic void–galaxy CCF became a precision AP/RSD probe. A joint fit to RSD and AP, combined with velocity-field reconstruction to remove void-centre RSD, yielded

$\xi^s(s,\mu)$1

at $\xi^s(s,\mu)$2, a $\xi^s(s,\mu)$3 AP measurement, and

$\xi^s(s,\mu)$4

a $\xi^s(s,\mu)$5 growth-rate measurement from voids alone. Combining void information with BAO and galaxy RSD in the same sample improved $\xi^s(s,\mu)$6, $\xi^s(s,\mu)$7, and $\xi^s(s,\mu)$8 constraints and reduced uncertainties on extended-model parameters such as $\xi^s(s,\mu)$9, $$w_{p,vg}(r_p)=2\int_0^{\Pi_{\max}}\xi_{vg}(r_p,\pi)\,d\pi$$0, and $$w_{p,vg}(r_p)=2\int_0^{\Pi_{\max}}\xi_{vg}(r_p,\pi)\,d\pi$$1 when combined with Planck (Nadathur et al., 2019).

Forecasts for Euclid spectroscopic voids push the same observable into the Stage-IV regime. With voids identified after RSD reconstruction, the expected precision is about $$w_{p,vg}(r_p)=2\int_0^{\Pi_{\max}}\xi_{vg}(r_p,\pi)\,d\pi$$2 on the ratio $$w_{p,vg}(r_p)=2\int_0^{\Pi_{\max}}\xi_{vg}(r_p,\pi)\,d\pi$$3 and between $$w_{p,vg}(r_p)=2\int_0^{\Pi_{\max}}\xi_{vg}(r_p,\pi)\,d\pi$$4 and $$w_{p,vg}(r_p)=2\int_0^{\Pi_{\max}}\xi_{vg}(r_p,\pi)\,d\pi$$5 on $$w_{p,vg}(r_p)=2\int_0^{\Pi_{\max}}\xi_{vg}(r_p,\pi)\,d\pi$$6 in each of four redshift bins covering $$w_{p,vg}(r_p)=2\int_0^{\Pi_{\max}}\xi_{vg}(r_p,\pi)\,d\pi$$7. In flat $$w_{p,vg}(r_p)=2\int_0^{\Pi_{\max}}\xi_{vg}(r_p,\pi)\,d\pi$$8CDM this translates to

$$w_{p,vg}(r_p)=2\int_0^{\Pi_{\max}}\xi_{vg}(r_p,\pi)\,d\pi$$9

and in flat $C^{vg}_{ij}(\ell)$0CDM to a dark-energy equation-of-state constraint of about $C^{vg}_{ij}(\ell)$1 from voids alone (Radinović et al., 2023).

Forward-model approaches reinforce the same conclusion but change the modelling strategy. An HOD-based DESI Year 5 forecast using the joint data vector $C^{vg}_{ij}(\ell)$2 predicts $C^{vg}_{ij}(\ell)$3 and $C^{vg}_{ij}(\ell)$4 constraints on $C^{vg}_{ij}(\ell)$5 and $C^{vg}_{ij}(\ell)$6, and shows that the combination of void and galaxy summary statistics breaks degeneracies in the galaxy–halo connection and cosmology relative to galaxy clustering alone (Salcedo et al., 11 Apr 2025). A neural-network emulator trained on AbacusSummit and fit to the BOSS void–galaxy CCF recovered

$C^{vg}_{ij}(\ell)$7

together with a $C^{vg}_{ij}(\ell)$8 reduction in errors for $C^{vg}_{ij}(\ell)$9 relative to a template-based method, while also showing that AP errors had previously been underestimated if void centres were assumed to respond to AP in the same way as galaxies (Fraser et al., 2024).

6. Variants, selection effects, and related extensions

The CCF is not determined by cosmology alone; it is also sensitive to how the tracer population is selected. A direct study of galaxy-property dependence found that void catalogues built from luminosity-selected galaxies and halos are consistent within errors in both the void size function and density profiles, whereas star-formation-rate-selected catalogues can differ, especially at tracer densities $$w_{p,vg}(r_p)=2\int_0^{\Pi_{\max}}\xi_{vg}(r_p,\pi)\,d\pi,$$0. In particular, SFR-selected voids can show a lower ridge amplitude in the density profile, consistent with the lower large-scale bias of SFR-selected galaxies relative to luminosity-selected galaxies at fixed number density (Panchal et al., 2020).

A broader unifying perspective comes from void–tracer cross-correlations. In void–lensing forecasts, the same large-scale machinery is written in terms of the void–matter cross-spectrum and its projection with a tracer kernel. Replacing the lensing kernel by a galaxy kernel reproduces the tomographic void–galaxy angular spectrum, and recent CSST work models the underlying void–matter cross-power with the Halo Void Dust Model and an HSW density profile before projecting to