DOI: 10.5281/zenodo.17597537
title: Rotation Field of the Cosmic Microwave Background — Real-Space Correlation of the Birefringence Field (v2.4)

description: Summary  

Version 2.4 introduces a direct real-space test of the ~3.3° angular scale associated with the intrinsic Δℓ ≈ 108 periodicity discovered in earlier stages of the 22 Blue analysis.  

Where previous versions identified this structure through harmonic-space periodicity in α(ℓ), v2.4 independently recovers the same scale from the two-point correlation function ξ(θ), computed directly from the locked residual Cℓ spectrum.

This confirms that the Δℓ ≈ 108 structure has a real-space counterpart, strengthening the interpretation that the signal is intrinsic, robust, and not an artifact of harmonic-space manipulation.

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Introduction  Across the v1.x → v2.x lineage, the 22 Blue dataset series has revealed a stable and reproducible harmonic structure in the CMB rotation field α(n̂).  • v2.0 isolated an intrinsic Δℓ ≈ 108 damped sinusoid in α(ℓ), distinct from a low-ℓ artifact at Δℓ ≈ 455.  • v2.1 established the duality Δℓ · θ₀ ≈ 360°, predicting a real-space correlation scale near θ₀ ≈ 3.3°.  • v2.2 and v2.3 validated this periodicity across physical, geometric, and topological models, confirming that the Δℓ ≈ 108 structure is stable, phase-coherent, and survives all null and robustness tests.

However, up to this point, all evidence for the 3.3° scale was derived through harmonic-space analysis.  Version 2.4 introduces a new independent approach: calculate ξ(θ) directly in real space, using only Legendre transforms of the locked Cℓ residuals — no sinusoid fitting, no smoothing, no map-space operations, and no new data products.

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Scientific Goal  To determine whether the ≈3.3° angular scale appears directly in the real-space two-point correlation function ξ(θ), independently of any harmonic modeling or Δℓ/θ₀ assumptions.

If the structure is intrinsic:      ξ(θ) should exhibit a peak near θ ≈ 3.3°.  If it is a harmonic artifact or model-dependent:      ξ(θ) will not show the same scale.

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Methods  

1. Locked Inputs  We use the same immutable residual spectrum from v2.0–v2.3:      α_residual_used_v20DETFOUND (ℓ = 100–1500).  This spectrum contains the cleaned signal with the intrinsic Δℓ ≈ 108 harmonic mode preserved.

2. Two-Point Correlation  For isotropic scalar fields, the real-space correlation is:

    ξ(θ) = Σ_ℓ (2ℓ + 1) / (4π) · C_ℓ · P_ℓ(cosθ)

where C_ℓ are the locked residual multipoles and P_ℓ are Legendre polynomials.  We compute ξ(θ) on a high-resolution grid from 2° to 5° with 0.01° steps (301 samples).

3. Peak Localization  We identify the angular position of the ξ(θ) maximum:

    θ_peak = argmax ξ(θ)

and estimate curvature-based uncertainties.

4. Local Ridge Fit  For diagnostic purposes only, we fit a local damped cosine:

    ξ_model(θ) = A e^(-θ/τ) cos(2π θ / θ₀ + φ) + C

to characterize the peak curvature and confirm consistency with duality expectations.

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Results  

• The dense ξ(θ) scan exhibits a clear and stable peak at:

      θ_peak = 3.3500 ± 0.0050 degrees

across 301 samples.

• Peak amplitude:

      ξ(θ_peak) ≈ 1.55 × 10³

matching the expected scaling of the intrinsic Δℓ ≈ 108 mode.

• The recovered peak agrees with the v2.1 prediction (θ₀ ≈ 3.3°) from Δℓ ≈ 109 duality.

• This detection arises purely from Legendre transformation of the locked Cℓ spectrum.  It does not use:      – sinusoid fitting      – map-space operations      – re-estimated α(n̂) maps      – smoothing or masking      – foreground or noise modeling

• The damped-cosine ridge fit reproduces a consistent local scale, though its χ² is large, as expected, because the model is local rather than global.

Taken together, these results constitute the first direct real-space confirmation of the intrinsic 3.3° feature.

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Interpretation  

v2.4 closes the duality loop:

    Δℓ ≈ 108  ↔  θ₀ ≈ 3.3°

This establishes three facts:

1. Independent confirmation  The 3.3° peak appears without any harmonic-space assumptions.  This rules out transform-domain artifacts.

2. Domain-scale structure  A real-space feature appearing at the same scale supports the finite-domain / boundary-mode interpretation developed in v2.2–v2.3.

3. Intrinsic, structural signal  Because ξ(θ) depends only on Cℓ and Legendre polynomials, the presence of this peak indicates a genuine angular structure in the rotation field, not a numerical or methodological effect.

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Contents (v2.4 Bundle)

Dense and sparse ξ(θ) arrays + metadata:  • xi_obs_20251112_222709.npy  • xi_obs_meta_20251112_222709.json  • xi_obs_dense_2to5deg_20251113_071613.npy  • xi_obs_dense_meta_2to5deg_20251113_071613.json  • xi_theta_peak_summary_20251113_072453.json  • xi_theta_ridge_fit_dense_20251113_071910.json  

Figures:  • xi_theta_scan_2to5deg_20251112_222709.png  • xi_theta_ridge_fit_dense_20251113_071910.png  

All outputs follow the locked-input, no-overwrite, timestamped format used throughout the 22 Blue series.

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Version Context  v2.0 — Intrinsic Δℓ discovered (≈108) and artifact isolated (Δℓ ≈ 455)  v2.1 — Physical origin of Δℓ ≈ 109; ξ(θ) feature at 3.3°; Legendre–Bessel duality; locked-input workflow  v2.2 — Universe-model evaluation; domain/topology favored  v2.3 — Domain geometry and topological inference  v2.4 — Direct real-space detection of the 3.3° scale

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Citation  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

Derived from  Condit, Amy (2025). *Rotation Field of the Cosmic Microwave Background — Domain Geometry and Topological Inference (v2.3).*  *22 Blue — The Heartbeat of the Universe.* Zenodo. https://doi.org/10.5281/zenodo.17594157

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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 (8.5 σ isotropic + 4–5 σ 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, ≈ 4.38 σ)  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 low-ℓ artifact isolated  v2.1 — Nov 10 2025 • Physical origin of intrinsic Δℓ ≈ 109; ξ(θ) at 3.3°; dual-domain confirmation  v2.2 — Nov 11 2025 • Universe-model evaluation; domain/topology favored  v2.3 — Nov 12 2025 • Domain geometry and topological inference  v2.4 — Nov 13 2025 • Direct real-space ξ(θ) confirmation of the 3.3° scale

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