Cosmology · Origins, primordial tests and the limits of knowledge

Primordial signals that could distinguish models

Trace two supplied primordial power shapes through ideal transfer and foreground responses, then use a predeclared third band to test their different predictions.

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

Separate a primordial spectrum from its observable response.
Fit common nuisance quantities before comparing linked predictions.
Explain why a discriminating signal needs more than an improved fit to selected data.

A primordial model must predict an observable

Primordial proposals can differ in scalar or tensor spectra, relative modes and higher correlations. These are predictions for specified initial fluctuations, rather than direct labels attached to observed maps.

To compare with observations, propagate a prediction through the relevant transfer, selection and measurement response. Foregrounds and calibration can introduce further quantities that must be modeled or constrained.

The power spectrum summarizes a two-point correlation. Higher correlations such as a bispectrum can contain additional information, but their interpretation also requires a predicted shape and treatment of nonprimordial contributions.

An original deterministic three-band example below makes the response chain explicit. It supplies synthetic reference values and algebraic nuisance fits, without a sky realization, empirical likelihood or discovery probability.

Define the domain, amplitudes and band response

Let k_*>0 be a reference comoving wavenumber in m⁻¹ and define u = ln(k/k_*) over −1≤u≤1. Supply dimensionless scalar curvature power 𝒫_ζ(k) = A[1+vu²], with A = aA_* and arbitrary positive dimensionless reference A_*.

Compare separate fixed-shape hypotheses v = 0 and v = 1/5. Both have positive power throughout the declared domain; their amplitude parameter is a∈[1/2,2].

At the three selected values u = −1,0,1, supply ideal band responses τ = (1,1/2,1/4) and a common additive foreground response bA_*, with dimensionless b∈[0,1/2].

The stipulated normalized outputs are y_i = C_i/A_* = τ_i a(1+vu_i²)+b. This additive power response is part of the model; it supplies no actual CMB window, cross-component correlation law or instrumental covariance.

Two bands leave different shape hypotheses indistinguishable

Supply synthetic references y = (3/2,4/5,3/5). They are generated by v = 1/5, a = 1 and b = 3/10 under the declared response.

Use only the first two bands for fitting. For the feature hypothesis v = 1/5, the two equations give a = 1 and b = 3/10, reproducing both training values exactly.

For the baseline v = 0, the equations are a+b = 3/2 and a/2+b = 4/5. Their solution is a = 7/5 and b = 1/10, also within the allowed parameter bounds.

The two exact training fits therefore use different primordial amplitudes and foreground contributions. Neither fit by itself identifies the shape; setting b to zero would change the specified comparison rather than preserve either fit.

Exact fits to the first two synthetic bands; all quantities in this table are dimensionless
Shape hypothesis	v	Fitted a	Fitted b	Training y₁	Training y₂
Baseline	0	7/5	1/10	3/2	4/5
Feature	1/5	1	3/10	3/2	4/5

A linked prediction can separate the responses

Predeclare the third band as held out and keep the training-fit parameters fixed. The feature hypothesis predicts y₃ = (1/4)(1)(6/5)+3/10 = 3/5.

The baseline instead predicts y₃ = (1/4)(7/5)+1/10 = 9/20. Against the supplied reference 3/5, its residual reference minus prediction is 3/20.

The different transfer response in the third band makes the fitted primordial and foreground contributions distinguishable in this ideal example. Its value is a linked prediction of each already fitted response.

No standard error or sampling distribution was supplied, so the residual has no assigned significance. Using the third band to refit the model would require another independent check before making a held-out prediction claim.

One predeclared third-band test after the two training fits; residual is reference minus prediction
Band u	Role	Reference y	Baseline prediction	Feature prediction
−1	Training	3/2	3/2	3/2
0	Training	4/5	4/5	4/5
1	Held out	3/5	9/20	3/5

Choose comoving k_* > 0 in inverse metres, u = ln(k/k_*) on −1 to 1 and positive dimensionless A_*. Supply dimensionless scalar curvature power 𝒫ζ/A_* = a(1+vu²), with fixed separate hypotheses v = 0 or 1/5, a between 1/2 and 2, and common additive foreground b between 0 and 1/2. The ideal three narrow selected-k outputs are y_i = C_i/A_* = τ_i a(1+vu_i²)+b at u = −1,0,1, with τ = 1,1/2,1/4. Supplied references are 3/2,4/5,3/5. Fit only the first two, giving baseline a = 7/5,b = 1/10 and feature a = 1,b = 3/10. Panel A plots the fitted positive primordial shapes, not foreground-added outputs. Its horizontal linear u axis is −1 to 1 and zero-based vertical 𝒫ζ/A_* is 0 to 1.6. Solid blue circles mark baseline 7/5 at u = −1,0,1; dashed orange squares mark feature 6/5,1,6/5. The continuous curves are these supplied shapes only. Panel B shows the three discrete output responses with horizontal linear u plotting domain −1.2 to 1.2 and zero-based vertical y from 0 to 1.7. Connecting segments are guides, not a real CMB transfer function. Large dotted light circles mark references 1.5,0.8,0.6; hollow orange squares mark feature predictions 1.5,0.8,0.6; smaller solid-blue-path circles mark baseline predictions 1.5,0.8,0.45. All first-two-band markers coincide at exactly the same coordinates and are nested without displaced values. The u = 1 held-out region is shaded. With both training fits fixed, baseline y₃ = 9/20 and feature y₃ = 3/5. The native table prints prediction and reference-minus-prediction residual: baseline 3/20, feature 0. All power is positive on the declared domain, but a supplied positive two-point spectrum alone specifies neither a full field distribution nor Gaussianity. Separate dimensional P(k) carries comoving volume, unlike dimensionless 𝒫ζ. The stipulated τ_i and foreground are not actual Planck windows, sky samples, bispectrum, covariance, likelihood or instrument responses. No error law, empirical bound, significance, observed origin feature or complete physical origin scenario is supplied. Direct fitted-parameter labels, different patterns and marker shapes plus exact held-out table provide noncolor interpretation. Original Ark teaching model.

A spectral distinction is not yet an origin theory

The dimensionless 𝒫_ζ used here is a different convention from a dimensional Fourier power P(k) with units of comoving volume. Keep the convention, epoch and physical quantity explicit when comparing reported spectra.

Actual primordial tests require the appropriate continuous transfer calculation and uncertainty model. Finite sampling and smoothing can also affect whether a proposed feature survives propagation into measured observables.

Complementary scalar, tensor and higher-correlation predictions may help distinguish mechanisms when their response and joint uncertainties are specified. Additional summaries should test coherent links rather than be selected only because they fit an interesting pattern.

The example supplies positive power shapes and a discriminating algebraic response. It does not produce those shapes from a dynamical origin model, establish primordial non-Gaussianity or rank real cosmological proposals.

Worked reasoning

Calculate the baseline prediction after fitting two bands

What does the baseline predict in the third band?

Subtract the training equations a+b = 3/2 and a/2+b = 4/5 to obtain a/2 = 7/10.
Then a = 7/5 and b = 1/10.
Use the third-band transfer: y₃ = a/4+b = 7/20+1/10 = 9/20.

The baseline predicts 9/20, below the supplied reference 3/5 by 3/20.

Try the reasoning

What third-band value follows from the feature fit a = 1,b = 3/10?
Show the answer and reasoning
3/5.
At u = 1 the primordial factor is 6/5, so the output is 6/20+3/10.
Do the equal first-two-band fits identify the primordial shape?
Show the answer and reasoning
No.
The differing amplitude and foreground parameters compensate exactly in those two supplied responses.
Does the residual 3/20 alone supply a discovery probability?
Show the answer and reasoning
No.
The deterministic example provides no measurement-error law, covariance or likelihood.

What to carry forward

Primordial predictions must be propagated through the stated observable response.
Nuisance fits can hide distinctions in selected summaries.
A predeclared linked prediction can distinguish responses while still requiring a physical model and uncertainty assessment.

References

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

Planck Collaboration; Y. Akrami and collaborators2 sources
1. Planck 2018 results. X. Constraints on inflationPlanck Collaboration; Y. Akrami and collaboratorsResearch paperarxiv.orgRead source
2. Planck 2018 results. IX. Constraints on primordial non-GaussianityPlanck Collaboration; Y. Akrami and collaboratorsResearch paperarxiv.orgRead source