Astronomy · Observations that support cosmology

Galaxy maps, clustering and survey selection

Separate measured sky positions and redshifts from modelled spatial maps, then normalize a finite angular pair-count example before interpreting its estimator.

Continue in the approved chapter order. Necessary vocabulary, quantities and assumptions are explained in the lesson.

Back to the chapter

After reading, you can

Identify the observing and geometry assumptions behind a galaxy map.
Count distinct within-catalogue and cross-catalogue pairs consistently.
Calculate a normalized pair estimator and state its selection, finite-reference and physical-interpretation limits.

Begin with what the survey measured

A galaxy catalogue can retain angular positions, photometry, spectra, redshift estimates, uncertainty and quality information. A three-dimensional picture then converts those quantities into coordinates using additional assumptions. A plotted radial coordinate is not automatically a directly measured geometric distance.

Redshift-to-distance conversion needs a chosen expansion and geometry model, while motions relative to the assumed background can change redshift-space positions. The distinction matters when interpreting shapes, separations and apparent concentrations in a spatial map.

Before that conversion, angular patterns are already affected by where the survey could obtain usable data. Foregrounds, observing depth, image quality, target selection, spectral success and masks can vary across the footprint. A region with fewer catalogue objects can reflect selection as well as the underlying source distribution.

Reid and colleagues’ BOSS catalogue paper records angular and radial selection information alongside data and reference catalogues. Its distinct completeness definitions illustrate why a correction needs a stated numerator, denominator and observing process. One generic completeness percentage cannot describe every catalogue quantity.

A reference catalogue represents the opportunities to observe

A pair statistic counts source separations in a specified bin and compares them with a reference distribution following the relevant survey selection. In a real analysis, a random reference catalogue samples where sources could enter the dataset without inserting the clustering signal being measured.

The reference must include the appropriate mask and selection variation. Merely placing many reference points uniformly inside a bounding rectangle does not describe holes, variable depth or target restrictions. More reference points reduce sampling error only within the selection model supplied; they cannot discover omitted selection information.

For an original finite arithmetic example, use one thin great-circle strip with coordinate 0–10 degrees. This is a one-dimensional angular construction, not a two-dimensional sky mask or a physical three-dimensional galaxy box. Define separations as absolute coordinate differences and retain the bin 0.2 ≤ θ < 1.0 degree.

Supply ten data coordinates and twenty reference coordinates as listed below. The evenly spaced reference list is a declared illustrative list, not a random draw or the exact expected pair distribution of a continuous uniform sky. Its finite structure is retained in the calculated result; no unbiased population estimate or significance is asserted.

Complete original angular input lists in degrees, within the same declared strip. The small reference list serves only the finite normalization exercise.
Catalogue	Coordinates (degrees)
Data D, nD = 10	0.1, 0.3, 0.6, 1.1, 1.4, 5.1, 5.3, 5.6, 6.1, 6.4
Reference R, nR = 20	0.2, 0.7, 1.2, 1.7, 2.2, 2.7, 3.2, 3.7, 4.2, 4.7, 5.2, 5.7, 6.2, 6.7, 7.2, 7.7, 8.2, 8.7, 9.2, 9.7

Within-catalogue and cross-catalogue totals differ

For data-data pairs DD, count distinct unordered pairs: each pair of different data objects appears once and no object is paired with itself. Ten data points have 10 × 9/2 = 45 possible such pairs. The listed bin contains 14.

The two five-point data groups each contribute seven pairs in the bin, while separations between the groups exceed its upper edge. For instance, the pair 0.1 and 0.3 has separation 0.2 and is included, but 0.1 and 1.1 has separation 1.0 and is excluded.

Twenty reference points have 20 × 19/2 = 190 possible unordered reference-reference pairs. Adjacent points are separated by 0.5 degree and the next-nearest are separated by 1.0 degree. Only the 19 adjacent pairs enter this bin, giving RR = 19.

For data-reference pairs DR, each data object can pair once with each reference object. There are 10 × 20 = 200 possible cross-catalogue pairs, of which 28 enter the bin. Those denominators differ because cross-catalogue counting is not the same unordered within-catalogue operation.

Different catalogue sizes must not change the comparison arbitrarily

Normalize each raw count by its number of possible pairs: dd = 14/45 ≈ 0.311111111, dr = 28/200 = 0.14 and rr = 19/190 = 0.1. Each describes a fraction of the corresponding possible pairs in this finite construction.

With these normalized quantities and rr > 0, form the Landy–Szalay angular estimator what = (dd − 2dr + rr)/rr. Substitution gives (0.311111111 − 0.28 + 0.1)/0.1 ≈ 1.311111111.

Inserting the raw counts into that normalized-count formula would instead give (14 − 56 + 19)/19 ≈ −1.210526316. That substitution is wrong because the denominators differ, not simply because its sign is negative. A correctly formed correlation estimator can itself be negative.

The result is dimensionless but is not a posterior probability or a direct measurement of matter density. Here it is the deterministic statistic of the supplied lists. Its scientific meaning would depend on selection accuracy, the reference-generation process, uncertainty and the population model.

Exact finite pair normalization for the bin 0.2 ≤ θ < 1.0 degree. DD and RR are unordered distinct pairs; DR counts each cross-catalogue pair once.
Pair type	Raw count in bin	Possible pairs	Normalized count
DD	14	45	14/45 ≈ 0.311111111
DR	28	200	0.14
RR	19	190	0.1

A visible pattern still needs a model of how it was observed

The original Landy–Szalay analysis explicitly treats pair normalization, survey geometry and fluctuations. Finite reference catalogues and finite windows contribute limitations. A small structured reference list like this teaching list must not be treated as a noise-free continuous random reference.

A survey also estimates mean density within an observed window rather than sampling an unlimited universe. That finite-window constraint affects correlation estimation, and mismodelled selection can produce apparent clustering. Uncertainty analysis needs the relevant source and reference sampling, dependencies and scales.

To interpret galaxy clustering physically, additional assumptions connect selected galaxies with matter and translate angular or redshift separations into model quantities. Geometry, galaxy populations, motions and observing response remain part of that inference. A clustering statistic alone does not identify a particular unseen component.

Retain the catalogue definition, mask, quality and selection information, coordinate convention, pair counting, reference generation, estimator normalization and uncertainty. The next lesson examines another observational link: gravitational lensing, whose image information constrains a projected gravitational field under its own geometry and degeneracies.

Worked examples

Check the reference bin count

Why does the twenty-point reference list give RR = 19 in the declared bin?

Consecutive points have separation 0.5 degree, within the bin.
There are 19 such adjacent pairs.
Next-nearest pairs have separation exactly 1.0 degree, excluded by θ < 1.0.

Only the 19 adjacent pairs enter, out of 190 possible unordered reference pairs.

Calculate the finite estimator

What follows from DD = 14, DR = 28 and RR = 19 with nD = 10 and nR = 20?

Normalize by 45, 200 and 190 respectively: dd = 14/45, dr = 0.14, rr = 0.1.
Use what = (dd − 2dr + rr)/rr.
Evaluate (14/45 − 0.28 + 0.1)/0.1.

Approximately 1.311111111 for these supplied finite lists, without an assigned population significance or physical-density interpretation.

Try the reasoning

Is a separation of exactly 1.0 degree included in 0.2 ≤ θ < 1.0 degree?
Show the answer and reasoning
No.
The upper edge is explicitly excluded.
Why is the DR denominator 200 rather than 45 or 190?
Show the answer and reasoning
Every one of ten data objects can pair with each of twenty reference objects.
Cross-catalogue pair counting has nD × nR possible pairs.
Can a large uniform reference catalogue repair an unmodelled hole or depth variation in a real selection function?
Show the answer and reasoning
No.
The reference must encode that selection information before its size helps.
Does a negative correlation estimate by itself show that the arithmetic is wrong?
Show the answer and reasoning
No.
The normalization, estimator and sampling conditions determine validity; the statistic is not a probability.

What to carry forward

Galaxy maps combine measured evidence with selection and coordinate assumptions.
Normalize pair counts according to the actual counting convention and catalogue sizes.
A finite statistic requires selection, uncertainty and physical models before supporting a cosmic interpretation.

References

Teaching resources and research supporting this lesson’s explanations and examples.