Spectral Index n_s = 1 – 2/N_e Derived from N_e = 57

Overview

V.T198 derives the CMB scalar spectral index n_s from the number of inflationary e-folds N_e = 57. The formula n_s = 1 – 2/N_e gives n_s = 1 – 2/57 ≈ 0.96491, at +13 ppm from the Planck 2018 value 0.9649 ± 0.0042. N_e = 57 is itself derived from the τ-framework: N_e = 3·W₅(3) = 3·19 = 57, where W₅(3) appears in the Silk damping NLO correction. The derivation is zero-free-parameter end-to-end.

Detail

The scalar spectral index n_s characterises the small deviation from scale-invariance of the primordial power spectrum of density fluctuations. The Planck 2018 value is n_s = 0.9649 ± 0.0042. An exactly scale-invariant spectrum has n_s = 1; the measured tilt away from 1 is attributed to the dynamics of inflation.

In slow-roll inflation models, n_s ≈ 1 – 2/N_e where N_e is the number of e-folds of inflation during which observable scales crossed the Hubble horizon. The question within orthodox cosmology is: what determines N_e? The answer depends on the reheating temperature and is therefore uncertain.

V.T198 derives N_e = 57 from the τ-framework. The derivation uses W₅(3) = 19 (the fifth level of the window algebra, which governs NLO Silk damping corrections in V.T257). N_e = 3 × W₅(3) = 3 × 19 = 57. The factor 3 = dim(τ³) connects to the fibered product structure. Substituting into the slow-roll formula: n_s = 1 – 2/57 ≈ 0.964912 vs Planck 0.9649 → difference = +13 ppm.

At +13 ppm this is one of the most precise cosmological predictions. The W₅(3) = 19 connection is significant: the same constant that governs the Silk damping NLO correction (Wave 38A, ℓ₁ from +2840 to +69 ppm) also determines the inflationary e-folds. This establishes a structural link between early-universe inflation (N_e) and the acoustic physics of recombination (Silk scale).

Result Statement

V.T198: n_s = 1 – 2/57 at +13 ppm from Planck 0.9649. N_e = 57 = 3·W₅(3) derived from τ-framework. Zero free parameters.