Cosmology · Simulations, surveys and validation

Synthetic catalogues and survey selection

Pass a known two-class population through a supplied footprint and classification response, keeping expected counts distinct from a realized catalogue.

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

Follow class-conditional probabilities into selected expected counts.
Distinguish footprint loss from selection and classification.
Explain the conditions behind mean recovery and correlated count errors.

A simulated population is not yet an observed catalogue

An observational catalogue includes the effects of a measurement response and a selection process. Comparing an intrinsic simulated population directly with selected observations can compare different quantities.

A synthetic survey applies a declared observational response to a supplied population before testing an analysis. Its usefulness depends on whether the response includes the dependencies relevant to the intended test.

For example, detection or classification may depend on object properties and observing conditions. A successful mock recovery remains conditional on the mock and its response model.

The following cohort and probabilities are original synthetic inputs. They are not DES measurements, observed efficiencies, galaxy populations or survey area estimates.

True classes and observed labels are different

Supply a fixed cohort containing 120 true-class A objects and 80 true-class B objects. Different objects have independent categorical outcomes under the stated model.

Before footprint loss, the probabilities of observed labels (bright,faint,rejected) are (0.6,0.2,0.2) for A and (0.1,0.4,0.5) for B. Bright and faint are observed labels, not the true classes.

Independently retain each object with footprint probability m = 3/4, regardless of its true class or response outcome. An object outside the footprint is rejected. This independence is a supplied simplification.

For each true class, multiply the bright and faint probabilities by m. The final rejection probability is 1−m(R_bright+R_faint), including both footprint loss and rejection within the footprint.

A count ledger makes the selected comparison explicit

For A, the final probabilities are (0.45,0.15,0.4). Multiplying by 120 gives expected contributions (54,18,48).

For B, they are (0.075,0.3,0.625). Multiplying by 80 gives expected contributions (6,24,50). Thus total expected bright, faint and rejected counts are (60,42,98).

The expected footprint loss is 50 of 200 objects. Of the expected 150 within the footprint, another 48 are rejected by the response, leaving 102 selected objects.

Every count just calculated is an expectation. A realized catalogue fluctuates. Even an integer expectation such as 60 bright objects does not prescribe an exact realized count.

Known synthetic response; all contributions are expected counts and every probability is conditional on true class
True class	Fixed cohort	Final bright probability	Final faint probability	Final reject probability	Expected bright	Expected faint	Expected reject
A	120	0.45	0.15	0.4	54	18	48
B	80	0.075	0.3	0.625	6	24	50
Total	200	—	—	—	60	42	98

Two true-class count bars share one count scale. A contributes expected bright 54, faint 18 and rejected 48; B contributes bright 6, faint 24 and rejected 50. The category labels and numbers appear directly on or beside each actual bar segment; hatching and dot patterns distinguish selected and rejection channels without colour alone. A separate two-hundred-object expected inventory separates bright 60, faint 42, off-footprint 50 and rejection within the footprint 48. These four channels are disjoint by definition, and their sum is 200. Selected total is 102 and total rejection is 98. All plotted counts are expectations under the stipulated independent-object categorical response and independent footprint probability 3/4; no realized catalogue is asserted. The known-response inverse only recovers the exact mean. Bright/faint covariance is negative under this fixed-cohort law, not an independent Poisson law. Known synthetic response; all contributions are expected counts and every probability is conditional on true class. Columns: True class; Fixed cohort; Final bright probability; Final faint probability; Final reject probability; Expected bright; Expected faint; Expected reject. Rows: A; 120; 0.45; 0.15; 0.4; 54; 18; 48 / B; 80; 0.075; 0.3; 0.625; 6; 24; 50 / Total; 200; —; —; —; 60; 42; 98.

Optional mathematical path: inversion uses a known response

The selected two-by-two response is P = [[0.45,0.075],[0.15,0.3]], mapping the true-class count column to expected bright and faint counts. Its determinant is 99/800 > 0.

Applying its inverse to the exact mean vector (60,42) recovers (120,80). This is conditional mean recovery with known probabilities; it does not reconstruct individual objects or guarantee exact, positive estimates for every realized count pair.

Dividing the selected counts by m alone would remove only the stipulated footprint factor. Misclassification and within-footprint selection still mix the two true classes.

The ratios 54/60 = 0.9 and 18/42 = 3/7 describe A contributions relative to the corresponding total expected counts. They are not claims about the expected random purity in every possible realized catalogue.

A forward test needs its uncertainties and limitations

For these fixed cohorts and independent-object categorical outcomes, Var(C_bright) = 141/4 = 35.25, Var(C_faint) = 321/10 = 32.1 and Cov(C_bright,C_faint) = −99/10 = −9.9, in count-squared units.

The covariance is negative because one object cannot enter both disjoint selected labels. These are not two independent Poisson variables; that would be a different sampling model.

Real footprint and detection dependencies can correlate with class, position or observing conditions. An exactly invertible supplied response does not make an uncertain or incomplete real response harmless.

Synthetic catalogues are useful for tracing selection, response and analysis choices. Recovery under the same known response is a conditional test, while comparison with actual observations and response uncertainties remains necessary.

Worked reasoning

Separate footprint loss and response rejection

How do 200 supplied objects lead to 102 selected objects in expectation?

The independent footprint retains 3/4 of the cohort, giving expected inside count 150 and outside count 50.
Within the footprint, response rejection contributes another 48 objects in expectation.
The expected selected total is 200−50−48 = 102, split into 60 bright and 42 faint.

Both footprint loss and the conditional response shape the expected catalogue.

Try the reasoning

Must a realized catalogue contain exactly 60 bright objects?
Show the answer and reasoning
No.
Sixty is the expectation of a random count under the supplied categorical responses.
Does dividing bright and faint counts by 3/4 recover the true A and B counts?
Show the answer and reasoning
No.
That removes footprint loss alone; selection and classification still mix the true classes.
Does invertibility establish that an uncertain real survey response is harmless?
Show the answer and reasoning
No.
The exact mean recovery used perfectly known synthetic probabilities; unmodeled dependencies require separate checks.

What to carry forward

A synthetic observed catalogue includes response and selection.
Expected counts and ratios of expectations are distinct from realized outcomes.
Mean recovery remains conditional on the response and sampling assumptions.

References

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

Dark Energy Survey Year 3 results: cosmology from combined galaxy clustering and lensing – validation on cosmological simulationsJ. DeRose; R. H. Wechsler; M. R. Becker; E. S. Rykoff; et al. (DES Collaboration)Research paperarxiv.orgRead source