spectralIndex
plain-language theorem explainer
The scalar spectral index n_s is defined as 1 minus six times the first slow-roll parameter plus twice the second slow-roll parameter, evaluated at positive field value φ. Cosmologists working in Recognition Science cite this when extracting the tilt of the primordial power spectrum from J-cost inflation. The definition is a direct algebraic substitution of the slow-roll parameters derived from the inflaton potential.
Claim. The scalar spectral index satisfies $n_s = 1 - 6ε + 2η$ for field value φ > 0, where ε = (V'/V)^2 / 2 and η = V''/V with potential V(φ) = (φ + φ^{-1})/2 - 1.
background
In the COS-001 module, cosmic inflation arises from the J-cost function J(x) = (x + x^{-1})/2 - 1, identified as the inflaton potential V(φ) with minimum at φ = 1. Slow-roll parameters are formed from its derivatives: ε quantifies the squared slope relative to height, while η quantifies curvature relative to height. Upstream definitions compute ε(φ) = ((1 - 1/φ²)/2 / V)^2 / 2 and η(φ) = (1/φ³) / V when V > 0.
proof idea
The definition is a one-line wrapper that applies slowRollEpsilon and slowRollEta to the supplied field value φ.
why it matters
This supplies the scalar spectral index n_s used by CMBObservable to enumerate cosmological observables and by CosmicRayCert to certify the spectral index band (2.61, 2.63). It completes the COS-001 derivation of the nearly scale-invariant spectrum n_s ≈ 0.96 from J-cost slow roll, linking the eight-tick octave and phi-ladder structure to early-universe observables.
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