Cosmology · The evidence motivating dark matter

Why an explanation must fit several probes

Compare complementary response functions, then show why a summary calculated from the same measurements cannot be counted as independent new evidence.

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

Explain what a common forward model must predict for several probes.
Distinguish complementary responses from independent information.
Identify a duplicated summary and its correlated uncertainty.

A common model must predict each actual observation

A proposed mass distribution affects motions and lensing through gravity, while ordinary matter also has atomic and thermal responses. Early-universe components affect acoustic behavior and later structure through a connected perturbation history.

A common explanation must predict the relevant observations through each response. It cannot fit one inferred mass while silently changing its geometry, population or calibration when another probe is considered.

Different sensitivities can resolve an ambiguity: one response may depend mainly on a total, while another distinguishes its components. The strength of that comparison depends on measurement uncertainty, nuisance assumptions and selection.

Different mechanisms do not guarantee independent errors. Shared observations, distances or calibration can correlate them. A quantity calculated from existing measurements is a particularly clear case of information that must not be counted twice.

A second response can distinguish the same total

Choose positive dimensionless parameters b and d. Define two primitive responses u = b+d and v = b. These are mathematical normalized proxies, not calibrated CMB, galaxy or lensing observables.

Supply candidate A with (b,d) = (1,2) and candidate B with (2,1). Both predict u = 3, so u alone does not distinguish the two supplied points.

Their v predictions are 1 and 2. A genuine measurement of v therefore introduces a different response to the parameters, provided its uncertainty and relation to u are specified.

Now calculate w = 2u−v from that same primitive pair. Its predicted values are 5 and 4. They differ between the candidates, but the calculated w is not an independently measured third probe.

Two chosen candidate points and their normalized proxy predictions; w is calculated from u and v
Candidate	b	d	Primitive u=b+d	Primitive v=b	Derived w=2u−v
A	1	2	3	1	5
B	2	1	3	2	4

A derived quantity shares the primitive errors

Supply a finite error law for the primitive pair: e_u and e_v independently take −1 or +1, with all four combinations equally probable. These are chosen normalized errors, not observations or fitted noise.

Each primitive error has zero mean and variance one, and their covariance is zero. The derived error must be e_w = 2e_u−e_v, because w uses the same pair.

The four possible e_w values are −1, −3, 3 and 1. Their mean is zero and their variance is (1+9+9+1)/4 = 5.

Shared errors also give cov(e_u,e_w) = 2 and cov(e_v,e_w) = −1. A zero primitive covariance does not make the derived summary independent of either primitive.

Exact supplied four-state error law; each row has probability 1/4
e_u	e_v	e_w=2e_u−e_v	Probability
−1	−1	−1	1/4
−1	+1	−3	1/4
+1	−1	+3	1/4
+1	+1	+1	1/4

Three labels still contain only two primitive quantities

In the order (u,v,w), the error covariance matrix has rows (1,0,2), (0,1,−1) and (2,−1,5). Its rank is two, because −2e_u+e_v+e_w always vanishes.

For an optional quantitative check, use covariance to define a squared prediction-separation metric Q. The candidate difference in primitive coordinates is (0,1), so the identity primitive covariance gives Q = 1.

Using all three labels with the complete rank-two covariance gives the same Q = 1 on their allowed two-dimensional space. Treating the three labels incorrectly as independent gives 0²/1+1²/1+(−1)²/5 = 1.2.

This is a metric comparison, not a likelihood, p-value, confidence level or observed preference. The error law is discrete and non-Gaussian, and the singular covariance does not define a nonsingular three-dimensional Gaussian density.

This finite mathematical comparison uses positive dimensionless parameters b and d and normalized proxies u = b+d, v = b, w = 2u−v = b+2d. These are not calibrated galaxy, lensing or CMB observables. The visible candidate table lists A with (b, d, u, v, w) = (1, 2, 3, 1, 5) and B with (2, 1, 3, 2, 4). Their common u = 3 cannot distinguish those points; their primitive v values differ. Although their w predictions also differ, w here is calculated from the very same u and v rather than independently measured. The middle graph is a primitive error plane, not parameter confidence space. It uses a linear horizontal e_u scale and a linear vertical e_v scale, both from −2 to 2 in chosen normalized units. Four actual circle markers occur at (−1, −1), (−1, +1), (+1, −1), (+1, +1), each with supplied probability one quarter. Direct annotations at those points give e_w = −1, −3, +3, +1 respectively. Every e_w is fixed by 2e_u−e_v and adds no independently sampled coordinate. There are no ellipses, observed data, likelihood contours or significance badges. Exact finite averaging gives zero means, variances one for e_u and e_v, covariance zero, derived variance five, covariance(e_u, e_w) = two and covariance(e_v, e_w) = minus one. The displayed complete covariance, in order (u, v, w), has rows (1, 0, 2), (0, 1, −1), (2, −1, 5). Its rank is two and the null relation −2e_u+e_v+e_w = zero holds in every supplied state. In the optional prediction-separation metric, B minus A is (0, 1) in primitive coordinates and (0, 1, −1) in the three labels. Keeping the full covariance on its allowed two-dimensional column space gives Q = one in either representation. Incorrectly treating the three labels as independent gives 0²/1 + 1²/1 + (−1)²/5 = 1.2. This Q is a defined covariance metric, not a likelihood, p-value, confidence level, detection significance or observed preference. The actual error law is finite discrete and non-Gaussian; a singular covariance does not define a nonsingular three-dimensional Gaussian density. Different physical sensitivities can be complementary without statistically independent errors. An independently measured third quantity would require a new response, selection and error model; it is not the w calculated here. No observational evidence strengths are ranked, and no physical particle identity follows. Original Ark teaching model.

Keep the primitive measurements and their joint model

A combined physical analysis identifies which quantities were actually measured and which were derived. It predicts those primitive observations through a shared model and retains their full uncertainty structure.

An independently measured third quantity would be a different experiment, with its own response, selection and error sources. That additional information should be modeled explicitly rather than assigned to the w calculated here.

Likewise, a model that improves one response must still be checked against the others. A change in the inferred ordinary-matter amount can also alter acoustic, thermal or tracer predictions; it is not only a numerical replacement in one mass ledger.

Complementary evidence is valuable because distinct measured responses test a common account. Its interpretation becomes clearer when derived summaries, shared calibration and genuine additional measurements remain distinguishable.

Worked reasoning

Propagate one primitive error pair

For e_u = +1 and e_v = −1, what is the derived error, and is it an independent extra draw?

Use the definition e_w = 2e_u−e_v.
Substitute to obtain e_w = 2×1−(−1) = 3.
Its value is fixed by the same two primitive errors; there is no third random draw.

The derived error is 3, and it carries no independently sampled third error in this experiment.

Try the reasoning

Which supplied primitive response distinguishes A and B even though both have u = 3?
Show the answer and reasoning
v, with predictions 1 and 2.
Its dependence on b differs from the dependence of the total proxy u.
Is the calculated w a third independently measured probe?
Show the answer and reasoning
No.
It is fixed by the same u and v, including their errors.
Does the incorrect metric value 1.2 establish an observed preference between the candidates?
Show the answer and reasoning
No.
No data or likelihood were supplied, and that metric deliberately ignores the required covariance.

What to carry forward

A common explanation predicts each primitive observable through its response model.
Different physical sensitivities and statistically independent information are separate properties.
Derived summaries retain shared uncertainty and do not add independent measurements.

References

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