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ZOBOV: a parameter-free void-finding algorithm

Mark C. Neyrinck

arXiv:0712.3049v2astro-ph

TL;DR

The paper addresses the absence of a standard, shape-independent definition for cosmic voids. It presents ZOBOV, a parameter-free Voronoi-based algorithm that identifies density minima, voids, and subvoids while estimating their Poisson-fluctuation probabilities. Applied to simulated dark matter, it also reveals a broad high-density peak near ρ ≈ 10^3ρ̄.

  • Problem

    No standard definition of cosmic voids exists, despite their value for studying cosmology and galaxy formation.

  • Method

    ZOBOV uses Voronoi-based density estimates and zoning around density minima without free parameters or shape assumptions.

  • Results

    A broad high-density peak appears in the logarithmically binned dark-matter particle-density distribution at ρ ≈ 10^3ρ̄.

  • Takeaways & Limitations

    ZOBOV provides statistical significance measurements and hierarchical void catalogues containing both voids and subvoids.

  • Takeaways & Limitations

    Testing the probability of rare, high-density contrasts is limited because the Poisson simulation used is not large enough.

Abstract

from arXiv · show

ZOBOV (ZOnes Bordering On Voidness) is an algorithm that finds density depressions in a set of points, without any free parameters, or assumptions about shape. It uses the Voronoi tessellation to estimate densities, which it uses to find both voids and subvoids. It also measures probabilities that each void or subvoid arises from Poisson fluctuations. This paper describes the ZOBOV algorithm, and the results from its application to the dark-matter particles in a region of the Millennium Simulation. Additionally, the paper points out an interesting high-density peak in the probability distribution of dark-matter particle densities.

1 INTRODUCTION

ZOBOV addresses the lack of a standard void definition with a parameter-free finder that places few restrictions on void shape and reports statistical significance. It returns voids and subvoids, trading straightforward catalogue analysis for a more inclusive description of density depressions.

  • Motivation: Voids probe cosmological parameters, dark-energy clustering, and galaxy formation through their structure and evolution.Their internal flows and ellipticities can carry information about cosmology.
  • Motivation: No consensus definition of a cosmic void exists, motivating another approach to void finding.The Aspen-Amsterdam comparison project explored differences among definitions but did not establish a consensus.
  • ZOBOV: ZOBOV seeks density depressions with no free parameters and without assuming a particular shape.Its conceptual void is a depression around a density minimum.
  • ZOBOV: ZOBOV returns many shallow voids, so users can apply statistical or physical significance criteria to select them.A physical criterion discussed is ρmin/ρ̄ < 0.2.
  • ZOBOV: Unlike most void finders, ZOBOV reports subvoids alongside larger voids, producing hierarchical catalogues that may be less straightforward to analyse.The algorithm’s relation to VOBOZ reverses the search from density maxima to minima.
  • Related work: Compared with WVF, ZOBOV analyses raw unsmoothed data rather than smoothing particle densities with mathematical-morphology techniques.Both methods use tessellation and watershed concepts.

2 METHOD

ZOBOV uses the first two steps of VOBOZ, but searches for density minima rather than maxima. This inversion adapts the procedure from halo finding to void finding.

  • Core algorithm: ZOBOV is identical to VOBOZ’s first two algorithmic steps except that it searches for density minima instead of maxima.VOBOZ is the VOronoi Bound Zones dark-matter-halo finder.

2.1 Particle density and adjacency measurement

ZOBOV estimates local particle densities and adjacency relationships from the Voronoi tessellation. Each particle’s density is the inverse volume of its Voronoi cell, and neighbouring cells define the next-step connectivity.

  • Density estimation: The Voronoi Tessellation Field Estimator assigns particle i a density of 1/V(i), where V(i) is its Voronoi-cell volume.The cell contains the region closer to particle i than to any other particle.
  • Adjacency measurement: Voronoi neighbours are particles whose cells share a boundary, providing ZOBOV’s natural adjacency set.This adjacency information is used in the subsequent zoning step.
  • Illustration: Figure 1b shows the tessellation of a two-dimensional particle set, with cells shaded according to area.The particle set corresponds to galaxies in a slice of the Millennium simulation.

2.2 Zoning

ZOBOV partitions particles into zones by following each particle toward its lowest-density neighbour until reaching a local minimum. Each zone consists of particles flowing to the same minimum, whose lowest-density particle forms the core.

  • Zone construction: A minimum is lower-density than every Voronoi neighbour, and particles repeatedly follow their lowest-density neighbour until reaching one.This downward-flow rule partitions the particle set around each density minimum.
  • Zone construction: A zone contains all particles that flow downward into one minimum, while its core is that zone’s minimum-density particle.Zoning both accelerates computation and compresses dataset information.
  • Interpretation: Because of discreteness noise, many zones are spurious or represent only central parts of larger voids.ZOBOV therefore requires a later step to combine zones into voids.

2.3 From zones to voids

ZOBOV grows voids from density-minimum zones by raising a watershed-like density level and annexing adjacent zones until a deeper zone is reached. This produces nested void structures but can also yield unexpected shapes under particular particle arrangements.

  • Void growth: ZOBOV raises a zone’s density level and adds adjacent zones reached by water until the flow enters a deeper zone.For the deepest void, flooding continues through the whole field except the zone separated by the highest-density link.
  • Void growth: The resulting void retains the original zone’s minimum-density particle as its core, while shallow zones may remain equal to themselves.The link density ρl(z) records the level at which water flows into a deeper zone.
  • Shapes and assumptions: ZOBOV can produce surprising topologies, including a single dumbbell-shaped void when a low-density particle connects two visually distinct voids.The density estimate involves about 16 particles, so such a connection would require a coordinated particle arrangement and might also appear significant by eye.
  • Figure 1: Figure 1 illustrates the pipeline from particles and Voronoi cells to zones and staged growth of the deepest void.Cell shading represents Voronoi-cell area, zone colours distinguish regions, and progressively lighter colours mark later additions.

2.4 Statistical significance of voids

ZOBOV evaluates void reality through the density contrast between a void minimum and the ridge leading to a deeper void, calibrated against Poisson simulations. The calibration supplies significance estimates, but its high-contrast tail is limited by sparse simulation data.

  • Density contrast: ZOBOV defines a void’s density contrast as r(v)=ρl(v)/ρmin, comparing the link density to the void’s minimum density.The link is the minimum-density particle on a ridge beyond which lies a deeper void.
  • Poisson calibration: The Poisson calibration uses cumulative contrast distributions from 128^3- and 256^3-particle simulations together with a fitted function.Figure 2 compares the simulation curves, the fit, and the AAVFCP void distribution.
  • Poisson calibration: P(r) estimates the likelihood that a void with contrast r arises from Poisson noise and is therefore fake.The estimate is obtained by comparing the contrast with a uniform Poisson particle distribution.
  • Limitations: The fit is trusted only to roughly r=3 because the underlying Poisson data do not adequately test rarer, higher-density contrasts.The fit was based on a relatively small 256^2-particle sample and extends to approximately P(r)≈10^-3.
  • Application to AAVFCP: r ≈ 1.5 is where the AAVFCP and Poisson curves intersect, corresponding to about 2-σ and seeming too low as an acceptance threshold.This provides a possible stopping point for accepting voids as real, but the paper regards its significance as low.
  • Poisson calibration: 335025 voids were detected in the 256^3-particle Poisson simulation, with an average of 50.1 particles per zone.A void can contain multiple zones, so its average particle count exceeds the average zone size.
  • Physical criterion: A physical criterion additionally requires ρmin<0.2 for dark-matter voids, though it may be inappropriate for galaxies.In the AAVFCP sample, statistically significant voids above about 3-σ already satisfy this cutoff.

2.5 Defining the edges of voids

ZOBOV offers three ways to define void edges: retain the hierarchy, excise significant subvoids, or select each void’s most-probable extent. The latter evaluates successive zone additions using statistical significance and can prevent growth into dense regions.

  • Alternative edge definitions: ZOBOV can retain a large void together with subvoids and sub-subvoids, preserving a hierarchical representation of structure.Zones may belong to multiple voids and subvoids.
  • Alternative edge definitions: Significant subvoids can instead be excised from parent voids, producing disjoint voids when that is desired.Removing a subvoid also removes zones joining the parent during the same or later accretion events.
  • Most-probable extents: The most-probable extent method compares the probability that separate zones are fake with the probability that their union is fake at each zone-adding event.The significance is updated as zones are successively added to a void.
  • Most-probable extents: For zones 3, 3′, and 3′′, the separate-fakeness probability is 0.30, whereas their union has probability 0.0013, favoring the union statistically.The union is therefore treated as a more significant structure than the three zones separately.
  • Most-probable extents: Void 1 reaches a local minimum after three additions, but its significance later changes as dense zones are included, allowing a density limit of ρl,max = 0.2 to prevent growth into haloes.An alternative is to accept the lowest minimum before the final downward ramp.
  • Most-probable extents: Figure 3b depicts nested extents from 1-σ through 4-σ, with the deepest 4-σ void lightest and 1-σ subvoids hatched.Particles in the darkest region belong to no void over 1 σ.

2.6 Selection functions, boundaries, and holes

Applying ZOBOV to observations requires handling selection functions, boundaries, and holes. The method can correct spatially varying sampling, but edge treatments and missing regions introduce practical ambiguity.

  • Selection functions: ZOBOV corrects a position-dependent selection function φ(x) by dividing each particle or galaxy density at x_i by φ(x_i).The VTFE is more natural than the DTFE for this correction because it estimates densities at particles.
  • Boundaries: ZOBOV is designed for periodic simulations, while isolated datasets require modifications to avoid spurious edge density minima.Unmodified edge particles can have arbitrarily large Voronoi volumes.
  • Boundaries: Two proposed boundary treatments are assigning edge particles densities above all interior values or adding a mean-density buffer zone around the dataset.The buffer can also suppress edge effects for particles slightly below the surface, but its construction is ambiguous.
  • Holes: Holes and significantly non-convex boundaries are identified as especially difficult, with possible remedies including mean-density Poisson sampling, interpolation, or constrained realizations.These approaches fill or estimate densities in otherwise missing regions.

3 RESULTS

In the Millennium Simulation application, ZOBOV found many low-significance voids and subvoids, while its highly significant void count was typical of other finders. The analysis also exposed boundary and probability-model issues for extreme void extents, alongside a distinct high-density particle population linked to collapsed structures.

  • 29 5-σ voids were detected in the AAVFCP region, a count typical of other void finders.ZOBOV found a couple of orders of magnitude more total voids because it also identified many low-significance voids and subvoids.
  • The deepest full-cube void had its core on the outer edge, possibly reflecting tessellation boundary conditions, so analysis focused on voids with cores in the inner cube.The largest significant void with a core in the inner AAVFCP cube did not encompass nearly the whole volume; the deeper edge-core void did.
  • The original probability fit placed the featured void’s most-likely extent at i = 2, whereas the 5-σ truncated extent occurred at i = 200 under a steeper fit.At i = 200, the density contrast reached r = 4.5, and the fakeness probability needed to fall by a factor of e351 for that extent to become most probable.
  • The Poisson simulation was too small to test the abundance of ZOBOV voids with density contrasts as high as the featured void’s.This limits confidence in calibrating probabilities for such rare, high-density-contrast cases.
  • 3.1 Lagrangian density distribution: Voids split into a low-density population with ρmin < 0.2 and a high-density population near ρmin ∼102.5; only one high-density void barely exceeded 3-σ, versus 10 expected from 3765 Poisson voids.All highly significant voids above roughly 3-σ were also physically significant, with ρmin < 0.2, supporting both significance measures.
  • 3.1 Lagrangian density distribution: The high-density void cluster coincided with a peak in P(ρ) near ρ ≈102.5, approximately the fiducial virialization density, indicating that these particles typically reside in collapsed structures.The peak was smaller in the inner cube than in the full cube, while the low-density peak near ρ ∼1 showed the opposite pattern.

4 DISCUSSION

ZOBOV’s parameter-independent, statistically calibrated, hierarchical void catalogue offers flexible structure but requires choices for interpretation and further methodological study. The discussion also identifies a high-density feature in the Millennium Simulation.

  • ZOBOV strengths: ZOBOV returns voids without free parameters, defining them as depressions around density minima, while implementation choices still act like parameters.The algorithm’s parameter-independence shifts ambiguity from tuning to interpreting many returned shallow voids.
  • ZOBOV strengths: ZOBOV assigns each void a probability of being real by comparing its density contrast with Poisson realizations.Users can impose a significance threshold or weight voids by their probability when analysing catalogues.
  • ZOBOV strengths: ZOBOV detects hierarchical voids and subvoids, but this structure is not straightforward to analyse with traditional methods.The algorithm also provides a mechanism for determining the most-probable extents of voids.
  • Possible improvements: The discussion proposes testing Delaunay-based density estimation because DTFE is more natural for minima, especially in sparse galaxy samples.For well-sampled N-body simulations, DTFE–VTFE differences are likely negligible; VTFE may better handle variable selection functions.
  • Possible improvements: The statistical-significance definition compares a void’s minimum density with the lowest ridge density beyond which lies a deeper void.The authors describe this measure as simple and easy to calculate, with probabilities that compare favourably to visual inspection.
  • Simulation result: A broad density peak appears near ρ ≈ 10^3 ρ̄ in the logarithmically binned dark-matter particle-density distribution.The paper suggests investigating whether the feature can be modelled with the halo model of large-scale structure.
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