Structure, averaging scale and homogeneity tests

A statistical symmetry does not erase local structure

Statistical homogeneity concerns whether a probability description changes under spatial translation. It does not require every small region in one realisation to contain the same matter or number of galaxies.

A homogeneous statistical ensemble can be clustered, and its conditional counts around selected objects can vary strongly with scale. Isotropy adds rotation symmetry; it is a different property from translation symmetry.

Observations provide finite samples, selection and a distance conversion rather than direct access to an entire ensemble. Operational homogeneity tests must therefore state their statistics, boundaries and tolerances.

The following model assumes stationarity and isotropy by construction. Its table illustrates scale-dependent clustering; it does not infer those symmetries for the Universe.

A positive probability construction defines the pair correlation

Choose arbitrary positive comoving length L at one epoch. Place mathematical parent points as a uniform three-dimensional Poisson process with intensity κ. Each parent has an independent Poisson number of daughter tracers with mean μ = 2.

Give each daughter an independent isotropic Gaussian displacement with per-axis standard deviation s = L/√2. Its three-dimensional density is f_s(x) = (2πs²)⁻³ᐟ²exp[−|x|²/(2s²)], and the mean tracer number density is n̄ = μκ.

Choose A = 1/2 and κL³ = 1/[A(2π)³ᐟ²] ≈ 0.12698727. The Gaussian displacement convolution gives ξ(r) = (f_s*f_s)(r)/κ = Aexp[−r²/(2L²)], with n̄L³ ≈ 0.25397454.

This is a supplied consistent clustered point process in three dimensions, newly derived with three-dimensional normalization. It is not a copied two-dimensional formula, a gravitational formation mechanism, a simulated observed catalogue or a fitted galaxy population.

Two scale statistics ask different questions

The expected number of other tracers within R around a selected tracer is N(<R) = 4πn̄∫₀ᴿ[1+ξ(r)]r²dr. The central tracer is excluded. These are ensemble expectations, not integer counts from a realised catalogue.

Define scaled counts C(R) = N/[n̄(4π/3)R³] = 1 + 3R⁻³∫₀ᴿξ(r)r²dr. A value above one indicates positive averaged pair excess relative to the supplied uncorrelated reference.

Define the logarithmic count slope D₂ = dln N/dln R = 3[1+ξ(R)]/C(R). This finite-scale statistic is not automatically a spatial dimension or a unique global fractal dimension.

At R/L = 0.25, D₂ is already near three even though C is about 1.49. At R/L = 2, D₂ is about 2.729 despite the assumed stationary ensemble. A local slope and a normalized count amplitude provide different information.

A chosen closeness criterion gives a chosen radius

For a teaching convention, require both |C−1| ≤ ε and |D₂/3−1| ≤ ε. These compare an amplitude and a relative slope departure; ε is a supplied tolerance, not an observational confidence level.

For ε = 0.02, a radius R/L ≈ 4.546626 satisfies both conditions, with C = 1.02 and D₂ ≈ 2.941224. Tightening to ε = 0.01 gives R/L ≈ 5.728624, C = 1.01 and D₂ ≈ 2.970297.

At R/L = 0.25, the relative slope departure is only about 0.00410, passing a one-percent slope criterion. But C−1 ≈ 0.490729 fails the one-percent amplitude criterion by a large margin.

The tolerance radii depend on the chosen correlation model and definition. They have units of the arbitrary L, with no conversion to an observed megaparsec scale or a physical wall separating two kinds of space.

Real data require a window, metric and uncertainty model

At large R/L in this supplied finite-correlation model, C approaches one and D₂ approaches three. This limiting behaviour does not turn the example into a proof of all higher-order statistical properties or uniformity across the entire Universe.

Observed counts require matched selection references, survey masks, edge corrections and treatment of shared objects and covariance. Complete ideal ensemble sampling in the model supplies none of those observational uncertainties.

Converting redshifts and angles into separations also uses a metric and cosmological distance assumptions. A test performed within that framework is a consistency test with stated conditions, rather than an automatically model-independent verdict.

Report the tracer population, reference normalization, scale range, tolerance, distance conversion and uncertainty treatment together. Local structure, an ensemble symmetry and an operational count criterion can then be compared without treating them as the same claim.

Worked reasoning

Try the reasoning

1. Does D₂ < 3 at an intermediate radius contradict stationarity in the supplied ensemble?

   Show the answer and reasoning

   No.

   Stationarity is imposed by the probability construction, while the conditional-count slope varies with clustering scale.

2. Is ε = 0.01 a 99-percent observational confidence statement?

   Show the answer and reasoning

   No.

   It is a chosen numerical closeness tolerance, with no observed sample or coverage model supplied.

3. Do the displayed tolerance radii measure a universal cosmic homogeneity scale?

   Show the answer and reasoning

   No.

   They are conditional outputs in arbitrary L units; real inference requires selection, geometry, covariance and a defined statistic.

What to carry forward

• Statistical homogeneity can coexist with substantial local clustering.
• Normalized conditional counts and their logarithmic slope provide different scale information.
• A finite tolerance test needs stated definitions and observational controls before it supports a cosmic inference.