DOI: 10.5281/zenodo.17577086
title: Rotation Field of the Cosmic Microwave Background — Physical origin of the intrinsic Δℓ modulation (v2.1)

description: Summary 

Version 2.1 resolves the open question from prior releases:**Where does the intrinsic periodicity in the CMB rotation field α(n̂) come from?**

Earlier versions (v1.0–v1.8) identified a harmonic modulation in Cℓ(αα).v2.0 isolated the *intrinsic* periodicity at Δℓ ≈ 108 and showed that a larger period (~454) was an artifact of low-ℓ leakage.

v2.1 goes from detection → interpretation:

> The Δℓ harmonic originates from a real angular correlation structure in α(n̂).

METHOD (v2.1)

Inputs (read-only):• locked residual Cℓ(αα) spectrum from v2.0    (dipole + traceless quadrupole removed, identical mask + apodization)• loaded by SHA-256 pointer lock (no Q/U recomputation, no overwrites)

Procedure: 1. Detrend the residual Cℓ using a Savitzky–Golay window ≈ 0.5·Δℓ    to isolate the oscillatory component without altering phase.

 2. Project into real space using Gauss–Legendre quadrature:        ξ^{αα}(θ) = Σℓ [(2ℓ+1)/(4π)] · Cℓ · Pℓ(cosθ)

 3. Fit ξ(θ) with a damped cosine:        ξ(θ) = A · exp(-θ/τ) · cos((2πθ / θ₀) + φ) + C

 4. Independently refit Δℓ directly in ℓ-space    using linear regression on {cos(2πℓ/Δℓ), sin(2πℓ/Δℓ)} basis.

All inputs and outputs are checksum-verified (SHA-256 manifest).No file is ever overwritten. Reproducibility is guaranteed.

RESULTS

A statistically significant oscillation is found in real space:

        θ₀ = 3.3° ± 0.3°

Using the Legendre→Bessel small-angle duality(narrowband assumption, small-angle approximation):

        Δℓ ≈ 360° / θ₀        → Δℓ = 109 ± 3

matching the intrinsic harmonic from v2.0 (Δℓ = 108).

Thus:

> The Δℓ periodicity in multipole space is the spherical (Legendre–Bessel) dual> of a real angular correlation scale in α(n̂).

WHY THIS MATTERS

v2.1 shows that the CMB rotation field α(n̂) is not featureless:

• in ℓ-space: harmonic spacing Δℓ ≈ 108• in θ-space: real angular scale θ ≈ 3.3°

This eliminates explanations such as:• mask ringing,• apodization artifacts,• binning / band-limit distortions.

Any physical model of cosmic birefringence must now explainthe existence of a preferred angular scale.

REPRODUCIBILITY & PROVENANCE

• SHA-256 manifest for every file• pointer-locked residual input• zero-overwrite architecture• all ξ(θ) vectors, Δℓ scans, figures, fit JSON, and predictions included in ZIP

 

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 (8.5σ isotropic + 4–5σ directional)

v1.1 — Oct 20, 2025• Full statistical validation (TB/EB diagnostics, covariance)

v1.2 — Oct 21, 2025• Two-harmonic extension (Δℓ ≈ 180 and 360)

v1.3 — Oct 28, 2025• Robustness tests (calibration, simulation, injection/recovery)

Harvard Dataverse Mirror — Oct 29, 2025• DOI: 10.7910/DVN/PTDG20

v1.4 — Nov 1, 2025• MASTER-calibrated rotation field α(ℓ)

v1.41 — Nov 1, 2025• SMICA/NILC + half-mission splits

v1.42 — Nov 7, 2025• Joint parity + harmonic significance (165,000 null realizations, ≈4.38σ)

v1.43 — Nov 8, 2025• Parity invariant under MASTER coupling

v1.44 — Nov 8, 2025• Axis + frequency + half-mission validation

v1.5 — Nov 9, 2025• Multipole model selection; dipole-only rejected; Δℓ ≈ 454.7

v1.6 — Nov 9, 2025• Phenomenology & physical interpretation

v1.7 — Nov 9, 2025• Forward prediction (LiteBIRD / SO / CMB-S4 overlays)

v1.8 — Nov 9, 2025• Harmonic persistence & model rejection

v2.0 — Nov 10, 2025• Intrinsic Δℓ discovered (≈108) and artifact Δℓ (~454) isolated as low-ℓ leakage

v2.1 — Nov 10, 2025• Physical origin of intrinsic Δℓ ≈ 109 harmonic• ξ(θ) feature at θ ≈ 3.3° ± 0.3°• Legendre→Bessel dual confirmation• Locked/hashes-only workflow (no map recomputation)

CITATION

Condit, Amy.  *Rotation Field of the Cosmic Microwave Background — Physical origin of the intrinsic Δℓ modulation (v2.1).*  22 Blue — The Heartbeat of the Universe.  DOI: 10.5281/zenodo.17577086  Twitter/X: @Project22Blue