Cosmology Primordial Spectrum Power Spectrum

The primordial spectrum

The cosmic microwave background (CMB) is the oldest light we can see, released about 380,000 years after the Big Bang. It is not perfectly smooth. Tiny temperature variations, roughly one part in 100,000, form a pattern across the sky, and that pattern is the seed of every galaxy and cluster we observe today. Cosmologists describe these ripples with a power spectrum, a function that says how much of the variation lives at each angular scale. The standard form is P(k) ∝ k^(n_s − 1), where k is the wavenumber, a measure of spatial frequency. The spectral index n_s tells whether the fluctuations are exactly the same on all scales (n_s = 1) or slightly stronger on large scales (n_s < 1).

Measurements from the Planck satellite, released in 2018, give n_s ≈ 0.9649, a value slightly less than 1, meaning the spectrum is a little "red": large-scale fluctuations are a bit stronger than small-scale ones. The overall amplitude A_s is about 2.1 × 10⁻⁹, a very small number that sets the size of the ripples. The pivot scale, the reference point where the amplitude is defined, is conventionally 0.05 Mpc⁻¹. This nearly scale-invariant spectrum is a central prediction of inflationary cosmology, the idea that the universe underwent a brief period of exponential expansion in its first fraction of a second, stretching quantum fluctuations to cosmic size.

In Recognition Science (RS), the framework models these primordial fluctuations as coming from the cost of recognition events. The framework's central object, the cost function J(x) = (x + 1/x)/2 − 1, measures the price of a recognition event, and the framework proposes that quantum fluctuations in this cost during inflation produce the density variations we observe. The framework's library defines a PowerSpectrum structure with amplitude, spectral index, and pivot scale, and provides a function to compute the power at any wavenumber. It also records the observed values as definitions: spectral_index_observed = 0.9649, scalar_amplitude_observed = 2.1 × 10⁻⁹, and a tensor-to-scalar upper bound of 0.06.

The framework makes a specific prediction for the spectral tilt. It derives that |n_s − 1| ≈ 1/(8φ³), where φ is the golden ratio, approximately 1.618. This gives n_s ≈ 0.970, which is within about 15% of the observed value. The framework also predicts a tensor-to-scalar ratio r ≈ 0.02 from (φ−1)⁴, though the theorem in the library only proves a weaker bound: 0.1 < (φ−1)⁴ < 0.2. The framework lists these as predictions, not theorems: n_s ≈ 0.970, r ≈ 0.02, running of n_s near zero, and non-Gaussianity f_NL near zero. A falsifier structure exists, stating that if n_s shows no φ connection or r contradicts the prediction, the framework's account is falsified.

What the PowerSpectrum declaration does not claim is as important as what it does. It does not prove that the observed spectrum matches the φ-based prediction; the spectral_tilt_phi_connection theorem is a placeholder with a True body, not a proof. It does not derive the amplitude 2.1 × 10⁻⁹ from first principles; the amplitude_derivation theorem is likewise a placeholder. The framework's contribution is a model with specific, testable predictions, not a completed derivation. The measured values are recorded as definitions, and the predictions are hypotheses awaiting confirmation or falsification.

/-- In RS, primordial fluctuations are J-cost fluctuations:

    1. During inflation, the ledger undergoes quantum fluctuations
    2. These manifest as J-cost variations: δJ ~ √(ℏ/τ₀)
    3. The fluctuations freeze out as the universe expands
    4. Later, they seed density perturbations -/
theorem fluctuations_from_jcost :
    -- δρ/ρ ∝ δJ / J
    -- Power spectrum P(k) ∝ ⟨δJ²⟩
    True := trivial

/-- The primordial power spectrum P(k) = A_s (k/k_*)^(n_s - 1).

    - k: wavenumber (inverse length scale)
    - k_*: pivot scale (0.05 Mpc⁻¹)
    - A_s: amplitude at pivot
    - n_s: spectral index -/
structure PowerSpectrum where
  amplitude : ℝ
  spectral_index : ℝ
  pivot_scale : ℝ
  amplitude_pos : amplitude > 0
  pivot_pos : pivot_scale > 0

/-- The scalar spectral index n_s ≈ 0.9649 (Planck 2018). -/
noncomputable def spectral_index_observed : ℝ := 0.9649

/-- The scalar amplitude A_s ≈ 2.1 × 10⁻⁹. -/
noncomputable def scalar_amplitude_observed : ℝ := 2.1e-9

/-- Analysis of n_s - 1 ≈ -0.035:

    Possible φ-connections:
    1. |n_s - 1| = (φ - 1)² = 0.382² = 0.146 (too large)
    2. |n_s - 1| = (φ - 1)³ = 0.236 × 0.382 = 0.090 (still large)
    3. |n_s - 1| = 1/(2φ³) = 1/(2 × 4.236) = 0.118 (too large)
    4. |n_s - 1| = 1/(8φ³) = 0.030 (close!)
    5. |n_s - 1| = 1/(φ⁸) = 1/46.98 = 0.021 (too small)

    Best fit: |n_s - 1| ≈ 1/(8φ³) ≈ 0.030 (vs observed 0.035) -/
noncomputable def phi_prediction_tilt : ℝ := 1 / (8 * phi^3)

/-- RS predictions for CMB observations:

    1. **n_s - 1 ≈ -1/(8φ³)**: Testable with Planck precision
    2. **r ≈ (φ-1)⁴ ≈ 0.02**: Testable by CMB-S4
    3. **Running ≈ 0**: Consistent with observations
    4. **f_NL ≈ 0**: Small non-Gaussianity -/
def predictions : List String := [
  "n_s ≈ 0.970 from φ-structure",
  "r ≈ 0.02 from (φ-1)⁴",
  "Running of n_s ~ 0",
  "Non-Gaussianity f_NL ~ 0"
]

What this page does not claim

The spectral_tilt_phi_connection theorem is a placeholder with a True body, not a proof of the φ connection. The amplitude 2.1 × 10⁻⁹ is not derived from first principles; the amplitude_derivation theorem is a placeholder. The framework does not claim its predictions are confirmed; they are hypotheses with a named falsifier structure.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Cosmology/PrimordialSpectrum.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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