GLOBAL KNOWLEDGE ARK

Cosmology · Galaxies as tracers of cosmic structure

Galaxy populations and samples

Compare an intrinsic population with a flux-selected sample, using explicit accessible volumes and expected counts.

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

  • Distinguish a galaxy population from the sample selected by an observation.
  • Calculate how luminosity changes flux reach and accessible volume in a defined ideal geometry.
  • Identify the extra selection and calibration information needed for real surveys.
01

An observed catalogue has conditions for entry

A galaxy population describes objects within a specified physical or statistical domain. A survey sample contains objects that pass particular observing and analysis conditions. These sets need not have the same mixture of properties.

Entry can depend on received flux, apparent size, surface brightness, angular coverage, image quality, colour selection and the success of a redshift measurement. A catalogue with many objects can still miss a class systematically.

Comparing samples therefore begins with a selection function: how likely is an object with given properties and location to enter the catalogue? A statement of the sky region and the intended population must accompany that function.

This lesson uses a deliberately simpler construction to isolate one effect. It supplies exact bolometric luminosities and a sharp flux threshold in a static Euclidean space, without treating these assumptions as a cosmological survey model.

02

More luminous objects can enter from a larger volume

Choose arbitrary positive units L_* of power and D_* of distance. Define F_* = L_*/(4πD_*²), V_* = 4πD_*³/3, dimensionless luminosity ℓ = L/L_* and distance r = D/D_*.

For isotropic radiation in this transparent static geometry, bolometric flux satisfies F/F_* = ℓ/r² for positive r. An ideal detection threshold F ≥ F_* permits r ≤ √ℓ.

Restrict the survey to D ≤ 2D_*. Its accessible radius is therefore r_max = min(2,√ℓ), and its accessible volume is V_max = V_*r_max³. The volume integral includes the regular limiting volume near the origin even though the point-source flux expression is not evaluated at D = 0.

Supply a faint class with ℓ = 1 and a bright class with ℓ = 4. Bright objects reach twice as far, but their accessible volume is eight times as large. Distance reach and volume reach are different factors.

03

A changed sample mixture need not mean changed intrinsic density

Give both classes the same uniform, non-evolving number density n = 10/V_*. Their expected selected counts are nV_max: 10 faint and 80 bright. These numbers are expectations, not observed catalogue counts.

The bright share of those expected counts is 80/(10 + 80) = 8/9, although the intrinsic bright fraction is 1/2. A ratio of expected counts is not generally the expected value of a fluctuating observed fraction.

In a common inner volume D ≤ D_*, both classes are fully included by this ideal threshold. Each then has expected count 10 and the bright share of expected counts is 1/2.

A volume-limited comparison can remove this particular luminosity-reach difference. It reduces the usable volume for the brighter class; it does not establish that every other survey selection effect has disappeared.

Two supplied uniform classes with equal intrinsic density 10/V_*; all counts are expected values
ClassL/L_*D_max/D_*V_max/V_*Expected flux-selected countExpected count inside D_*
Faint1111010
Bright4288010
04

A correction needs a known inclusion rule

Dividing each expected selected count by its accessible volume returns 10/V_* for both classes: 10/V_* and 80/(8V_*). Under these supplied assumptions, inverse-volume weighting recovers the equal density inputs.

That arithmetic does not prove a universally unbiased estimator for a real clustered, evolving catalogue. Detection probabilities, boundaries and the intended redshift interval must be known and must fit the statistical calculation.

A class with no chance of entering the sample cannot be reconstructed merely by giving observed objects larger weights. Objects near a poorly known threshold may also create large sensitivity to calibration and measurement errors.

Actual cosmological counts require the volume element and luminosity-distance relation of a specified model. Replacing Euclidean distance with a cosmological quantity while keeping the same volume formula would silently change the problem.

Flux reach changes expected sample countsThis original supplied comparison uses static transparent Euclidean space, isotropic bolometric radiation and a sharp threshold F >= F_*. Positive arbitrary units satisfy F_* = L_*/(4 pi D_*²) and V_* = 4 pi D_*³/3. For r = D/D_* > 0 and ell = L/L_*, flux is F/F_* = ell/r². Panel A plots the exact threshold ell = r² only on 0.25 <= r <= 2; both axes are linear dimensionless ratios. At a given positive r, a supplied object passes the threshold if its luminosity is at or above the blue curve. Orange circle faint class ell = 1 meets it at r = 1; green diamond bright class ell = 4 meets it at r = 2. The outer survey radius is 2 D_* and r_max = min(2,sqrt(ell)). No point-source flux is evaluated at zero distance. The volume integral includes the regular limit toward the origin, giving V_max/V_* = r_max³: one for faint and eight for bright. Each uniform non-evolving class has supplied density 10/V_*. Panel B has a separate linear zero-baseline expected-count scale. Blue circle bars give flux-selected expected counts 10 faint and 80 bright. Orange square bars give expected counts 10 and 10 inside the same inner radius D_*. The class labels and marker shapes supplement color. Bright share of flux-selected expected counts is 8/9, whereas the intrinsic and inner-volume share is 1/2. This is a ratio of expectations, not an observed fraction or a general expectation-of-ratios identity. Dividing the supplied counts by each accessible volume returns equal density 10/V_*. It does not prove an unbiased estimator for real clustered or evolving populations. No zero-inclusion class is recovered by weights. No cosmological distance/volume, bandpass correction, surface-brightness selection, observed galaxy counts, measured density or noise model is inferred.Flux reach changes expected sample countsStatic Euclidean toy; bolometric F / F_* = ell / r² for r > 0.r = D / D_*; ell = L / L_*; outer survey radius = 2 D_*.Both uniform classes have chosen density 10 / V_*.threshold ell = r²A · Luminosity ratio ell · linear scale012340.250.511.52Positive distance ratio r · finite domain 0.25 to 2Faint ell = 1 reaches r = 1; bright ell = 4 reaches r = 2.Passing the flux rule means ell ≥ r², within the survey cap.B · Expected counts · linear scale, zero baselineflux selectedcommon r ≤ 1020406080Same supplied classes · expected counts, not observationsFaintBrightAccessible volumes: 1 and 8 V_*; common inner volume: 1 V_*.Counts / own V_max return the same input density 10 / V_*.The bright share 8/9 is a ratio of expected selected counts.No observed catalogue, cosmological volume or noise law.
This original supplied comparison uses static transparent Euclidean space, isotropic bolometric radiation and a sharp threshold F >= F_*. Positive arbitrary units satisfy F_* = L_*/(4 pi D_*²) and V_* = 4 pi D_*³/3. For r = D/D_* > 0 and ell = L/L_*, flux is F/F_* = ell/r². Panel A plots the exact threshold ell = r² only on 0.25 <= r <= 2; both axes are linear dimensionless ratios. At a given positive r, a supplied object passes the threshold if its luminosity is at or above the blue curve. Orange circle faint class ell = 1 meets it at r = 1; green diamond bright class ell = 4 meets it at r = 2. The outer survey radius is 2 D_* and r_max = min(2,sqrt(ell)). No point-source flux is evaluated at zero distance. The volume integral includes the regular limit toward the origin, giving V_max/V_* = r_max³: one for faint and eight for bright. Each uniform non-evolving class has supplied density 10/V_*. Panel B has a separate linear zero-baseline expected-count scale. Blue circle bars give flux-selected expected counts 10 faint and 80 bright. Orange square bars give expected counts 10 and 10 inside the same inner radius D_*. The class labels and marker shapes supplement color. Bright share of flux-selected expected counts is 8/9, whereas the intrinsic and inner-volume share is 1/2. This is a ratio of expectations, not an observed fraction or a general expectation-of-ratios identity. Dividing the supplied counts by each accessible volume returns equal density 10/V_*. It does not prove an unbiased estimator for real clustered or evolving populations. No zero-inclusion class is recovered by weights. No cosmological distance/volume, bandpass correction, surface-brightness selection, observed galaxy counts, measured density or noise model is inferred. Original Ark teaching model.
05

Luminosity, stellar mass and observed band are different selections

At different redshifts, one observed band samples different emitted wavelengths. Rest-frame comparisons need spectral and bandpass calibration, commonly expressed through a stated K-correction convention. Bolometric flux and flux density in one band are distinct quantities.

A flux-complete sample is not automatically complete above a stellar mass threshold. Stellar mass is inferred using assumptions about stellar populations, dust and other properties; different populations can produce different light per inferred mass.

Star formation, gas consumption, mergers and feedback can also change galaxy properties across time. Selection and physical evolution may operate together, so a changed catalogue mixture alone does not identify the cause.

A useful galaxy statistic therefore states the population definition, observing window, inclusion rule, calibration and uncertainty treatment. The next lesson keeps galaxy number contrast separate from total matter contrast even when the sample has been defined.

Worked reasoning

Recover the supplied class densities

Do expected counts of 80 bright and 10 faint objects establish an intrinsic density ratio of eight?

  1. The faint class is visible in V_*; the bright class is visible in 8V_*.
  2. Compute the densities from the expected counts: 10/V_* and 80/(8V_*).
  3. Both equal 10/V_*, consistent with the equal-density inputs.

No. The count ratio reflects different accessible volumes in this example; the intrinsic densities are equal.

Try the reasoning

  1. If one class is four times as luminous, what are its relative distance and volume reaches under this threshold before the survey cap?

    Show the answer and reasoning

    Twice the distance and eight times the volume.

    Flux gives D_max proportional to √L, while Euclidean volume is proportional to D_max³.

  2. Is 8/9 here an observed fraction or a general expectation-of-a-ratio identity?

    Show the answer and reasoning

    Neither; it is the bright share of the supplied expected counts.

    The model supplies no realised count draw or noise law.

  3. Can inverse-volume weighting alone recover a class with zero inclusion probability?

    Show the answer and reasoning

    No.

    There are no observed members to weight, and its unobserved abundance requires additional information.

What to carry forward

  • A survey sample inherits an inclusion rule and can differ from its intended population.
  • In a defined ideal geometry, luminosity-dependent reach changes expected counts through accessible volume.
  • Real comparisons need selection, rest-frame calibration, population definitions and uncertainty treatment.

References

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

  1. Spectroscopic Target Selection in the Sloan Digital Sky Survey: The Main Galaxy SampleMichael A. Strauss; David H. Weinberg; Robert H. Lupton; Vijay K. Narayanan; James Annis; Mariangela Bernardi; Michael Blanton; Scott Burles; A. J. Connolly; Julianne Dalcanton; Mamoru Doi; Daniel Eisenstein; Joshua A. Frieman; Masataka Fukugita; James E. Gunn; Željko Ivezić; Stephen Kent; Rita S. J. Kim; G. R. Knapp; Richard G. Kron; Jeffrey A. Munn; Heidi Jo Newberg; R. C. Nichol; Sadanori Okamura; Thomas R. Quinn; Michael W. Richmond; David J. Schlegel; Kazuhiro Shimasaku; Mark SubbaRao; Alexander S. Szalay; Dan Vanden Berk; Michael S. Vogeley; Brian Yanny; Naoki Yasuda; Donald G. York; Idit ZehaviResearch paperarxiv.orgRead source
  2. The K correctionDavid W. Hogg; Ivan K. Baldry; Michael R. Blanton; Daniel J. EisensteinTeaching resourcearxiv.orgRead source
  3. The Evolution of the Stellar Mass Functions of Star-Forming and Quiescent Galaxies to z=4 from the COSMOS/UltraVISTA SurveyAdam Muzzin; Danilo Marchesini; Mauro Stefanon; Marijn Franx; Henry J. McCracken; Bo Milvang-Jensen; James S. Dunlop; J. P. U. Fynbo; Gabriel Brammer; Ivo Labbé; Pieter G. van DokkumTeaching resourcearxiv.orgRead source