Cosmology · Galaxies as tracers of cosmic structure

Correlation functions and clustering statistics

Count six distinct pairs on a supplied periodic line and compare their separation bins with an exact geometric reference.

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

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After reading, you can

Explain why a pair count needs a reference geometry and normalization.
Calculate a finite binned pair excess without confusing it with an ensemble correlation function.
Recognise the information and uncertainty lost when separations are grouped into coarse bins.

Clustering asks about excess pairs at a separation

A map with dense regions can suggest clustering, but a numerical statistic must state its reference. How many pairs would occur at the same separation if objects followed the chosen null distribution within the same observing window?

For a statistically homogeneous ensemble, a two-point correlation describes the fractional excess of finding neighbours at a separation relative to an uncorrelated reference with the stated mean density. A finite catalogue provides an estimator, rather than direct access to that ensemble.

Geometry and selection affect the reference pair distribution. Survey edges, holes, angular masks and distance-dependent inclusion can change pair counts without any new physical clustering.

Use a small one-dimensional construction to expose these steps. Its periodic geometry is supplied for arithmetic; it is not a claim that the Universe is a circle or that a four-object sample represents a three-dimensional survey.

Four supplied points make six pairs

Take a periodic line of circumference 10D_*, with arbitrary positive length unit D_*. Assign four labelled positions x/D_* = 0, 1, 2 and 6.

For each unordered pair use the shorter periodic separation d/D_* = min(|x_i−x_j|/D_*, 10−|x_i−x_j|/D_*). Its range is zero to five; no object is paired with itself.

There are 4 × 3/2 = 6 distinct unordered pairs. Listed by position pairs (0,1), (0,2), (0,6), (1,2), (1,6) and (2,6), the separations are 1, 2, 4, 1, 5 and 4 in units of D_*.

Choose two bins: 0 ≤ d/D_* < 1.5 and 1.5 ≤ d/D_* ≤ 5. Two assigned pairs are in the first bin and four in the second. The first upper endpoint is excluded and the last endpoint is included.

Normalize the pair counts before taking an excess

For two independent uniform positions on the same periodic line, the probability that their shorter separation is below t is 2t/(10D_*) for 0 ≤ t ≤ 5D_*. This follows from the interval of total length 2t around either point.

The two null-bin probabilities are therefore 3/10 and 7/10. For a catalogue of four independent uniform points, expected bin counts would be six times those probabilities: 1.8 and 4.2.

Normalize the assigned counts by six to obtain DD = 1/3 and 2/3. Define the finite binned excess ê = DD/p − 1 using the exact geometric null probability p. The results are 1/9 and −1/21.

The positive first-bin excess means more assigned close pairs than this geometric reference expects. The negative second-bin excess means fewer in the wider bin. It is not a negative number of objects or an attractive force measurement.

Finite assigned pair counts compared with the exact uniform-periodic-line reference; no observed data or error model
Separation bin d/D_*	Assigned pair count	Normalized DD	Null probability p	Null expected count	Finite excess ê
[0,1.5)	2	1/3	3/10	1.8	1/9
[1.5,5]	4	2/3	7/10	4.2	−1/21

This original mathematical example assigns four labelled positions 0,1,2,6 on a periodic one-dimensional line of circumference 10 D_*. It asserts no cosmic topology. Unordered non-self pairs are (0,1),(0,2),(0,6),(1,2),(1,6),(2,6), with shorter periodic separations 1,2,4,1,5,4 D_*. There are exactly six such pairs. The first bin is [0,1.5) and the second is [1.5,5] in separation/D_*; the first upper boundary is excluded, the final boundary included. Two assigned pairs fall in the first bin and four in the second. Panel A uses linear zero-based normalized pair probabilities, not counts or an unnormalized random catalogue. Solid blue circle bars show DD = 2/6 = 1/3 and 4/6 = 2/3. Orange square outline bars show the exact reference probabilities 3/10 and 7/10. For two independent uniform points on this periodic line, the shorter separation falls below t with probability 2t/(10 D_*) for 0 <= t <= 5 D_*; this geometric identity determines the reference, without finite random-catalogue noise. Null expected counts for four independent uniform points would be 1.8 and 4.2 out of six pairs. Panel B uses a separate signed dimensionless excess scale with explicit zero baseline. Its bars give DD/p - 1 = 1/9 and -1/21. Negative excess means fewer assigned pairs than the reference expectation, not negative objects or a force. The weighted sum (3/10)(1/9) + (7/10)(-1/21) is zero because the two bins exhaust the normalized pairs. Moving the last point from 6 to 5 D_* keeps these two coarse bin counts but changes full separations. Shared objects and total normalization do not make pairs or bins independent observations. No observed catalogue, three-dimensional ensemble correlation function, error interval, Poisson covariance, significance or physical clustering fit is supplied. Original Ark teaching model.

A few bins retain only part of the point pattern

The normalization imposes a relation between these two excesses: (3/10)(1/9) + (7/10)(−1/21) = 0. All six assigned pairs must appear somewhere in the two exhaustive bins.

Moving the point at 6D_* to 5D_* leaves the coarse counts at two and four. The full list of separations changes, so equal values in these two bins do not establish an identical point pattern or identical statistics at every scale.

Pair indicators can share an object, and bin estimates share the finite catalogue and its normalization. Treating every pair or every bin as an independent observation would need justification.

No uncertainty interval or significance is assigned here. A Poisson-looking count does not automatically supply the correct covariance, and a small excess does not establish a detection without a stated repeated-sampling model.

A real correlation estimate must follow the survey window

In real surveys, a reference catalogue can represent the angular and radial selection. Data–data, data–reference and reference–reference pair terms need their respective total-pair normalizations and an estimator suited to the problem.

An arbitrary regular grid or a correlated reference distribution cannot silently replace independent random points in the same formula. Changing the reference correlations can change the estimator unless the method accounts for them.

The periodic example uses a known analytic bin probability, so it does not introduce finite random-catalogue noise. Real boundaries and selection usually require a more complete treatment, including finite-window constraints and covariance.

Finally, galaxy clustering traces a selected population. Relating its correlation to matter or cosmological parameters also needs bias, distance-coordinate and redshift-space models. A pair statistic is one stage of inference, not a complete physical explanation.

Worked reasoning

Compare close pairs with the same geometry

What is the supplied first-bin pair excess when two of six pairs fall below 1.5D_*?

Normalize the assigned count: DD = 2/6 = 1/3.
The exact periodic-line reference is p = 2 × 1.5/10 = 3/10.
Compute ê = (1/3)/(3/10) − 1 = 1/9.

The finite first-bin excess is 1/9 relative to the stated reference; it is not an observed three-dimensional correlation or a significance.

Try the reasoning

How many distinct unordered non-self pairs do four objects have?
Show the answer and reasoning
Six.
Each of four objects has three partners, and division by two removes double counting.
Does ê = −1/21 mean there are negative pairs?
Show the answer and reasoning
No.
The assigned bin count is positive but lies below its normalized reference expectation.
If two patterns have the same two coarse bin counts, must their full separation distributions be identical?
Show the answer and reasoning
No.
Binning discards information; the supplied point move from 6D_* to 5D_* is an explicit counterexample.

What to carry forward

Clustering statistics compare normalized pairs with a stated geometric and selection reference.
A finite pair estimator differs from the ensemble correlation it may be designed to estimate.
Binning, shared objects and the survey window matter for interpretation and uncertainty.

References

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