Cosmology · Inflation and primordial perturbations

Primordial fluctuations and their statistical spectrum

Distinguish dimensional Fourier power from variance per logarithmic wavenumber, using three supplied finite-band spectra with a common pivot.

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

Interpret a primordial power spectrum as an ensemble statistic.
Compare amplitude and tilt using a stated Fourier convention and pivot.
Calculate finite-band variance without extending a supplied spectrum beyond its domain.

A spectrum describes statistics across scales

A smooth background expansion is only part of a cosmological model. Spatial fluctuations provide another part, describing departures from that background and supplying initial patterns for later evolution.

A power spectrum characterizes two-point statistics of those patterns as a function of scale. It does not list the locations of every peak in one realized sky, and by itself it does not specify the complete probability distribution.

Use a dimensionless scalar curvature perturbation ℛ(x) with a statistically homogeneous and isotropic ensemble. Comoving wavenumber k labels inverse comoving length. Larger k refers to shorter comoving wavelengths under the same convention.

The spectra in this lesson are supplied mathematical examples. Their amplitudes and slopes are not estimates from CMB data, and specifying them does not select a unique field dynamics or quantum origin.

Dimensional power and dimensionless power are different

For the optional Fourier account, use ℛ(x) = ∫ d³k/(2π)³ ℛ_k exp(ik·x), and ⟨ℛ_k ℛ_k′⟩ = (2π)³δ³(k + k′)P_dim(k). This real-field convention uses no complex conjugate in the displayed correlation.

Since ℛ is dimensionless and k has inverse-length units, P_dim has units of comoving length cubed. Define dimensionless power Δ_ℛ²(k) = k³P_dim(k)/(2π²). It gives variance contributed per logarithmic interval d ln k under this convention.

A constant Δ_ℛ² therefore does not mean constant P_dim. It instead requires P_dim ∝ k⁻³. Statements about a flat spectrum need to identify which power quantity is plotted.

Readers can compare the table without Fourier calculations: all three columns show Δ_ℛ² divided by the same chosen amplitude. The Fourier definition explains why integrating those columns uses logarithmic wavenumber rather than an unweighted interval of k.

A common pivot separates amplitude from slope

Choose A = 10⁻⁶ and a positive arbitrary comoving length L_ref, with k_* = 1/L_ref and κ = k/k_*. Within the finite domain −2 ≤ ln κ ≤ 2, prescribe Δ_ℛ² = Aκ^(n − 1) for n = 0.9, 1 and 1.1.

All three spectra equal A at the pivot κ = 1. The n = 0.9 example places more dimensionless power at smaller k within the band; n = 1.1 places more at larger k. The n = 1 example is constant per logarithmic interval.

At ln κ = 2, the n = 0.9 and n = 1.1 powers are about 0.818731A and 1.221403A. At ln κ = −2, those values are exchanged. The common pivot makes their opposite slopes visible without changing their pivot amplitude.

No running of the tilt or additional window is included. Power outside the supplied band is left unspecified. The examples do not authorize extrapolation to arbitrarily small or large wavelengths.

Supplied finite-band dimensionless primordial power; A = 10⁻⁶ and k_* is an arbitrary inverse comoving reference length
ln(k/k_*)	Δ_ℛ²/A for n = 0.9	Δ_ℛ²/A for n = 1	Δ_ℛ²/A for n = 1.1
−2	1.221403	1	0.818731
−1	1.105171	1	0.904837
0	1	1	1
1	0.904837	1	1.105171
2	0.818731	1	1.221403

Equal band variance does not imply equal spectra

The variance contributed by this sharp finite logarithmic band is σ_band² = ∫₋₂² Δ_ℛ² d ln κ. This is a band-limited variance; it is not a claim about the total variance of an unspecified spectrum outside the band.

For n = 1, the integrand is constant A and the logarithmic width is four. Thus σ_band² = 4A = 4 × 10⁻⁶. The variance is dimensionless because both Δ_ℛ² and d ln κ are dimensionless.

For n ≠ 1, integration gives σ_band² = 2A sinh[2(n − 1)]/(n − 1). Both n = 0.9 and n = 1.1 give approximately 4.026720A. Opposite tilts have equal integrated variance over this symmetric band.

The equal totals conceal different scale distributions: one spectrum emphasizes the lower-k end and the other the higher-k end. A single integrated number therefore loses information that remains visible in the spectrum.

Three supplied statistically homogeneous and isotropic curvature-perturbation spectra have chosen amplitude A = 10^(−6), κ = k/k_*, k_* = 1/L_ref for an arbitrary positive comoving length L_ref, and Δ_ℛ² = Aκ^(n−1). The plot is confined to −2 ≤ ln κ ≤ 2; power outside that band is unspecified. The horizontal axis is natural logarithmic wavenumber, and the vertical axis is the linear dimensionless Δ_ℛ²/A, displayed from 0.7 to 1.3 rather than zero. Solid blue circles show n = 0.9, dashed green squares n = 1, and dotted orange diamonds n = 1.1. At ln κ = −2, −1, 0, 1, 2 the n = 0.9 values are 1.221403, 1.105171, 1, 0.904837, 0.818731; the n = 1 values are all one; n = 1.1 reverses the first sequence. They share pivot κ = 1, ln κ = 0 and dimensionless power A. The apparent overlapping pivot symbols represent the same value. The separate table integrates only this sharp finite log band: σ_band²/A = 4.026720, 4, 4.026720 for n = 0.9, 1, 1.1, giving dimensionless variances approximately 4.026720×10^(−6), 4×10^(−6), 4.026720×10^(−6). The exact integral is 2 sinh[2(n−1)]/(n−1) times A when n differs from one, with the continuous n = 1 limit 4A. Under ℛ(x) = ∫d³k ℛ_k exp(ik·x)/(2π)^3 and ⟨ℛ_kℛ_k′⟩ = (2π)^3δ³(k+k′)P_dim(k), Δ_ℛ² = k³P_dim/(2π²) is variance per d ln k, whereas P_dim has comoving length-cubed units. A constant dimensionless spectrum therefore has P_dim proportional to k^(−3). Equal band integrals conceal opposite distributions across scale. No running, outside-band extension, random sky, observed parameter estimate, CMB angular spectrum, transfer calculation or uniquely selected origin model is supplied. Original Ark teaching model.

Observed patterns require evolution and inference

A primordial spectrum supplies statistical initial information for later calculations. Relating it to an observed CMB angular spectrum or a matter distribution requires the relevant evolution, transfer and projection, together with the observation model.

One observed realization also has phases and finite sampling. An ensemble power law does not specify those phases or reproduce one sky pixel by pixel. Instrument response, foregrounds and uncertainty enter the inference from data.

Different mechanisms can yield similar two-point spectra over the scales being tested. Higher-order statistics, tensor information and other predictions can help discriminate proposals, but a chosen tilt or near-constant dimensionless spectrum is not a unique mechanism label.

When reading a spectrum, record its variable, units, normalization, pivot, domain and statistic. Those details distinguish a supplied statistical model, an evolved observable and an inferred parameter estimate.

Worked reasoning

Integrate constant power per logarithmic interval

What variance does n = 1 contribute on −2 ≤ ln κ ≤ 2 when A = 10⁻⁶?

For n = 1, Δ_ℛ² = A everywhere in the stated band.
The interval width in ln κ is 2 − (−2) = 4.
Integrate the constant to obtain σ_band² = 4A = 4 × 10⁻⁶.

The supplied finite-band variance is 4 × 10⁻⁶, dimensionless.

Try the reasoning

If Δ_ℛ² is constant, is P_dim also constant?
Show the answer and reasoning
No.
With the stated Fourier convention, P_dim = 2π²Δ_ℛ²/k³, so it decreases as k⁻³.
Do the equal band variances for n = 0.9 and 1.1 mean their spectra are identical?
Show the answer and reasoning
No.
They weight opposite ends of the symmetric logarithmic band differently while integrating to the same total.
Does this power spectrum specify a particular observed sky or uniquely identify inflation?
Show the answer and reasoning
No.
It is an ensemble two-point statistic. Realization, transfer to observables and mechanism discrimination need additional information.

What to carry forward

Dimensionless power is variance per logarithmic wavenumber under a stated convention.
A pivot amplitude and a tilt describe different aspects of a finite-band spectrum.
Two-point statistics, observed realizations and claims about an origin mechanism require separate reasoning.

References

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

TASI Lectures on InflationDaniel BaumannCourse notesarxiv.orgRead source