CMB First Peak ℓ₁: LO at +2840 ppm, NLO at +69 ppm
The CMB first acoustic peak position ℓ₁ = 220.6 is derived at LO to +2840 ppm and at NLO to +69 ppm using δ_h = ι_τ/W₅(3).
Overview
V.T192 derives ℓ₁ = 220.6 at +2840 ppm (LO). V.T257 gives the NLO correction δ_h = ι_τ/W₅(3) = ι_τ/19, bringing ℓ₁ to +69 ppm. W₅(3) = 19 is the fifth windowing number at depth 3, which also equals N_e/3 where N_e = 57 is the number of e-foldings. The same CF window governs both the CMB peak and inflation, giving a structural connection between pre- and post-recombination physics.
Detail
The CMB acoustic peaks encode the physics of baryon-photon oscillations before recombination. The first peak position ℓ₁ ≈ 220 is the most prominent CMB feature and is measured by Planck to be ℓ₁ = 220.0 ± 0.4. Book V derives ℓ₁ from the τ-framework. At LO (V.T192), the peak position comes from the angular diameter distance to the last scattering surface divided by the sound horizon: ℓ₁ = π·D_A/r_s. The τ-framework values for D_A and r_s (using Ω_m = ι_τ²·W₃(4)/(2π) and other τ-determined quantities) give ℓ₁ at +2840 ppm. The NLO correction (V.T257, Wave 38A sprint) introduces δ_h = ι_τ/W₅(3) = ι_τ/19 as a dimensionless correction to the sound horizon. This brings ℓ₁ to +69 ppm. The key structural connection: W₅(3) = 19 = N_e/3 where N_e = 57 (Wave 14 result) — the same windowing number that governs inflationary e-foldings also governs the CMB peak position. This self-consistency is a structural fingerprint of the framework.
Result Statement
V.T192: ℓ₁ at +2840 ppm (LO). V.T257: NLO with δ_h = ι_τ/W₅(3) = ι_τ/19 gives +69 ppm. W₅(3) = 19 = N_e/3 connects inflation and CMB peak.