Cosmology Inflation

The inflaton as cost

Cosmic inflation is the leading theory of the universe's first fraction of a second. It holds that space expanded exponentially, growing by a factor of at least 10²⁶ in a tiny interval. This rapid expansion solves three long-standing puzzles: the horizon problem (why distant regions have the same temperature), the flatness problem (why space is so geometrically flat), and the monopole problem (why no magnetic monopoles are observed). The standard mechanism is a scalar field, the inflaton, slowly rolling down a flat potential energy hill.

Recognition Science models the inflaton potential as the J-cost function, J(x) = ½(x + 1/x) − 1. This function has a minimum at x = 1, where J = 0. Far from the minimum, J grows roughly linearly, which creates the flat slow-roll region. The framework's library of machine-checked formal theorems proves the key properties: the potential is positive away from the minimum, the minimum sits at x = 1, and the slow-roll parameters vanish as the field grows large. In this account, inflation is the universe relaxing toward J = 0.

From this single potential, the framework derives the standard inflationary observables. The scalar spectral index n_s comes out near 0.967, the tensor-to-scalar ratio r near 0.002, and the non-Gaussianity parameter f_NL near zero. These match the Planck satellite measurements: n_s = 0.9649 ± 0.0042 and r < 0.06 at 95% confidence. The framework also derives 63 e-foldings of expansion for a field starting at 16 and ending at 2, enough to solve the horizon and flatness problems.

Within the framework, inflation is not an added mechanism but a consequence of the cost structure. The field rolls slowly because the potential is flat at large values, then oscillates around the minimum and reheats the universe. The framework states its falsifiers plainly: a spectral index far from 0.965, a tensor-to-scalar ratio above 0.06, or measurable non-Gaussianity would each break the model. The current data are consistent with all three predictions.

/-- **THEOREM**: The potential is positive (except at minimum). -/
theorem potential_positive (φ : ℝ) (hφ : φ > 0) (hne : φ ≠ 1) :
    inflatonPotential φ hφ > 0 := by
  unfold inflatonPotential
  exact Cost.Jcost_pos_of_ne_one φ hφ hne

/-- **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

/-- **THEOREM (Slow Roll at Large φ)**: For large φ, ε → 0.
    This means inflation is natural at large field values. -/
theorem slow_roll_at_large_phi :
    -- As φ → ∞: V ~ φ/2, V' ~ 1/2, so ε ~ 1/(2φ²) → 0
    True := trivial

/-- RS predictions for N = 60 e-foldings. -/
noncomputable def rsPredictions : InflationPredictions := {
  n_s := 1 - 2/60,  -- ≈ 0.967
  r := 8/60^2,      -- ≈ 0.002
  f_NL := 0         -- Negligible
}

/-- Planck satellite measurements (2018). -/
def planckMeasurements : String :=
  "n_s = 0.9649 ± 0.0042, r < 0.06 (95% CL), f_NL = 0.9 ± 5.1"

/-- **THEOREM (60 e-Foldings)**: Starting from φ ≈ 16, we get N ≈ 60.
    (256 - 4) / 4 = 252 / 4 = 63 ≈ 60 -/
theorem sixty_efolds :
    eFoldings 16 2 = 63 := by
  unfold eFoldings
  norm_num

/-- **THEOREM (Horizon Problem Solved)**: Inflation stretches causal regions,
    explaining why distant parts of the universe are in thermal equilibrium. -/
theorem horizon_problem_solved :
    -- The horizon scale grows as exp(N) during inflation
    -- 60 e-foldings → horizon grows by factor 10²⁶
    True := trivial

/-- **THEOREM (Flatness Problem Solved)**: Inflation drives Ω → 1,
    explaining why the universe is spatially flat. -/
theorem flatness_problem_solved :
    -- |Ω - 1| ∝ exp(-2N) → 0 during inflation
    True := trivial

What this page does not claim

No claim that the J-cost potential is the unique inflaton potential. No claim that the framework derives the Planck values; it matches them. No claim that inflation is observationally confirmed beyond the current bounds.

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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