tensor_to_scalar_upper_bound
Recognition Science cosmology adopts 0.06 as the upper limit on the tensor-to-scalar ratio r. Researchers comparing RS-derived primordial spectra to CMB observations would cite this bound when testing J-cost fluctuation models. The definition is a direct assignment of the combined Planck and BICEP/Keck constraint.
claimThe tensor-to-scalar ratio satisfies $r < 0.06$, with the numerical value taken from Planck and BICEP/Keck observations of the cosmic microwave background.
background
The PrimordialSpectrum module derives the CMB power spectrum P(k) = A_s (k/k_*)^(n_s - 1) from Recognition Science principles. Primordial fluctuations arise from J-cost quantum fluctuations during inflation, with the phi-ladder fixing the spectral tilt n_s - 1. The module imports the scale function defined as phi^k and the power function that multiplies amplitude by the scaled wavenumber term. Upstream amplitude definitions from S-matrix and double-slit contexts supply the underlying transition probabilities used in fluctuation calculations.
proof idea
The definition is a direct assignment of the observational upper bound 0.06 for the tensor-to-scalar ratio.
why it matters in Recognition Science
This constant supplies the observational anchor for the COS-009 derivation of the primordial spectrum from J-cost fluctuations. It supports the target paper proposition on the CMB spectral index from the golden ratio and connects to the nearly scale-invariant spectrum with n_s ≈ 0.965. The bound is referenced against the amplitude and power functions in the same module but has no downstream uses yet.
scope and limits
- Does not derive the numerical bound from RS first principles or the phi-ladder.
- Does not include observational error bars or confidence intervals.
- Does not define the tensor-to-scalar ratio explicitly in terms of J-cost or defectDist.
- Does not specify how the bound enters the power spectrum formula.
formal statement (Lean)
52noncomputable def tensor_to_scalar_upper_bound : ℝ := 0.06
proof body
Definition body.
53
54/-! ## The Power Spectrum -/
55
56/-- The primordial power spectrum P(k) = A_s (k/k_*)^(n_s - 1).
57
58 - k: wavenumber (inverse length scale)
59 - k_*: pivot scale (0.05 Mpc⁻¹)
60 - A_s: amplitude at pivot
61 - n_s: spectral index -/