Cosmology · From density fluctuations to growing structure

Density contrast and initial inhomogeneities

Compare densities with a stated reference and calculate how pairing cells changes contrast variance while preserving the same matter inventory.

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

Define dimensionless density contrast and distinguish it from density or density ratio.
Calculate volume-weighted reference density and contrasts in a supplied finite inventory.
Explain how spatial averaging can reduce contrast variance without removing mass or estimating an observational uncertainty.

Contrast measures a departure from a reference

Structure studies compare the distribution of matter with an explicitly defined reference. Here ρ_m is physical mass density, in kilograms per cubic metre, and ρ̄_m is a positive reference mass density at the same chosen epoch.

Define density contrast δ = (ρ_m − ρ̄_m)/ρ̄_m. A positive contrast denotes an overdensity relative to that reference; a negative contrast denotes an underdensity. Contrast is dimensionless, while the density difference in its numerator has mass-per-volume units.

The density ratio is ρ_m/ρ̄_m = 1 + δ. It should not be confused with δ itself. For a nonnegative mass density, δ cannot be less than −1; δ = −1 corresponds to zero density in this particular mass account.

A reference may be a specified cosmological background or a defined finite-region average. Those choices need not coincide. The example below uses an exactly supplied finite inventory and makes no measurement of the cosmic mean.

Volume weighting determines the reference density

For cells with physical volumes V_i, define ρ̄_m = ∑ρ_m,iV_i/∑V_i. This volume-weighted mean equals total mass divided by total volume. Unequal cell volumes would prevent replacing it by an unweighted average of cell densities.

Supply eight equal cells, each of volume V_unit, with densities 6, 10, 14, 10, 10, 6, 10 and 14 times an arbitrary positive ρ_unit. Their total mass is 80ρ_unitV_unit and total volume is 8V_unit.

The chosen reference is therefore 10ρ_unit. The two 6ρ_unit cells have δ = −0.4, the two 14ρ_unit cells have δ = 0.4, and the four remaining cells have δ = 0.

These are simultaneous cell values at one epoch. The inventory is not an observed density map or a set of isolated cells whose gravitational evolution has been solved. Its ordering supplies adjacent pairs for the next comparison.

One supplied eight-cell inventory at a single epoch; every physical volume is V_unit and the reference is 10ρ_unit
Cell	ρ_m/ρ_unit	Density ratio ρ_m/ρ̄_m	Contrast δ
1	6	0.6	−0.4
2	10	1	0
3	14	1.4	0.4
4	10	1	0
5	10	1	0
6	6	0.6	−0.4
7	10	1	0
8	14	1.4	0.4

Pairing changes the resolution rather than the epoch

Combine adjacent cells into pairs (1,2), (3,4), (5,6) and (7,8). Each pair has physical volume 2V_unit. Its average density is the mean of its two original densities because those two volumes are equal.

The four pair densities are 8, 12, 8 and 12 times ρ_unit. Their contrasts against the same 10ρ_unit reference are −0.2, 0.2, −0.2 and 0.2.

Total mass remains [2(8 + 12 + 8 + 12)]ρ_unitV_unit = 80ρ_unitV_unit, and total physical volume remains 8V_unit. The reference density has not changed. No matter has been removed or transported by calculating these averages.

The fine and coarse descriptions thus represent the same supplied distribution with different spatial resolution. An arrow between the two inventories would denote an averaging operation, rather than a later stage of physical growth or smoothing by a specified force.

Averaging hides some variation within pairs

Use the volume-population variance σ² = ∑V_iδ_i²/∑V_i for this inventory, whose mean contrast is zero. It is a deterministic spatial statistic of the supplied cells, not an uncertainty assigned to a survey estimate.

At the fine resolution, four squared contrasts are 0.16 and four are zero, giving σ_fine² = 4(0.16)/8 = 0.08. At the paired resolution, every squared contrast is 0.04, giving σ_coarse² = 0.04.

Within each pair, the two original contrasts differ by ±0.2 from their pair mean. The volume-averaged within-pair variance is therefore 0.04. In this example, 0.08 = 0.04 + 0.04 separates the between-pair variation from the variation hidden inside pairs.

A lower coarse variance does not show that an instrument became more precise or that fluctuations evolved away. It follows from averaging the same fixed inventory. A physical growth calculation would need dynamics and initial motion as well as density.

At one supplied epoch eight equal physical-volume cells have densities 6, 10, 14, 10, 10, 6, 10, 14 times an arbitrary positive mass density ρ_unit, in kilograms per cubic metre. Each volume is V_unit in cubic metres. Total mass is 80 ρ_unit V_unit and total volume is 8 V_unit, giving the defined finite-region reference ρ̄_m = 10 ρ_unit. Density contrast δ = (ρ_m−ρ̄_m)/ρ̄_m is dimensionless; density ratio is 1+δ, not δ. Panel A has a linear contrast scale from −0.4 to +0.4 and fine cell indices one to eight. Blue signed bars with circle endpoints show −0.4, 0, +0.4, 0, 0, −0.4, 0, +0.4. Explicit zero-height bars retain endpoint markers. Panel B separately averages adjacent pairs (1, 2), (3, 4), (5, 6), (7, 8), each with volume 2 V_unit. Its orange bars have twice the fine bar width and square endpoints; the same linear contrast scale gives −0.2, +0.2, −0.2, +0.2. Pair densities are 8, 12, 8, 12 times ρ_unit. Total mass, physical volume and reference remain unchanged. Bar height represents contrast and width records relative cell volume in this illustration; horizontal placement is cell ordering, not a distance or trajectory. The volume-population variance is 0.08 at fine resolution and 0.04 after pairing, with within-pair variance 0.04. Thus fine variance is between-pair plus within-pair variance. These are deterministic spatial statistics, not survey errors. Pairing is a calculation on the same inventory, not physical time evolution, mass transport, observed smoothing, isolated-cell dynamics or a gravitational growth solution. The reference is not an estimate of the cosmic mean. No observation or physical scale is inferred. Original Ark teaching model.

Scale and background belong with the quoted contrast

Density contrast becomes useful for structure evolution when its background, epoch, matter component and spatial scale are specified. Changing the smoothing scale can change the contrast statistic even before any dynamics is considered.

The reference background itself can evolve in an expanding cosmology. A later density and contrast must be compared at their own stated epoch, rather than against an earlier background without explanation.

Small contrasts can support a linear perturbation approximation under the relevant physical conditions. A chosen contrast value alone does not supply the pressure, particle motion, scale regime or omitted-term estimate needed to justify a particular error bound.

The next lessons separate pressure-driven sound, collisionless motion, linear gravitational growth and nonlinear collapse. The finite cell account supplies the definitions needed to distinguish those changes in physical state from a change in how a distribution is averaged.

Worked reasoning

Use one reference at two resolutions

What are the contrasts of cell 1 and the pair (1,2) against the supplied reference?

The reference is ρ̄_m = 10ρ_unit, so cell 1 gives δ_1 = (6 − 10)/10 = −0.4.
The two equal-volume cells average to (6 + 10)/2 = 8ρ_unit.
The paired contrast is (8 − 10)/10 = −0.2, while the pair still contains the combined mass of both cells.

Cell 1 has contrast −0.4 and its pair has contrast −0.2; the difference comes from spatial averaging at the same epoch.

Try the reasoning

If δ = 0.4, is the density ratio ρ_m/ρ̄_m also 0.4?
Show the answer and reasoning
No. It is 1.4.
Density ratio equals 1 + δ, while δ is the fractional departure from the reference.
Does pairing the cells remove mass or change the defined reference density?
Show the answer and reasoning
Neither.
Total mass remains 80ρ_unitV_unit and total volume remains 8V_unit, so the mean stays 10ρ_unit.
Is the change from variance 0.08 to 0.04 a measured reduction in survey uncertainty?
Show the answer and reasoning
No.
It is a deterministic change in the spatial variance of one supplied inventory under an averaging operation.

What to carry forward

Density contrast is a dimensionless departure from a specified reference at a stated epoch.
Volume weighting and smoothing scale are part of its definition and interpretation.
Spatial averaging can hide variance while preserving the same total mass and mean density.

References

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