VoidFinder Voids: Cosmic Underdensities

• VoidFinder Voids are contiguous, macroscopic underdense regions identified through a systematic wall/field classification and sphere-growing methodology.
• They exhibit sharply bucket-shaped density profiles with effective radii typically around 15–17 h⁻¹ Mpc and fill about 60–62% of surveyed volumes.
• VoidFinder catalogs provide a robust framework for studying environmental galaxy properties and testing cosmological models, validated by simulations.

Cosmic voids are dominant underdense regions shaping the large-scale structure of the Universe. The concept of "VoidFinder Voids" specifically refers to statistically significant, galaxy-defined cosmic voids identified using the galaxy-based VoidFinder algorithm, developed by Hoyle, Vogeley, El-Ad, and collaborators. VoidFinder Voids are crucial for quantifying the topology of the cosmic web, probing the growth of structure, and testing gravitational theories and cosmological parameters.

1. Definition and Identification of VoidFinder Voids

VoidFinder Voids are defined as contiguous, macroscopic underdense regions in the Universe, statistically identified through the spatial distribution of galaxies in redshift surveys. The VoidFinder algorithm proceeds through a sequence of discrete, quantitative steps:

1. Wall/Field Classification: Galaxies are classified as either "wall" galaxies—those tracing filaments and clusters—or "field" galaxies—candidates for the interiors of voids. In practice, a galaxy is deemed to be "field" if its third-nearest neighbor distance $d$ exceeds the threshold $d_3 + 1.5\sigma_{d_3}$, where $d_3$ is the mean third nearest-neighbor distance and $\sigma_{d_3}$ its standard deviation. For SDSS DR7, this is $d > 6.3\, h^{-1}\mathrm{Mpc}$, corresponding to local density contrasts $\delta\rho/\rho < -0.47$ (1103.4156).
2. Sphere-Growing and Merging: After wall galaxies are selected, the survey volume is gridded (typically with $5\,h^{-1}$ Mpc cubes). Spheres are grown from empty cells until each is bounded by four wall galaxies. Spheres with radii $>10\,h^{-1}$ Mpc are retained. Overlapping spheres are merged into a void if they overlap by more than 10% of their volume; otherwise, they seed a new void region.
3. Effective Radius: The effective radius $R_\mathrm{eff}$ of a void is defined by the volume of the merged region: $$R_\mathrm{eff} = \left(\frac{3V}{4\pi}\right)^{1/3}$$ where $V$ is the total void volume.

This process yields a catalog of voids that are nearly spherical, have well-defined boundaries, and isolate the most dynamically distinct underdense regions (1103.4156, Douglass et al., 2022, Zaidouni et al., 29 Mar 2024, Rincon et al., 31 Oct 2024).

2. Statistical Properties and Density Profile

VoidFinder Voids exhibit characteristic properties in real and simulated catalogs:

• Filling Factor: In SDSS DR7, VoidFinder identifies 1,054 voids in the northern galactic hemisphere (for $R_\mathrm{eff} > 10\,h^{-1}\mathrm{Mpc}$), occupying 62% of the survey volume (1103.4156). In updated samples (e.g., DESI DR1), similar filling factors ($\sim$60%) are recovered (Rincon et al., 31 Oct 2024).
• Size Distribution: The median effective radius is typically $15$–$17\,h^{-1}\mathrm{Mpc}$; the largest observed void slightly exceeds $30\,h^{-1}\mathrm{Mpc}$ (1103.4156, Douglass et al., 2022). The distribution is sharply truncated at large radii and shows a narrow spread—unlike watershed-based catalogs which can extend to much larger voids.
• Density Profile: The radial density profiles are sharply "bucket-shaped," showing a flat, underdense interior ($\delta\sim-0.9$) with a steep wall at the void boundary. At the void's edge, the density contrast is $\delta < -0.85$ (1103.4156). Interior densities may be below 10% of the cosmic mean; the density rises rapidly near the walls and transitions to the cosmic mean beyond.
• Profile Parameterization: The density jump at the edge is quantified as: $$\delta = \frac{\rho(r) - \bar{\rho}}{\bar{\rho}} < -0.85$$ where $\bar{\rho}$ is the cosmic mean.

3. Void Galaxies and Environmental Dependence

VoidFinder catalogs not only delineate the cosmic underdensities but identify the galaxy population residing within them:

• Galaxy Fraction: Within the typical SDSS DR7 volume-limited sample ($M_r<-20.09$), about 7% of galaxies fall inside voids, while the voids occupy 62% of the volume (1103.4156). This highlights the extreme emptiness of these regions.
• Galaxy Properties: Void galaxies—those with $M_r<-20.09$ found in void interiors—tend to be brighter (because of the magnitude limit), but more generally, studies using VoidFinder voids demonstrate that void galaxies are bluer, fainter, lower mass, and exhibit higher specific star formation rates compared to wall galaxies (Zaidouni et al., 29 Mar 2024, Curtis et al., 4 Jan 2024). The distinct properties are robust to the void definition, provided that misclassification (e.g., leaking of wall galaxies into the void sample, as is common in watershed/filling algorithms) is minimized.

4. Dynamical Distinctiveness and Comparison with Other Algorithms

A defining feature of VoidFinder Voids is their close correspondence to dynamically distinct underdense regions:

• Shell-Crossing Diagnostic: Voids identified by VoidFinder correspond to regions with very low or zero shell-crossing in the dark matter field, as quantified by the crossing-number (CN) formalism of the ORIGAMI algorithm (Veyrat et al., 2023). Almost half the matter in VoidFinder voids is CN=0, compared to $\sim$30% in ZOBOV/Voronoi-defined voids. This dynamically distinct signature factually distinguishes VoidFinder voids from those found by watershed algorithms, which often include significant wall material.
• Contamination and Boundary Effects: Watershed VoidFinder alternatives (e.g., ZOBOV/VIDE, V2, REVOLVER) identify voids through density topology; however, their boundaries tend to extend into walls, leading to a "leakage" problem—up to 23% of galaxies classified as wall galaxies by VoidFinder are classified as void galaxies by watershed finders (Zaidouni et al., 29 Mar 2024). This misclassification dilutes the environmental contrasts in galaxy properties and reduces the dynamical purity of the void sample.

The following table summarizes key differences:

Algorithm	Void Shape	Dynamical Purity	Boundary Control
VoidFinder	Spherical	High (CN=0)	Sharp (no leakage)
ZOBOV/VIDE, V2	Irregular	Low/Moderate	Fuzzy, wall leakage

5. Robustness, Simulations and Cosmological Consistency

The statistical properties of VoidFinder Voids are matched in cosmological simulations, demonstrating algorithm robustness and the insensitivity to sample specifics within the volume-limited, wall-field separation approach:

• Simulation Agreement: In $\Lambda$CDM N-body and SPH halo model simulations matched to the SDSS geometry, VoidFinder recovers void fractions (62–69% of volume) and median radii ($17\,h^{-1}$Mpc) substantially similar to those in the SDSS data (1103.4156).
• Boundary Effects: The algorithm's efficiency is verified through the inclusion of consistent treatments for survey boundaries—excluding spheres overlapping by more than 10% with the boundary ensures no spurious growth along survey edges (Douglass et al., 2022).
• Environmental Control: Volume–filling factor, size function, and galaxy occupation are all reproducible using realistic mock catalogs (Douglass et al., 2022).
• Public Catalogs: VoidFinder-produced catalogs are publicly available for both SDSS and DESI, including maximal spheres, overlapping spheres/hole catalogs (to paper void geometry and substructure), merged void regions, and explicit void galaxy lists (1103.4156, Douglass et al., 2022, Rincon et al., 31 Oct 2024).

6. Cosmological and Astrophysical Applications

VoidFinder Voids are foundational for a range of precision cosmology and galaxy evolution analyses:

• Environmental Studies: The robust classification of void environments enables quantitative studies of galaxy evolution, AGN incidence, star formation, and baryon content in underdense regions (Watson et al., 2022, Curtis et al., 4 Jan 2024).
• Cosmological Constraints: The geometrical and statistical properties of VoidFinder Voids (size function, profile, filling factor) are consistent with $\Lambda$CDM predictions, providing independent tests of cosmological models.
• Complementarity: The sharp separation between walls and voids augments other segmentation schemes and enables cross-correlation with, for example, Ly$\alpha$ absorbers and weak lensing maps, allowing precise environmental studies (Watson et al., 2022).

VoidFinder Voids serve as a benchmark for algorithmic, dynamical, and observational definitions of cosmic voids, providing a robust and interpretable framework for large-scale structure science. They align closely with theoretical models of underdensity evolution, but empirical properties—such as the density jump at the edge and the narrow size distribution—are set by their stringent wall/field-based definition.