Cosmology · Dark-matter candidates and discriminating tests

Small-scale structure and competing explanations

Separate a halo inventory from a luminous catalogue using a finite example in which two different inventories predict the same detected 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 gravitating halos, luminous occupants and detected objects.
Calculate a conditional selection response without double counting.
Identify which extra observation could break a supplied degeneracy.

Explain the route from matter to an observed object

Small-scale tests compare predictions for matter structure with observations of tracers. The connection includes how visible objects form and how the survey detects them.

A gravitationally defined halo is not automatically a detected galaxy. A halo can have no selected luminous occupant, and an existing occupant can fall below a survey’s threshold.

Momentum distributions and dark-sector interactions can change a predicted matter population. Visible-matter processes and selection can also change the observed catalogue, through distinct parts of the prediction.

The example below supplies two hypothetical inventories and their observation rules. Neither inventory is a calculated cold- or warm-dark-matter prediction, and no observed catalogue is supplied.

Keep the mass bins and volume common

Use the same finite volume and the same lower and upper halo-mass bin definitions for models A and B. Supply exact halo counts A: 100 and 40; B: 50 and 20.

In each bin, a halo either has or lacks one luminous occupant of the selected class. Supply luminous probabilities A: 0.1 and 0.5; B: 0.2 and 1. These probabilities are chosen, not inferred from observations.

Conditional on a luminous occupant existing, supply detection probabilities 0.5 in the lower bin and 0.8 in the upper bin for both models.

The expected detected count is N_halo × P(luminous|halo) × P(detected|luminous,halo). The last probability is conditional, so this product does not require an additional assumption that luminosity and detection are independent.

Supplied finite inventories with one possible selected occupant per halo
Model and bin	Halo count	P(luminous)	P(detected\|luminous)	Expected luminous	Expected detected
A lower	100	0.1	0.5	10	5
A upper	40	0.5	0.8	20	16
B lower	50	0.2	0.5	10	5
B upper	20	1	0.8	20	16

A catalogue alone need not fix the inventory

Both models predict expected luminous counts 10 and 20, and expected detected counts 5 and 16. The common detection response then preserves the degeneracy already present in the luminous population.

For the lower bin, A has twice as many halos as B but half the luminous probability. Their products are equal. Knowing only that product does not identify either factor separately.

These are expectations, not promises that two realized catalogues contain exactly the same integers. No correlation model, uncertainty distribution or likelihood has been supplied for comparing realizations.

Nor may the labeled mass bins simply be treated as directly measured galaxy masses. A real mapping from observable properties to a halo-mass definition must be modeled and checked.

An extra observable must carry different information

Stipulate a separate ideal mass-sensitive probe that selects each lower-bin halo with probability 0.5 regardless of whether it has a luminous occupant. This is a hypothetical response, not an implemented observing method.

Its expected selected lower-bin count would be 50 for A and 25 for B. Unlike the luminous count, this response directly retains the difference in the supplied lower-bin halo inventories.

A real probe proposed for this role would need its own forward response, calibration, contaminants and uncertainty model. It might involve gravitational effects, but calling a method gravitational does not make its response known.

The example shows what a degeneracy-breaking observation must accomplish: constrain a factor or combination that the original observation leaves undetermined.

This original finite example supplies the same volume and the same lower/upper halo-mass bins for two hypothetical models A and B. Neither is assigned to a real cold or warm dark-matter calculation. Every displayed selected count is an expectation, not a realized catalogue. The table columns are model and bin, supplied halo count, expected luminous count and expected detected count. Rows are A lower (100, 10, 5), A upper (40, 20, 16), B lower (50, 10, 5), B upper (20, 20, 16). Each halo either has one selected luminous occupant or lacks it. Chosen luminous probabilities are A: 0.1, 0.5 and B: 0.2, 1 in the lower and upper bins. Chosen conditional detection probabilities given that occupant are 0.5 lower and 0.8 upper for both models. Thus expected luminous counts are N_halo times P(luminous|halo); expected detections multiply by P(detected|luminous, halo). The product is conditional and does not require a separate assumption of unconditional independence between luminosity and detection. Panel A deliberately draws one shared pair of expectations for BOTH A and B in each categorical bin, rather than artificially separating identical A/B predictions. It has a zero-based linear expected-count scale from 0 to 25. Blue luminous bars with circle category cues have heights 10 and 20; orange detected bars with square cues have heights 5 and 16. Direct counts, shapes and the legend avoid reliance on color. Equal luminous/detected products leave halo number and luminous probability degenerate. They establish neither identical halos nor equality of the entire sampling distribution; no uncertainty or correlation model was supplied. Panel B is a separately stipulated ideal mass-sensitive response selecting each lower-bin halo with probability 0.5 irrespective of its luminous state. Its own zero-based linear expected-count scale runs from 0 to 60, with directly labelled Model A and Model B bars at 50 and 25. This explicitly hypothetical response retains the supplied halo-count difference. It is not an implemented, calibrated or demonstrated observing method, nor an assertion that a real gravitational probe sees every dark halo. A real method needs its forward response, contaminants, calibration and uncertainty. Count quantities and probabilities are dimensionless; no mass-bin numerical boundaries or physical volume units are invented. No observed missing-galaxy percentage, real selection completeness, statistical significance, preferred dark-sector theory or candidate viability is inferred. Original Ark teaching model.

Test both matter physics and the visible response

Real small-scale predictions can depend on formation history, baryonic dynamics, numerical resolution and the definition of a selected system. These are parts of the explanation, not optional corrections added after a mismatch.

An interaction model also needs its velocity and angular dependence. A particle momentum model needs a production distribution and evolution. Neither can be inferred uniquely from a shortfall in luminous counts.

Useful comparisons therefore predict multiple observations with a common matter model and explicit nuisance assumptions. Independent calibration can constrain the luminous and selection responses rather than allowing arbitrary adjustments.

The finite example establishes an exact conditional degeneracy and one stipulated way to break it. It supplies no significance, empirical deficit or preference among physical dark-matter theories.

Worked reasoning

Recover luminous counts without assuming a halo count

An expected lower-bin detected count is five and the conditional detection probability is 0.5. What is determined?

Dividing five by 0.5 gives an expected luminous count of ten.
This is N_halo × P(luminous|halo), not N_halo alone.
Both 100 × 0.1 and 50 × 0.2 equal ten.

The expected luminous count is determined under the supplied detection rule; the halo inventory remains degenerate with its luminous probability.

Try the reasoning

Why do both supplied upper bins predict sixteen detected objects in expectation?
Show the answer and reasoning
Each has expected luminous count twenty, multiplied by conditional detection probability 0.8.
A uses 40 × 0.5, while B uses 20 × 1 before applying detection.
Must every realization of model A contain exactly five detected lower-bin objects?
Show the answer and reasoning
No.
Five is an expectation. A sampling model is needed to describe realization-to-realization variation.
What expected lower-bin counts distinguish A and B under the separately stipulated ideal mass-sensitive probe?
Show the answer and reasoning
Fifty for A and twenty-five for B.
The response selects half the halos regardless of luminous occupation; its real implementation has not been supplied.

What to carry forward

A halo inventory reaches a catalogue through formation and selection.
Different inventories can predict identical luminous and detected expectations.
An additional test needs a known response to a different combination of the unknowns.

References

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