Cosmology Primordial Spectrum Tensor Spectrum

Tensor spectrum in the framework

The cosmic microwave background (CMB) is the oldest light in the universe, a faint glow that carries a map of density variations from the first moments after the Big Bang. These variations, called primordial fluctuations, seeded the clumping of matter that became galaxies and clusters. Cosmologists describe them with a power spectrum, a function that states how much fluctuation exists at each spatial scale. The spectrum is nearly scale-invariant: the amplitude is almost the same at every scale, with a slight tilt toward more power on large scales. Measurements from the Planck satellite give a spectral index n_s ≈ 0.965, where exactly 1 would mean perfect scale invariance. The overall amplitude is about 2.1 × 10⁻⁹.

Beyond the scalar fluctuations that become galaxies, inflation also predicts a background of gravitational waves, ripples in spacetime itself. These produce a separate tensor spectrum. The standard parameter for its strength is the tensor-to-scalar ratio r, the relative amplitude of the tensor spectrum to the scalar one. Current upper bounds place r below about 0.06, and no detection has yet been confirmed. A detection would be a major confirmation of inflation and a window into physics at energies far beyond the reach of particle accelerators.

In Recognition Science, the framework models primordial fluctuations as arising from fluctuations in its fundamental cost function J, the discrete ledger of recognition events. The framework's library, a machine-checked collection of formal theorems, defines a structure called TensorSpectrum with two fields: an amplitude and a tensor index. This is a definitional choice, not a derivation. The framework also defines the tensor-to-scalar ratio as the quotient of the tensor amplitude by the scalar amplitude, and it records a prediction for r based on the golden ratio φ: the value (φ − 1)⁴, which is about 0.146. The library notes that the correct value of about 0.021 would come from (2 − φ)⁴, and proves only a weaker bound that (φ − 1)⁴ lies between 0.1 and 0.2.

The framework's prediction list states r ≈ 0.02, which is consistent with current upper limits but not yet detected. This is a hypothesis with a named falsifier: the framework's SpectrumFalsifier structure declares that if the spectral index shows no connection to φ, or if the tensor-to-scalar ratio contradicts the prediction, the framework is falsified. The framework also predicts a spectral index n_s ≈ 0.970, close to but slightly higher than the observed 0.9649, and a running of the spectral index near zero. These are predictions to be tested, not established results.

What TensorSpectrum does not claim is a derivation of the tensor spectrum from first principles. The structure is a definition, and the connection between the golden ratio and the tensor amplitude is a prediction, not a theorem. The framework's own library notes that the comment linking (φ − 1)⁴ to 0.02 is incorrect, and that the actual value of (φ − 1)⁴ is about 0.146, far above the observed upper bound. The honest statement is that the framework defines the object, makes a testable prediction, and awaits data.

/-- Inflation also predicts tensor modes (primordial gravitational waves).

    Tensor power spectrum: P_T(k) = A_T (k/k_*)^(n_T)

    Consistency relation: n_T = -r/8 (single-field slow-roll)

    Current bound: r < 0.06 (Planck + BICEP/Keck)
    Future: CMB-S4 will probe r ~ 0.001 -/
structure TensorSpectrum where
  amplitude : ℝ
  tensor_index : ℝ

/-- The tensor-to-scalar ratio r = A_T / A_s. -/
noncomputable def tensor_to_scalar (ps_s ps_t : ℝ) (hs : ps_s > 0) : ℝ :=
  ps_t / ps_s

/-- RS prediction for r:

    r may be φ-related. Possible predictions:
    - r = (φ - 1)⁴ = 0.021 (testable by CMB-S4!)
    - r = 1/(8φ⁵) = 0.011
    - r = 1/φ⁸ = 0.021

    All these are in the observable range! -/
noncomputable def rs_prediction_r : ℝ := (phi - 1)^4

theorem r_prediction :
    -- r ≈ 0.02 is a testable RS prediction
    -- NOTE: The comment "(φ-1)⁴ = 0.382⁴" is incorrect.
    -- φ - 1 ≈ 0.618 (the golden ratio conjugate), so (φ-1)⁴ ≈ 0.146.
    -- The correct value 0.021 would be (2-φ)⁴ = 0.382⁴.
    -- For now, we prove a weaker bound: 0.1 < (φ-1)⁴ < 0.2
    0.1 < rs_prediction_r ∧ rs_prediction_r < 0.2 := by
  unfold rs_prediction_r
  -- φ - 1 ≈ 0.618, so (φ-1)⁴ ≈ 0.146
  -- Using bounds: 1.61 < φ < 1.62, so 0.61 < φ-1 < 0.62
  have h_phi_gt : phi - 1 > 0.61 := by
    have h := phi_gt_onePointSixOne
    linarith
  have h_phi_lt : phi - 1 < 0.62 := by
    have h := phi_lt_onePointSixTwo
    linarith
  -- 0.61^4 ≈ 0.138 > 0.1, 0.62^4 ≈ 0.148 < 0.2
  have h_low : (0.61 : ℝ)^4 > 0.1 := by norm_num
  have h_high : (0.62 : ℝ)^4 < 0.2 := by norm_num
  have h_phi_pos : phi - 1 > 0 := by linarith [one_lt_phi]
  constructor
  · calc 0.1 < (0.61 : ℝ)^4 := h_low
       _ < (phi - 1)^4 := by
           apply pow_lt_pow_left₀ h_phi_gt (by norm_num) (by norm_num)
  · calc (phi - 1)^4 < (0.62 : ℝ)^4 := by
           apply pow_lt_pow_left₀ h_phi_lt (le_of_lt h_phi_pos) (by norm_num)
       _ < 0.2 := h_high

/-- 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 tensor spectrum amplitude is not derived from the framework's axioms; it is a defined quantity. The value (φ − 1)⁴ is not claimed to match the observed upper bound of 0.06; the library itself notes this discrepancy. No claim is made that the tensor spectrum has been measured; current data only provides an upper limit.

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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