Magnetic Misalignment of Interstellar Dust Filaments

The dust TB was reported in Planck Collaboration et al. (2016c), where it was noted that a positive signal in the multipole range 60 < ℓ < 130 became more significant as the sky area was increased. The investigation was continued in Planck Collaboration et al. (2020a) with the observation that TB/TE ∼ 0.1 for 40 < ℓ < 600. The EB signal was reported to be consistent with null.

In Weiland et al. (2020), the TB signal was further confirmed by using Wilkinson Microwave Anisotropy Probe (WMAP) K-band polarization (Page et al. 2007) in place of the Planck B modes and also by using the magnetic-field template from Page et al. (2007) that is based on optical starlight-polarization catalogs (Heiles 2000; Berdyugin et al. 2001, 2004; Berdyugin & Teerikorpi 2002). The K-band measurement is dominated by synchrotron rather than dust emission but is also a probe of Galactic magnetic fields. The starlight measurements largely probe the same magnetic dust-grain alignment that produces polarized millimeter-wave emission. Both choices are independent of the Planck polarization calibration, and both show positive TB.

Magnetic misalignment is a discrepancy between the orientation of filamentary density structures and the polarization-inferred magnetic-field lines. In the case of perfect magnetic alignment, we expect TE > 0 and TB = 0 (Zaldarriaga 2001). A misalignment of 45° would produce TE = 0 and TB ≠ 0, where the sign depends on the chirality of the misalignment. The robustly positive TE measured by Planck can be interpreted as supportive evidence for magnetic alignment of dust filaments (Clark et al. 2015; Planck Collaboration et al. 2016b; Kalberla et al. 2016). In Planck Collaboration et al. (2020a), the TE correlation over sky regions and multipoles is reported as ${r}_{{\ell }}^{{TE}}={D}_{{\ell }}^{{TE}}/\sqrt{{D}_{{\ell }}^{{TT}}{D}_{{\ell }}^{{EE}}}=0.357\pm 0.003$, where ${D}_{{\ell }}^{{XY}}$ denotes the cross-spectrum of X and Y; the TB correlation is reported as ${r}_{{\ell }}^{{TB}}={D}_{{\ell }}^{{TB}}/\sqrt{{D}_{{\ell }}^{{TT}}{D}_{{\ell }}^{{BB}}}\approx 0.05$. Since the TB correlation is much smaller than the TE correlation, the magnetically aligned model need only be perturbed a small but coherent amount in order to produce the observed TB, and this perturbation would also produce a positive EB (Huffenberger et al. 2020), though this EB would be obscured by Planck noise (Clark et al. 2021).

In Clark et al. (2021), it was suggested that an H i-based filamentary polarization template could be used as a comparison point in the search for magnetic misalignment. The template of Clark & Hensley (2019) is constructed by (1) quantifying the orientation of linear H i structures with the rolling Hough transform (RHT; Clark et al. 2014) in velocity-channel maps from Hi4PI (Hi4PI Collaboration et al. 2016), (2) assuming perfect alignment between the RHT-measured H i orientation and the POS magnetic-field orientation and thereby obtaining a prediction for the dust polarization angle, (3) applying weights based on the H i intensity, and (4) integrating the channel maps to form a template that can be compared to the measured millimeter-wave dust polarization. A strong correlation with the Planck 353 GHz maps is detected in both E and B modes up to the Hi4PI beam scale of 162 (ℓ ≲ 1000). The Hi4PI-based template was used in Clark et al. (2021), but Clark & Hensley (2019) also constructed polarization templates with observations from the Galactic Arecibo L-Band Feed Array H i Survey (Peek et al. 2018), which has higher angular resolution ($\mathrm{FWHM}=4\buildrel{\,\prime}\over{.} 1$) but smaller sky coverage (32% of the celestial sphere).

Given the alignment between H i and dust filaments, a difference between the Planck-measured dust polarization angles and the H i-inferred angles is a potential indication of magnetic misalignment and could be used as a tracer of dust TB and EB (Clark et al. 2021). In extending the work of Clark et al. (2021), we measure the aggregate misalignment angle in different sky regions by using the H i template as a reference. We study the observed properties of this misalignment angle and its correlation with the measured dust TB and EB.

Clark et al. (2021) also introduced a scale-dependent effective misalignment angle ψ_ℓ , which is a function of multipole ℓ. This effective misalignment angle is given explicitly by (cf. Equation (11) of Clark et al. 2021)

$$\begin{eqnarray}&&{\psi }_{{\ell }}\equiv \displaystyle \frac{1}{2}\arctan \displaystyle \frac{{D}_{{\ell }}^{{TB}}}{{D}_{{\ell }}^{{TE}}},\end{eqnarray} \tag{ 1 }$$

where we see that the ratio TB/TE is the controlling quantity. ⁴ As noted in Planck Collaboration et al. (2020a), the ratio TB/TE is approximately constant across a broad range of multipoles at high Galactic latitudes, and this is related to the observation of Clark et al. (2021) that ψ_ℓ ∼ 5° in the range 100 ≲ ℓ ≲ 500 on a similar sky area.

Equation (1) provides an estimator for the effective misalignment angle. Because it is formed from the TB and TE cross-spectra, we will call this type of estimator "spectrum-based." In this work, we present several additional spectrum-based estimators by considering the auto- and cross-spectra of the Planck dust maps and H i templates. We also present a map-based estimator that is similar to the projected Rayleigh statistic of Jow et al. (2017). We test for consistency among these estimators.

Although Clark et al. (2021) allowed for scale dependence in the misalignment angle, we find in this work that ψ_ℓ tends to display a scale independence even when measured on small regions of the sky. Equivalently, we find that ψ_ℓ is roughly constant with ℓ, which we will occasionally refer to as "harmonic coherence."

It is important to note that the dust is likely organized only partially in filaments, which are in turn only partially captured by the H i template. We expect, therefore, that there are contributions to the dust polarization that are unrelated to the H i template and, more generally, unrelated even to the true underlying filamentary structure. An estimator like that of Equation (1) may be influenced by these nonfilamentary contributions, since it depends only on the TB and TE cross-spectra of the full dust maps. Some of the estimators we will introduce in later sections will be defined by reference to the H i template, which will partially but imperfectly restrict the analysis to modes that are related to filaments.

In contrast to the previous paragraph, the DUSTFILAMENTS code of Hervias-Caimapo & Huffenberger (2022) constructs a phenomenological dust model, which is composed entirely of filaments and reproduces the main features of the angular power spectra measured by Planck. Using this model, it was recently shown in Huang (2022) that the measured TB is unlikely to be a statistical fluctuation of an underlying parity-even distribution if the assumptions of the DUSTFILAMENTS code represent the true sky.

Cosmic birefringence is an observable consequence of certain types of parity-violating physics beyond the Standard Model and manifests as a rotation of the plane of linear polarization of photons (Carroll et al. 1990; Harari & Sikivie 1992; Carroll 1998). A popular source of cosmic birefringence is an electromagnetically coupled axion-like field, which can behave as both dark matter and dark energy (Marsh 2016). In the CMB, the polarization rotation can be detected as an EB correlation (Lue et al. 1999; Feng et al. 2005, 2006; Liu et al. 2006). A TB correlation should also be produced, but it is typically a less sensitive observable on account of the large cosmic variance in T.

There are several species of cosmic birefringence that have been investigated in the literature. An isotropic, static cosmic birefringence manifests as an overall polarization rotation by the same angle along every line of sight. This observable is, unfortunately, degenerate with a miscalibration of the instrumental polarization orientation (Yadav et al. 2010). The degeneracy is sometimes exploited as a means of "self-calibration" by assuming a standard cosmology in which the true EB vanishes (Keating et al. 2013). Although this type of calibration removes sensitivity to an isotropic, static cosmic birefringence, it is still possible to search for cosmic birefringence that is anisotropic (Ade et al. 2015; BICEP2 Collaboration et al. 2017; Bianchini et al. 2020; Namikawa et al. 2020) or time-variable (BICEP/Keck et al. 2021, 2022; Ferguson et al. 2022). Through a campaign of modeling and calibration, it is possible to account for instrumental systematics and measure the isotropic, static cosmic-birefringence angle. Recent measurements of this kind are consistent with a standard cosmology (Kaufman et al. 2014; Planck Collaboration et al. 2016d; Gruppuso et al. 2016; Choi et al. 2020).

A new technique was proposed in Minami et al. (2019), who exploited the fact that the Galactic foregrounds are subject only to polarization miscalibration and not to cosmic birefringence. The observed CMB is rotated by both miscalibration and a possible cosmic birefringence. With measurements at multiple frequencies, the calibration angles and the cosmic-birefringence angle can be extracted simultaneously. Applied to Planck 2018 polarization data (Planck Collaboration et al. 2020b), a cosmic-birefringence angle β = 035 ± 014, a discrepancy with the null hypothesis with a significance of 2.4σ, was reported in Minami & Komatsu (2020) under the assumption of a vanishing dust EB. With the newer Planck maps produced by the NPIPE pipeline (Planck Collaboration et al. 2020c), the same prescription produced β = 030 ± 011, as reported in Diego-Palazuelos et al. (2022). Recently, a similar analysis that includes WMAP polarization data (Bennett et al. 2013) produced the consistent but stronger result $\beta ={0\buildrel{\circ}\over{.} \,342}_{-0\buildrel{\circ}\over{.} \,091}^{+0\buildrel{\circ}\over{.} \,094}$ (Eskilt & Komatsu 2022). In these two recent cosmic-birefringence analyses, the impact of a possible foreground EB correlation was incorporated by two different approaches, one of which was based on the filamentary misalignment paradigm of Huffenberger et al. (2020) and Clark et al. (2021). When accounting for a possible foreground EB, the birefringence angle varies as a function of sky fraction but remains positive. Diego-Palazuelos et al. (2022) refrained from an estimate of statistical significance due to the currently limited understanding of foreground polarization, while Eskilt & Komatsu (2022) quoted a significance of 3.6σ but acknowledged that the foreground polarization must be better understood to be confident that the measured EB is cosmological rather than Galactic. The study of magnetic misalignment is, therefore, of central importance in the search for cosmic birefringence.

In Section 2, we describe the data products used throughout the analysis. In Section 3, we introduce a new filamentary polarization template that relies on the Hessian matrix of H i intensity maps. In Section 4, we present our misalignment ansatz, i.e., our assumptions of how misalignment perturbs the dust polarization in both map space and harmonic space. We derive misalignment estimators in terms of the auto- and cross-spectra of the Planck dust maps and the H i-based polarization templates, and we test some immediate consequences of these relations. In Section 5, we describe a set of mock skies that we have used to check our estimators. These mock skies are constructed to match the two-point statistics of the Planck dust maps, including cross-spectra with the H i template. In Section 6, we introduce a map-based misalignment estimator and present tentative evidence for a global misalignment angle of ∼2°. In Section 7, we divide the sky into small patches and present evidence for scale independence (harmonic coherence) of magnetic misalignment, as well as evidence of spatial coherence. In Section 8, we present evidence for a scale-independent relation between magnetic misalignment and parity-violating cross-spectra such as TB and EB. We close in Section 9 with suggestions for improvements in our analysis and new directions to further the investigation of parity violation in Galactic dust polarization.

We use the Galaxy masks provided by the Planck Legacy Archive. ⁵ These masks are constructed to limit Galactic emission to varying levels. The masks with a smaller sky fraction f_sky restrict the analysis to relatively high latitudes. The masks with larger f_sky allow more contributions from nearer to the Galactic plane. The set of Galaxy masks is shown in Figure 1. Our fiducial mask in much of the analysis is defined by f_sky = 70%, and we will refer to it as the "70% Galaxy mask."

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We use the subscript "H i" to denote quantities derived from the H i-based polarization template. For example, the H i-based prediction for dust E modes is denoted by E_{H i} . It is important to note that these quantities are describing H i-based predictions for the polarization of dust rather than the polarization properties of the H i itself. The H i is measured in total (unpolarized) intensity, and prescriptions like the Hessian method of Section 3 convert those intensity maps into dust polarization templates.

We use the subscript "d" to denote quantities related to Galactic dust. Usually, this will refer specifically to the Planck Commander maps described above.

Much of our analysis is restricted to ℓ > 100, and we often form maps that are bandpass-filtered. We filter by applying an ℓ-dependent Tukey window to the spherical-harmonic representation of the maps. We use a taper of length t_ℓ = 50, which produces a flat-topped passband when the window width is larger than 2t_ℓ . We test these filters on full-sky Planck dust maps and find the out-of-band response to be suppressed by a factor of more than 10⁴. In particular, the out-of-band leakage is below the level of the high-latitude dust power (computed on the Planck 70% Galaxy mask), even when the filtered power spectra are computed on the full sky, i.e., including the Galactic plane.

We use the Hessian matrix to identify filament orientations. To construct H i-based templates for dust polarization, we form weights from the Hessian eigenvalues.

We analyze the H i maps in individual velocity bins. Our final polarization template is produced by summing over velocities. The H i intensity in velocity channel v is denoted I_v . We work in spherical coordinates with polar angle θ and azimuthal angle ϕ. The local Hessian matrix is given by

$$\begin{eqnarray}H\equiv \left(\begin{array}{cc}{H}_{{xx}} & {H}_{{xy}}\\ {H}_{{yx}} & {H}_{{yy}}\end{array}\right),\end{eqnarray} \tag{ 2 }$$

where

$$\begin{eqnarray}&&{H}_{{xx}}=\displaystyle \frac{{\partial }^{2}{I}_{v}}{\partial {\theta }^{2}},\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&{H}_{{yy}}=\displaystyle \frac{1}{{\sin }^{2}\theta }\displaystyle \frac{{\partial }^{2}{I}_{v}}{\partial {\phi }^{2}},\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&{H}_{{xy}}={H}_{{yx}}=-\displaystyle \frac{1}{\sin \theta }\displaystyle \frac{{\partial }^{2}{I}_{v}}{\partial \phi \partial \theta }.\end{eqnarray} \tag{ 5 }$$

The eigenvalues are

$$\begin{eqnarray}&&{\lambda }_{\pm }=\displaystyle \frac{1}{2}\left({H}_{{xx}}+{H}_{{yy}}\pm \alpha \right),\end{eqnarray} \tag{ 6 }$$

where

$$\begin{eqnarray}&&\alpha \equiv \sqrt{{\left({H}_{{xx}}-{H}_{{yy}}\right)}^{2}+4{H}_{{xy}}^{2}}.\end{eqnarray} \tag{ 7 }$$

The candidate polarization angle is then

$$\begin{eqnarray}&&{\theta }_{v}=\arctan \left(\displaystyle \frac{{H}_{{xx}}-{H}_{{yy}}+\alpha }{2{H}_{{xy}}}\right),\end{eqnarray} \tag{ 8 }$$

but we will enforce conditions below to ensure this identification is sensible.

First, for the local curvature to be negative along at least one axis, we need λ₋ < 0. Second, we want this negative curvature to be the dominant local morphology, so we require λ₋ to be the larger of the two eigenvalues in magnitude. Define

$$\begin{eqnarray}&&{\rm{\Delta }}\lambda \equiv | {\lambda }_{-}| -| {\lambda }_{+}| .\end{eqnarray} \tag{ 9 }$$

Then we define the velocity-dependent weight

$$\begin{eqnarray}&&{w}_{v}\equiv | {\lambda }_{-}| [{\rm{\Delta }}\lambda \gt 0][{\lambda }_{-}\lt 0],\end{eqnarray} \tag{ 10 }$$

where the Iverson brackets on the right-hand side enforce conditions on Δλ and λ₋. ⁶

Our velocity-dependent Stokes templates are given by

$$\begin{eqnarray}&&{T}_{v}(\hat{{\boldsymbol{n}}})\equiv {w}_{v}(\hat{{\boldsymbol{n}}}),\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&{Q}_{v}(\hat{{\boldsymbol{n}}})\equiv {w}_{v}(\hat{{\boldsymbol{n}}})\cos [2{\theta }_{v}(\hat{{\boldsymbol{n}}})],\end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray}&&{U}_{v}(\hat{{\boldsymbol{n}}})\equiv {w}_{v}(\hat{{\boldsymbol{n}}})\sin [2{\theta }_{v}(\hat{{\boldsymbol{n}}})].\end{eqnarray} \tag{ 13 }$$

The Hessian method is susceptible to small-scale noise and scan artifacts, so we restrict our analysis to the Hi4PI velocity bins with greatest sensitivity. We start with the binning of Clark & Hensley (2019). As a proxy for noisiness, we search for pixels with intensities that are reported to be negative. We remove any velocity slice that contains negative-intensity pixels on the Planck 70% Galaxy mask. This leaves a continuous range between −15 and 4 km s⁻¹. The velocity selection is intended to avoid numerical pathologies and should be revisited in a future iteration of the Hessian algorithm. Most of the H i emission is at low velocities (see, e.g., Figure 1 of Clark & Hensley 2019), so we are retaining the dominant contributions even with the current velocity cuts. The velocity cut affects our analysis mainly in terms of sensitivity, since there is potentially useful information about dust filaments in the velocity slices that are discarded. In general, sensitivity is greater when the H i template correlates more strongly with Planck dust polarization. We defer to future work an investigation of the potential improvements from differently chosen velocity cuts (G. Halal et al. 2022, in preparation). We repeated a majority of the following calculations with the RHT-based template (Appendix A) that uses the much broader velocity range of −90 to 90 km s⁻¹ as in Clark & Hensley (2019), and we find consistent results. The full templates are given by

$$\begin{eqnarray}&&{X}_{{\rm{H}}\,{\rm\small{I}}}(\hat{{\boldsymbol{n}}})\equiv \displaystyle \sum _{v}{X}_{v}(\hat{{\boldsymbol{n}}})\end{eqnarray} \tag{ 14 }$$

for X ∈ {T, Q, U}.

An illustration of our Hessian method is provided in Figure 2, where we analyze a region with area 10° × 10° centered on (l, b) = (12°, 45°). The velocity bin is centered on −4.4 km s⁻¹ with a width of 1.3 km s⁻¹. The panels of Figure 2 show how the raw Hi4PI intensity map is transformed into a filamentary intensity w_v and how the filament orientations determine the inferred magnetic-field orientations.

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Additional material related to our Hessian method is provided in Appendix A.

The H i-based filamentary model of dust polarization assumes perfect alignment. We observe in Appendix A.3 that the H i model displays no intrinsic parity violation. We therefore assume that E_{H i} correlates with ${\tilde{E}}_{{\rm{d}}}$ but not with ${\tilde{B}}_{{\rm{d}}}$, and we make a symmetric assumption for B_{H i} . Our assumptions are summarized by

$$\begin{eqnarray}&&{D}_{{\ell }}^{{E}_{{\rm{H}}\,{\rm\small{I}}}{\tilde{B}}_{{\rm{d}}}}={D}_{{\ell }}^{{B}_{{\rm{H}}\,{\rm\small{I}}}{\tilde{E}}_{{\rm{d}}}}={D}_{{\ell }}^{{T}_{x}{\tilde{B}}_{{\rm{d}}}}=0\end{eqnarray} \tag{ 16 }$$

for x ∈ {H i, d}.

With Equations (15) and (16), we can derive the following relations between observable cross-spectra in terms of the misalignment angle ψ_ℓ :

$$\begin{eqnarray}&&{D}_{{\ell }}^{{E}_{{\rm{H}}\,{\rm\small{I}}}{B}_{{\rm{d}}}}=\tan (2{\psi }_{{\ell }}){D}_{{\ell }}^{{E}_{{\rm{H}}\,{\rm\small{I}}}{E}_{{\rm{d}}}},\end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray}&&{D}_{{\ell }}^{{B}_{{\rm{H}}\,{\rm\small{I}}}{E}_{{\rm{d}}}}=-\tan (2{\psi }_{{\ell }}){D}_{{\ell }}^{{B}_{{\rm{H}}\,{\rm\small{I}}}{B}_{{\rm{d}}}},\end{eqnarray} \tag{ 18 }$$

$$\begin{eqnarray}&&{D}_{{\ell }}^{{T}_{x}{B}_{{\rm{d}}}}=\tan (2{\psi }_{{\ell }}){D}_{{\ell }}^{{T}_{x}{E}_{{\rm{d}}}},\end{eqnarray} \tag{ 19 }$$

and

$$\begin{eqnarray}&&{D}_{{\ell }}^{{E}_{{\rm{d}}}{B}_{{\rm{d}}}}=\displaystyle \frac{1}{2}\tan (4{\psi }_{{\ell }})\left({D}_{{\ell }}^{{E}_{{\rm{d}}}{E}_{{\rm{d}}}}-{D}_{{\ell }}^{{B}_{{\rm{d}}}{B}_{{\rm{d}}}}\right).\end{eqnarray} \tag{ 20 }$$

From the known positive T_d B_d and T_d E_d measured by Planck (Section 1.1), we expect ψ_ℓ to be mostly positive in the range 100 ≲ ℓ ≲ 500 on large sky areas away from the Galactic plane, e.g., with a 70% Galaxy mask (Clark et al. 2021). We also know that E_{H i} E_d > 0, B_{H i} B_d > 0 (Clark & Hensley 2019), and E_d E_d > B_d B_d (Planck Collaboration et al. 2016c) across the same multipole range and on the same sky area. Our qualitative expectations, then, are to find E_{H i} B_d, T_{H i} B_d, and E_d B_d to be positive but B_{H i} E_d to be negative.

We can make simple estimates of ψ_ℓ with Equations (17)–(19), though each is potentially biased by noise in the denominator:

$$\begin{eqnarray}&&\tan (2{\psi }_{{\ell }})=\displaystyle \frac{{D}_{{\ell }}^{{E}_{{\rm{H}}\,{\rm\small{I}}}{B}_{{\rm{d}}}}}{{D}_{{\ell }}^{{E}_{{\rm{H}}\,{\rm\small{I}}}{E}_{{\rm{d}}}}}=-\displaystyle \frac{{D}_{{\ell }}^{{B}_{{\rm{H}}\,{\rm\small{I}}}{E}_{{\rm{d}}}}}{{D}_{{\ell }}^{{B}_{{\rm{H}}\,{\rm\small{I}}}{B}_{{\rm{d}}}}}=\displaystyle \frac{{D}_{{\ell }}^{{T}_{{\rm{H}}\,{\rm\small{I}}}{B}_{{\rm{d}}}}}{{D}_{{\ell }}^{{T}_{{\rm{H}}\,{\rm\small{I}}}{E}_{{\rm{d}}}}}=\displaystyle \frac{{D}_{{\ell }}^{{T}_{{\rm{d}}}{B}_{{\rm{d}}}}}{{D}_{{\ell }}^{{T}_{{\rm{d}}}{E}_{{\rm{d}}}}}.\end{eqnarray} \tag{ 21 }$$

We could form a similar estimate from Equation (20), but the E_d B_d measurement from Planck is especially noisy, so we ignore it for the remainder of this section. The four cross-spectrum ratios in Equation (21) allow for tests of the misalignment ansatz without explicit calculation of ψ_ℓ .

While positive T_{H i} B_d and E_{H i} B_d might be anticipated on account of the known positive T_d B_d, T_d T_{H i} , T_d E_d, and E_{H i} E_d, it is, in principle, possible for the T_d B_d signal to be entirely decoupled from the H i-correlated components of the dust maps. In Section 5, we describe how to construct mock skies with exactly this property. These mock skies show positive T_d B_d but zero T_{H i} B_d and zero E_{H i} B_d. While we consider T_{H i} B_d, E_HI B_d > 0 to be the most plausible expectation, it is formally nontrivial.

The negativity of B_{H i} E_d is a new prediction of the misalignment ansatz. We can make quantitative predictions for this signal (Equation (21)), and comparisons are shown in Figure 4 for our fiducial 70% Galaxy mask (Section 2.1). We mentioned above that we expect ψ_ℓ to be smooth over the multipole range 100 ≲ ℓ ≲ 500, so we expect B_{H i} E_d to be smooth over similar multipoles. We can, therefore, gain in per-bandpower sensitivity by using the relatively large bin width of Δℓ = 100.

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In Figure 4, we find that B_{H i} E_d tends negative and is broadly consistent with the expectations of Equation (21) over the full mask and in the northern and southern hemispheres independently. Due to the unavailability of suitable dust and H i simulations, we do not attempt a statistical evaluation of the consistency. The plotted error bars are derived from Gaussian variances. As the dust field displays both non-Gaussianity and statistical anisotropy, these variances are meant only as a rough indication of the fidelity of the measurements.

We expect B_{H i} E_d to be noisier than E_{H i} B_d and T_{H i} B_d because ${r}_{{\ell }}^{{B}_{{\rm{H}}\,{\rm\small{I}}}{B}_{{\rm{d}}}}$ is smaller (by roughly a factor of 2–3 for ℓ > 100) than ${r}_{{\ell }}^{{E}_{{\rm{H}}\,{\rm\small{I}}}{E}_{{\rm{d}}}}$ and ${r}_{{\ell }}^{{T}_{{\rm{H}}\,{\rm\small{I}}}{T}_{{\rm{d}}}}$; i.e., B_{H i} is a less accurate representation of B_d than E_{H i} or T_{H i} is of E_d or T_d, respectively. The H i-based polarization template is, therefore, more sensitive to ${\tilde{E}}_{{\rm{d}}}$ modes mixed into the observed B_d than to ${\tilde{B}}_{{\rm{d}}}$ modes mixed into E_d (Equation (15)).

We consider the results of Figure 4 to be a first step in confirming that the misalignment ansatz of Equation (15) is at least a partial description of the true sky. These results avoided an explicit calculation of the misalignment angle ψ_ℓ . In later sections, we will compute ψ_ℓ directly.

We present maps of $\hat{\psi }$ in Figure 7. We compute $\hat{\psi }$ on masks defined by HEALPix pixels with various values of N_side. For the low-pass-filtered (ℓ < 702) results in the left column of Figure 7, the misalignment angles are partially correlated between patches due to the presence of large-scale polarization modes. Part of the motivation for the bandpass filtering (101 < ℓ < 702) implemented for the right column of Figure 7 is to remove these correlations and acquire approximately independent estimates in each patch. For N_side = 32, the smallest patch size we consider, the side length of each mask is 18, which means that the above-mentioned bandpass filtering suppresses modes with wavelengths larger than a single patch. Most of the patch-to-patch correlations are removed by the bandpass filtering. (We will show in Section 7.2, however, that there is evidence for nontrivial spatial coherence of $\hat{\psi }$ that cannot be simply attributed to large-scale modes.)

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As the patch size decreases, regions of higher and lower variance emerge at all latitudes in a pattern that is similar to that of Δθ in Figure 3 of Clark et al. (2021). The above observations are broadly consistent between the low-pass-filtered (ℓ < 702) and bandpass-filtered (101 < ℓ < 702) maps, but the former are visually smoother.

To check for biases in our misalignment estimator, we measure $\hat{\psi }$ on masks defined by HEALPix pixels as described above, artificially rotate the angles of the Planck polarization map by a known amount ψ₀ ∈ [−90°, 90°], and then recompute $\hat{\psi }$. We track the median of the distribution and find that it follows ψ₀. We conclude that $\hat{\psi }$ is an unbiased estimator of misalignment angle.

We observe a tendency toward positive misalignment angles in Figure 7. To estimate the statistical significance, it is tempting to appeal to the central limit theorem. Unfortunately, the values of $\hat{\psi }(\hat{{\boldsymbol{n}}})$, where $\hat{{\boldsymbol{n}}}$ represents a particular patch, are neither completely independent nor identically distributed. By bandpass filtering to 101 < ℓ < 702 as in the right column of Figure 7, we can achieve approximate independence of the estimates for different $\hat{{\boldsymbol{n}}}$. We cannot, however, guarantee that the estimates are identically distributed.

Nevertheless, because the calculation is simple, we estimate a mean and standard error by appealing to the central limit theorem. For N_side = 32, we find ${\hat{\psi }}_{{\ell }\lt 702}=1\buildrel{\circ}\over{.} 9\pm 0\buildrel{\circ}\over{.} 3$, but we caution that the patches are nontrivially correlated with each other by the bright, large-scale polarization modes and, therefore, refrain from claiming any statistical significance. Restricting to the sky area allowed by our fiducial 70% Galaxy mask (see Figure 8), we find ${\hat{\psi }}_{{\ell }\lt 702}=1\buildrel{\circ}\over{.} 7\pm 0\buildrel{\circ}\over{.} 3$. After bandpass filtering, the patches are more (but not completely) independent, and we find ${\hat{\psi }}_{101\lt {\ell }\lt 702}=0\buildrel{\circ}\over{.} 9\pm 0\buildrel{\circ}\over{.} 3$, which implies a statistical significance of 3σ. Restricting to our fiducial 70% Galaxy mask, we find ${\hat{\psi }}_{101\lt {\ell }\lt 702}=0\buildrel{\circ}\over{.} 8\pm 0\buildrel{\circ}\over{.} 4$, which implies a significance of 2σ. We have deliberately limited ourselves to a single significant figure because we consider these calculations to be crude estimates.

We note that the large-scale features of $\hat{\psi }$ are similar to those of the dust polarization fraction ${p}_{{\rm{d}}}\equiv \sqrt{{Q}_{{\rm{d}}}^{2}+{U}_{{\rm{d}}}^{2}}/{T}_{{\rm{d}}}$. For visual comparison, we provide in Figure 9 maps displaying misalignment angle $\hat{\psi }$, dust intensity T_d, and dust polarization fraction p_d. We omit an estimate of the correlation strengths because the bright, large-scale modes induce covariances that are difficult to model. The low-column (small-T_d) sky regions in the northeast and southwest are also regions of increased variance in $\hat{\psi }$, as evidenced by the large fluctuations between neighboring patches, but a much clearer visual correspondence appears between $\hat{\psi }$ and p_d. The regions of larger p_d show smaller variance in $\hat{\psi }$, and those regions also tend to $\hat{\psi }\gt 0$. For both T_d and p_d, the correspondence with regions of lower variance may be related to the signal-to-noise ratio of the misalignment measurement. In regions with higher polarized intensity, there is less variance in $\hat{\psi }$.

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There may also be a connection to S_d, the angle dispersion of the dust polarization, and S_{H i} , that of the H i polarization template. The former anticorrelates with p_d; i.e., regions of greater polarization-angle coherence have larger polarization fractions (Planck Collaboration et al. 2020e). The variation in polarization-angle coherence may be related to the magnetic-field orientation relative to the line of sight (e.g., Hensley et al. 2019). The H i-based dispersion S_{H i} and polarization fraction p_{H i} also anticorrelate, and the alignment of the dust polarization angle to the H i template anticorrelates with S_{H i} (Clark & Hensley 2019). We would therefore expect that regions of larger polarization fraction correlate with regions of coherent magnetic misalignment, which is indeed what we observe.

A major motivation for this study is to understand the origin of the parity-violating TB correlation measured by Planck on large fractions of the high-latitude sky (Clark et al. 2021). In addition to measuring the variation in misalignment angle across relatively small patches of sky (Figure 7), we can apply our estimator (Equation (29)) to large sky areas and compare to expectations based on measured cross-spectra (Equation (21)). On large sky areas, the variation in $\hat{\psi }$ is suppressed, and we can safely make a small-angle approximation. Then we expect

$$\begin{eqnarray}&&{\hat{\psi }}_{{\ell }}\approx \displaystyle \frac{{D}_{{\ell }}^{{E}_{{\rm{H}}\,{\rm\small{I}}}{B}_{{\rm{d}}}}}{2{D}_{{\ell }}^{{E}_{{\rm{H}}\,{\rm\small{I}}}{E}_{{\rm{d}}}}}=-\displaystyle \frac{{D}_{{\ell }}^{{B}_{{\rm{H}}\,{\rm\small{I}}}{E}_{{\rm{d}}}}}{2{D}_{{\ell }}^{{B}_{{\rm{H}}\,{\rm\small{I}}}{B}_{{\rm{d}}}}}=\displaystyle \frac{{D}_{{\ell }}^{{T}_{{\rm{H}}\,{\rm\small{I}}}{B}_{{\rm{d}}}}}{2{D}_{{\ell }}^{{T}_{{\rm{H}}\,{\rm\small{I}}}{E}_{{\rm{d}}}}}=\displaystyle \frac{{D}_{{\ell }}^{{T}_{{\rm{d}}}{B}_{{\rm{d}}}}}{2{D}_{{\ell }}^{{T}_{{\rm{d}}}{E}_{{\rm{d}}}}}.\end{eqnarray} \tag{ 30 }$$

Noise in the denominators may bias these expressions, but we are here looking only for a broad consistency and the approximate level of aggregate misalignment on large sky areas. Since ${\hat{\psi }}_{{\ell }}$ is estimated by reference to the H i template, we expect greatest consistency with the dust–H i cross-spectra, e.g., E_{H i} B_d, as opposed to the Planck-only T_d B_d and E_d B_d.

In Figure 10, we compare the misalignment estimates from Equation (30) for a 70% Galaxy mask (Section 2.1). We find a misalignment angle of ∼2° that is coherent in the range 101 < ℓ < 702. As expected, $\hat{\psi }$ tends to be more consistent with the dust–H i cross-spectra, especially E_{H i} B_d and T_{H i} B_d, which are more sensitive than B_{H i} E_d (Section 4.2). The Planck-only T_d B_d is more discrepant (though not dramatically so) but reproduces the coherently positive behavior for ℓ < 500.

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The $\hat{\psi }$ estimates are broadly consistent between the northern and southern Galactic hemispheres. In particular, the magnitude of the misalignment and its scale (multipole) independence are consistent. The Planck-only T_d B_d/2T_d E_d (red in Figure 10) also shows a similar consistency between the hemispheres. The positive $\hat{\psi }$, the approximate scale independence of $\hat{\psi }$, and the consistency of $\hat{\psi }$ between hemispheres are robust to the choice of Galaxy mask; we checked this for f_sky ∈ {40%, 60%, 70%, 80%, 90%} (Section 2.1) and present some related results in Section 6.7. The consistency between hemispheres begs for an explanation, which should be a target for future investigations.

The uncertainties on $\hat{\psi }$ in Figure 10 are derived from the scatter of our mock skies (Section 5) and used only for visualization purposes, namely, to give a rough indication of the expected variance. We do not rely on these uncertainties for statistical inference.

Because the large-scale (low-ℓ) modes are difficult to reproduce in our mock-sky framework (Section 5), we have restricted Figure 10 to 101 < ℓ < 702. We can calculate $\hat{\psi }$ for ℓ < 101, but we cannot form a reliable uncertainty based on mock skies. Nevertheless, it is interesting to report the values for ${\hat{\psi }}_{{\ell }\lt 101}$. We find 14 when using both hemispheres, 29 in the northern hemisphere, and 07 in the southern hemisphere. The statistical weight cannot be evaluated in the present analysis, but it is noteworthy that the large scales show the same tendency for misalignment to be positive.

An intriguing possibility is that there is an aggregate global misalignment of ∼2°. An aggregate misalignment would appear as an isotropic, multipole-independent rotation of the dust polarization relative to the filamentary structures, i.e., ${\psi }_{{\ell }}=\mathrm{constant}$. The implied magnetic-field structure relative to the dust intensity field would be qualitatively similar to that depicted by the green pseudovectors in Figure 3. A global polarization rotation can also be produced by a miscalibration of the absolute polarization angle or in the CMB by the phenomenon of cosmic birefringence (Section 1.3).

We consider miscalibration to be unlikely, since Planck estimates a systematic uncertainty of 028 (Planck Collaboration et al. 2016d), nearly an order of magnitude smaller than our measured misalignment. In the following sections, we measure $\hat{\psi }$ in small sky regions and search for correlated variations with other interesting quantities. The relative variation from region to region is insensitive to an overall miscalibration.

A global misalignment signal in the dust, which acts as a foreground for CMB measurements, would need to be accounted for in searches for cosmic birefringence, especially with methods that rely on the symmetry properties of the dust polarization, e.g., Minami et al. (2019).

As a consistency check, we modified our H i template (Section 3) by imposing a global polarization rotation of 2°. This rotation mixes the E and B modes. Because the H i template is dominated by E modes (E_{H i} E_{H i} /B_{H i} B_{H i} ∼ 5 for ℓ > 100), the effect is fractionally stronger in the B modes. We can estimate the expected impact of this modification by considering that 2° ≈ 0.03 rad, so this should produce a percent-level change in the correlations. We correlate with the Planck dust maps and find that the B-mode correlation for 100 < ℓ < 700 increases by 0.1%–0.5% in addition to the original correlation of 10%–25%, which is indeed a fractional increase of ${ \mathcal O }(1 \% )$. We performed the same exercise with the opposite rotation, i.e., by −2°, and we find an approximately symmetric decrease in the H i–Planck correlation. These results are consistent with the estimates of Figure 10 and increase our confidence in a true on-sky aggregate misalignment of approximately 2°.

The T_d B_d-based estimate of ψ_ℓ (red in Figure 10) is coherent only over the range 100 ≲ ℓ ≲ 500. Where it is nonzero, it also tends to be larger than $\hat{\psi }$. The discrepancy may be an indication that T_d B_d and T_d E_d are affected by additional polarization sources that are missed by the filamentary misalignment model, and it is conceivable that the simultaneous positivity of T_d B_d and $\hat{\psi }$ is merely coincidental. In Section 8, we seek further evidence of a relationship by analyzing small sky patches.

We track the dependence of these misalignment estimates on the sky fraction f_sky. In Figure 10, we consider only f_sky = 70%, whereas we now allow f_sky ∈ {20%, 40%, 60%, 70%, 80%, 90%, 97%} (Figure 1). In Figure 10, we consider multiple estimators for ψ_ℓ . For simplicity, we now downselect to only two. One is ${\psi }_{{\ell }}^{{\rm{d}}\times {\rm{d}}}\equiv \mathrm{atan}2({T}_{{\rm{d}}}{B}_{{\rm{d}}},{T}_{{\rm{d}}}{E}_{{\rm{d}}})/2$ (red in Figure 10), which uses only the dust field (see Figure 9 of Clark et al. 2021). We contrast the dust-only estimator with one that includes some H i filamentary information: ${\psi }_{{\ell }}^{{\rm{H}}\,{\rm\small{I}}\times {\rm{d}}}\,\equiv \mathrm{atan}2({T}_{{\rm{H}}\,{\rm\small{I}}}{B}_{{\rm{d}}},{T}_{{\rm{H}}\,{\rm\small{I}}}{E}_{{\rm{d}}})/2$ (green in Figure 10). Whereas ${\psi }_{{\ell }}^{{\rm{d}}\times {\rm{d}}}$ includes information from the entire dust field, ${\psi }_{{\ell }}^{{\rm{H}}\,{\rm\small{I}}\times {\rm{d}}}$ collapses the misalignment estimate onto the filamentary modes. When the two are in agreement, the filamentary magnetic misalignment is representative of the parity violation in the full dust field. When they deviate from each other, the H i template may be incomplete or inaccurate, or the full dust field may contain parity-violating contributions that are nonfilamentary.

In Figure 11, we show ${\psi }_{{\ell }}^{{\rm{d}}\times {\rm{d}}}$ and ${\psi }_{{\ell }}^{{\rm{H}}\,{\rm\small{I}}\times {\rm{d}}}$ for a variety of sky fractions f_sky.

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We show the estimates for individual multipole bins (see Figure 10), and we also high pass filter to form the broadband ψ_ℓ>100, which may potentially average away signal but is less noisy. We find that ${\psi }_{{\ell }}^{{\rm{H}}\,{\rm\small{I}}\times {\rm{d}}}$ is consistently positive over all ℓ and remains in the range of 0°–5°, while ${\psi }_{{\ell }}^{{\rm{d}}\times {\rm{d}}}$ is much more variable, especially at large f_sky. We note that the two estimates display closer agreement at small f_sky, i.e., when restricting to high Galactic latitudes. At the same time, we find that ${\psi }_{{\ell }\gt 100}^{{\rm{H}}\,{\rm\small{I}}\times {\rm{d}}}$ steadily decreases from ∼3° to ∼1° as f_sky increases from 20% to 97% (right panels of Figure 11), a phenomenon that is observed in both hemispheres independently. The decline may be related to the fact that the H i becomes a less robust tracer of dust at low Galactic latitudes where the column densities are relatively large (e.g., Lenz et al. 2017), so it may be that the H i template is simply less representative of the dust field for large f_sky.

Interestingly, all of the variations considered in Figure 11 produce ${\psi }_{{\ell }}^{{\rm{H}}\,{\rm\small{I}}\times {\rm{d}}}$ in the range of 0°–5°. This behavior persists for all of the considered multipoles and sky fractions and in both hemispheres independently. Furthermore, we performed this analysis with the RHT-based H i template (Clark & Hensley 2019; Appendix A.1) instead of the Hessian, and we find consistent results. These observations lend more weight to the speculations of Section 6.6 about a possible global misalignment angle of ∼2°.

The relative constancy of the Planck TB/TE at high latitudes (see, e.g., Figure 9 of Clark et al. 2021) is perhaps a hint that magnetic misalignment, if it is to address the mystery of the positive dust TB, ought to display a harmonic coherence in the multipole range 100 ≲ ℓ ≲ 500. Indeed, we find that a direct calculation of the misalignment angle ${\hat{\psi }}_{{\ell }}$ yields an apparent coherence over an even larger multipole range, tentatively across all ℓ < 702 (Figure 10 and Section 6.5). Is the apparent harmonic coherence an emergent phenomenon that appears only when aggregating large sky areas? In this section, we divide the sky into smaller patches and test whether harmonic coherence is a generic feature of magnetic misalignment.

To look for coherence in harmonic space, we bandpass filter the maps into two disjoint multipole ranges. For all of our results, 101 < ℓ < 702, so ${{\ell }}_{\min }=101$ and ${{\ell }}_{\max }=702$ can be taken as lower and upper limits, respectively, on the multipole ranges. We form a set of maps with ℓ < ℓ_c − Δℓ/2 and a set with ℓ > ℓ_c + Δℓ/2, where ℓ_c is a transition multipole and Δℓ is a multipole buffer between the two ranges. We allow Δℓ ∈ {0, 100, 200}, and we sweep ℓ_c across the range [101, 702].

Let ${\hat{\psi }}_{\mathrm{lo}}(\hat{{\boldsymbol{n}}})$ be the misalignment angle estimated in the patch centered on sky coordinate $\hat{{\boldsymbol{n}}}$ after filtering to ℓ < ℓ_c − Δℓ/2, and let ${\hat{\psi }}_{\mathrm{hi}}(\hat{{\boldsymbol{n}}})$ be similarly defined after filtering to ℓ > ℓ_c + Δℓ/2. We will refer to ${\hat{\psi }}_{\mathrm{lo}}(\hat{{\boldsymbol{n}}})$ and ${\hat{\psi }}_{\mathrm{hi}}(\hat{{\boldsymbol{n}}})$, respectively, as the "low-pass-filtered" and "high-pass-filtered" misalignment estimates. Recall, however, the multipole limits ${{\ell }}_{\min }$ and ${{\ell }}_{\max }$ mentioned above, so these estimates are, in fact, products of bandpass filtering.

Our correlation calculations must consider the circularity of the misalignment angle. We expect $\hat{\psi }$ to cluster around $0^\circ \,\mathrm{mod}\,180^\circ$. Even for the smallest masks of Figure 7, which are defined by N_side = 32, the majority of $\hat{\psi }$ values lie within $[-45^\circ ,45^\circ ]\,\mathrm{mod}\,180^\circ$. In our correlations, therefore, we ignore the circularity of $\hat{\psi }$ and instead force the values to their physical equivalents in the range [−90°, 90°]. In a small minority of cases, we will miss correlations between angles that lie at opposite extremes of this range. In testing for a correlation, our choice is conservative. Since we will be using Spearman correlations, which operate on rank variables, a convenient ordering strategy is to form $\tan \hat{\psi }$.

We form the Spearman cross-power (Spearman version of Equation (B2))

$$\begin{eqnarray}&&{\hat{\psi }}_{\mathrm{lo}}\times {\hat{\psi }}_{\mathrm{hi}}\equiv {s}_{s}(\tan {\hat{\psi }}_{\mathrm{lo}},\tan {\hat{\psi }}_{\mathrm{hi}}),\end{eqnarray} \tag{ 31 }$$

where the sum is taken over patches labeled by $\hat{{\boldsymbol{n}}}$. Note that these Spearman cross-powers are not correlation coefficients, so the numerical values range outside of [−1, 1]. Correlation coefficients are less numerically stable in the presence of noise, so we prefer cross-powers for the purposes of establishing a relationship. In Figure 12, we show ${\hat{\psi }}_{\mathrm{lo}}\times {\hat{\psi }}_{\mathrm{hi}}$ for several choices of ℓ_c and Δℓ for the patches of Figure 8 (masks defined by N_side = 8 and f_sky = 70%). We find a positive cross-power for ℓ_c ≲ 450 for all choices of Δℓ. The noise in these measurements is mainly in the high-pass-filtered misalignment estimates ${\hat{\psi }}_{\mathrm{hi}}$.

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We do not expect our mock skies (Section 5) to show a coherence over ℓ because the Gaussian modes are resampled independently of each other and the H i template. One concern might be that the masking creates mode correlations, and this was part of the motivation for introducing the multipole buffer Δℓ. As Δℓ increases, the two multipole passbands are further separated, and spurious correlations between the two are less likely.

As a null test, we calculate ${\hat{\psi }}_{\mathrm{lo}}\times {\hat{\psi }}_{\mathrm{hi}}$ for an ensemble of mock skies (Section 5), and we find the mean values to be consistent with zero. Recall that these mock skies are simplified in the sense that they are designed to reproduce only the two-point statistics of the dust field, both in correlation with itself and with the H i template. Nonetheless, they are helpful in checking that our estimators produce sensible results. The H i template appears in these mock skies with the observed amplitude, and the only magnetic misalignment that has been input is due to random scatter. We see no positive bias in the mock-sky cross-powers. The positive signal seen in the real map (Figure 12) must be due to a feature that it is absent in the mock skies.

Due to noise in the $\hat{\psi }$ estimates, it is nontrivial to determine the fraction of the misalignment signal that is harmonically coherent. In computing a correlation coefficient, noise tends to dilute the true signal. Using half-mission splits as in Equation (B3) may lead to numerical pathologies when the denominators are small. From the decay of ${\hat{\psi }}_{\mathrm{hi}}^{(1)}\times {\hat{\psi }}_{\mathrm{hi}}^{(2)}$ in Figure 12, we see that the high-pass-filtered estimate ${\hat{\psi }}_{\mathrm{hi}}$ is especially noisy. The Spearman cross-powers ${\hat{\psi }}_{\mathrm{lo}}^{(1)}\times {\hat{\psi }}_{\mathrm{lo}}^{(2)}$ (red in Figure 12) and ${\hat{\psi }}_{\mathrm{hi}}^{(1)}\times {\hat{\psi }}_{\mathrm{hi}}^{(2)}$ (purple) are limited only by noise and set rough upper limits on the cross-power ${\hat{\psi }}_{\mathrm{lo}}\times {\hat{\psi }}_{\mathrm{hi}}$. Even if the misalignment angles were perfectly coherent across multipoles, noise would suppress the cross-power. That ${\hat{\psi }}_{\mathrm{lo}}\times {\hat{\psi }}_{\mathrm{hi}}$ is of the same order as ${\hat{\psi }}_{\mathrm{lo}}^{(1)}\times {\hat{\psi }}_{\mathrm{lo}}^{(2)}$ and ${\hat{\psi }}_{\mathrm{hi}}^{(1)}\times {\hat{\psi }}_{\mathrm{hi}}^{(2)}$ is an indication that, within the limits of the noise, the harmonically coherent component is contributing a nonnegligible fraction of the misalignment signal.

We estimate the statistical significance of the apparently positive signal in Figure 12. For each choice of ℓ_c and Δℓ, we construct permutation tests (Appendix B.1), where covariances are preserved by using the same permutations for all choices. We combine the results with weights based on the half-mission cross-powers (bands in Figure 12) and produce a single overall estimate of the statistical significance. For the case of Figure 12, we estimate the statistical significance to be 2.3σ, where most of the sensitivity comes from the cross-powers with smaller ℓ_c and smaller Δℓ. This is because ${\hat{\psi }}_{\mathrm{hi}}$ quickly becomes noise-dominated as ℓ_c increases, and increasing Δℓ pushes the filter cutoff even higher. The data points in Figure 12 are computed from the same maps but with different filtering parameters, so we expect them to be highly correlated. As such, combining the data points increases the overall significance only modestly. To a rough approximation, the overall significance can be estimated from the lowest-ℓ_c data point.

The results of Figure 12 are based on the patches shown in Figure 8, which are defined by N_side = 8 and f_sky = 70%. We can compute similar quantities for other values of N_side and f_sky, and the results are compiled in Table 1, where we see that the significances are generally between 2σ and 4σ for N_side ∈ {2, 4, 8, 16, 32} and f_sky ∈ [60%, 90%]. With smaller f_sky, the significances tend to be smaller, but this may be simply a consequence of a decreased signal-to-noise ratio. On the full sky, the significances also tend to decrease, and this may be due to the inclusion of longer, denser sight lines at low Galactic latitudes. We do not attempt to combine the results from different choices of N_side and f_sky because the covariances are difficult to capture.

Table 1. Statistical Significance (in Units of σ) of Measurements of the Harmonic Coherence of $\hat{\psi }$, e.g., Those Presented in Figure 12, for Different Values of N_side (Rows with Side Lengths Provided Parenthetically) and f_sky (Columns)

	40%	60%	70%	80%	90%	100%
2 (293)	1.1	2.6	3.0	3.8	3.6	0.2
4 (147)	3.7	3.3	3.5	3.2	2.8	0.9
8 (73)	1.9	2.4	2.3	2.5	3.9	3.1
16 (37)	1.2	2.8	2.7	3.3	2.2	2.2
32 (18)	3.8	3.9	4.2	2.5	1.5	1.1

Note. All of the results are positive with little dependence on N_side and f_sky.

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We consider the results of Figure 12 and Table 1 to represent tentative evidence for the harmonic-space coherence of misalignment angles measured in small regions of sky away from the Galactic plane.

We additionally search for spatial coherence of misalignment angles by considering neighboring pairs of sky masks. Although we bandpass-filtered to 101 < ℓ < 702 in order to include only modes with wavelengths smaller than each patch, there is still a residual correlation between neighboring patches, which we detect with the mock skies of Section 5. To avoid the coherence due to common modes between neighboring patches, we again construct high- and low-pass-filtered maps as in Section 7.1. We correlate the low-pass-filtered estimate from each patch with the high-pass-filtered estimate from each of its neighbors, and we simultaneously correlate with the opposite application of filters. The misalignment estimates that enter the correlation calculations are separated in both harmonic and map space.

We utilize the Spearman version of the four-variable cross-power (Equation (B5))

$$\begin{eqnarray}\begin{array}{rcl}{\hat{\psi }}_{\mathrm{lo}}(\hat{{\boldsymbol{n}}})\times {\hat{\psi }}_{\mathrm{hi}}(\hat{{\boldsymbol{n}}}^{\prime} ) & \equiv & {S}_{s}\left(\tan {\hat{\psi }}_{\mathrm{lo}}(\hat{{\boldsymbol{n}}}),\tan {\hat{\psi }}_{\mathrm{hi}}(\hat{{\boldsymbol{n}}}^{\prime} ),\right.\\ & & \quad \quad \quad \left.\tan {\hat{\psi }}_{\mathrm{hi}}(\hat{{\boldsymbol{n}}}),\tan {\hat{\psi }}_{\mathrm{lo}}(\hat{{\boldsymbol{n}}}^{\prime} \right),\end{array}\end{eqnarray} \tag{ 32 }$$

where $\hat{{\boldsymbol{n}}}$ and $\hat{{\boldsymbol{n}}}^{\prime}$ are the central sky coordinates of neighboring patches. As in Section 7.1, we consider several choices for N_side and f_sky, where Figure 8 shows one example. The sum is taken over all pairs of neighboring patches. Each patch appears multiple times in this sum, but each pair appears only once. Equation (32) measures a simultaneous correlation between ${\hat{\psi }}_{\mathrm{lo}}(\hat{{\boldsymbol{n}}})$ and ${\hat{\psi }}_{\mathrm{hi}}(\hat{{\boldsymbol{n}}}^{\prime} )$ and between ${\hat{\psi }}_{\mathrm{hi}}(\hat{{\boldsymbol{n}}})$ and ${\hat{\psi }}_{\mathrm{lo}}(\hat{{\boldsymbol{n}}}^{\prime} )$. The entire multipole range is being used in both patches but in two splits. Without multipole separation, we cannot pass a null test based on our mock skies (Section 5). With multipole separation, however, the mock skies show no significant cross-power.

The results for ${\hat{\psi }}_{\mathrm{lo}}(\hat{{\boldsymbol{n}}})\times {\hat{\psi }}_{\mathrm{hi}}(\hat{{\boldsymbol{n}}}^{\prime} )$ (Equation (32)) are shown in Figure 13 for patches defined by N_side = 16. This patch area is four times smaller than that used for the measurement of harmonic coherence (Figure 12). Measuring neighbor correlations at N_side = 16 probes the spatial coherence within patches defined by N_side = 8, so we are approximately measuring the spatial coherence within the patches of Figure 12. For the particular example of Figure 13, we find positive spatial coherence for ℓ_c ≲ 500.

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We estimate the statistical significance of the positive signal shown in Figure 13 by following a prescription similar to that of Section 7.1. Combining all of the measurements in a manner that accounts for covariances, we estimate the statistical significance to be 3.6σ, where most of the sensitivity, as for harmonic coherence, comes from the low-ℓ_c, low-Δℓ cross-powers.

We can compute similar quantities with other values of N_side and f_sky, and the results are compiled in Table 2, where we see that the spatial coherence tends to be stronger as the resolution is made finer.

Table 2. Statistical Significance (in Units of σ) of Measurements of the Spatial Coherence of $\hat{\psi }$ (Equation (32)), e.g., Those Presented in Figure 13, for Different Values of N_side (Rows with Side Lengths Provided Parenthetically) and f_sky (Columns)

	40%	60%	70%	80%	90%	100%
2 (293)	−0.1	−2.0	−2.4	−1.0	−0.3	−2.2
4 (147)	2.9	1.7	1.8	1.7	2.0	−0.4
8 (73)	1.3	0.8	1.0	0.5	0.0	−0.7
16 (37)	3.8	3.9	3.6	3.0	2.7	3.4
32 (18)	2.9	2.9	4.0	5.1	4.7	4.1

Note. The significances are mostly positive and tend to increase with N_side.

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For N_side ∈ {16, 32}, the significances are mostly between 2σ and 5σ. As the side length associated with N_side = 32 is 18, these results may imply that magnetic misalignment displays a coherence length of ${ \mathcal O }(1^\circ )$.

We consider the results of Figure 13 and Table 2 to represent tentative evidence for spatial coherence of misalignment angles. One perspective, however, is to consider the spatial coherence to be a necessary implication of the harmonic coherence that was established in Section 7.1. We explained at the beginning of this section that the multipole split in Equation (32) helps to evade correlations between neighboring patches, which appear even in our statistically aligned mock skies (Section 5). The claim is that ${\hat{\psi }}_{\mathrm{lo}}(\hat{{\boldsymbol{n}}})$ is correlated with ${\hat{\psi }}_{\mathrm{lo}}(\hat{{\boldsymbol{n}}}^{\prime} )$ even in the mock skies. So we chose to correlate ${\hat{\psi }}_{\mathrm{lo}}(\hat{{\boldsymbol{n}}})$ with ${\hat{\psi }}_{\mathrm{hi}}(\hat{{\boldsymbol{n}}}^{\prime} )$, and the mock skies show no correlation in this case. But the mock skies are also lacking harmonic coherence. In the real maps, harmonic coherence appears to correlate ${\hat{\psi }}_{\mathrm{lo}}(\hat{{\boldsymbol{n}}})$ with ${\hat{\psi }}_{\mathrm{hi}}(\hat{{\boldsymbol{n}}})$ and ${\hat{\psi }}_{\mathrm{lo}}(\hat{{\boldsymbol{n}}}^{\prime} )$ with ${\hat{\psi }}_{\mathrm{hi}}(\hat{{\boldsymbol{n}}}^{\prime} )$ (Section 7.1), but then we should expect, on the basis of the residual neighbor-to-neighbor correlations in the mock skies, a correlation between ${\hat{\psi }}_{\mathrm{lo}}(\hat{{\boldsymbol{n}}})$ and ${\hat{\psi }}_{\mathrm{hi}}(\hat{{\boldsymbol{n}}}^{\prime} )$ and between ${\hat{\psi }}_{\mathrm{hi}}(\hat{{\boldsymbol{n}}})$ and ${\hat{\psi }}_{\mathrm{lo}}(\hat{{\boldsymbol{n}}}^{\prime} )$. So there may indeed be a spatial coherence, but the crucial ingredient might be the harmonic coherence.

As described in Section 7, we are interested in distinguishing between random and harmonically coherent misalignment. In both cases, we expect to find that the misalignment angle is correlated with TB and EB, but we can impose additional constraints to isolate harmonically coherent correlations.

Random misalignment is exemplified by the mock skies of Section 5. The mock skies show deviations from the H i template, but the deviations are incoherent across multipoles, and there is no aggregate misalignment on large sky areas (see Section 6.5). We find that the mock skies show significant correlations between ${\hat{\psi }}_{{\ell }}$ and ${D}_{{\ell }}^{{T}_{{\rm{d}}}{B}_{{\rm{d}}}}$ and between ${\hat{\psi }}_{{\ell }}$ and ${D}_{{\ell }}^{{E}_{{\rm{d}}}{B}_{{\rm{d}}}}$. If we instead correlate between disjoint multipole bins, e.g., ${\hat{\psi }}_{{\ell }\lt {{\ell }}_{c}}$ with ${D}_{{\ell }\gt {{\ell }}_{c}}^{{T}_{{\rm{d}}}{B}_{{\rm{d}}}}$ for some cutoff multipole ℓ_c, we find that the correlations vanish. The real data, as we will show in Sections 8.3 and 8.4, display both types of correlations.

The mock skies display correlations between ${\hat{\psi }}_{{\ell }}$ and ${D}_{{\ell }}^{{T}_{{\rm{d}}}{B}_{{\rm{d}}}}$ and between ${\hat{\psi }}_{{\ell }}$ and ${D}_{{\ell }}^{{E}_{{\rm{d}}}{B}_{{\rm{d}}}}$ as a direct consequence of the known strong correlation between the Planck dust maps and the H i templates (e.g., Clark & Hensley 2019). The H i–dust correlation is maintained in the mock skies. The H i component contributes nonnegligibly to the dust polarization, and the Planck maps can be viewed as perturbed versions of the H i templates. From Figure 6, we see that the perturbations need not be especially small; in fact, the Gaussian-dust component dominates over the H i component, though only modestly. If the perturbations are random, the dust polarization angles are symmetrically distributed relative to the H i template, and $\hat{\psi }=0$. If, however, there is a region of sky in which the dust polarization angles are distributed asymmetrically relative to the H i template, then $\hat{\psi }\ne 0;$ in this case, there will be a net chirality, which will in turn produce nonzero contributions to T_d B_d and E_d B_d. Even in the mock skies, there are regions of sky that fluctuate to nonzero $\hat{\psi }$, and these regions tend to contribute nonzero T_d B_d and E_d B_d with a corresponding sign. Our estimators avoid noise biases, so the relevant fluctuations are likely due to on-sky dust components that deviate from the H i template. This is the expected contribution of magnetic misalignment to the parity-violating dust polarization quantities, but we aim to investigate whether the observed T_d B_d and E_d B_d are consistent with random fluctuations away from the filament orientations—as exemplified by the mock skies—or show evidence for harmonic or spatial coherence, which might be expected from a physical misalignment between the magnetic field and dusty filaments.

It is important to note that, while $\hat{\psi }\gt 0$ implies a tendency toward T_d B_d, E_d B_d > 0, the converse is not guaranteed. It is possible to have T_d B_d, E_d B_d > 0 but $\hat{\psi }=0$. Our mock skies (Section 5) illustrate this point. They are constructed to retain the T_d B_d spectrum of the true dust maps, but this property is placed entirely in the Gaussian component, which is statistically independent of the H i component. As such, the mock skies display no aggregate misalignment (beyond realization-dependent scatter). For example, on a 70% Galaxy mask, the ensemble mean of $\hat{\psi }$ is zero, although the mean T_S B_S spectrum is positive for 100 ≲ ℓ ≲ 500, as for the true T_d B_d (Section 1.1).

To the extent that the observed T_d B_d is related to magnetic misalignment, an outstanding question is whether the real dust TB is a consequence of physical misalignment or random scatter. Thus, we search for harmonically coherent relationships between misalignment angle and parity-violating cross-spectra. The mock skies will help us to make the distinction, since harmonic coherence is not included in them.

We begin the investigation with sky areas that are only modestly smaller than those of Sections 4 and 6.5. In this limit, the aggregate misalignment angles are small, and the expected relationship to parity-violating cross-spectra can be approximated as (see Equation (30))

$$\begin{eqnarray}&&\hat{\psi }\approx \displaystyle \frac{{D}_{{\ell }}^{{T}_{{\rm{d}}}{B}_{{\rm{d}}}}}{2{D}_{{\ell }}^{{T}_{{\rm{d}}}{E}_{{\rm{d}}}}}\approx \displaystyle \frac{{D}_{{\ell }}^{{E}_{{\rm{d}}}{B}_{{\rm{d}}}}}{2\left({D}_{{\ell }}^{{E}_{{\rm{d}}}{E}_{{\rm{d}}}}-{D}_{{\ell }}^{{B}_{{\rm{d}}}{B}_{{\rm{d}}}}\right)}.\end{eqnarray} \tag{ 33 }$$

We will focus more on the dust-only spectra T_d B_d and E_d B_d, as opposed to the dust–H i spectra T_{H i} B_d, E_{H i} B_d, and B_{H i} E_d, but similar operations can be performed for either set.

We divide the sky into patches defined by N_side = 2 within an overall 70% Galaxy mask. In each patch, we measure $\hat{\psi }$ as in Figure 7 (with an additionally imposed Galaxy mask). We form a combined mask from the patches with $\hat{\psi }$ larger than the median value, and we form an analogous mask for the patches with $\hat{\psi }$ smaller than the median; the masks are shown in the bottom right of Figure 14, where the input $\hat{\psi }$ values are from maps that have been filtered to 101 < ℓ < 702 (as in the right column of Figure 7). We repeat for maps that have been low-pass-filtered to 101 < ℓ < ℓ_c (which will be labeled by the subscript "lo") and for maps that have been high-pass-filtered to ℓ_c < ℓ < 702 (subscript "hi") for ℓ_c ∈ {202, 302, 402}.

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We calculate auto- and cross-spectra for the full combination of patches for the large- and small-$\hat{\psi }$ samples. To avoid sharp mask features at shared vertices of the HEALPix-defined patches, we use a conservative apodization scale of 5° for the results in this subsection. Note that, due to the HEALPix pixelization and the increased apodization scale, the full combination of patches represents a smaller overall sky area than produced by the fiducial 70% Galaxy mask of Sections 4 and 6.5. The reduction is somewhat severe and leaves a sky fraction of only 37%.

We convert the auto- and cross-spectra to misalignment estimates according to Equation (33). The results are shown in the panel of spectra in Figure 14.

We find that the T_d B_d- and E_d B_d-based misalignment estimates increase and decrease in a manner consistent with the $\hat{\psi }$-based mask definition. The large-$\hat{\psi }$ masks tend to produce larger T_d B_d/T_d E_d and E_d B_d/(E_d E_d − B_d B_d), though the latter is much noisier. The small-$\hat{\psi }$ masks tend to produce smaller spectrum-based estimates; interestingly, the resulting T_d B_d/T_d E_d (blue in the top row of Figure 14) is broadly consistent with zero rather than negative. This may suggest that the positive T_d B_d measured at high Galactic latitudes is due to a few regions of sky with positive misalignment and that the rest of the sky respects parity.

When $\hat{\psi }$ is estimated over the same multipole range as the spectra, as in the leftmost column of Figure 14, we cannot distinguish between the case of random fluctuations and that of harmonically coherent misalignment (Section 8.1). To isolate the harmonically coherent signal, we compare estimates from disjoint multipole ranges as in Sections 7.1 and 7.2. The three rightmost columns Figure 14 show the results when estimating $\hat{\psi }$ from restricted multipole ranges. Because the dust is brighter at low multipoles, the selections based on ${\hat{\psi }}_{\mathrm{lo}}$ tend to be similar to those based on the unfiltered $\hat{\psi }$. For T_d B_d and E_d B_d to respond to the $\hat{\psi }$ selection in the disjoint multipole range is an indication of the harmonic coherence of magnetic misalignment, which was demonstrated in Section 7.1 but has now been explicitly connected to T_d B_d. With multipole splits, the data are too noisy to make a confident claim about E_d B_d. The T_d B_d results, which are less vulnerable to noise fluctuations, are similar for all choices of multipole filtering; furthermore, the small-${\hat{\psi }}_{\mathrm{lo}}$ and small-${\hat{\psi }}_{\mathrm{hi}}$ results tend to track each other, as do the large-${\hat{\psi }}_{\mathrm{lo}}$ and large-${\hat{\psi }}_{\mathrm{hi}}$ results. This may be yet another indication of the harmonic coherence of $\hat{\psi }$; i.e., the $\hat{\psi }$-based patch selections are broadly similar in all multipole ranges.

We estimate the statistical significance of the multipole-split results by making random patch selections to define the masks. We preserve the covariances by using the same randomization for all ℓ_c. In combining the results from all ℓ_c, we weight by a measure of signal-to-noise ratio (see Section 7.1), and we estimate the overall significance of the harmonic coherence to be 2.2σ for T_d B_d and 0.8σ for E_d B_d.

We search for correlations between the misalignment angle $\hat{\psi }$ and the parity-violating cross-spectra T_d B_d, E_d B_d, T_{H i} B_d, E_{H i} B_d, and B_{H i} E_d. We outlined our expectations in Equations (17)–(20). For small angles, ${\hat{\psi }}_{{\ell }}$ is expected to track the parity-violating cross-spectra, but there are additional scaling factors. Rather than correlating the spectra with ${\hat{\psi }}_{{\ell }}$ directly, we transform ${\hat{\psi }}_{{\ell }}$ according to Equations (17)–(20).

We compute Spearman half-mission correlation coefficients (see Equation (B4)) because both variables entering each of the following calculations are derived from the same Planck dust modes and therefore subject to covariant noise fluctuations. Due to the aforementioned transformations, the variables are different for each correlation calculation. We compute (see Equations (17)–(20))

$$\begin{eqnarray}&&{\tilde{r}}_{s}^{(\mathrm{HM})}\left({D}_{{\ell }}^{{E}_{{\rm{H}}\,{\rm\small{I}}}{B}_{{\rm{d}}}},{D}_{{\ell }}^{{E}_{{\rm{H}}\,{\rm\small{I}}}{E}_{{\rm{d}}}}\tan \left(2{\hat{\psi }}_{{\ell }}\right)\right),\end{eqnarray} \tag{ 34 }$$

$$\begin{eqnarray}&&{\tilde{r}}_{s}^{(\mathrm{HM})}\left({D}_{{\ell }}^{{B}_{{\rm{H}}\,{\rm\small{I}}}{E}_{{\rm{d}}}},{D}_{{\ell }}^{{B}_{{\rm{H}}\,{\rm\small{I}}}{B}_{{\rm{d}}}}\tan \left(2{\hat{\psi }}_{{\ell }}\right)\right),\end{eqnarray} \tag{ 35 }$$

$$\begin{eqnarray}&&{\tilde{r}}_{s}^{(\mathrm{HM})}\left({D}_{{\ell }}^{{T}_{x}{B}_{{\rm{d}}}},{D}_{{\ell }}^{{T}_{x}{E}_{{\rm{d}}}}\tan \left(2{\hat{\psi }}_{{\ell }}\right)\right)\end{eqnarray} \tag{ 36 }$$

for x ∈ {d, H i} and

$$\begin{eqnarray}&&{\tilde{r}}_{s}^{(\mathrm{HM})}\left({D}_{{\ell }}^{{E}_{{\rm{d}}}{B}_{{\rm{d}}}},\left({D}_{{\ell }}^{{E}_{{\rm{d}}}{E}_{{\rm{d}}}}-{D}_{{\ell }}^{{B}_{{\rm{d}}}{B}_{{\rm{d}}}}\right)\tan \left(4{\hat{\psi }}_{{\ell }}\right)\right),\end{eqnarray} \tag{ 37 }$$

where Equation (35) is expected to be negative and all others positive. We show these correlations and maps of the parity-violating quantities in Figure 15 for patches defined by N_side = 8 (as in Figure 8). The correlations have the expected sign in all cases.

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Half-mission cross-correlations (Equation (B4)) avoid noise covariance but not sample variance in the dust measurements. The map features that produce positive $\hat{\psi }$ also produce positive T_d B_d, E_d B_d, T_{H i} B_d, and E_{H i} B_d and negative B_{H i} E_d (Section 8.1). While the effective mode weighting in calculating ${\hat{\psi }}_{{\ell }}$ is different than in the cross-spectra, we nevertheless find a correlation between the two in our mock skies (Section 5), for which the non-H i component is statistically independent of the H i component. In particular, the mock skies approximately reproduce the results of Figure 15.

That our mock skies show correlations between misalignment angle and parity-violating cross-spectra is an indication that the correlations of Figure 15 could be attributed to random fluctuations away from the H i template (Section 8.1). This is yet another motivation to restrict the search to signals that are coherent in either harmonic or map space rather than correlating identical patches with identical multipole bins.

Independent of the distinction between random and harmonically coherent misalignment, the correlations of Figure 15 disfavor the presence of significant confounding contributions to parity violation in the polarization field. A priori, we might have expected nonfilamentary contributions to dilute the relationship between H i-based misalignment angle and parity-violating cross-spectra, especially when we consider that the H i-correlated component is a minority contributor to the dust field (Figure 6). Results like those of Figure 15 and the leftmost panel of Figure 14 suggest that the H i template is sufficiently significant and representative to provide a reference for searches for parity violation.

Instead of directly correlating ${\hat{\psi }}_{{\ell }}$ with, e.g., ${D}_{{\ell }}^{{T}_{{\rm{d}}}{B}_{{\rm{d}}}}$, we define disjoint multipole ranges and correlate the low-pass-filtered quantities with the high-pass-filtered. This is similar to the multipole splits described in Section 7.1 and better extracts a signal that is coherent across multipoles. For the misalignment angle, we low or high pass filter the map to form ${\hat{\psi }}_{\mathrm{lo}}$ or ${\hat{\psi }}_{\mathrm{hi}}$, respectively. For the spectra, we simply bin the lower or higher multipoles to form ${D}_{\mathrm{lo}}^{{XY}}$ or ${D}_{\mathrm{hi}}^{{XY}}$, respectively. As in Section 7.1, the low-pass-filtered multipole range is 101 < ℓ < ℓ_c − Δℓ/2, and the high-pass-filtered is ℓ_c + Δℓ/2 < ℓ < 702, where ℓ_c is a transition multipole, and Δℓ is a multipole buffer between the two ranges.

We now modify Equations (34)–(37) to correlate across multipole splits. We filter the spectra in the same way and the misalignment angle in the opposite way; e.g., we correlate ${D}_{\mathrm{lo}}^{{T}_{{\rm{d}}}{B}_{{\rm{d}}}}$ with ${D}_{\mathrm{lo}}^{{T}_{{\rm{d}}}{E}_{{\rm{d}}}}\tan \left(2{\hat{\psi }}_{\mathrm{hi}}\right)$. We are seeking a relationship between ${D}_{\mathrm{lo}}^{{T}_{{\rm{d}}}{B}_{{\rm{d}}}}$ and ${D}_{\mathrm{lo}}^{{T}_{{\rm{d}}}{E}_{{\rm{d}}}}$, and our hypothesis is that the connection is provided by ${\hat{\psi }}_{\mathrm{hi}}$, even though the latter is estimated in a disjoint multipole range. We look for a simultaneous correlation when the multipole ranges are switched.

For this purpose, we use the Spearman version of the four-variable cross-power (Equation (B5))

$$\begin{eqnarray}\begin{array}{rcl}\hat{\psi }\times {T}_{{\rm{d}}}{B}_{{\rm{d}}} & \equiv & {S}_{s}\left({D}_{\mathrm{lo}}^{{T}_{{\rm{d}}}{B}_{{\rm{d}}}},{D}_{\mathrm{lo}}^{{T}_{{\rm{d}}}{E}_{{\rm{d}}}}\tan \left(2{\hat{\psi }}_{\mathrm{hi}}\right),\right.\\ & & \quad \quad \quad \left.{D}_{\mathrm{hi}}^{{T}_{{\rm{d}}}{B}_{{\rm{d}}}},{D}_{\mathrm{hi}}^{{T}_{{\rm{d}}}{E}_{{\rm{d}}}}\tan \left(2{\hat{\psi }}_{\mathrm{lo}}\right)\right)\end{array}\end{eqnarray} \tag{ 38 }$$

and similar combinations for the other four parity-violating cross-spectra. The cross-powers are presented in Figure 16 for patches defined by N_side = 8 (as in Figure 8). We also form these cross-powers for an ensemble of mock skies. As noted earlier, our mock skies cannot be used for null-hypothesis testing, but they are useful for testing the basic properties of our correlation metrics and misalignment estimators. We find that the mock skies produce null results within the realization-dependent scatter.

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To assess the noise level in the parity-violating cross-spectra, we consider the half-mission cross-powers (Spearman version of Equation (B2)),

$$\begin{eqnarray}&&{{XY}}_{\mathrm{lo}}^{(1)}\times {{XY}}_{\mathrm{lo}}^{(2)}\equiv {s}_{s}\left({D}_{\mathrm{lo}}^{{X}^{(1)}{Y}^{(1)}},{D}_{\mathrm{lo}}^{{X}^{(2)}{Y}^{(2)}}\right),\end{eqnarray} \tag{ 39 }$$

where X⁽ⁱ⁾ and Y⁽ⁱ⁾ are the X and Y fields, respectively, from the ith half-mission. We form a similar quantity for the high-pass-filtered observables:

$$\begin{eqnarray}&&{{XY}}_{\mathrm{hi}}^{(1)}\times {{XY}}_{\mathrm{hi}}^{(2)}\equiv {s}_{s}\left({D}_{\mathrm{hi}}^{{X}^{(1)}{Y}^{(1)}},{D}_{\mathrm{hi}}^{{X}^{(2)}{Y}^{(2)}}\right).\end{eqnarray} \tag{ 40 }$$

Both ${{XY}}_{\mathrm{lo}}^{(1)}\times {{XY}}_{\mathrm{lo}}^{(2)}$ and ${{XY}}_{\mathrm{hi}}^{(1)}\times {{XY}}_{\mathrm{hi}}^{(2)}$ are limited only by noise. When noise is subdominant to the sky components, they will show strong positive signals; when noise is significant, they will decay to zero. We plot ${{XY}}_{\mathrm{lo}}^{(1)}\times {{XY}}_{\mathrm{lo}}^{(2)}$ and ${{XY}}_{\mathrm{hi}}^{(1)}\times {{XY}}_{\mathrm{hi}}^{(2)}$ as red and purple bands, respectively, in Figure 16, where we find that, in general, the high-ℓquantities are substantially noisier than the low-ℓ quantities.

As discussed in Section 7.1, half-mission cross-powers like ${\hat{\psi }}_{\mathrm{hi}}^{(1)}\times {\hat{\psi }}_{\mathrm{hi}}^{(2)}$ and ${{XY}}_{\mathrm{hi}}^{(1)}\times {{XY}}_{\mathrm{hi}}^{(2)}$ set rough upper limits on the observable strength of the signals we are seeking. In the case of Figure 16, we must consider the fidelity of both $\hat{\psi }$ (red/purple bands in Figure 12) and XY (red/purple in Figure 16). When $\hat{\psi }\times {XY}$ is of the same order as the half-mission cross-powers, a nonnegligible fraction of the variation in XY is associated with harmonically coherent misalignment. In Figure 16, this is the case for E_{H i} B_d, B_{H i} E_d, T_{H i} B_d, and T_d B_d. For E_d B_d, the half-mission cross-powers, especially $({E}_{{\rm{d}}}{B}_{{\rm{d}}}{)}_{\mathrm{hi}}^{(1)}\times ({E}_{{\rm{d}}}{B}_{{\rm{d}}}{)}_{\mathrm{hi}}^{(2)}$, are too noisy to make a reliable comparison.

We estimate the statistical significance of measurements like those of Figure 16 by following a prescription similar to those of Sections 7.1 and 7.2. The weights used in combining the measurements now account for noise in both $\hat{\psi }$ (bands in Figure 12) and the cross-spectra (bands in Figure 16). For the example of Figure 16, we find that $\hat{\psi }\times {T}_{{\rm{d}}}{B}_{{\rm{d}}}$ has a significance of 2.9σ, while $\hat{\psi }\times {E}_{{\rm{d}}}{B}_{{\rm{d}}}$ yields −0.8σ, i.e., consistency with null. The cross-powers with T_{H i} B_d, E_{H i} B_d, and B_{H i} E_d yield, respectively, 2.8σ, 2.8σ, and −1.75σ, where the last value is expected to be negative (Equation (18)). Since the Planck–H i cross-spectra are approximate measures of the dust rotation relative to the H i template, the latter correlations can be considered further confirmation of the harmonic coherence of $\hat{\psi }$ (Section 7.1).

In Table 3, we compile estimates of statistical significance for $\hat{\psi }\times {T}_{{\rm{d}}}{B}_{{\rm{d}}}$ and $\hat{\psi }\times {E}_{{\rm{d}}}{B}_{{\rm{d}}}$ using different choices for N_side and f_sky. The estimates are correlated with each other, and we have not attempted to estimate a global significance. What can be gleaned, however, is a tendency for positive correlations with T_d B_d and mostly insignificant correlations with E_d B_d. A few variations show a significance above 2σ for E_d B_d, and a majority are positive. But the overall picture is less compelling than in the case of T_d B_d. The $({E}_{{\rm{d}}}{B}_{{\rm{d}}}{)}_{\mathrm{lo}}^{(1)}\times ({E}_{{\rm{d}}}{B}_{{\rm{d}}}{)}_{\mathrm{lo}}^{(2)}$ and $({E}_{{\rm{d}}}{B}_{{\rm{d}}}{)}_{\mathrm{hi}}^{(1)}\times ({E}_{{\rm{d}}}{B}_{{\rm{d}}}{)}_{\mathrm{hi}}^{(2)}$ cross-powers (Equations (39) and (40); shown as red and purple bands in Figure 16) are two to three times smaller than the corresponding T_d B_d quantities at low ℓ_c and have fractionally larger uncertainties. Furthermore, the $({E}_{{\rm{d}}}{B}_{{\rm{d}}}{)}_{\mathrm{hi}}^{(1)}\times ({E}_{{\rm{d}}}{B}_{{\rm{d}}}{)}_{\mathrm{hi}}^{(2)}$ signal becomes consistent with zero for ℓ_c ≳ 300; i.e., ${({E}_{{\rm{d}}}{B}_{{\rm{d}}})}_{\mathrm{hi}}$ becomes noise-dominated. Given these considerations, it is consistent with our expectations that the ∼3σ results for $\hat{\psi }\times {T}_{{\rm{d}}}{B}_{{\rm{d}}}$ weaken to mostly null results for $\hat{\psi }\times {E}_{{\rm{d}}}{B}_{{\rm{d}}}$.

Table 3. Statistical Significance (in Units of σ) of Measurements of the Harmonically Coherent Correlations between $\hat{\psi }$ and T_d B_d (E_d B_d), e.g., Those Shown in Figure 16, for Different Values of N_side (Rows with Side Lengths Provided Parenthetically) and f_sky (Columns)

	40%	60%	70%	80%	90%	100%
2 (293)	1.8 (1.4)	2.1 (1.2)	2.2 (1.8)	1.8 (1.9)	2.5 (1.3)	2.6 (1.5)
4 (147)	1.3 (−0.5)	2.5 (0.6)	2.8 (1.3)	3.0 (3.0)	2.3 (2.4)	1.7 (1.2)
8 (73)	1.9 (−0.9)	1.5 (−1.5)	2.9 (−0.8)	3.7 (−0.9)	3.9 (0.8)	4.4 (0.2)

Note. The $\hat{\psi }\times {T}_{{\rm{d}}}{B}_{{\rm{d}}}$ significances are all positive and tend to increase with N_side and f_sky. The $\hat{\psi }\times {E}_{{\rm{d}}}{B}_{{\rm{d}}}$ significances are mostly positive but generally smaller.

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The real data, within the limits of the noise, are broadly consistent with our expectations, namely, positive correlations between $\hat{\psi }$ and T_d B_d, E_d B_d, T_{H i} B_d, and E_{H i} B_d and negative between $\hat{\psi }$ and B_{H i} E_d, though these signals disappear for some choices of N_side, ℓ_c, and Δℓ. In particular, the signal tends to decay as ℓ_c increases, which we expect due to increased noise in the high-pass-filtered quantities. For N_side = 4, the expected negative correlation with B_{H i} E_d appears only for ℓ_c ≳ 350 and is fairly weak. For N_side = 8, the expected correlation with E_d B_d disappears.

These results build confidence in our picture of harmonically coherent magnetic misalignment. Alternatively, these correlations can be considered necessary implications of the harmonic coherence of $\hat{\psi }$ (Section 7.1) coupled with the expected relationship between $\hat{\psi }$ and the parity-violating cross-spectra (Sections 8.1 and 8.3). With this perspective, the cross-powers in Figure 16 are merely tests for consistency.

We draw special attention to the correlations between $\hat{\psi }$ and T_d B_d, as the positive T_d B_d measured by Planck has been recently discussed in the literature (Huffenberger et al. 2020; Weiland et al. 2020; Clark et al. 2021; Huang 2022). Although correlations between $\hat{\psi }$ and E_d B_d yielded only weak results, we note the positivity of $({E}_{{\rm{d}}}{B}_{{\rm{d}}}{)}_{\mathrm{lo}}^{(1)}\times ({E}_{{\rm{d}}}{B}_{{\rm{d}}}{)}_{\mathrm{lo}}^{(2)}$ (Equation (39); shown as the red band in Figure 16), which indicates on-sky variation in E_d B_d that rises above the noise level. Variation in E_d B_d may be attributable to sample-variance fluctuations of underlying parity-even statistical processes, but the particular dust realization that we observe is a foreground that must be mitigated for, e.g., measurements of the CMB. Spatial variation in E_d B_d may be of relevance for measurements of cosmic birefringence (Minami et al. 2019; Minami & Komatsu 2020; Diego-Palazuelos et al. 2022; Eskilt & Komatsu 2022). If future measurements can more confidently establish a relationship between $\hat{\psi }$ and E_d B_d, foreground removal could be performed more robustly by, e.g., relating E_d B_d to T_d B_d and other observables (e.g., Equations (17)–(20)).

The RHT is another filament-finding algorithm (Clark et al. 2014) from which we can produce polarization templates that correlate strongly with the dust polarization measured by Planck (Clark et al. 2015; Clark & Hensley 2019).

We find that the Hessian correlates more strongly with the Planck dust B modes than the RHT for ℓ ≳ 100, and we show a comparison in Figure 17. In the E modes, the two perform similarly. In these comparisons, the RHT is constructed from velocities spanning −90 to 90 km s⁻¹ (as in Clark & Hensley 2019), while the Hessian template is constructed from the restricted range of −15 to 4 km s⁻¹, (Section 3.1).

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For the T template, we consider two choices for the RHT. One might use I_Hi4PI, i.e., the Hi4PI intensity map (Hi4PI Collaboration et al. 2016) without any processing. Since we are especially interested in the filamentary component of the dust intensity, it may be preferable to high pass filter the Hi4PI intensity as in the first step in the RHT algorithm of Clark & Hensley (2019). The filter is implemented as an unsharp mask with $\mathrm{FWHM}=30^{\prime}$. Denote the high-pass-filtered intensity by T_RHT. In Figure 17, we find that I_Hi4PI correlates more strongly than T_RHT with the Planck T modes. This is not necessarily the relevant metric, however, since we are specifically targeting filaments. We also present in Figure 17 the correlation with the Planck E modes, since the TE correlation is a signature of filamentary polarization. We find that ${r}_{{\ell }}^{{T}_{\mathrm{RHT}}{E}_{{\rm{d}}}}$ is generally larger than ${r}_{{\ell }}^{{I}_{{\rm{H}}\,{\rm\small{I}}4\mathrm{PI}}{E}_{{\rm{d}}}}$. For this reason, we prefer T_RHT as a template for filamentary dust intensity.

The Hessian intensity T_H (Equations (11) and (14)) is defined mainly by the Hessian eigenvalues. In Figure 17, we find that ${r}_{{\ell }}^{{T}_{H}{T}_{{\rm{d}}}}$ is smaller than ${r}_{{\ell }}^{{T}_{\mathrm{RHT}}{T}_{{\rm{d}}}}$ but that ${r}_{{\ell }}^{{T}_{H}{E}_{{\rm{d}}}}$ is similar to and, at high ℓ, slightly larger than ${r}_{{\ell }}^{{T}_{\mathrm{RHT}}{E}_{{\rm{d}}}}$.

On account of the greater correlation in the B modes, we have selected the Hessian-based template as our baseline. The T templates considered above all correlate strongly with both T_d and E_d and at roughly the same level. The E templates for both algorithms correlate with E_d at roughly the same level.

We defer to future work a more detailed investigation of these and related filament-finding algorithms for the construction of polarization templates (G. Halal et al. 2022, in preparation). Each can be modified and tuned by making different choices for, e.g., velocity binning, weighting, spatial filtering, etc.

The H i-based polarization templates have different mode structures than the Planck dust maps. For example, the Hessian method upweights small-scale features; the E_H and B_H power spectra increase with ℓ. The RHT also upweights small-scale features but especially emphasizes the multipole range 300 ≲ ℓ ≲ 500 for E_RHT and 150 ≲ ℓ ≲ 350 for B_RHT (with RHT parameters set to those of Clark & Hensley 2019).

Correlations, e.g., those presented in Clark & Hensley (2019), are insensitive to differences in mode structure because they are evaluated in individual multipole bins. The upweighting of one multipole bin relative to another is normalized out of the calculation. The difference in mode structure can be viewed as a multipole-dependent transfer function.

For the purposes of converting our H i-based polarization templates into quantities that are directly comparable to the Planck dust maps, we assume a transfer function that depends only on multipole ℓ:

$$\begin{eqnarray}&&{k}_{{\ell }}^{(X)}\equiv \displaystyle \frac{{D}_{{\ell }}^{{X}_{{\rm{H}}\,{\rm\small{I}}}{X}_{{\rm{d}}}}}{{D}_{{\ell }}^{{X}_{{\rm{H}}\,{\rm\small{I}}}{X}_{{\rm{H}}\,{\rm\small{I}}}}}.\end{eqnarray} \tag{ A1 }$$

This transfer function converts an H i-based quantity into dust-intensity units with a mode structure that is directly comparable to the observed dust field. We find that k_ℓ (X) rises strongly at low multipoles, which is an indication that the H i templates tend to underpredict the amplitude of large-scale dust polarization relative to small-scale. In spite of this underprediction, the correlations, which normalize out the ℓ dependence, are actually stronger at low ℓ.

There is no guarantee that the transfer function ${k}_{{\ell }}^{(X)}$ provides a representative estimate of the amplitude of the H i-based modes in the real dust maps. The amplitudes may depend on both ℓ and m, the spherical-harmonic eigenvalues. We use ${k}_{{\ell }}^{(X)}$ as a rough conversion factor to make direct comparisons between the H i templates and the real dust maps.

Because the Hessian-based template correlates nonvanishingly with Planck up to at least ℓ = 750, the transfer function remains usable across the entire multipole range considered in our analysis.

The H i templates are produced under the assumption of perfect magnetic alignment. Even so, chirality in the filament morphology could produce parity-violating signatures such as nonzero T_{H i} B_{H i} and E_{H i} B_{H i} .

We computed these parity-violating cross-spectra for both the Hessian and the RHT templates. ⁸ To determine if the results are significant, we compare to the T_d B_d and E_d B_d spectra from the Planck dust maps. To make this comparison, we applied the transfer function ${k}_{{\ell }}^{(X)}$ introduced in Appendix A.2, which converts the H i templates into dust-intensity units. The results are shown in Figure 18.

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The Planck T_d B_d displays a positive signal, and T_{H i} B_{H i} is negligible in comparison. The Planck E_d B_d appears to be consistent with noise, and we find that E_{H i} B_{H i} is negligible in comparison with the fluctuations. The above comparisons are restricted to ℓ > 100, which is the target multipole range of the analysis presented in this work.

Based on these observations, the intrinsic (or morphological) H i-based T_{H i} B_{H i} and E_{H i} B_{H i} are assumed to vanish in our analysis.

Intriguingly, however, the H i-based E_{H i} B_{H i} shows a rise for ℓ < 150. With finer multipole binning Δℓ, we find that this rise persists down to ℓ = 17 with Δℓ = 10, the lowest bin center with the finest binning that we checked. There is a corresponding rise in T_{H i} B_{H i} that persists down to ℓ = 27. In all cases, the H i-based predictions are subdominant to the expected noise in the Planck measurements but only by a factor of ∼10. We defer to future work an investigation of these low-ℓH i-based predictions, which could represent a source of parity violation independent of magnetic misalignment.

For statistical inference regarding our correlation metrics, we use permutation tests. For two-variable metrics, we randomly permute Y to obtain π(Y), where the function π defines a random permutation that here acts on the data vector Y. We then compute, e.g., s_s (X, π(Y)). With a large number of permutations, we can build a null-hypothesis distribution for s_s (X, Y). Similar permutation tests can be formed for the other two-variable correlation metrics. In this work, each ensemble of permutations contains 200 realizations. Our results change only negligibly with larger permutation ensembles.

For the four-variable correlation metrics, we also appeal to permutation tests, but we coordinate the permutations of X and Z. We form many realizations of, e.g., S_s (W, π(X), Y, π(Z)).

We will often estimate uncertainties on our correlation coefficients by converting to z scores and taking half the difference between the value at 1σ and the value at −1σ. For s_s (X, Y), call this uncertainty estimate σ[s_s (X, Y)], and we maintain similar notational conventions for the other correlation metrics.