Cosmology Primordial Spectrum

The primordial spectrum

The primordial power spectrum describes how the strength of density fluctuations in the early universe varies with scale. In standard cosmology, it is written as P(k) ∝ k^(n_s - 1), where k is the wavenumber (inverse wavelength) and n_s is the spectral index. A perfectly scale-invariant spectrum would have n_s = 1, meaning equal fluctuation power on all scales. Measurements from the cosmic microwave background (CMB), the oldest light in the universe, give n_s ≈ 0.9649, slightly less than 1, a "red" tilt where larger scales have slightly more power. The overall amplitude A_s is measured at about 2.1 × 10⁻⁹, and the spectrum is conventionally evaluated at a pivot scale of 0.05 Mpc⁻¹. These ripples, imprinted when the universe was about 380,000 years old, are the seeds from which all cosmic structure grew.

The standard account attributes this spectrum to quantum fluctuations stretched across the sky during cosmic inflation, a period of exponential expansion in the first fraction of a second. The near-scale-invariance is a hallmark of inflation, but the precise value of n_s and the amplitude are inputs, not outputs, of the theory. The tensor-to-scalar ratio r, which measures the relative strength of gravitational waves versus density fluctuations, is only bounded from above, at about 0.06. These are the observed parameters that any deeper theory must reproduce.

In Recognition Science, the framework models primordial fluctuations as arising from fluctuations in its core quantity, the cost, a measure of recognition effort that the framework asserts must take a specific functional form. The framework's library, a machine-checked collection of formal theorems, states that density perturbations δρ/ρ are proportional to cost fluctuations δJ/J, and that the power spectrum is proportional to the variance ⟨δJ²⟩. The framework connects the spectral tilt to the golden ratio φ, the number satisfying φ² = φ + 1, which emerges from the cost function's structure. Its prediction for the tilt, 1/(8φ³) ≈ 0.0118, is within 15 percent of the measured |n_s - 1| ≈ 0.0351. The framework also defines a prediction for the tensor-to-scalar ratio as (φ - 1)⁴, though the library notes this expression evaluates to about 0.146, not the 0.02 listed in its own predictions, a discrepancy the framework acknowledges.

The framework's central statement, fluctuations_from_jcost, is a formal placeholder that is trivially true, not a derived result. The same holds for the amplitude derivation and the spectral tilt connection. The framework does establish a falsifier: a structure that would refute its claims if the spectral index showed no φ-connection or if the tensor-to-scalar ratio contradicted expectations. The observed values n_s ≈ 0.9649, r < 0.06, and f_NL ≈ -2 are recorded as definitions, and a list of predictions states n_s ≈ 0.970, r ≈ 0.02, and negligible running and non-Gaussianity.

What this means in plain language: the framework has a story for where the CMB ripples come from, tying their slight redness to the golden ratio. But the formal derivation of that story is not yet written. The framework is a scaffold of definitions, observed values, and testable predictions, with the actual derivation marked as a target. The honest verdict is that the framework proposes a mechanism and a falsifiable prediction, but does not yet establish the spectrum's shape from its axioms.

/-- 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

theorem spectral_tilt_phi_connection :
    -- |n_s - 1| ≈ 1/(8φ³) within 15%
    -- This connects spectral tilt to 8-tick and φ
    True := trivial

/-- 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)

/-- 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 derivation would be falsified if:
    1. n_s has no φ-connection
    2. r contradicts (φ-1)⁴ prediction
    3. Large non-Gaussianity found -/
structure SpectrumFalsifier where
  ns_no_phi : Prop
  r_contradicts : Prop
  large_nongaussianity : Prop
  falsified : ns_no_phi ∧ r_contradicts → False

What this page does not claim

The framework proves the primordial spectrum's shape from its axioms; the central statement is a placeholder. The framework derives the observed amplitude or the exact value of the spectral index. The framework's tensor-to-scalar prediction matches the observed upper bound, given the library's own noted arithmetic issue.

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.

Derived articles

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