Cosmology · Clusters and gravitational lensing

Cluster populations, selection and mass proxies

Follow a finite supplied mass population through detection and proxy-bin assignment, separating true mass counts from catalogue expectations.

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

State why a cluster abundance requires a mass definition and survey selection.
Calculate expected counts after detection and imperfect proxy assignment.
Explain why one completeness correction cannot generally remove migration between mass bins.

A population count depends on what qualifies as a cluster

Galaxy clusters can connect the abundance of massive structures with cosmological growth. An abundance prediction and a catalogue count can be compared only after defining the objects, their mass convention, observing volume and selection.

One common operational convention specifies a spherical radius R_Δ,ref within which mean enclosed mass density is Δ times a chosen reference density ρ_ref. The associated mass is M_Δ,ref = (4π/3)Δρ_refR_Δ,ref³.

The reference may be a background mean matter density or a critical density, and Δ must also be stated. Two masses using different radii or reference conventions are not interchangeable measurements of one fixed aperture.

Mass is usually inferred through observables or gravitational models rather than directly read from a catalogue. Detection, the mass–observable relation, scatter and calibration therefore enter the same population calculation.

A supplied mass function gives expected counts

Choose one common finite volume V₀ at one reference epoch and one consistent mass convention. Let m = M/M₀ for arbitrary positive mass unit M₀. Supply dN/dln m = (800/3)m⁻² only for 1 ≤ m ≤ 4.

Here ln is the natural logarithm, and dN/dln m describes expected counts in V₀ per logarithmic mass interval. Dividing by V₀ would give a number-density function. This is not a fitted halo mass function or an observed count distribution.

Since dln m = dm/m, integration over [a,b] gives N[a,b] = (400/3)(a⁻² − b⁻²). The bins 1–2 and 2–4 therefore have expected counts 100 and 25. The shared endpoint has zero measure in this continuous expectation.

There are no objects outside the supplied mass range in this construction. No redshift evolution, blends, unrelated contaminants or counting-noise model is added.

Two stages change the catalogue mixture

Supply detection probabilities 0.8 for the lower true-mass bin and 0.5 for the upper bin. Expected selected counts are then 80 and 12.5. Expected counts need not be integers.

Conditional on being selected and on true bin, assign lower-bin objects to the lower proxy bin with probability 0.9 and upper proxy bin with probability 0.1. For selected upper true-bin objects, use probabilities 0.2 and 0.8 respectively.

The lower proxy expectation is 0.9(80) + 0.2(12.5) = 74.5. The upper is 0.1(80) + 0.8(12.5) = 18. Each conditional pair sums to one, so proxy assignment preserves the total selected expectation 92.5.

These probabilities are a deliberately supplied joint response. In real surveys, selection acts on observables and can correlate with proxy or follow-up residuals; independently attaching unrelated correction factors would need justification.

Chosen expected-count flow in one finite volume; true mass bin, selection and proxy assignment are all supplied
True mass/M₀	True expectation	Detection probability	Selected expectation	To lower proxy bin	To upper proxy bin
1–2	100	0.8	80	72	8
2–4	25	0.5	12.5	2.5	10
Total	125	—	92.5	74.5	18

A high proxy bin contains contributions from both true bins

Of the 18 expected upper-proxy objects, 10 come from the upper true-mass bin and 8 from the lower bin. The upper-proxy true-upper share is 10/18 = 5/9 under this response.

Dividing 18 by the upper true-bin detection probability 0.5 would give 36, rather than the supplied true upper expectation 25. That operation ignores upward migration and the upper-bin objects assigned downward.

If the entire response is exactly known, the two equations C_low = 0.72N_low + 0.10N_high and C_high = 0.08N_low + 0.40N_high can be inverted. Inserting 74.5 and 18 recovers N_low = 100 and N_high = 25.

This exact inversion is a property of the supplied two-bin system. Uncertain response, finite sampling, additional bins and correlated selection can make a real inference unstable or require a full likelihood. The calculation supplies no observational confidence interval.

At one chosen epoch and in one finite volume V₀, use one consistent mass convention and m = M/M₀. The supplied expected-count function dN/dln m = (800/3)m⁻² is restricted to 1 ≤ m ≤ 4. Its integral N[a, b] = (400/3)(a⁻²−b⁻²) gives 100 objects in true bin 1–2 and 25 in true bin 2–4. These are expectations, not an observed catalogue or a fitted halo mass function. Detection probabilities 0.8 and 0.5 give selected expectations 80 and 12.5. Conditional on selection and true bin, lower true objects enter lower or upper proxy bins with probabilities 0.9 and 0.1; upper true objects use 0.2 and 0.8. Four equally weighted schematic arrows are explicitly labelled with expected counts: lower true to lower proxy 72, lower true to upper proxy 8, upper true to lower proxy 2.5, upper true to upper proxy 10. Arrow length and stroke width do not encode probability or count. The separate horizontal stacked bars use a linear expected-count scale from zero to 80. Solid blue segments represent contributions from the lower true bin and orange segments contributions from the upper true bin. The lower proxy bar is 72 + 2.5 = 74.5; the upper proxy bar is 8 + 10 = 18. Both share the same zero baseline and count scale. The selected total is 92.5. Missed expectations 20 and 12.5 sum to 32.5 and are not proxy-bin contributions. The total true expectation remains 125. In the upper proxy bin the fraction of expected contributions from the upper true bin is 10/18 = 5/9. This is a ratio of supplied expectations, not a claim about the expectation of a random ratio. Dividing its mixture 18 by 0.5 gives 36, not the true upper count 25. Using the full known response C_low = 0.72N_low + 0.10N_high and C_high = 0.08N_low + 0.40N_high instead recovers the two true counts. No objects outside the finite mass range, redshift evolution, contamination, blends, counting-noise model or observational confidence interval is supplied. Each actual stacked-bar segment carries a centered circle for the lower true-bin contribution or a centered square for the upper true-bin contribution, matching the legend and the visible explanation. These symbols encode category only; the unchanged segment lengths encode expected counts. Lower true contributions are on the left and upper true contributions on the right in each bar. Original Ark teaching model.

The complete forward model carries the cosmological information

A cosmological cluster analysis connects a predicted mass population and volume to observable selection and calibrated mass proxies. Its inference can depend strongly on how those stages are connected.

A steep population means numerous lower-mass objects can contribute to a high-proxy sample through scatter. The resulting change cannot be assessed by considering the scatter distribution without the underlying population slope.

Independent mass information and follow-up observations can help constrain calibration, with their own geometric, astrophysical and selection assumptions. A lensing estimate also begins with projected information before any deprojection to a spherical mass convention.

State the mass definition, redshift and volume interval, observable selection, conditional response and uncertainty model together. That makes an abundance comparison a test of a specific physical and statistical account rather than a raw count of bright systems.

Worked reasoning

Trace both contributions to one proxy bin

Why does dividing the expected upper-proxy count 18 by 0.5 fail to recover the true upper count?

Selected lower-bin objects contribute 0.1 × 80 = 8 to the upper proxy bin.
Selected upper-bin objects contribute 0.8 × 12.5 = 10, giving a total expectation of 18.
Dividing the mixture by 0.5 gives 36; recovering the true upper expectation 25 requires the full two-bin response.

The upper proxy bin is a mixture with migration in both directions, so one completeness factor does not undo the whole response.

Try the reasoning

What is the total selected expectation before and after proxy assignment?
Show the answer and reasoning
92.5 in both cases.
Detection reduces 125 to 92.5; the conditional proxy probabilities redistribute those selected objects without losing them.
Can masses defined relative to mean matter density and critical density be exchanged without specifying their radii and conversion?
Show the answer and reasoning
No.
The reference density and overdensity convention define different apertures and associated masses.
Does an expected count of 12.5 assert that a realised catalogue contains half an object?
Show the answer and reasoning
No.
An expectation is a population-model average; realised counts would be integers under a specified sampling model.

What to carry forward

Cluster abundance uses a stated mass convention, volume and selection.
Detection and conditional proxy migration jointly transform expected true counts into catalogue counts.
Mass calibration and response uncertainty belong in the cosmological forward model.

References

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

Cosmological Parameters from Observations of Galaxy ClustersSteven W. Allen; August E. Evrard; Adam B. MantzTeaching resourcearxiv.orgRead source