Cosmology · Galaxies as tracers of cosmic structure

Evolution and selection across cosmic time

Compare two supplied populations with changing sample quotas, showing how a pooled trend can reverse while both within-category fractions rise.

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

Separate changing population properties from changing sample composition.
Calculate expected sample fractions and a common-reference comparison under a specified design.
Distinguish cross-sectional population evolution from individual descendant histories.

Earlier and later catalogues need comparable definitions

Looking farther into the past does not automatically produce a sample comparable to a nearby one. Flux reach, rest-frame band, mass completeness and classification can change with redshift and observing conditions.

At the same time, galaxy populations can physically evolve through star formation, gas consumption, mergers and feedback. Selection changes and real population changes can occur together.

A comparison therefore needs a defined target population, property categories, observing selection and calibration. Stellar mass and a category such as quiescent are inferred quantities with model and classification assumptions.

The following finite sampling model supplies perfect category labels and all inclusion probabilities. It illustrates a composition effect; its invented fractions are not measurements of star-forming or quiescent galaxies.

Two mass categories have different type fractions

At each of two hypothetical epochs, supply 100 objects in a lower-mass category and 100 in a higher-mass category. The category boundaries are held conceptually comparable; no physical mass thresholds or numerical redshifts are assigned.

Each object has a perfect binary type tag Q. Earlier, the Q counts are 20 in the lower category and 80 in the higher category, giving fractions 0.2 and 0.8. Later they are 40 and 90, giving 0.4 and 0.9.

At the earlier epoch, take fixed sample quotas of 10 lower-category and 90 higher-category objects. At the later epoch, reverse the quotas to 90 and 10. Within each category sample uniformly without replacement, independently of Q.

The sample size is exactly 100 at each epoch. Both categories have positive inclusion probability, and selection within a category is representative by construction. These assumptions make the following expectation calculation valid.

The pooled trend reverses despite two rising fractions

For a category with population size N, Q count K and fixed sample size n, the expected selected Q count is nK/N. Earlier, the two contributions are 10 × 0.2 = 2 and 90 × 0.8 = 72.

The earlier expected pooled Q fraction is (2 + 72)/100 = 0.74. Later, the contributions are 90 × 0.4 = 36 and 10 × 0.9 = 9, so the expected pooled fraction is 0.45.

The sample expectation falls even though the type fraction rises in both categories. More weight has moved to the category with the lower fraction. This is an explicit composition effect, not a contradiction.

Here the denominator 100 is fixed by design, so dividing the expected Q count by 100 does give the expected sample fraction. A realised sample may differ from that expectation; the table does not pretend to be one observed draw.

Invented two-epoch finite populations and fixed-quota sampling; Q is a supplied binary tag and counts are dimensionless
Epoch	Population Q fraction: lower / higher	Sample quota: lower / higher	Expected selected Q: lower / higher	Expected pooled fraction	Half/half reference fraction
Earlier	0.2 / 0.8	10 / 90	2 / 72	0.74	0.50
Later	0.4 / 0.9	90 / 10	36 / 9	0.45	0.65

A standardized comparison asks a different question

Choose a common reference that gives each category weight 1/2 at both epochs. Its Q fraction is (0.2 + 0.8)/2 = 0.50 earlier and (0.4 + 0.9)/2 = 0.65 later.

This comparison rises by 0.15, while the raw expected sample fraction falls by 0.29. Neither arithmetic result needs to be hidden: they summarize different mixtures under fully stated definitions.

Because the supplied populations have equal category sizes, the chosen half/half reference also matches their actual pooled population fractions. In general a standardized reference need not equal the true population mixture; its weights must be stated.

Estimating within-category fractions from data would bring sampling and classification uncertainties. The known population fractions used here avoid that inference step; standardization alone cannot correct unknown selection that depends on Q within a category.

This original finite construction supplies two separate cross-sectional populations labelled earlier and later, with no numerical cosmic time, redshift or descendant matching. Each epoch has 100 lower-category and 100 higher-category objects and a perfect binary tag Q. In the lower category Q fractions rise from 0.2 to 0.4, while higher-category fractions rise from 0.8 to 0.9. Panel A shows those four supplied fractions as discrete zero-baseline linear bars, with blue circles for the lower category and orange squares for the higher. At the earlier epoch fixed sample quotas are 10 lower and 90 higher; later the quotas are 90 and 10. Sampling within each category is uniform without replacement and independent of Q, with positive inclusion probability in both categories. Sample size is exactly 100 at each epoch. Expected selected Q counts are 2 and 72 earlier, then 36 and 9 later. Panel B has a separate linear zero-baseline fraction axis. Blue circle bars give expected pooled sample fractions 0.74 then 0.45, whereas orange square bars use the same chosen half/half reference at both epochs, giving 0.50 then 0.65. The category labels are ordered alternatives, not a continuous fitted time function. Because the denominator is fixed by the design, the expected selected count divided by 100 equals the expected sample fraction here; no general expectation-of-ratios identity is asserted. Both within-category fractions rise, but shifting sample weight toward the lower-fraction category reverses the pooled trend. The reference comparison rises by 0.15 while the expected raw sample comparison falls by 0.29. Equal supplied category sizes make the half/half reference match the supplied pooled population fraction in this construction; a reference need not equal a real population mixture in general. No realised sample draw, observed galaxy fraction, classification uncertainty, confidence interval, individual transition, quenching mechanism or identified descendant history is inferred. Unknown within-category selection or zero inclusion would require additional information rather than being fixed by these weights. Original Ark teaching model.

A population trend does not identify individual descendants

The earlier and later populations are separate cross-sectional sets. The construction provides no matching of individual objects across epochs, no transition rates and no record of which galaxies formed, merged or changed type.

A rising standardized Q fraction therefore does not by itself establish a particular quenching mechanism or individual evolutionary pathway. Several physical histories can give the same population summaries.

Real analyses also face changing inferred masses, uncertain rest-frame colours, measurement scatter and incomplete categories. If a category has zero inclusion probability, its missing fraction cannot be repaired from that sample by weighting alone.

A useful evolutionary result distinguishes the raw catalogue trend, the selected reference comparison and the physical interpretation. It carries the selection and uncertainty model through each step before attributing a change to cosmic evolution.

Worked reasoning

Keep the reference weights fixed

How can the expected sample fraction fall from 0.74 to 0.45 while both category fractions rise?

Earlier quotas give expected selected Q = 10(0.2) + 90(0.8) = 74 out of the fixed sample size 100.
Later quotas give 90(0.4) + 10(0.9) = 45 out of 100; the lower-fraction category now dominates the sample.
With the same half/half reference at both epochs, the fractions instead change from 0.50 to 0.65.

Changing sample composition reverses the pooled expected trend. The fixed-reference comparison isolates a different population summary, not an individual history.

Try the reasoning

Which within-category Q fractions increase in the supplied populations?
Show the answer and reasoning
Both: 0.2 to 0.4 in the lower category and 0.8 to 0.9 in the higher category.
The pooled sample decline arises from changed quotas, despite both increases.
Why can the expected selected Q count be divided by 100 to obtain the expected sample fraction here?
Show the answer and reasoning
The sample denominator is fixed at 100 by design.
This does not invoke a general identity equating expectations of arbitrary ratios with ratios of expectations.
Does the common-reference increase identify which earlier galaxies became later Q objects?
Show the answer and reasoning
No.
The model contains separate populations and no descendant matching or physical transition mechanism.

What to carry forward

Selection and physical evolution can change a catalogue at the same time.
A pooled trend can reverse when category weights change, even if every within-category fraction rises.
A fixed-reference population comparison needs explicit inclusion assumptions and does not reconstruct individual galaxy histories.

References

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