DOI: 10.5281/zenodo.17604982
title: Rotation Field of the Cosmic Microwave Background — Spectral Surgery on the Δℓ ≈ 109 Harmonic (v2.5)

description: Summary

Version 2.5 performs a targeted “spectral surgery” on the locked birefringence spectrum Cℓ(αα) to determine whether the intrinsic harmonic at Δℓ ≈ 109—identified in v2.0–v2.1 and confirmed in v2.4—is the source of the 3.3° real-space correlation ridge observed in ξ(θ). This workflow constructs and compares four strictly ℓ-space spectra (observed, Δℓ-removed, Δℓ-only, and phase-scrambled), and evaluates their ξ(θ) peaks using only ℓ-domain and θ-domain operations (no maps, no alm, no healpy). The goal is to test whether the Δℓ ≈ 109 structure is necessary, nearly sufficient, and phase-coherent for producing the 3.3° feature.

Scientific Motivation

v2.0 revealed an intrinsic Δℓ ≈ 109 harmonic in the birefringence spectrum.  v2.1 showed that this same harmonic naturally produces a 3.3° feature in ξ(θ).  v2.4 directly measured that 3.3° peak in real space using Legendre projection.

v2.5 asks the structural question:  Does the 3.3° ridge genuinely originate from the Δℓ ≈ 109 harmonic—or could it be incidental, diffuse, or dependent on non-harmonic structure?

Locked Inputs

Residual spectrum (unchanged since v2.0–v2.4): alpha_residual_used_v20DETFOUND_20251110_235343.npy  ℓ range: 100–1500  Locked priors: Δℓ = 109.0, θ₀ = 3.3°  All methods restricted to ℓ-space and θ-space.  No α(n̂), no alm, no healpy, no recalibration.

Methods

1. Harmonic Model  Hℓ(A, φ, ℓ_d, C) = A cos(2πℓ/109 + φ) exp(−ℓ/ℓ_d) + C

Fitted parameters (from v2.5):  A = 0.7164216634  φ = −0.0983750160  ℓ_d = 513.0066225  C = 0.3264039017  

2. Constructed Spectra  Observed spectrum:        Cℓ^obs  Δℓ-removed spectrum:      Cℓ^noΔℓ = Cℓ^obs − Hℓ  Δℓ-only spectrum:         Cℓ^harm = Hℓ  Phase-scrambled spectrum: Cℓ^scrambled = Hℓ with randomized φℓ (fixed seed)

3. Real-Space ξ(θ) Reconstruction  ξ(θ) = Σℓ (2ℓ+1)/(4π) · Cℓ · Pℓ(cos θ)  θ grid: 2°–5° with 0.01° spacing.

Findings

Observed ξ(θ):         θ = 3.350°, ξ = 1551.36  Δℓ-removed:            θ = 2.840°, ξ = 206.781  Δℓ-only:               θ = 3.350°, ξ = 1451.36  Scrambled phases:      θ = 2.250°, ξ = 258.65  

Amplitude ratios (relative to the observed ξ peak):  R_noΔℓ      = 0.133  R_Δℓ-only   = 0.936  R_scrambled = 0.167  

Interpretation

Necessary: Removing the Δℓ ≈ 109 harmonic collapses the ξ(θ) amplitude to 13% and shifts the peak away from 3.3°, demonstrating that the ridge cannot form without this mode.

Nearly Sufficient: Keeping only the Δℓ ≈ 109 harmonic reproduces 94% of the observed ξ(θ) amplitude at exactly the same angular location, showing that the intrinsic standing wave accounts for nearly the full structure.

Phase-Coherent: Using the same power spectrum but scrambling the phases destroys the coherence, reducing the signal to 17% and shifting the peak to 2.25°. The ridge is therefore a phase-coherent standing-wave, not a random power arrangement.

These three results together show that the Δℓ ≈ 109 harmonic is the structural origin of the 3.3° real-space feature.

Citation

Condit, Amy (2025). *Rotation Field of the Cosmic Microwave Background — Spectral Surgery on the Δℓ ≈ 109 Harmonic (v2.5).*  *22 Blue — The Heartbeat of the Universe.* Zenodo. https://doi.org/10.5281/zenodo.17604982

Derived from  Condit, Amy (2025). *Rotation Field of the Cosmic Microwave Background — Real-Space Correlation of the Birefringence Field (v2.4).*  *22 Blue — The Heartbeat of the Universe.* Zenodo. https://doi.org/10.5281/zenodo.17597537

Version History (22 Blue Dataset Series)

Separate publication — Sep 20 2025 • Evidence for Harmonic Phase Alignments in Planck 2018 CMB polarization data (Δℓ ≈ 320)  v1.0 — Oct 10 2025 • First quantitative detection of anisotropic cosmic birefringence  v1.1 — Oct 20 2025 • Statistical validation and reproducibility  v1.2 — Oct 21 2025 • Two-harmonic extension  v1.3 — Oct 28 2025 • Robustness tests and injections  Dataverse Mirror — Oct 29 2025 • DOI: 10.7910/DVN/PTDG20  v1.4 — Nov 1 2025 • MASTER-calibrated α(ℓ)  v1.41–1.44 — Nov 1–8 2025 • SMICA/NILC + HM splits; parity and axis checks  v1.5 — Nov 9 2025 • Model selection; Δℓ ≈ 454.7  v1.6 — Nov 9 2025 • Phenomenology and physical interpretation  v1.7 — Nov 9 2025 • Forward prediction  v1.8 — Nov 9 2025 • Harmonic persistence and rejection tests  v2.0 — Nov 10 2025 • Intrinsic Δℓ ≈ 108 discovered  v2.1 — Nov 10 2025 • Physical origin; ξ(θ) at 3.3°  v2.2 — Nov 11 2025 • Universe-model evaluation  v2.3 — Nov 12 2025 • Domain geometry inference  v2.4 — Nov 13 2025 • Direct real-space ξ(θ) confirmation  v2.5 — Nov 13 2025 • Spectral surgery: Δℓ ≈ 109 shown necessary (13%), nearly sufficient (94%), and phase-coherent (17%) for the 3.3° ridge

22 Blue — The Heartbeat of the Universe  Twitter/X: @Project22Blue