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The paper argues that map-space E/B decomposition of CMB polarization creates apparent spectral complexity even when the sky is a simple power law in P=Q+iU, so foreground modelling should be done in the spin-preserving fields P_E and P_B r

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load-bearing objection A coherent formal framework for spectral complexity in E/B-separated fields, but the practical recommendation is only demonstrated on full-sky noiseless maps — exactly the regime real B-mode analyses leave behind. the 2 major comments →

arxiv 2603.02177 v2 pith:R6S7M7GS submitted 2026-03-02 astro-ph.CO astro-ph.GA

Interpreting map-based E/B spectral properties of CMB foregrounds

Gilles Weymann-Despres , L\'eo Vacher , Michael E. Jones , Angela C. Taylor , Carlo Baccigalupi , A.J. Banday , Richard D.P. Grumitt , Nicoletta Krachmalnicoff This is my paper
classification astro-ph.CO astro-ph.GA
keywords CMB polarizationB-mode foregroundsE/B decompositionspectral complexitysynchrotron emissioncomplex log-Taylor expansionmoment expansionFaraday rotation
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper tries to establish that the map-space E/B transform is a non-local (finite-width convolution) operation, so it mixes pixels with different spectral indices and thereby manufactures apparent spectral curvature, tilt variation, and polarization-angle rotation in the projected fields even when the underlying sky is a rigid power law in P=Q+iU. It then argues that among all field representations, the spin-2 complex fields P_E and P_B are the most useful: they satisfy the exact closure relation P=P_E+P_B at every frequency and at every spectral order, their amplitudes and angles remain physically interpretable, and they suffer only moderate induced distortions. By contrast, scalar combinations |E| and |B| acquire the largest induced spectral complexity, while S=E+iB is less complex but has no interpretable polarization amplitude or angle. The paper develops complex log-Taylor and moment expansions to quantify this, links the coefficients to physical mechanisms (line-of-sight mixing, spatially varying spectral index, ageing, Faraday rotation), and gives three-frequency diagnostics that can decide whether the data prefer a single power law in P or independent power laws in (P_E,P_B). If correct, these tools give practical guidance for foreground extrapolation, masking, and cleaning in CMB B-mode searches.

Core claim

The central discovery is that spectral simplicity is not an invariant of the E/B decomposition. Even when the total polarization P follows a rigid, angle-stable power law, the non-fully-local E/B projectors — which are short-range convolutions on the sphere — superpose contributions from neighbouring pixels with different spectral indices, generating effective curvature and angle evolution in the E- and B-projected fields. Because the projectors are linear, their action commutes with the moment expansion: the complex moments of P_E and P_B are just the projections of the moments of P, and the closure relation P=P_E+P_B extends to all orders, w_n = w_{n,E} + w_{n,B}. The paper verifies this c

What carries the argument

The load-bearing objects are the complex spin-2 field P=Q+iU and its spin-preserving E/B projections P_E=L_E[P] and P_B=L_B[P], defined through harmonic-space operations whose real-space kernels are finite-width convolutions. The central technical instruments are the complex log-Taylor expansion, with parameters (log A, beta, gamma) equal to derivatives of log X with respect to s=log(nu/nu0), and the complex moment expansion around a reference spectral index, with moments w_n. Linearity of L_E and L_B makes the two expansions transform trivially: moments project field-wise, w_{n,E}=L_E[w_n], w_{n,B}=L_B[w_n], and the closure P=P_E+P_B becomes w_n=w_{n,E}+w_{n,B} at every order. The non-local

Load-bearing premise

All main conclusions assume noiseless, full-sky, band-limited maps, because the E/B-separating transforms are strictly well defined only on the full sky; real CMB analyses use masks and partial-sky data, so the ranking of field representations may not survive on a cut sky.

What would settle it

Take a full-sky, noiseless simulation where P is exactly a rigid power law in every pixel, apply the map-space E/B transform, and fit the first three log-Taylor parameters in P_E and P_B from a few channels. If the fitted curvature and angle derivatives in P_E/P_B are not systematically larger than in P — or if the closure relation fails on a masked sky — the induced-complexity claim collapses.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Foreground spectral extrapolation and component separation should be performed in P_E/P_B rather than in Q,U,P,|E|,|B|, because these fields retain closure and geometric interpretability with moderate induced spectral complexity.
  • Masking for CMB B-mode searches can be built by thresholding a |P_B|-based foreground tracer instead of the full P map, isolating the curl-like component that directly contaminates tensor-mode searches.
  • With three frequency channels, comparing prediction errors under a power-law-in-P hypothesis versus independent power-laws-in-(P_E,P_B) can tell which effective spectral model the sky supports.
  • Induced spectral complexity from E/B projection is largely a geometric effect of the transform, not a numerical artifact: it persists when the multipole cutoff of the transform is changed.
  • Angles and amplitudes in P_E/P_B maps, although they come after E/B filtering, are meaningful in map space, unlike the phase of S=E+iB.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: the same non-locality argument applies to any linear filter that mixes pixels (beams, needlets, smoothing); the complexity-ranking between spin-2 and scalar representations may transfer to those filters, making this framework a general diagnostic for filtered polarization.
  • Editorial inference: if partial-sky E/B reconstruction breaks the closure relation in masked regions, the practical preference for P_E/P_B may weaken exactly where CMB analyses operate; testing the three-frequency diagnostics on cut skies is a natural next step.
  • Editorial inference: the ratio |w_{n,E}|/|w_{n,B}| at higher orders could become a new observational probe of Faraday turbulence and line-of-sight mixing, since it isolates parity-dependent spectral gradients.
  • Editorial inference: when the sky is truly power-law in P, modelling P_E/P_B independently overfits and degrades extrapolation; conversely a rigid P model fails when the sky is power-law in P_E/P_B — so the choice of field is data-driven, not a priori.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper develops a map-space spectral framework for E/B-separated polarized foregrounds, centered on complex log–Taylor and moment expansions of the complex spin-2 field P=Q+iU, its spin-preserving projections P_E and P_B, and the scalar S=E+iB. The authors derive analytic predictions for spectral moments generated by line-of-sight superposition, spatially varying spectral indices, intrinsic curvature, synchrotron ageing, and Faraday rotation/depolarization. They validate these predictions on a controlled toy model to ~1e-4/1e-5 accuracy near the pivot, and then use a PySM s5 synchrotron model to quantify the spectral complexity induced by the non-fully-local E/B transform. The central claims are that P=P_E+P_B closure extends to all moment orders, P_E/P_B retain interpretable amplitudes and angles with only moderate induced distortions, while |E|, |B| acquire the largest induced complexity and S lacks direct geometric meaning. The paper concludes with three-frequency diagnostics and recommends modelling foregrounds directly in P_E/P_B for CMB B-mode analyses.

Significance. If the central results hold, the paper provides a principled field representation for map-space foreground modelling, with analytic transformation rules that are internally consistent and validated on controlled simulations. The closure relation P=P_E+P_B and its moment-order generalization are useful formal results, and the explicit comparison of P_E/P_B versus |E|, |B|, and S is a practical contribution. The paper is honest that the validation is a self-consistency check on noiseless, full-sky, band-limited simulations rather than proof that real foregrounds behave this way. The main strength is the internally coherent analytic framework; the main limitation is that the practical CMB recommendation rests on full-sky, noiseless demonstrations, whereas CMB analyses are necessarily masked.

major comments (2)
  1. [Sec. 2 (footnote after Eq. 5), Sec. 4.2.2 (Fig. 11), Sec. 4.2.4 (Fig. 13), Sec. 5] The practical recommendation in Sec. 5 to model frequency dependence in P_E/P_B for CMB B-mode analyses is not supported by the demonstrations, because all simulations and analytic identities assume full-sky maps. The paper itself states that the transforms are strictly well defined only on the full sky. On a masked sky, E/B separation is non-unique, and any practical reconstruction (apodized full-sky transform, pseudo-inverse, inpainting, truncated kernel) will not exactly satisfy the closure P=P_E+P_B (Eq. 9) or its moment generalization (Eq. 22). The ranking in Fig. 11 and the diagnostic in Fig. 13 are computed from full-sky noiseless maps; mask edges and non-local kernel tails could alter the induced complexity of P_E/P_B relative to |E|/|B| and change the preferred-field conclusions. Please either restrict the recommendations to full-sky analyses or add partial-sky tests or an analy
  2. [Sec. 4.2.2, paragraph on l_max insensitivity] The text states that repeating the analysis for l_max=95 and l_max=11 gives essentially unchanged distributions, but this result is not shown. This claim is used to argue that the induced spectral complexity is mainly geometric and not a numerical artefact of kernel truncation, which is load-bearing for the central interpretation. Without the histograms or a quantitative summary (e.g., widths or tail percentiles for each l_max), the reader cannot verify the robustness claim. Please include the supporting figure or table.
minor comments (5)
  1. [Sec. 4.1.2, footnote 6; Sec. 4.1.3.1] Typographical errors: 'perfectely' should be 'perfectly', and 'heareafter' should be 'hereafter'.
  2. [Eq. (23) and surrounding text] The symbol P is used both for the complex field P=Q+iU and for the polarization amplitude |P|. Please clarify notation, e.g., using bold or an explicit modulus, to avoid confusion in equations such as Eq. (23).
  3. [Sec. 4.2.2, Fig. 11] The histograms are informative but do not define a quantitative 'spectral complexity' metric. Reporting widths, tail quantiles, or a scalar measure for each field would strengthen the claimed hierarchy and make it reproducible.
  4. [Sec. 4.1.4 and Sec. 4.2.3, Figs. 9 and 12] The comparison between log-Taylor and moment expansions depends on the chosen reference index beta_bar. The paper briefly states that the results are weakly affected by small variations, but a quantitative sensitivity test would be useful, especially because Eq. (17) makes w1 directly dependent on beta_bar.
  5. [Data Availability] The data availability statement says maps are available from the corresponding author upon request. Making the toy-model and PySM analysis scripts and maps publicly available would improve reproducibility.

Circularity Check

0 steps flagged

No significant circularity: derivations are algebraic from stated inputs; validations are self-consistency checks; the full-sky restriction is a scope limitation, not a circular step.

full rationale

The paper's central claims are derived algebraically from stated physical mechanisms. The moment combination rules (Eqs. 31-34, 38, 45, 49-53) are Taylor expansions of sums/products of power laws and multiplicative corrections; the E/B projection relations (Eqs. 21-22) follow exactly from linearity of the projectors L_E, L_B, which the paper explicitly defines as orthogonal projections with L_E+L_B=I (Sec. 2.1, Eq. 9). No parameter is fitted to a target and then renamed a prediction. The Sec. 4.1.2 'verification' is a closed-loop simulation check: maps are generated from the same component SEDs used in Eqs. 89-91, so agreement is a consistency test of the algebra/numerics, not independent empirical evidence; this is not a hidden equivalence in the paper's argument. The PySM s5 comparison is an external, parameter-free benchmark relative to the paper's own fitted values, and the Fig. 13 experiment is a controlled model-selection sanity check in which each hypothesis is the generative model, so the winner matching the truth is by design. Minor self-citations to Vacher et al. 2023a,b for the spin-moment expansion are acknowledged prior art; the paper rederives the relations, so they are not load-bearing. The admitted limitation that the transforms are 'strictly well defined only on the full sky' (footnote after Eq. 5) and that partial-sky complications are deferred is a scope restriction, not circularity: on a cut sky the closure relation may break, which threatens external validity but not the internal derivation. No circular step is exhibited.

Axiom & Free-Parameter Ledger

1 free parameters · 5 axioms · 0 invented entities

The central derivation is a parameter-free mathematical framework plus domain-specific SED models; no new physical entities are introduced. The only hand-chosen expansion parameter is the reference spectral index beta_bar=-3, not fitted to data. Physical SED prescriptions (ageing, Faraday) are taken from cited literature.

free parameters (1)
  • Reference spectral index beta_bar = -3
    Hand-picked expansion origin for the moment formalism (Sec. 4.1.2 footnote), close to the average-sky beta but not fitted to data. Affects w_n values; authors state weak sensitivity but do not show the test.
axioms (5)
  • standard math E/B decomposition operators L_E, L_B are linear projection operators on the full sky (Rotti & Huffenberger 2019)
    Used throughout (Sec. 2.1, Appendix B) to derive closure and moment projection relations.
  • domain assumption The spin-moment expansion formalism from Vacher et al. 2023a,b and Chluba et al. 2017 is valid for polarized complex fields
    The complex moment expansion (Eq. 15) is assumed to describe SED departures; central to all derivations.
  • domain assumption Physical SED models: broken power-law ageing and Faraday screen/internal Faraday expressions from Sokoloff 1998 and Burn 1966
    Used in Secs. 3.5-3.6 to derive curvature/moment predictions; not re-derived.
  • domain assumption Reference index beta is fixed across the sky to preserve linearity
    Required for Eq. 21 linear projection of moments; a spatially varying beta would break it (Sec. 2.2.3).
  • domain assumption Second-order truncation of log-Taylor/moment expansions is adequate for the three-channel tests
    Used in Secs. 4.1.4 and 4.2.3; accuracy is checked via D maps on toy/PySM models, but not guaranteed generally.

pith-pipeline@v1.3.0-alltime-deepseek · 37217 in / 14278 out tokens · 135568 ms · 2026-08-02T19:24:00.073068+00:00 · methodology

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read the original abstract

Map-space $E$/$B$ decompositions of linear polarization are attractive for foreground and CMB analyses because they separate parity families: $B$-family patterns directly contaminate primordial tensor searches, while $E$-family patterns trace coherent Galactic structures. However, the $E$/$B$ transform is not fully local and can induce apparent spectral complexity even when the underlying sky is spectrally simple in $\underline{P}=Q+iU$. We quantify this effect for synchrotron emission using complex log--Taylor and moment expansions for $\underline{P}$, its spin-preserving projections $\underline{P}_E$ and $\underline{P}_B$, and its more standard scalar projections $E$ and $B$. We relate the coefficients of these expansions to physical mechanisms such as line-of-sight mixing, synchrotron ageing, and Faraday effects. Using simple sky models, we show how $\underline{P}_E$ and $\underline{P}_B$ reorganize the spectral behaviour of $\underline{P}$ into parity families with a clear geometric meaning. They retain interpretable amplitudes and angles and satisfy the closure relation $\underline{P}=\underline{P}_E+\underline{P}_B$, which extends to all moment orders. By contrast, scalar quantities such as $|E|$ and $|B|$ show larger induced variability of effective spectral parameters and enhanced spectral complexity, while $E+iB$ lacks interpretable polarization amplitude and angle. Finally, we present simple CMB-oriented applications: three-frequency diagnostics to test whether the sky is better described by a power law in $P$ or by separate effective laws in $(P_E,P_B)$, and idealized ILC and masking examples showing that the preferred map-space field depends on where foreground simplicity and residual contamination reside. This framework provides practical diagnostics for choosing foreground modelling, cleaning, and masking strategies in Galactic and CMB $B$-mode analyses.

Figures

Figures reproduced from arXiv: 2603.02177 by A.J. Banday, Angela C. Taylor, Carlo Baccigalupi, Gilles Weymann-Despres, L\'eo Vacher, Michael E. Jones, Nicoletta Krachmalnicoff, Richard D.P. Grumitt.

Figure 1
Figure 1. Figure 1: Example of the power law superposition effect (Sec. 3.2): Eq. 37 with two components, 𝑃𝜈 = 𝑃𝑎,𝜈 + 𝑃𝑏,𝜈 with 𝑃𝑎,0 = 𝑒 2𝑖 𝜋/8 , 𝛽𝑎 = −3.5, 𝑃𝑏,0 = 𝑒 −2𝑖 𝜋/8 and 𝛽𝑏 = −2.5. Upper panel: SED, lower panel: polarization angle, both as functions of frequency. The blue curves describe the baseline polarization spectrum, which is modified into the total black curves by the superimposition effect (whose correction 𝐶L… view at source ↗
Figure 2
Figure 2. Figure 2: Example of a single-component intrinsic positive curvature effect (Sec. 3.4): Eq. 48 with 𝑃0 = 𝑒 2𝑖 𝜋/8 , 𝛽 = −3.2 and 𝛾 = 1. The panels and curves are otherwise similar to the ones in [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Example of the ageing effect (Sec. 3.5): Eq. 56 with 𝛽 = −3.2, 𝜎 = 0.5, Δ𝛼 = 1 and 𝜈𝑏 = 7 GHz. The panels and curves are otherwise similar to the ones in [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Example of the combination of internal and external Faraday effects (Sec. 3.6): Eq. 72 with 𝑃0 = 𝑒 2𝑖 𝜋/8 , 𝛽 = −3.2, 𝜎RM,int = 𝜎RM,ext = 0.5/𝑐 2 , RMint = −2/𝑐 2 and RMext = 4/𝑐 2 , with RMs expressed in units of GHz2 /𝑐 2 . The panels and curves are otherwise similar to the ones in [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Toy-model polarization sky at a single frequency. The numbered structures in the total polarization amplitude panel (top row, centre) correspond to the components defined in [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Two-dimensional distributions of several pairs of map-level quan￾tities derived from the toy model. The first row shows the (𝑄𝐵, 𝑈𝐵 ) and (𝑄𝐸 , 𝑈𝐸 ) planes, while the second row displays the planes (𝑃𝐸/𝑃, 𝑃𝐵/𝑃), (𝜓𝐸 , 𝜓𝐵 ), (𝜓, 𝜓𝐵 ), and (𝜓, 𝜓𝐸 ), all evaluated at the reference frequency of 10 GHz. The grey-scale density displays the pixel counts in log scale, in each parameter space [PITH_FULL_IMAGE:figu… view at source ↗
Figure 7
Figure 7. Figure 7: Maps of complex log–Taylor parameters for the toy model, for 𝑃 (top row), 𝑃𝐸 (second row), 𝑃𝐵 (third row) and 𝑆 (bottom row). Columns show log 𝐴𝑋, 𝜓𝑋, 𝛽𝑋, 𝜓 ′ 𝑋 , 𝛾𝑋 and 𝜓 ′′ 𝑋 (from left to right), where primes denote derivatives with respect to 𝑠 (cf. Eq. 10). The colorbar of a given column is common to all rows and provided at the bottom of the figure. Angular fields are given in radians. numerical effe… view at source ↗
Figure 8
Figure 8. Figure 8: Generalized E/B diagnostics for the toy model. Upper panels: maps of 𝜌𝑛 = |𝑤𝑛,𝐸/𝑤𝑛,𝐵 | and |Δ𝜓𝑛 | = 1 2 | arg(𝑤𝑛,𝐸/𝑤𝑛,𝐵 ) | for 𝑛 = 0, 1, 2. Lower panels: differences between 𝐸 and 𝐵 log–Taylor parameters, namely log 𝐴𝑃𝐸 − log 𝐴𝑃𝐵 , |𝜓𝑃𝐸 − 𝜓𝑃𝐵 |, 𝛽𝑃𝐸 − 𝛽𝑃𝐵 , 𝜓 ′ 𝑃𝐸 − 𝜓 ′ 𝑃𝐵 , 𝛾𝑃𝐸 − 𝛾𝑃𝐵 , and 𝜓 ′′ 𝑃𝐸 − 𝜓 ′′ 𝑃𝐵 . The scalar field 𝑆 (lower panels) shows yet another behaviour: its angle and angle derivatives v… view at source ↗
Figure 9
Figure 9. Figure 9: Maps of the decimal logarithm of the ratio between the log-Taylor and moment reconstruction errors, evaluated at 𝜈 = {5, 10, 15} GHz with pivot 𝜈0 = 10 GHz. From left to right the panels show log10 𝐷log(𝑃) /𝐷mom(𝑃) [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Similar to [PITH_FULL_IMAGE:figures/full_fig_p015_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Pixel histograms (logarithmic 𝑦-axis) of fitted spectral parameters in the PySM s5 model. Top-left: spectral tilt 𝛽𝑋. Top-right: curvature 𝛾𝑋. Bottom￾left: first angle derivative 𝜓 ′ 𝑋 ≡ 𝑑𝜓𝑋/𝑑𝑠. Bottom-right: second angle derivative 𝜓 ′′ 𝑋 ≡ 𝑑 2𝜓𝑋/𝑑𝑠2 . Curves are shown for 𝑋 ∈ {𝑃, 𝑃𝐸 , 𝑃𝐵, 𝑆, |𝐸|, |𝐵| }; the 𝑃 curve also overlaps the corresponding distributions for |𝑄| and |𝑈| in this model. The vertical… view at source ↗
Figure 12
Figure 12. Figure 12: Same as [PITH_FULL_IMAGE:figures/full_fig_p016_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Mean-squared prediction error in the intermediate-frequency channel (10 GHz) for 𝑄, 𝑈, 𝑃, 𝑄𝐸 , 𝑈𝐸 , 𝑃𝐸 , 𝑄𝐵, 𝑈𝐵, 𝑃𝐵 when extrap￾olating from two surrounding frequencies under the two hypotheses. Blue curves: assuming a single power law in 𝑃 (Hypothesis P). Orange curves: assuming independent power laws in 𝑃𝐸 and 𝑃𝐵 (Hypothesis EB). Solid lines: true sky is the standard s5 model (power law in 𝑃). Dashed li… view at source ↗
Figure 14
Figure 14. Figure 14: Transformation of a two-dimensional positive Gaussian feature in 𝑄 into the various 𝐸- and 𝐵-mode fields. The panels show, from left to right, the input Stokes fields (𝑄, 𝑈, 𝑃, 𝜓) followed by the derived 𝐸-family fields (𝑄𝐸 , 𝑈𝐸 , 𝑃𝐸 , 𝐸) and the 𝐵-family fields (𝑄𝐵, 𝑈𝐵, 𝑃𝐵, 𝐵). The color maps is the same for all panels (blue = negative, red = positive). 𝑖𝑈𝐵). Furthermore, under a local rotation by 𝛼 the … view at source ↗
Figure 15
Figure 15. Figure 15: Full-sky Mollweide maps of spectral quantities for the toy-model total polarization field 𝑃. From left to right, columns show log 𝐴𝑃, 𝛽𝑃, 𝛾𝑃, 𝜓𝑃, 𝜓 ′ 𝑃 and 𝜓 ′′ 𝑃 (first and second columns of panels) and, in the lower part, |𝑤0 |, |𝑤1/𝑤0 |, |𝑤2/𝑤0 |, 1 2 arg(𝑤0 ), 1 2 arg(𝑤1/𝑤0 ) and 1 2 arg(𝑤2/𝑤0 ). The first and third rows display the quantities predicted from the analytic relations of Sec. 3, while the… view at source ↗
Figure 16
Figure 16. Figure 16: Similar to [PITH_FULL_IMAGE:figures/full_fig_p022_16.png] view at source ↗

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Galactic Science with the LiteBIRD satellite: Spectral characterization of diffuse Galactic polarized emission at the angular power spectrum level

    astro-ph.GA 2026-07 conditional novelty 6.0

    LiteBIRD is forecast to measure Galactic polarized dust and synchrotron spectral parameters with errors as low as σ(T_d)≈0.2 K, σ(β_d)≈0.006, σ(β_s)≈0.04, and to detect E/B and T/P spectral discrepancies in the diffuse ISM.

Reference graph

Works this paper leans on

3 extracted references · cited by 1 Pith paper

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    Because𝑄±𝑖𝑈are spin-∓2fields they admit the spin-weighted harmonic expansion (𝑄±𝑖𝑈)(n)= ∑︁ ℓ𝑚 𝑎±2 ℓ𝑚 ±2𝑌ℓ𝑚(n),(A2) where 2𝑌ℓ𝑚 denotes the spin-2spherical harmonics

    A local rotation of this tangent basis by angle𝛼,(ˆ𝑒𝜃,ˆ𝑒𝜙)↦→(ˆ𝑒 ′ 𝜃,ˆ𝑒′ 𝜙)=𝑅(𝛼)(ˆ𝑒 𝜃,ˆ𝑒𝜙)changes the Stokes parameters according to (𝑄±𝑖𝑈) ′(n)=𝑒 ∓2𝑖𝛼(𝑄±𝑖𝑈)(n).(A1) Thisisthedefiningpropertyofaspin-∓2fieldonthesphere.Equiv- alently,P𝑎𝑏 transforms under the action of the rotation on its tensor indices as a rank-2 symmetric traceless object. Because𝑄±𝑖𝑈are ...

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