Cosmology · Complementary probes of cosmic history

Shared assumptions and genuinely complementary data

Compare a shared calibration mode, a deterministic duplicate and an explicitly new primitive response in a finite repeated-error experiment.

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

Identify how a common calibration affects a sum and a difference.
Explain why a derived output is not independent new data.
Propagate uncertainty for a new response under stated conditional-independence assumptions.

Different labels do not establish independent evidence

Two probes can respond differently to physical parameters while sharing calibration, foregrounds, samples or model assumptions. Complementarity therefore requires looking at the primitive observations and their dependencies.

A computed summary of existing data may make a useful pattern clearer, but it does not become an independent measurement merely by being given another name.

Use two dimensionless physical directions p and q, treated as fixed unknown values in a finite repeated-error model. Define outputs A = p+q+c+e₁ and B = p−q+c+e₂.

The shared calibration draw c affects both outputs equally. The separate e₁ and e₂ terms describe additional errors under a supplied probability law, rather than measured instrumental uncertainties.

A covariance needs an underlying experiment

Let c take −0.4, 0 or +0.4 with probabilities 1/4, 1/2 and 1/4. Let e₁ take ±0.1 with equal probabilities, and e₂ take −0.2, 0 or +0.2 with probabilities 1/4, 1/2 and 1/4.

The three draws are independent conditional on fixed p and q, giving 18 weighted outcomes. Each has zero mean, with Var(c) = 0.08, Var(e₁) = 0.01 and Var(e₂) = 0.02.

It follows that Var(A) = 0.09, Var(B) = 0.10 and Cov(A,B) = 0.08. The covariance is created by the common c, which is not reduced by pretending the two outputs use unrelated calibrations.

A fixed unknown systematic bias is not automatically a zero-mean repeated draw. The finite law is an explicit teaching assumption, not a universal probability description of every calibration uncertainty.

A shared mode survives in one direction and cancels in another

Define p̂ = (A+B)/2 and q̂ = (A−B)/2. Their errors are c+(e₁+e₂)/2 and (e₁−e₂)/2 respectively.

The shared c survives fully in p̂ and cancels from q̂. This is an algebraic response property; it does not mean that every calibration problem disappears from every inferred parameter.

The variances are Var(p̂) = 7/80 = 0.0875 and Var(q̂) = 3/400 = 0.0075, with Cov(p̂,q̂) = −1/400 = −0.0025. The asymmetry comes from the shared mode and the different individual-error variances.

Dropping Cov(A,B) would instead assign 0.0475 to each inferred variance, misallocating uncertainty between the two directions. The actual finite distributions also need not have Gaussian interval coverage.

Conditional second moments of the supplied finite repeated-error model; all units are squared dimensionless response units
Quantity	Exact value	Decimal
Var(A)	9/100	0.09
Var(B)	1/10	0.10
Cov(A,B)	2/25	0.08
Var(p̂)	7/80	0.0875
Var(q̂)	3/400	0.0075
Cov(p̂,q̂)	−1/400	−0.0025

An additional primitive measurement needs its own error assumptions

Suppose another reported output is S = A+B, computed from the same A and B. Knowing A and B already fixes S exactly, so it supplies no additional independent information.

Its joint covariance with A and B is singular because of that deterministic relation. Counting the three as independent would reuse the same information.

Instead supply a genuinely new primitive response R = p+η, where η takes ±0.1 with equal probabilities and is independent of c,e₁,e₂ conditional on the same fixed p,q. Then Var(R) = 0.01 and its conditional error covariance with A or B is zero.

The shared physical signal p does not make those declared error draws identical. Conversely, calling a real instrument new would not prove this conditional independence; it would need its own calibration and response evidence.

Optional mathematical path: one unbiased weighted estimate

Choose p_new = (4/39)p̂+(35/39)R. The weights sum to one, so its repeated-error mean equals p under the supplied zero-mean error laws.

Because the errors of p̂ and R are independent here, Var(p_new) = (4/39)²(7/80)+(35/39)²(1/100) = 7/780 ≈ 0.008974359.

The new primitive response reduces this chosen estimator’s variance in the constructed experiment. Enumerating all 36 joint weighted outcomes gives the same result; no prior or posterior was introduced.

A real comparison must test whether the extra response contributes a new physical direction or independent uncertainty under its actual assumptions. The calculation establishes neither real cosmological precision improvement nor independence simply from a list of probe names.

This original finite repeated-error experiment fixes unknown dimensionless physical p,q and supplies A = p+q+c+e₁, B = p−q+c+e₂. The independent conditional draws are c = −0.4,0,+0.4 with probabilities 1/4,1/2,1/4; e₁ = ±0.1 equally; e₂ = −0.2,0,+0.2 with probabilities 1/4,1/2,1/4. Panel A plots all 18 weighted outcomes of errors p̂−p = c+(e₁+e₂)/2 horizontally and q̂−q = (e₁−e₂)/2 vertically, where p̂ = (A+B)/2 and q̂ = (A−B)/2. Both axes are linear: horizontal −0.6 to 0.6, vertical −0.2 to 0.2, in dimensionless response units. Their pixel scales are separately stated by those axes and do not assert equal aspect. A filled blue circle marks each outcome. Circle area is proportional to probability, not probability radius; the three legend probabilities 1/32,1/16,1/8 have radii 4,4√2,8 pixels. Positions are determined by the listed product law, not measured scatter. No Gaussian ellipse, posterior contour, sample frequency or coverage is implied. All probabilities sum to one. The common c spreads the horizontal sum direction and cancels from the vertical difference direction. Var(A) = 0.09, Var(B) = 0.10, Cov(A,B) = 0.08, Var(p̂) = 7/80 = 0.0875, Var(q̂) = 3/400 = 0.0075 and Cov(p̂,q̂) = −1/400 = −0.0025. Panel B uses horizontal zero-based linear bar lengths from 0 to 0.1 in squared dimensionless response units. Labeled blue, orange, green and violet bars give respectively Var(p̂) = 0.0875, Var(q̂) = 0.0075, Var(R) = 0.01 and Var(p_new) = 7/780 ≈ 0.008974359. Each bar has its name and exact or decimal value, so color is unnecessary. R = p+η is a new primitive only by the explicit assumption that η = ±0.1 is equally likely and independent of c,e₁,e₂ conditional on fixed p,q. Then p_new = (4/39)p̂+(35/39)R is unbiased and has the displayed variance; all 36 joint weighted outcomes reproduce it. S = A+B is instead fixed by A,B, so its joint covariance is singular and it adds no primitive information. Dropping Cov(A,B) would wrongly assign 0.0475 to each original direction. A fixed unknown calibration bias is not automatically a draw from this finite zero-mean law. These are repeated-error second moments, not posterior content, normal-distribution coverage, empirical uncertainties or a real cosmological precision improvement. Original Ark teaching model.

Worked reasoning

Add the same calibration displacement to A and B

Which inferred direction retains that displacement?

Adding c to each output changes A+B by 2c.
Thus p̂ = (A+B)/2 shifts by c.
The difference A−B is unchanged, so q̂ receives no common-c contribution.

The shared displacement remains in p̂ and cancels from q̂ under the specified responses.

Try the reasoning

Does S = A+B provide a third independent probe?
Show the answer and reasoning
No.
It is fixed by the first two outputs, giving a singular joint covariance.
Can every unknown fixed calibration bias be assumed to follow the finite zero-mean c law?
Show the answer and reasoning
No.
That probability law was explicitly supplied for this repeated-error experiment and requires justification in an actual analysis.
What variance does the stated p_new estimate have under the supplied independence assumptions?
Show the answer and reasoning
7/780 ≈ 0.008974359.
Propagate the two independent variances using the squared weights; this is a repeated-error variance, not posterior content.

What to carry forward

Shared calibration can affect different parameter directions very differently.
A deterministic summary of existing observations is not independent new information.
A new response adds information only through its stated physical response and supported uncertainty dependencies.

References

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

Measurements of Omega and Lambda from 42 High-Redshift SupernovaeS. Perlmutter; and the Supernova Cosmology ProjectResearch paperarxiv.orgRead source