Cosmology Inflaton Potential Structural Slow Roll Eta Pos

The slow-roll parameter

In cosmology, the slow-roll approximation describes a scalar field that drives a period of accelerated expansion, called inflation, by rolling down its potential energy very gradually. Two parameters, epsilon and eta, quantify how slowly the field moves and how flat the potential is. In the standard notation, epsilon is proportional to the square of the field's velocity, and eta measures the curvature of the potential relative to the expansion rate. A positive eta means the potential is not curving downward steeply enough to accelerate the field; the field stays in a slow roll.

The Recognition Science framework defines its own inflaton potential with five structural regimes: a slow-roll plateau, a slow-roll slope, a hilltop decline, reheating, and the post-reheating radiation era. Within that model, the slow-roll parameters take specific values: epsilon equals 1 divided by twice the fifth power of the golden ratio, and eta equals 1 divided by the fifth power of the golden ratio. The declaration slowRollEta_pos establishes that this defined eta is strictly greater than zero. In plain terms, the framework's model of the inflaton potential has a positive eta, consistent with a slow-roll phase rather than a fast-roll or tachyonic instability.

The declaration is a formal theorem in the framework's machine-checked library of formal theorems, proved without any unproved assumptions beyond the standard logical axioms. It does not, however, assert that the actual universe's inflaton field has a positive eta. It only proves a property of the framework's chosen definition. The model's parameters are set by the framework's internal structure, not by a fit to cosmological data. The declaration also does not claim that the slow-roll approximation is valid for the real universe; that is a separate physical question.

What the result offers is a consistency check: within the Recognition Science model, the slow-roll phase is internally coherent. A reader who accepts the framework's definitions can rely on this positivity as a proved fact, not a guess. For a cosmologist outside the framework, the declaration is a statement about a particular model's parameters, not an observational constraint.

theorem slowRollEta_pos : 0 < slowRollEta := by
  unfold slowRollEta
  exact div_pos one_pos (pow_pos phi_pos 5)

inductive InflatonRegime where
  | slowRollPlateau
  | slowRollSlope
  | hilltopDecline
  | reheating
  | radiationEra
  deriving DecidableEq, Repr, BEq, Fintype

/-- Slow-roll parameter η = 1/φ⁵. -/
noncomputable def slowRollEta : ℝ := 1 / phi ^ 5

What this page does not claim

This declaration does not claim that the actual universe's inflaton field has a positive eta. This declaration does not claim that the slow-roll approximation is observationally verified. This declaration does not claim that the framework's model parameters are derived from first principles without any definitional choices.

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/InflatonPotentialStructural.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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