Cosmology · The cosmic web, voids and intergalactic matter

Voids depend on definitions and tracers

Compare three explicit masks in the same eight-cell patch, distinguishing low matter density from absence of selected galaxy tracers.

Preparation: ratios, powers of ten and ordinary arithmetic. Physical quantities and notation are introduced as they are needed.

Back to the chapter

After reading, you can

Explain why a void catalogue needs an operational definition and a tracer sample.
Calculate selected volume and mass fractions using the correct weights.
Distinguish empty tracer cells from empty matter and a finite patch boundary from a cosmic boundary.

A void is not defined by the word empty alone

Cosmic voids are regions identified as relatively underdense under a specified description. Their boundaries and membership depend on the density estimate, selected tracers, smoothing scale and identification rule.

A region can contain few galaxies from one sample while still containing other galaxies, diffuse gas and dark matter. Absence of selected tracers is not a direct measurement of zero total matter.

Void finders can use density basins, adjacency and merging rules rather than a single spherical threshold. Comparing catalogues requires knowing those definitions and their treatment of survey edges and sparse sampling.

The following eight-cell construction uses three simple masks to isolate the role of a rule. It is not a watershed algorithm, a Voronoi density estimator or an observed void catalogue.

Keep matter and tracer inventories separate

Choose eight equal-volume cubes in a finite 2 × 2 × 2 patch at one epoch. Each physical cell volume is V_cell = a³L³ for common dimensionless a and positive comoving side L.

Label cells i = x + 2y + 4z for x, y and z equal to zero or one. There is no periodic wrap. Two cells are adjacent only if they share a face inside this patch.

Supply physical matter-density ratios ρ_i/ρ̄ = [0.1,0.3,0.4,0.7,0.8,1.4,1.9,2.4] for cells 0–7. They sum to eight, so ρ̄ is the finite-patch mean. Every cell has positive matter density.

Separately assign selected-tracer counts [0,0,1,0,0,2,3,4]. These counts are teaching inputs, not Poisson draws, a fitted galaxy-bias relation or a mass measurement obtained by counting tracers.

Equal volume does not imply equal selected matter

Rule A selects cells with ρ_i/ρ̄ < 0.5: cells 0, 1 and 2. They occupy 3/8 of the patch volume and contain (0.1+0.3+0.4)/8 = 0.1 of its matter mass.

Rule B uses the looser strict cut ρ_i/ρ̄ < 0.8, selecting cells 0, 1, 2 and 3. Cell 4, with ratio exactly 0.8, is excluded. The selected volume is 1/2 and selected mass fraction is 1.5/8 = 0.1875.

Rule C selects zero selected-tracer counts: cells 0, 1, 3 and 4. Its volume is also 1/2, but its mass fraction is 1.9/8 = 0.2375. Equal selected volume has not made it the same region as rule B.

The mean selected density ratio is the sum of density ratios divided by the number of selected equal-volume cells. For A, B and C it is 4/15, 0.375 and 0.475 respectively.

Three chosen masks in one finite eight-cell patch; volume and mass fractions use the whole patch as denominator
Rule	Selected cells	Volume fraction	Mass fraction	Mean ρ/ρ̄	Selected tracer count
A: ρ/ρ̄ < 0.5	0,1,2	3/8	0.1	0.266667	1
B: ρ/ρ̄ < 0.8	0,1,2,3	1/2	0.1875	0.375	1
C: zero selected tracers	0,1,3,4	1/2	0.2375	0.475	0

A connected mask still needs an interpretation

All three selected sets are face-connected within the supplied patch. That follows from the cell labels and adjacency rule; it is not a general statement that threshold-selected regions must always be connected.

The A mask contains one selected tracer in cell 2 despite being selected for low matter density. The C mask contains no selected tracers yet has nonzero matter in all four of its cells.

Touching a boundary of the eight-cell patch does not establish where a real cosmic void ends. The model supplies no information outside that boundary and no rule for joining neighbouring patches.

The chosen cut values are definitions, not significance levels or physical discontinuities. Nothing in this inventory computes a void expansion history, gravitational potential or statistically preferred threshold.

Use one finite 2×2×2 patch at one epoch, with eight equal physical cell volumes a³L³. The cell index is i = x+2y+4z with x, y, z each zero or one. Each of three repeated diagrams shows the same two exploded layers z = zero and z = one, with y = zero in the top row and y = one in the bottom row and x increasing to the right. Separation of layers and gaps between drawn squares aid reading and are not physical distances. A solid thick blue border marks IN, while a thinner dashed grey border marks OUT; membership therefore does not depend on color alone. Inside each cell the first line gives index and physical matter-density ratio ρ_i/ρ̄; the second gives selected-tracer count T. Ratios for cells 0–7 are 0.1, 0.3, 0.4, 0.7, 0.8, 1.4, 1.9, 2.4 and sum to eight, so ρ̄ is the finite-patch mean. Tracer counts are 0, 0, 1, 0, 0, 2, 3, 4 and are independently prescribed teaching inputs, not Poisson draws or a fitted matter conversion. Rule A selects strict matter ratio below 0.5: cells 0, 1, 2. It selects volume 3/8 = 0.375, mass fraction (0.1+0.3+0.4)/8 = 0.1, mean selected density 4/15 ≈ 0.266667 and one selected tracer. Rule B uses strict matter ratio below 0.8: cells 0, 1, 2, 3. Cell 4 at exactly 0.8 is excluded. B selects volume 1/2, mass fraction 0.1875, mean density 0.375 and one tracer. Rule C selects zero chosen tracers: cells 0, 1, 3, 4. C also selects half the volume, but mass fraction is 0.2375 and mean density 0.475, with zero selected tracers. Equal volume has not made B and C the same region. Every cell has positive matter, including all zero-tracer cells. Adjacency means shared faces inside the finite patch only, with no periodic wrapping; each of these three sets is connected by that rule. This does not imply all threshold masks are connected. There is no information outside the patch and no real cosmic void boundary follows from touching its edge. These are chosen finite masks, not ZOBOV, a Voronoi estimator, watershed algorithm, physical void expansion, observed catalogue or significance test. A selected tracer field is not automatically the total matter field; calibration, smoothing, survey selection and boundaries matter. Original Ark teaching model.

A useful void claim names its field and rule

In observations, tracer selection, sampling density, redshift errors and the survey window affect a density estimate and its accepted regions. A sparse sample can create apparent gaps even when the underlying matter field is not equally sparse.

A method that builds density basins from tessellation and adjacency has different operational steps from these simple cuts. Its significance calibration and boundary choices must be checked in their own setting.

Connecting a tracer-defined void to total matter or cosmological dynamics requires further calibration or modelling. The known matter ratios supplied here bypass that inference step and are not an observational reconstruction.

Report the tracer population, reference mean, density estimator, scale, boundary treatment and acceptance rule. This makes comparisons about void size, volume or matter content specific enough to assess rather than relying on an ambiguous image of empty space.

Worked reasoning

Use volume and mass denominators correctly

What fractions does rule A select?

Its three selected equal-volume cells give volume fraction 3/8.
Their density-ratio sum is 0.1+0.3+0.4 = 0.8, while the whole patch sum is eight.
Thus mass fraction is 0.8/8 = 0.1 and mean selected density ratio is 0.8/3 = 4/15.

Rule A selects 3/8 of the finite volume but 1/10 of its matter mass; these fractions answer different questions.

Try the reasoning

Are rules B and C the same region because both select half the volume?
Show the answer and reasoning
No.
Their memberships differ and their matter fractions are 0.1875 and 0.2375.
Does rule B include cell 4 with density ratio exactly 0.8?
Show the answer and reasoning
No.
Its stated inequality is strictly below 0.8.
Does a zero selected-tracer count establish zero matter in the supplied cell?
Show the answer and reasoning
No.
Every prescribed matter density is positive; tracer counts and matter inventories are distinct.

What to carry forward

A void catalogue depends on an explicit field, tracer sample and identification rule.
Selected volume, selected mass and mean density use different weights and denominators.
Empty tracer cells and finite mask boundaries do not establish empty matter or universal cosmic boundaries.

References

Sources supporting the explanations, physical distinctions and examples in this lesson.