Appendix U — Derivation 21: CMB Interference Structure from Modal Origins

Appendix U — CMB Harmonics as Coherence-Shell Interference

(concept sketch • simulation notebook to follow)

Status banner
This note outlines how Phase-Biased Geometry could reproduce the CMB
acoustic-peak spectrum without plasma sound waves.
No numerical fit is claimed; a dedicated notebook
cmb-coherence-shell.ipynb will generate a full $C_{ℓ}$ curve once the
spherical-harmonic mode sum is implemented.

1 The CMB in PBG

In PBG the early universe reaches a saturation radius
$R_{s}$ beyond which modal anchoring fails.
Modes that decouple at this boundary leave a coherence shell whose
residual phase pattern is observed today as the microwave background.
Temperature anisotropies therefore map interference of phase
surfaces, not plasma oscillations.

2 Boundary harmonics

Each marginal mode on the sphere is approximated by

ψ_{ℓ m} (θ, ϕ) = A_{ℓ m} e^{i m ϕ} P_{ℓ} (\cos θ),

with $ℓ$ the angular quantum number and $m = - ℓ \dots ℓ$ .
Interference between many such modes gives an angular power

C_{ℓ} = ⟨ | \sum_{m} a_{ℓ m} |^{2} ⟩, a_{ℓ m} = \int ψ (θ, ϕ) Y_{ℓ m}^{*} (θ, ϕ) d Ω .

3 Peak positions from geometry

Phase alignment enforces

ℓ_{p e a k} λ_{c} ≃ R_{s},

so the first peak appears at the largest wavelength
( $ℓ ≃ R_{s} / λ_{c}$ ).
Higher peaks correspond to harmonic numbers $2 ℓ_{p e a k}$ ,
$3 ℓ_{p e a k}$ , etc., with damping as constructive interference
falters at small scales.

4 No photon–baryon plasma required

Standard ΛCDM	PBG coherence shell
Standing sound waves	Boundary-anchored phase modes
Sound horizon sets $ℓ$	$R_{s} / λ_{c}$ sets $ℓ$
Damping from Silk viscosity	Damping from phase-decoherence
Peaks need baryon loading	Peaks need only mode geometry

Both pictures can yield similar $C_{ℓ}$ spectra; the difference lies in
underlying physics, not mathematics.

5 Next-step simulation (notebook plan)

Choose $λ_{c}$ from the locked constants
$k = \sqrt{β / α}$ (Appendix AA).
Populate random phase coefficients $A_{ℓ m}$ with a
blue-tilted spectrum (saturation bias).
Compute $C_{ℓ}$ up to $ℓ = 2500$ .
Compare with Planck likelihood (plot only; no parameter fit).

Notebook stub: notebooks/cmb-coherence-shell.ipynb.

Take-away

In PBG, the microwave background is interpreted as
harmonic interference on a coherence boundary, not as fossil sound
waves. Quantitative agreement with the observed peak pattern will be
tested once the harmonic-sum notebook is complete.

Appendix T | [Index](./Appendix Master) | Appendix V - Maxwell Equations