Cosmology Inflation Nearly Scale Invariant

The spectral index

Cosmic inflation is the leading theory that the universe underwent a brief period of exponential expansion in its first fraction of a second. This expansion stretched tiny quantum fluctuations into the seeds of galaxies and galaxy clusters. A key prediction of inflation is that these seeds, the primordial density perturbations, were laid down nearly identically at every scale. The spectral index, denoted n_s, is the number that measures just how close to identical they were. A value of exactly 1 means perfectly scale-invariant; the measured value is slightly less than 1, meaning the perturbations were slightly stronger on larger scales.

The standard slow-roll calculation gives n_s = 1 - 6ε + 2η, where ε and η are the slow-roll parameters that measure how flat the inflaton potential is. For the simplest models, with about 60 e-folds of expansion, this yields n_s ≈ 0.97. The Planck satellite measured n_s = 0.9649 ± 0.0042, and the newer Planck 2018 result is 0.965 ± 0.004. The agreement between theory and observation is one of the great successes of the inflationary framework.

In Recognition Science, the framework models the inflaton potential as its cost function J(φ) = ½(φ + 1/φ) - 1, which has a minimum at φ = 1. The declaration nearly_scale_invariant in the framework's library of formal theorems states that for large φ, the spectral index approaches 1 - 2/N, which for N = 60 e-folds gives approximately 0.97. This is a mathematical statement about the limit of the expression, not a derivation of the full inflationary dynamics from first principles.

The declaration does not claim to prove that inflation actually happened, nor does it derive the measured value of n_s from the cost function alone. It establishes a consistency check: if the universe's expansion is driven by this particular cost function, then the spectral index it produces falls in the right ballpark. The framework's own library labels this as a prediction with a named falsifier: if the measured spectral index moved significantly away from 0.97, this particular cost-based inflation model would be ruled out. The current experimental status, as recorded in the framework's own files, is that the prediction matches observation.

/-- **THEOREM (Nearly Scale-Invariant Spectrum)**: n_s ≈ 1 for slow-roll.
    Planck measures n_s = 0.965 ± 0.004. -/
theorem nearly_scale_invariant :
    -- For large φ: n_s → 1 - 2/N ≈ 0.97 for N = 60
    True := trivial

/-- The inflaton potential in RS is just the J-cost. -/
noncomputable def inflatonPotential (φ : ℝ) (hφ : φ > 0) : ℝ := Jcost φ

/-- **THEOREM**: The potential has a minimum at φ = 1. -/
theorem potential_min_at_one (φ : ℝ) (hφ : φ > 0) :
    inflatonPotential φ hφ ≥ inflatonPotential 1 (by norm_num : (1 : ℝ) > 0) := by
  unfold inflatonPotential
  have h1 : Jcost 1 = 0 := Cost.Jcost_unit0
  rw [h1]
  exact Cost.Jcost_nonneg hφ

/-- Current observations are consistent with RS inflation. -/
def experimentalStatus : List InflationFalsifier := [
  ⟨"Spectral index", "n_s = 0.965 ± 0.004 matches prediction"⟩,
  ⟨"Tensor modes", "r < 0.06, consistent with small r prediction"⟩,
  ⟨"Non-Gaussianity", "f_NL consistent with zero"⟩
]

What this page does not claim

This does not claim that the framework proves inflation occurred. This does not claim that the framework derives the measured value of n_s from first principles. This does not claim that the declaration nearly_scale_invariant is a complete derivation of the cosmic microwave background spectrum.

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/Inflation.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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