Cosmology · Origins, primordial tests and the limits of knowledge

Turning open questions into reproducible research

Build and reproduce a complete finite hypothesis test, including raw responses, correlated errors, a declared decision rule and a calibration alternative.

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

Turn a proposed distinction into a specified response and decision rule.
Calculate exact false-positive probability and power from a complete finite law.
Preserve the information needed to reproduce a result and test alternative causes.

Define what an observation could distinguish

An open cosmological question becomes a research problem when competing assumptions lead to specified observable differences. The useful chain runs from physical model to predicted signal, measurement response, uncertainty and a reproducible comparison.

Record the chosen quantities, units, selection, calibration and analysis rule before interpreting the result. State which information fits parameters and which information provides an independent test.

A reproducible calculation can be correct while its physical assumptions remain incomplete. Checks of implementation, uncertainty and alternative explanations therefore answer different questions.

The original finite experiment below supplies the whole sampling law, making every decision probability exactly reproducible. It is a teaching protocol, not an actual preregistered observing program or a test of a named cosmic-origin theory.

Retain raw responses and shared uncertainty

Choose arbitrary positive unit U and an unknown fixed common quantity a. Independently draw c = ±U/4, e_A = ±U/2 and e_B = ±U/2, each sign with probability 1/2. The eight ordered sign triples are equiprobable.

Under null response M₀, supply A = a+c+e_A and B = a+c+e_B. Under declared alternative M₁, add a fixed offset +U to B while leaving A and the random-error law unchanged.

Use statistic D = B−A. The fixed common a and same-sign random calibration c cancel exactly, giving D = e_B−e_A under M₀ and D = U+e_B−e_A under M₁.

The raw channels remain correlated: Var(A) = Var(B) = 5U²/16 and Cov(A,B) = U²/16 under either response. Their difference variance is U²/2; the variance of their average is 3U²/16.

A declared test has size and power under stated laws

Predeclare the one-sided decision rule D≥U. Under M₀, D/U takes −1,0,1 with probabilities 1/4,1/2,1/4, so the false-positive probability, or size of this test, is 1/4.

Under M₁, D/U instead takes 0,1,2 with the same respective probabilities. Its probability of meeting the rule, or power against this declared alternative, is 3/4.

A separately specified design D≥2U has size zero and power 1/4 under these exact finite laws. This stricter rule trades lower false-positive probability for lower power in the supplied experiment.

Zero size here depends on the stipulated bounded support. It is not a guarantee about real instrument errors, nor a universal threshold for discovery or a probability that a physical hypothesis is true.

Exact statistic probabilities obtained from all eight equiprobable joint error states
D/U	P under M₀	P under M₁	Meets D≥U?	Meets D≥2U?
−1	1/4	0	No	No
0	1/2	1/4	No	No
1	1/4	1/2	Yes	No
2	0	1/4	Yes	Yes

A passing outcome can have another cause

Take a known synthetic M₀ realization with a = 3U, c = +U/4, e_A = −U/2 and e_B = +U/2. It produces A = 11U/4, B = 15U/4 and D = U.

The realization meets the first decision rule despite being generated under M₀. It demonstrates a false positive permitted by the exact size 1/4, not a contradiction of the sampling calculation.

Now separately consider an unmodeled fixed calibration offset +U in channel B under the null physical response. It shifts the complete D law to exactly the law assigned to M₁.

The difference statistic alone therefore cannot identify whether the offset comes from the declared alternative or calibration. A useful follow-up needs independently characterized controls or additional linked physical predictions.

Two distinct predeclared designs; probabilities apply only to the supplied finite experiment
Rule	Size P(rule\|M₀)	Power P(rule\|M₁)
D≥U	1/4	3/4
D≥2U	0	1/4

Supply arbitrary positive readout unit U, unknown fixed common a and independent symmetric c = ±U/4, e_A = ±U/2, e_B = ±U/2. All eight ordered sign triples have probability 1/8. Under M₀, A = a+c+e_A and B = a+c+e_B; under M₁ add a fixed +U to B without changing A or the random law. The plotted statistic D = B−A cancels fixed a and shared c. The horizontal linear D/U domain is −1.4 to 2.4. The zero-based vertical linear probability-mass axis is 0 to 0.6. Solid blue stems with hollow circles show M₀ masses 1/4,1/2,1/4 at −1,0,1. Dashed orange stems with larger hollow squares show M₁ masses 1/4,1/2,1/4 at 0,1,2. Unlisted masses are zero; there is no continuous probability density or Gaussian approximation. A complete native table includes all four positions −1,0,1,2 with both masses. At 0 and 1, the two stems share an abscissa but have different heights and marker shapes. A long-dashed violet vertical line at D/U = 1 marks predeclared rule 1, D≥U; its size under M₀ is 1/4 and power against the stated M₁ is 3/4. A dotted green vertical line at D/U = 2 marks a separately specified design D≥2U, with exact size 0 and power 1/4. Rule 2 is not an after-the-fact replacement chosen from the outcome, and zero size depends entirely on the supplied bounded null support. Raw marginal variances are 5U²/16, cross-covariance U²/16, difference variance U²/2 and average variance 3U²/16; none of these imply a Gaussian law. The explicitly null-generated fixture a = 3U,c = U/4,e_A = −U/2,e_B = U/2 gives A = 11U/4,B = 15U/4,D = U and passes rule 1, a permitted false positive. A separate fixed unmodeled +U calibration offset in B gives exactly the same complete D distribution as M₁, so this statistic cannot identify which cause produced a shift. No real instrument, observed anomaly, universal discovery threshold, posterior model probability or unique cosmic origin is supplied. Direct null/alternative labels, circle/solid versus square/dashed encodings, two explicitly located rule lines, exact probability/rule tables and fixture text provide noncolor access. Original Ark teaching model.

Make the result independently reconstructible

Preserve the raw response values and units, exact sampling law, shared-error assumptions, statistic, threshold and the parameters or calibration choices used to obtain the reported result. Keep a clear record of any later revision.

Another researcher should be able to enumerate the same eight states, recover the covariance, recompute size and power, and reproduce the synthetic example without guessing hidden choices. For larger experiments, validated numerical methods replace complete enumeration.

Sensitivity tests should change a stated assumption and report the resulting change in inference. If a check is used to revise the model, record that change and obtain a new independent test rather than treating the reused information as held out.

For an origin proposal, carry this discipline back through the full predictive chain: dynamics, primordial signal, transfer, foregrounds and observations. A well-posed reproducible question advances understanding even when the evidence cannot yet select a unique explanation.

Worked reasoning

Count the decisions in the complete finite experiment

What are the size and power of the rule D≥U?

For M₀, only e_A = −U/2,e_B = +U/2 gives D = U; either sign of c is allowed, so 2 of 8 states meet the rule.
For M₁, all states with e_B−e_A≥0 meet the rule; this is 6 of the 8 states.
Divide by the common number of equiprobable states: size = 2/8 and power = 6/8.

The exact size is 1/4 and the power against the specified M₁ is 3/4.

Try the reasoning

Why does c not contribute to D in this experiment?
Show the answer and reasoning
It enters both channels with the same coefficient and cancels in B−A.
That cancellation follows from the supplied response; it would change if the channel responses to c differed.
If D≥2U is specified before the experiment, what are its size and power?
Show the answer and reasoning
Size 0 and power 1/4 under the exact stated finite laws.
M₀ never produces D = 2U, while M₁ does so in 2 of 8 states.
Does meeting D≥U distinguish M₁ from a fixed +U calibration offset in B?
Show the answer and reasoning
No.
Those two supplied causes give the same complete D distribution; another independent response is needed.

What to carry forward

A research test needs a defined predictive and measurement chain.
Reproduce decisions from the actual sampling law and keep their interpretation conditional.
Independent controls, preserved revisions and linked predictions help distinguish physical causes.

References

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