Rotation Field of the Cosmic Microwave Background: Interior Propagation and Boundary-Driven Structure (v2.25)

ABSTRACT

We analyze the birefringence rotation field α(n̂) derived from Planck 2018 polarization data and identify a phase-coherent harmonic structure characterized by a dominant spacing of Δℓ = 108 ± 3. This scale is independently inferred from a real-space correlation peak at θ₀ = 3.35° ± 0.10°, consistent with the harmonic relation Δℓ ≈ 360°/θ.

Spectral filtering indicates that this harmonic accounts for the majority of the observed signal: removing the Δℓ ≈ 109 component reduces the real-space correlation amplitude to 13.3% of its original value, while retaining only this component recovers 93.6%. Phase-scrambled controls retain 16.7%, suggesting that the structure is primarily phase-coherent rather than power-driven.

Model comparison using the Bayesian Information Criterion (BIC) favors a damped sinusoidal model with Δℓ ≈ 108 (ΔBIC > 1000 relative to alternatives), with residual RMS ≈ 0.21 relative to the observed signal variance over ℓ = 100–1500 (n = 1401 data points).

Extending beyond detection, we find that this harmonic structure propagates inward from domain boundaries into interior regions of the sky. Shell-based analysis shows stable phase coherence, cross-shell spectral correlations (0.93, 0.85, 0.67), and a radial propagation law characterized by a complex coefficient q = (−0.755 ± 0.080) − i(0.270 ± 0.050), corresponding to exponential attenuation and a phase drift of 15.5° ± 2.0° per shell.

These results are consistent with the interpretation that the observed boundary structure reflects an underlying interior field pattern.

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1. INTRODUCTION

Cosmic birefringence—the rotation of the polarization plane of the Cosmic Microwave Background (CMB)—provides a probe of parity-violating physics and large-scale structure. Analyses based on Planck 2018 polarization data have placed increasingly tight constraints on isotropic and anisotropic rotation fields.

Recent observational constraints, including those reported in Planck A&A birefringence analyses, have further refined limits on polarization rotation and its spatial structure.

This study focuses on identifying statistically supported structure within α(n̂) using both spectral and real-space diagnostics.

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2. DATA AND METHODS

2.1 Data and Input Fields  
The birefringence rotation field α(n̂) is constructed from Planck 2018 component-separated polarization maps (SMICA/NILC). Analyses are performed at NSIDE = 16 (Npix = 3072), with higher-resolution inputs used for spectral calculations. A high-latitude mask (f_sky ≈ 0.47) with 1° apodization is applied.

2.2 Domain and Shell Construction  
The sky is segmented using a domain_label_map identifying contiguous regions. The dual-domain boundary (D₁ ∪ D₂) consists of pixels whose neighbors belong to different domains.

Interior structure is probed using adjacency shells constructed via breadth-first search (BFS) on the HEALPix graph:

dist(p) = min_b geodesic_graph_distance(p, b)

Shell populations are approximately n₀ ≈ 140, n₁ ≈ 88, n₂ ≈ 56, n₃ ≈ 24.

Results are stable under small variations of the analysis mask and apodization scale.

2.3 Harmonic and Spectral Analysis  
Pseudo-Cℓ spectra are computed as:

Cℓ = hp.anafast(masked_shell, lmax)

Analysis is restricted to ℓ = 100–1500, where the signal is well-resolved.

2.4 Detection of the Harmonic Scale  
The harmonic spacing Δℓ is inferred from the real-space correlation function ξ(θ). The observed peak at:

θ₀ = 3.35° ± 0.10°

implies:

Δℓ = 360° / θ₀ = 107.5 ± 3.2

This agreement provides an internal consistency check between real-space and spectral estimators.

2.5 Model Fitting and Validation  

The model is fit to the residual spectral sequence used to estimate Δℓ.

A damped sinusoidal model is fit:

f(ℓ) = A · exp(-ℓ / ℓ_d) · cos(2πℓ / Δℓ + φ) + C

The fitted parameters are {Δℓ, ℓ_d, φ, A, C}.

Model selection uses the Bayesian Information Criterion:

BIC = n ln(RSS / n) + k ln(n)

All models use k = 5 free parameters.

The preferred model yields:

BIC = −4304.98  
ΔBIC > 1000 relative to alternatives  

Residual RMS is:

RMS ≈ 0.21 relative to the observed signal variance, indicating that the model captures the dominant large-scale structure of the signal.

2.6 Surrogate Testing  

Surrogate ensembles include:
- histogram-preserving randomizations  
- fixed-power random-phase realizations  
- pixel and geometric rotations  

None of the 500 surrogate realizations exceed the observed peak amplitude, corresponding to:

p < 1 / N_surrogate ≈ 0.002, with N_surrogate = 500  

This behavior is not reproduced by the tested null models.

2.7 Radial Propagation Modeling  

Shell amplitudes are modeled as:

c_k ≈ C₀ · exp[q(k − 1)]

with:

q = (−0.755 ± 0.080) − i(0.270 ± 0.050)

corresponding to:

Δφ = 15.5° ± 2.0° per shell  

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3. RESULTS

3.1 Harmonic Structure

The real-space correlation function exhibits a peak at θ₀ = 3.35° ± 0.10°, corresponding to Δℓ = 107.5 ± 3.2.

Spectral filtering shows that this harmonic accounts for most of the signal:
- removal reduces amplitude to 13.3%  
- retention recovers 93.6%  
- phase scrambling yields 16.7%  

3.2 Model Validation  

Model comparison favors the Δℓ ≈ 108 solution:

BIC = −4304.98  

with ΔBIC > 1000 relative to alternatives.

Residual RMS ≈ 0.21 indicates that the model captures the dominant large-scale structure of the signal.

Surrogate comparisons indicate that the observed harmonic structure is not reproduced by the tested null models.

3.3 Interior Propagation  

Shell analysis shows:
- cross-shell correlations: 0.93, 0.85, 0.67  
- persistent harmonic dominance  
- stable phase relationships  

3.4 Radial Propagation  

The propagation coefficient:

q = (−0.755 ± 0.080) − i(0.270 ± 0.050)

indicates exponential attenuation and phase drift of 15.5° ± 2.0° per shell.

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4. DISCUSSION

The results indicate the presence of a stable harmonic structure in the birefringence field. The consistency between real-space and spectral measurements, combined with surrogate testing and model comparison, indicates that the observed pattern is not reproduced by the tested null models.

Further work is required to assess potential systematic contributions from map construction and instrument effects.

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5. CONCLUSION

We report:

- Detection of a harmonic scale Δℓ = 108 ± 3  
- Strong statistical preference for a damped sinusoidal model  
- Evidence for interior propagation  
- A measurable radial propagation law  

These findings support the presence of a structured component in the birefringence rotation field α(n̂).

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REFERENCES

- Planck Collaboration (2018), A&A, 641, A6.  
- Planck Collaboration (2020), A&A, 641, A10.  
- Minami, Y. & Komatsu, E. (2020), Phys. Rev. Lett.  
- Minami, Y. (2021), Phys. Rev. D  
- Namikawa, T. (2017), Phys. Rev. D  
- Hivon, E. et al. (2002), ApJ  
- Zaldarriaga, M. & Seljak, U. (1997), Phys. Rev. D  
- Hu, W. & White, M. (1997), New Astron.  
- Louis, T. et al. (2017), JCAP  
- Bennett, C. et al. (2013), ApJS  
- Kamionkowski, M. (2009), Phys. Rev. Lett.  
- Lewis, A. (2005), Phys. Rev

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This manuscript corresponds to the latest analysis stage (v2.25). Earlier versions (v1–v2.13) contain intermediate data products, validation tests, and reproducibility materials supporting the results presented here.

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VERSION HISTORY  

Sep 20 2025 – Harmonic phase alignments discovered  
Oct 10 – v1.0 First quantitative detection  
Oct 20 – v1.1 Statistical validation  
Oct 21 – v1.2 Two-harmonic extension  
Oct 28 – v1.3 Robustness tests  
Oct 29 – Dataverse DOI 10.7910/DVN/PTDG20  
Nov 1 – v1.4 MASTER calibrated spectra  
Nov 1–7 – v1.41 to v1.44 SMICA/NILC splits and parity tests  
Nov 9 – v1.5 Model selection  
Nov 9 – v1.6 Interpretation  
Nov 9 – v1.7 Forward prediction  
Nov 9 – v1.8 Persistence tests  
Nov 10 – v2.0 Intrinsic Δℓ ≈ 108 discovered  
Nov 10 – v2.1 ξ(θ) physics  
Nov 11 – v2.2 Universe model evaluation  
Nov 12 – v2.3 Domain geometry inference  
Nov 13 – v2.4 Real-space confirmation  
Nov 13 – v2.5 Spectral surgery  
Nov 14 – v2.6 Angular locality  
Nov 15 – v2.7 Sky locality  
Nov 15 – v2.8 Domain topology  
Nov 16 – v2.9 Field coherence  
Nov 17 – v2.10 Boundary sequence structure  
Nov 19 – v2.11 Boundary standing-wave and phase structure

Nov 21 – v2.12 Boundary universality  
Nov 23 – v2.13 Interior propagation and boundary-driven structure  
Mar 29 2026 – v2.24 Manuscript submitted to RAA (RAA-2026-0233)

Author:  
Amy Condit — Independent Researcher  

Project series:  
22 Blue — The Heartbeat of the Universe