correlation: insights from the spin-moment expansion

The change of physical conditions across the turbulent and magnetized interstellar medium (ISM) induces a 3D spatial variation of the properties of Galactic polarized emission. The observed signal results from the averaging of different spectral energy distributions (SED) and polarization angles, along and between lines of sight. As a consequence, the total Stokes parameters Q and U will have different distorted SEDs, so that the polarization angle becomes frequency dependent. In the present work, we show how this phenomenon similarly induces a different distorted SED for the three polarized angular power spectra DEE ` , DBB ` and DEB ` , implying a variation of the DEE ` /DBB ` ratio with frequency. We demonstrate how the previously introduced spin-moment formalism provides a natural framework to grasp these effects, allowing us to derive analytical predictions for the spectral behaviors of the polarized spectra, focusing here on the example of thermal dust polarized emission. After a quantitative discussion based on a model combining emission from a filament with its background, we further reveal that the spectral complexity implemented in the dust models commonly used by the cosmic microwave background (CMB) community produce such effects. This new understanding is crucial for CMB component separation, in which an extreme accuracy is required in the modeling of the dust signal to allow for the search of the primordial imprints of inflation or cosmic birefringence. For the latter, as long as the dust EB signal is not measured accurately, great caution is required about the assumptions made to model its spectral behavior, as it may not simply follow from the other dust angular power spectra.


Introduction
Understanding Galactic foregrounds is a critical challenge for the success of cosmic microwave background (CMB) experiments searching for primordial B-modes leftover by inflationary gravitational waves (see e.g. Kamionkowski & Kovetz 2016) and signatures of cosmic birefringence (see e.g. Planck Collaboration 2016b; Diego-Palazuelos et al. 2022a). In these quests, both the structure on the sky and the frequency-dependence of the foreground signal need to be modeled.
The two-point statistical properties of a polarized signal are described by angular auto-and cross-power spectra hereafter simply written as XY, where X and Y refer to E− and B−mode polarization or the total intensity T . Thermal dust polarization is the main polarized foreground at frequencies above ∼ 70 GHz (Krachmalnicoff et al. 2016). From observations of the Planck satellite at 353 GHz, dust power spectra in polarization are found to be well fitted by power-laws in of similar indices, with a EE/BB power ratio of about 2. A positive T E and a weaker, parity violating, T B signal have been significantly detected (Planck Collaboration 2016c. The dust EB signal remains however compatible with zero at the Planck sensitivity.
Send offprint requests to: leo.vacher@irap.omp.eu The EE/BB asymmetry and T E correlation relate to the anisotropic structure of the magnetized interstellar medium with filamentary structures in total intensity preferentially aligned with the Galactic magnetic field (Clark et al. 2015;Planck Collaboration 2016e). This interpretation has opened the path to empirical and phenomenological models (Ghosh et al. 2017;Huffenberger et al. 2020;Hervías-Caimapo & Huffenberger 2022;Konstantinou et al. 2022). Within this framework, a coherent misalignement between filamentary dust structures and the magnetic field can account for the dust T B signal, and should also imply a positive EB Cukierman et al. 2022). The possibility of a non-zero dust EB signal is at the heart of recent analyses of Planck data that seek to detect cosmic birefringence because it complicates attempts to measure a CMB EB correlation (Minami et al. 2019;Diego-Palazuelos et al. 2022a;Eskilt & Komatsu 2022;Diego-Palazuelos et al. 2022b).
While evaluating the amplitudes of the foreground EE/BB ratio and EB correlation are central subjects in the literature, their frequency dependence are rarely discussed. In the present work, we intend to open this discussion. The frequency dependence of dust polarization is a critical issue for CMB component separation, given the level of accuracy targeted by future exper-Article number, page 1 of 13 arXiv:2210.14768v1 [astro-ph.CO] 26 Oct 2022 A&A proofs: manuscript no. main iments (e.g. LiteBIRD Collaboration 2022; CMB-S4 Collaboration 2019), which relates CMB experiments to the modelling of dust emission (Guillet et al. 2018;Hensley & Draine 2022). The Planck data have also been crucial in building our current understanding of this topic (Planck Collaboration 2015b, 2016c. The mean SED, derived from EE and BB power spectra, is found to be well fitted by a single modified blackbody (MBB) law (Planck Collaboration 2020), but this empirical law does not fully characterize the frequency dependence of dust polarization. Indeed, integration along the line of sight and within the beam of multiple polarized signals with different spectral parameters and polarization angles must induce departures from the MBB coupled to variations of the total polarization angle with frequency (Tassis & Pavlidou 2015;Planck Collaboration 2017;Vacher et al. 2022b). These phenomena have been detected in Planck data by Pelgrims et al. (2021) for a discrete set of lines of sight selected based on H I data and by Ritacco et al. (2022) using power spectra. These results emphasize the need to account for variations of polarization angles in order to model the dust foreground to the CMB.
The moments expansion formalism, introduced in intensity by Chluba et al. (2017), proposes to treat the distortions coming from averaging over different emission points by Taylor expanding the canonical SED -the MBB for dust polarization -with respect to its spectral parameters. This framework has proven to be a powerful tool for component separation and Galactic physics (Remazeilles et al. 2016(Remazeilles et al. , 2021Sponseller & Kogut 2022). When carried out at the power-spectrum level, as in Mangilli et al. (2021), the expansion can be applied directly to the B-mode signal as an intensity (Azzoni et al. 2020;Vacher et al. 2022a;Ritacco et al. 2022). A recent generalization of this formalism to polarization in Vacher et al. (2022b), the "spin-moment" expansion, provides a natural framework to treat for the spectral dependence of the polarization angle. In the present paper, we provide the missing links between the spinmoments maps and the treatment of E-and B-modes. We hence connect the statistical studies of the sky maps with the modeling of the frequency dependence of dust polarization angular power spectra, allowing us to predict new consequences unique to polarization. In particular, we will discuss how the assumption of a common SED for EE, BB (and EB) breaks down.
This article is organized as follows. In Sec. 2, we introduce the specificity of the polarized signal and explain qualitatively why we expect the frequency dependence of EE, BB and EB to differ. In Sec. 3, we establish how the formalism of the spinmoments can naturally describe these effects and give an ana-lytical expression for their frequency dependence. In Sec. 4, we illustrate the derived formalism on a filament model. Then, in Sec. 5, we show that the frequency dependence of the EE/BB ratio and the non-trivial dependence of EB are already present in dust models extensively used by the CMB community. This brings us to stress the need for caution when inferring the spectral properties of the dust EB signal from other angular power spectra. We finally reach our conclusions in Sec. 6.

Combination of polarized signals
2.1. Mixing of polarized signals in the Q-U plane The frequency-dependent linear polarization 1 of a signal is described by a polarization spinor P ν 2 . It is a complex valued object that can be expressed both in Cartesian or exponential form as where i 2 = −1 is the imaginary unit. The spinor components (Q ν ,U ν ) are two of the Stokes parameters, proportional to the difference of intensities in two orthogonal directions rotated from one to another by an angle of 45 • . P ν is said to be a spin-2 object due to its transformation properties under rotations. It rotates in the complex plane by an angle −2θ under a right handed rotation around the line of sight by an angle θ of the coordinates in which the Stokes parameters are defined. The modulus of the spinor in Eq. (1) is the linear polarized intensity encoding the spectral energy distribution (SED) of the total polarized signal. Depending on the physics of emission, P ν can be a function of various parameters p, called the spectral parameters. Both the total intensity and the polarized intensity of the thermal dust grains at a given point of the Galaxy are expected to follow a modified blackbody (MBB) law (see e.g. Planck Collaboration 2015a) the corresponding spectral parameters being p = {β, T }, with β the spectral index, related to dust grain properties, and T the temperature associated to the black-body law B Pl ν . ν 0 is an arbitrary reference frequency and the dust polarization amplitude A = p 0 τ cos(γ) 2 , can be expressed in terms of the intrinsic degree of polarization p 0 , the dust opacity τ and the angle between the Galactic magnetic field and the plane of the sky γ (see e.g. Planck Collaboration (2015a)). The polarized emissivity function ε P ν (β, T ) encodes all the spectral dependence of the MBB law 3 .
The spinor phase in Eq. (1) is the so called polarization angle This definition explicitly shows that if Q ν and U ν have the same SED, ψ is a constant number defining a single orientation at all frequencies. This assumption is usually made when modeling locally the dust signal, ψ being orthogonal to the local orientation of the Galactic magnetic field 4 , which is well motivated by the behavior of the elongated dust grains. If however Q ν and U ν behave differently, the polarization orientation defined by ψ will change with frequency, so that ψ → ψ ν . In observational conditions, the mixing of multiple signals coming from emission points with different physical conditions are unavoidable along the line of sight n and between the lines of sight, inside the instrumental beam or over patches of the sky when using spherical harmonic transformations. As discussed in Chluba et al. (2017), this mixing will induce departures from the canonical SED of the total intensity known as SED distortions, that can be properly modeled using a Taylor expansion of the signal with respect to the spectral parameters themselves known as moment expansion. The mixing also has unique consequences in polarization. In the rest of this work, we will refer to the combination of individual polarized signals having different spectral parameters and polarization angles as the polarized mixing. The resulting spinor obtained through polarized mixing will inherit both SED distortions and a spectral dependence of its polarization angle which can be modeled using a complex moment expansion of P ν itself (Vacher et al. 2022b). In principle, it is even possible to predict the value of ψ ν from the distribution of polarization angles and spectral parameters. For example, considering a sum of MBBs with different polarization angles and spectral indices, one expects at first order that with ∆β ∈ C, the complex spectral index correction, and . . . indicating sums/integrals along the line of sight or in the beam of the A, β and ψ distributions.β is the pivot spectral index around which the expansion is performed.

Mixing of polarized signals in the E-B plane
In analogy with electromagnetism, the E− and B−modes are scalar and pseudo-scalar fields quantifying respectively the existence of curl-free and divergence-free patterns of the ψ-field over the sky. As such, the E-and B-fields are non local quantities, equivalent to convolutions of the Q and U fields around each point of the sky. From a generalization of the Helmotz decomposition theorem, a linearly polarized signal can be fully split into E and B components. The E-and B-modes of the polarized dust emission are frequency dependent quantities. When all the emission points share the same spectral parameters, E ν and B ν will have the same SED. In that case, the EE/BB ratio is constant with frequency and the EB correlation has the same SED as EE and BB. As we will show, polarized mixing will impose a different SED for E ν and B ν (as Q ν and U ν ) and in that case, the three angular power spectra should inherit a different distorted SED while the EE/BB ratio will become frequency dependent. In order to get a first grasp at this effect, let's start with a very simple illustration inspired from Zaldarriaga (2001). As in Fig. 1, consider an infinite filament 5 with a constant ψ ν 1 field at a given frequency ν 1 in front of a null background. Choosing a (x, y) reference frame such that y is in the direction of the filament, one can analytically derive the values of the E-and B-fields at the center of the filament: E ν ∝ Q ν and B ν ∝ U ν . The case showed on the left side of the figure at frequency ν 1 with ψ ν 1 = 90 • should then clearly have B ν = 0. It is now obvious that, if due to polarized mixing ψ becomes a frequency dependent quantity, the B-field will not be null anymore at another frequency ν 2 ν 1 (no matter if ψ ν rotates uniformly or not) , leading to an unavoidable frequency dependence of the signal's EE/BB ratio. For this to be possible, one should hence conclude that E ν and B ν do not share the same SED anymore, nor do EE, BB and EB.

Insights from the spin-moments
We now try to find a way to quantify and model the frequency dependence of the EE/BB ratio and the distortions of the EB correlation, predicted in Sec. 2.2. As we will see, the spin-moment expansion formalism provides a natural framework in order to do so. To keep the incoming discussion as simple as possible, we will consider only the case of the averaging of MBB with different A, β and ψ, displaying the expansions only up to first order. All the following derivations can however be straightforwardly generalized at any order and considering variations of temperatures. All our conclusions about the behavior of the spectra will however remain true for any choice of SED as e.g. for the synchrotron signal.

Q/U spin-moment expansion
As presented in sec. 2.1, let's consider that the dust grain polarized signal is given locally by the spinor P ν in every point of the Galaxy, with P ν = |P ν | being a MBB and ψ a constant. The average spinor over different emission points centered on a given line of sight n, will be given by the spin-moment expansion around an arbitrary pivot valueβ 6 for the spectral index as The spin-moments W β k of order k associated to the spectral index can be estimated directly from the distribution of ψ, A and β as 7 W 0 = Ae 2iψ plays the role of a total complex amplitude, that satisfies |W 0 | ≤ A . Purely geometrical phenomenon as cancellation of the phases e 2iψ , known as depolarization, or inclination of the galactic magnetic field towards the plane of the sky (cos(γ) ∼ 0), can greatly decrease the value of W 0 . The condition |W 0 | |W β k | defines the perturbative regime , such that one can consider the total signal as a perturbed MBB. However, as discussed in Vacher et al. (2022b), one expects the existence of configurations where the canceling effects are strong enough, such that the total signal is mostly or fully given by its moments thus loosing its MBB behavior. In general, the phase weighting 8 in Eq. (8), unique to polarization, is breaking the expected hierarchy between the moments and one cannot ascertain that W β k > W β m solely because k > m. The expansion can equivalently be split into Q and U coordinates as 9 Different moments in Q and U, expected in the general case, will necessarily imply a frequency dependence of the polarization angle ψ → ψ ν from the definition given in Eq. (4). In the perturbative regime, the leading order W β 1 /W 0 can be interpreted as a complex correction to the pivot spectral index, leading to giving back the result mentioned in Eq. (5). Note again that both variations of the polarization angles and the spectral indices are required for the second term to be non zero.

E ν and B ν in map space
As for Q and U in Eq. (1), the E and B fields can be grouped in a single complex scalar field (Zaldarriaga & Seljak 1997) of modulus and phase (which we will refer to as the E-B angle) The frequency dependence of the EE/BB ratio discussed in Sec. 2.2 would then translate itself into the rotation of S ν in the complex plane and the E-B angle will become frequency dependent, i.e. ϑ → ϑ ν . A straightforward way to obtain the S ν field from the previously introduced polarization spinor field P ν is to use the spinraising operatorð -the conjugate of the spin-lowering operator ð -as (see e.g. Goldberg et al. (1967) ð is acting both like an angular momentum ladder operator and as a covariant derivative on the sphere. As such it contains derivatives with respect to the spherical coordinates θ and ϕ. More technically, for a spin-s field η on the spherē 8 While the two angles ψ and γ have a common geometrical origin in the Galactic magnetic field, they play a very different role here. ψ is a complex phase, allowing for "interference"-type cancellation of the moment terms while cos(γ) 2 (as p 0 ) plays the role of a real and positive weight. 9 By treating the expansions of Q and U independently (instead of considering them together in P ν ), one would loose the information on their correlation. This operator is mixing the real and imaginary part of η in a nontrivial way. It is however linear and does not act on the spectral dependence, such that the moment expansion will keep the same structure where the scalar moment maps are extracted from the spinmoments as As in Remazeilles et al. (2016Remazeilles et al. ( , 2021 one can split the E and B expansions to treat them as two scalar spin-0 fields 10 (thus loosing their correlation) ð induces a mixing of the Q and U moments into E and B, such that different moments for Q and U imply necessarily different moments for E and B. A clear way to explicit the action ofð is to consider the flat sky approximation, for whichð = ∂ x + i∂ y , allowing to express the E and B moments map in term of the two Q and U spin-moment maps as E and B moments are expressed as linear combinations of second spatial derivatives of the Q and U spin-moments. Thus, ψ → ψ ν implies immediately ϑ → ϑ ν , introducing the EE/BB frequency dependence discussed in Sec. 2.2, and similarly modeled at first order as 3.3. E ν and B ν in harmonic space The spherical harmonic transformation of a spin s field X 11 creates additional mixing over angular scales . As such it is expected to increase the moment amplitudes. Averaging over patches of the sky with a different β in each pixel but no variation along the line of sight is then still expected to create SED distortions (see e.g. Vacher et al. (2022a)). As such, it is also expected to make the EE/BB ratio frequency dependent and distort the EB correlation. Since the transformation is linear, one can simply derive for ≥ 2 With Eq. (17), the expansion becomes Note however that the pivotβ maximizing the convergence of the expansion in harmonic space might be different than in real space. One can hence split the expansion in E and B separately as 3.4. Power spectra 3.5. The EE and BB power spectra Carrying the moment expansion at the power-spectrum level, we find comparable expressions as the ones introduced in intensity by Mangilli et al. (2021) and applied to B-modes in Azzoni et al. (2020); Vacher et al. (2022a). The cross angular power spectra D XX of two fields X and X are defined as For the EE and BB power spectra, replacing E and B by the moment expansion of the real or imaginary parts of S ν , one obtains with XX ∈ {EE, BB}. Hence, knowing the spin-moment maps W β k , one can in principle derive there E-and B-spectra to obtain the D W β k,X W β m,X . Just as Q and U, those two expansions are not expected to be independent as they are the expressions of the real and imaginary parts of the same complex number S ν 12 . While the Planck Collaboration (2016c) analysis found no significant difference between the EE(ν) and BB(ν) SEDs, a recent analysis in Ritacco et al. (2022) detected such a difference in the Planck data. As discussed before, this detection would be a direct indication for the existence of polarized mixing, leading to a spectral phase rotation ϑ ν , that is (W β k ) E (W β k ) B , such that one expects to find a frequency dependence of the EE/BB ratio in sky observations. The EE/BB ratio can be expressed at first order as Note that this expression does not depend on the modified blackbody, and gives a pure ratio of moments. As such, looking for the EE/BB ratio will probe the existence of differences between the SEDs of E and B due to polarized mixing, independently of any choice of canonical SED to model dust at the voxel 12 To keep this link explicit, one could for example consider D SS = D EE + D BB . level. Variations of spectral parameters alone would distort identically E and B, leading to the same moments and leaving the EE/BB ratio constant. As such, (r E/B ν ) provides, at the powerspectra level, an observable equivalent to tan(ψ ν ) or tan(ϑ ν ) at the map level.
As in Mangilli et al. (2021) and Vacher et al. (2022a), one can consider Eq. (29) as two independent moment expansions for EE and BB and interpret the order 1 term as a leading-order correction to the spectral index. A scale-dependent pivot can be obtained by canceling the first order term and replacinḡ Hence, in the presence of polarized mixing, the leading order is different in EE and BB and it is impossible to find a common pivot canceling simultaneously the leading order for EE and BB, i.e.β EE β BB . In the perturbative regime, D 1,X , one can approximate Eq. (30) as As such, while the amplitude of the power-law term indicates the value of the EE/BB ratio at ν = ν 0 , its exponent 2∆β E/B = 2β EE − 2β BB provides an indication of how the pivot spectral indices of EE and BB are expected to differ. The moment term quantifies the difference between the auto-correlation of the order one spin-moments in EE and BB and can not be strictly equal to zero if the EE/BB ratio is spectral dependent.

The EB power spectrum
A similar calculation can be done for the cross EB spectra, leading to The zeroth order term is quantifying solely by the structure of the magnetized interstellar medium. From parity considerations, this term is expected to be very small (Planck Collaboration 2020; Clark et al. 2021). Even if the leading term in Eq (36) is null, a non-zero frequency dependent D EB can be generated by the two other terms, which follow from variations of dust emission properties.
Isolating the EB moments is not a trivial task (as, for example, the EB/EE quantity could be dominated by the EE distortions). The favorable option would be to correct analytically for the scale dependent pivot as Article number, page 5 of 13 A&A proofs: manuscript no. main assuming that the first term is indeed the leading order, which should always be true if the mean signal is in the perturbative regime. Here again, in the presence of polarized mixing, we will observe thatβ EB β EE β BB , such that the three polarized spectra will have a different effective SED. This highlights that observing a spectral dependence of the EE/BB ratio guarantees the existence of EB distortions at some level.
However, in observational conditions, we cannot compute analytically the pivot defined in Eq. (35), as we do not have access to the 3D distribution of spectral parameters and polarization angles. Therefore, in order to highlight the EB SED distortions one can choose anyβ EB and consider the ratio The amplitude of the variations of (r E×B ν ) will depend strongly on the choice ofβ EB : on the first hand, one can imagine to fitβ EB directly on the EB data (but as the EB Galactic signal is very low, this fit might dramatically increase the Galactic modelling uncertainty) and minimize the amplitude of the distortions. On the second hand, one can use a proxy forβ EB (for example from the high signal-to-noiseβ EE ) but this will result in enhancing the EB distortions if |β EB −β EE | 0. Still, since EE, BB and EB have a common physical origin, they should rather be treated together with a sharedβ in the spin-moment formalism.

The toy-model filament
In order to refine the toy-model of the infinite filament presented in Sec. 2.2, let's now follow Planck Collaboration (2016e,d) and consider an infinite filament in front of a polarized background, both having a MBB emission law with different spectral indices and polarization angles. The frequency dependence of the polarization angle ψ-field in the filament will arise naturally from the polarized mixing described in Sec. 2.1 (See Fig. 2). We choose ψ bg = 0 • and consider various cases for ψ fl (where the exponents bg and fl stands hereafter respectively for background and filament). From astrophysical considerations, one would expect filaments to be colder than the diffuse background, that is T fl < T bg and the temperatures being anti-correlated with the spectral indices, we expect β fl > β bg . Here again, for the sake of simplicity, we will not consider temperature variations, fixing β fl = 1.8, β bg = 1.5 andT = T fl = T bg = 20 K. We also use A fl = A bg = 1, assuming that the background and the filament share the same opacity and the same inclination of the Galactic magnetic field. In order to keep the analysis easily interpretable we also ignore the impact of the size and orientation of the filament. Changing these parameters is expected to change the relative amplitudes of the spectra but should not impact our conclusions regarding the moment expansion formalism and the impact of polarized mixing on the angular power-spectra.
The map is a 32 × 32 flat pixel grid on which the filament represents a 11 × 32 vertical rectangle (see Fig. 2). The filament is still assumed to be infinite, as the power-spectra computation assumes periodic boundary conditions. We treat this example numerically using the Namaster library (Alonso et al. 2019) to evaluate the polarized power-spectra D XX of the flat-sky maps in a single multipole bin containing the 15 first values of . The frequency range is chosen to be an array from 100 to 400 GHz with intervals of 10 GHz, spanning a frequency interval relevant for CMB missions, under which the effect of the spectral index variations is expected to be dominant over possible temperature variations (LiteBIRD Collaboration 2022). The reference frequency is chosen to be ν 0 = 400 GHz.

Single pixel analysis
We now try to evaluate how the geometrical and spectral aspects of the signal are intertwined to produce the resulting angular power spectra. In a first step, we focus on the benefits of the spin-moment approach in a single filament pixel.
We will first consider only the two frequencies of ν 1 = 100 GHz and ν 2 = 400 GHz and test how the results change under a variation of ψ fl in the range [−90 • , 90 • ]. The modulus |P ν | and phase ψ ν of the total polarization spinor in the filament are displayed in Fig. 3. On the left panel, we display the departures from the pivot modified blackbody of the total signal modulus in the filament at 100 GHz, |P 100 |/|ε P 100 (β pix ,T )| with β pix = (β fl + β bg )/2 = 1.65. These departures are well modeled by the spin-moment expansion up to order 2 that one can derive analytically using equations (7) and (8), for every value of ψ fl .
One witnesses that the modulus of the signal is getting smaller when ψ fl goes away from 0 • . This is due to a corresponding vanishing of the amplitude of W 0 = A fl e 2iψ fl + A bg e 2iψ bg , due to a progressive depolarization. Indeed, at ψ fl = 0 • , the phases are aligned and the complex amplitude reaches its maximal value W 0 = A bg + A fl = 2. Moving away from ψ fl = 0 • , the phases cancel each other down to W 0 = A bg e 2i0 + A fl e ±2iπ/2 = 0 (this is a pure geometrical "spin-2" effect, independent of the values of the spectral parameters). On the other hand, the first order moment increases when going away from ψ fl = 0 • , traducing an increase of the distortion amplitudes and the change of polarization angle with frequency ∆ψ = ψ 400 − ψ 100 , as showed on the right panel. The behavior of ∆ψ with ψ fl is well modeled by the complex pivot correction (Eq. (5) and (11)), except in the nonperturbative regime, when W β 1 becomes large compared to W 0 , as discussed in Vacher et al. (2022b). In this regime, a pivot correction is impossible and all the terms of the expansion must be kept. Note that the second moment W β 2 is still required to get a good model for |P ν | around ψ fl = 0 • , where it is greater than the first order (W β 2 > W β 1 0). Indeed, as already discussed in Sec. 2.1, it is impossible to always guarantee the hierarchy between the moments due to the complex nature of the expansion and its non-perturbative treatment of polarization angles. Overall, both the modulus and the phase of the complex signal in the pixel is fully predicted by the spin-moment expansion.

Power spectra analysis
In a second step, we study the behavior of the power-spectra, displayed in Fig. 4, for this same filament model. In order to compare both frequencies having orders of magnitudes of differences, we normalize all the spectra by the maximal value of at each frequency. In the case without distortions, when β fl = β bg = 1.5, all the polarized angular power-spectra display an identical behavior between the two frequencies as E-and B-modes share the same SED. Hence no moments are expected neither in P ν , nor in S ν . The overall behavior of the angular power-spectra with respect to the filament's angle displayed here is very similar to the magnetic misalignment phenomenon (see Fig. 2 of Clark et al. (2021)), considering the extra depolarization effect. At ψ fl = 0 • , the sum of MBBs in the filament is aligned with the MBB in the background. The E-modes are hence maximal and there are no B-mode or EB correlation. For ψ fl = ±45 • , B-modes are maximal but lower than the maximum of the E-modes, due to the progressive depolarization and corresponding amplitude loss discussed above. At ψ fl = ±90 • , the E-modes are expected to peak again, but they are not due to depolarization that makes the signal minimal, and B-modes are back to zero. The absolute value of EB is maximal when √ EE 2 + BB 2 is maximal. Considering now our example with β fl β bg , distortions appear, as indicated by the rotation of ψ between 100 and 400 GHz in Fig. 3. No matter what the value of ψ fl is, ψ will drift away from alignment between the two frequencies (accordingly to the Figure, with a positive angle for ψ fl > 0 and negative for ψ fl < 0), leading to an increase of the E-modes and a corresponding decrease of the B-modes from 100 to 400 GHz, such that one expects r E/B ν to decrease with frequency. The distortion is greater as ψ fl goes away from 0 • , which is in agreement with the values of W β 1 in Fig. 3. Distortions are also witnessed for the EB spectra, illustrating how the mixing of polarized signal can increase the amplitude of this parity violating spectra from one frequency to another. Changing the value of ψ bg will not change the above conclusions but will change the relative amplitudes of the EE, BB and EB angular power spectra. In order to remain concise, other cases with ψ bg = 45 • (B-modes maximum) and ψ bg = 22.5 • (equipartition of E-and B-modes) are displayed in Appendix A.
4.3. Predicting the spectral dependence of the power-spectra.
The spectral behavior of the angular power spectra discussed above can be predicted by the spin-moment expansion. From Eq. (8), one can build maps of the spin-moments, knowing the ψ and β distributions in each pixel and using now, as a common pivot, the mean spectral parameter over the whole map β = Aβ / A ∼ 1.55. It is then straightforward to compute the corresponding D W k,X W k,X for each pair of moments. To do this, we use again Namaster on the flat sky moment maps. On Fig. 5, the spin-moment spectra are inserted in Eqs. (29) and (34). The moment prediction up to order 3 is compared to the signal over the whole frequency range for the special case ψ bg = 30 • . These examples demonstrate that the expansion derived above is correct and that it is possible to infer the polarized power-spectra from the spin-moment maps, themselves derived from the spectral parameter and polarization angle distributions. In experimental conditions however, one can not access directly the dis- tributions of spectral parameters and polarization angles, making this derivation impossible. We show however that the spinmoments and their expansion at the power-spectra level provide robust models for an accurate characterization of the polarized signal, regardless of the distributions of β and ψ. A detailed study of how far one can go by proceeding the other way around i.e. inferring dust properties from the power-spectra and/or spin moment maps, is left for future work.
In order to assess the validity of the EE/BB ratio approximation in Eq. (32), we fit the EE/BB ratio using the lmfit software (Newville et al. 2016) with a weighting proportional to the signal itself. In Fig. 5, we observe a good agreement between the fit and the data points. The best-fit values of A E/B , 2∆β E/B and δ 11 can be found in Tab. 1. The values are compared to the ones predicted using the moment maps and computing the pivotsβ EE . Values are close but not equal, because Eq. (32) stops at order 1 and the fit compensates for the higher-order moments. Still, the expression provides a simple and intepretable model to characterize the spectral dependence of the EE/BB ratio. Finally, as discussed in Sec. 3.6, we can find a pivotβ EB in order to evaluate r E×B ν . Both fitting and analytical derivations allow to findβ EB ∼ 1.69 β . The ratio r E×B ν is displayed in Fig. 6, quantifying the amplitude of the residual distortions of higher orders. The variations are at the percent level. Once again, the prediction from moment expansion is overlapping with the signal, validating our methodology.

PySM 3 models
We now propose to illustrate the previously discussed phenomenon on the PySM 3 models (Thorne et al. 2017;Zonca et al. 2021) on the sphere using the healpy package (Górski et al. 2005). We are considering the four following models: -d0, built with polarization Q and U map templates from the Planck 2015 data at 353 GHz (Planck Collaboration 2016a), extrapolated at all frequencies using a single MBB with fixed β = 1.54 and T = 20 K over the sky. -d1, with the same Q and U map templates as d0, extrapolated to other frequencies with a MBB with a varying β and T between pixels across the sky. -d10, a refined version of d1, where the extrapolation is performed with a MBB with spectral parameters in each pixel coming from templates of the GNILC needlet-based analysis of Planck data, including a color correction and random fluctuations of β and T on small scales. -d12, built out of six overlapping MBBs, as detailed in Martínez-Solaeche et al. (2018). This is the only model considered here that has variations of the spectral parameters and polarization angles along the line of sight, as in the toymodel filament presented in Sec. 4.
We choose again the frequency interval ν ∈ {100, 400} GHz, with steps of 50 GHz replacing the value at 350GHz by the reference frequency of the models ν 0 = 353 GHz. Power spectra are computed again using Namaster in a single multipole bin from 0 = 2 to max = 200, at a healpy resolution of N side = 128. In order to observe a large patch of the sky while still avoiding the central Galactic region, we use the Planck GAL080 raw mask with f sky = 0.8 available on the Planck Legacy Archive. We subsequently perform a Namaster C2 apodization with a scale of 2 • . Both the E-and B-modes are purified during the spectra computations.
In principle, knowing the A, ψ, β and T templates of the PySM maps, it is possible to compute analytically the spectra expansion as we did in Sec. 4.3. This would however require to consider the temperature effects and the β − T correlations, expected to have a significant impact on the modeling, as discussed in Vacher et al. (2022a) and Sponseller & Kogut (2022).

The EE/BB ratio
First, let us focus on the EE/BB ratio. Results are displayed in the left panel of Fig. 7. As expected, no departure from constancy is observed for the EE/BB ratio in the case of d0. This is also the case for the d8 PySM model, even though the canonical SED is not represented by a MBB but by an adjusted version of the model proposed in Hensley & Draine (2017), with constant spectral parameters across the sky. A frequency dependent EE/BB ratio is expected only in the presence of polarized mixing, between or along line of sights. As already stressed out, this makes the EE/BB ratio a powerful probe of these variations, independently of the SED effectively used to describe locally the signal.
For d1 and d10, we find variations of few percents. Even if the SEDs in each pixel are non distorted MBBs, variations between line of sights are enough to produce a frequency dependence of the EE/BB power spectra. d12, containing both variations of the spectral properties along and between the lines of sights, displays stronger variations up to ∼ 15%.
As an indication, we realized a fit of Eq. (32) for each model. One can witness that the resulting curve appears to be in good agreement with the signal in all cases, ensuring that the perturbative expression proposed in Sec. 3.4 remains a good way to quantify the departures from constancy of EE/BB on realistic dust templates.
In all cases, the observed amplitudes of variations are expected to change widely depending on the sky fraction and the range of multipoles considered, as averages are made over different Galactic regions. The impact of such an effect on cosmological analysis is left for future work, but we expect it to be substantial for CMB B-mode analyses, as a few percent error on the dust component is significant for measuring e.g. r = 0.001 (Planck Collaboration 2020).

The EB spectra and cosmic birefringence
As discussed in Sec. 3.6, highlighting the EB distortions is not trivial. In order to do so, we first perform a fit ofβ EB andT EB directly on the signal and consider r E×B ν . The results are displayed on the right panel of Fig. 7. One can witness changes with frequency at the percent level for all model but d0, indicating the presence of distortions at a level that could be neglected for these sky models in contemporary birefringence analysis.
As discussed in Sec. 3.6, however, as the dust EB signal is currently very small compared to the measurement errors in real observational conditions (and will probably remain modest in the future), it is impossible to access these quantities directly as we did here, and one could instead be tempted to use the high signalto-noiseβ EE andT EE as proxys for the EB spectra. On the left panel of Fig. 8, we show r E×B ν using the best fits of the EE spectral parameters as a pivot. One can now witness the existence of spectral variations from few percents for d1, to ∼ 40% for d10 and up to a factor 2 for d12. As such, the choice of spectral parameters used to highlight the EB SED is extremely important and requires particular caution.
It is also relevant to consider the quantity Article number, page 9 of 13 which can provide a higher signal-to-noise estimator of the foreground EB signal Diego-Palazuelos et al. 2022a). A E×B is a scale dependent amplitude, frequency independent by construction. In order to quantify the deviations to this approximation, we will consider the ratio Results are presented in Fig. 8. Large departures from Eq. 38 for all the models but d0 are observed, away from ν 0 . According to the moment formalism presented above, the SEDs of EE, T B and T E present distinct distortions in the presence of polarized mixing, in turn different from that of EB, explaining why this approximation breaks with frequency. For an accurate modeling of the parity violating foreground signal, in order to probe the existence of cosmic birefringence, great care must therefore be taken with spectral distortions that may be induced by the polarized mixing.

Conclusion
In the present work, we discussed how the combination of multiple polarized signals having different spectral parameters and polarization angles (referred to as polarized mixing) leads unavoidably to a different spectral behavior for the polarized angular spectra EE, BB and EB, thus implying spectral dependence of the EE/BB ratio and non trivial distortions of the EB correlation. We showed how this phenomenon can be understood and tackled using the spin-moment expansion, deriving formally all the analytical expressions at order 1 in the case of Galactic dust modified black bodies with varying spectral indices, keeping in mind that this would be straightforward to generalize for any polarized SED (e.g. synchrotron) and at any order. We then discussed thoroughly the toy-model example of a dust-emitting filament in front of a background. A careful understanding of the geometrical and spectral properties of the signal in the filament itself in pixel space allowed us to explain the shape of the total polarization angular power spectra when changing the value of the polarization angle of the filament ψ fl , as well as the amplitude of the observed distortions between 100 and 400 GHz. Moreover, we showed how one could recover accurately the spectral dependence of the polarization angular power spectra from the spin-moment maps, validating our previous theoretical considerations. Finally we considered some of the PySM models on the sphere. We showed that these models contain intrinsically variations of the EE/BB ratio with frequency and distortions of the EB correlation, which amplitudes are expected to strongly depend on the sky fraction and multipole range considered. This allowed us to stress that seeking for a spectral dependence of EE/BB provide a probe of the existence of polarized mixing, and thus independently of the canonical SED used to model the signal. In these PySM models, simple assumptions about the frequency dependence of the dust EB signal used in CMB cosmic birefringence analysis break due to the polarized mixing.
Further studies need to be done in order to assess precisely the expected amplitude of both these effects on real sky data. Meanwhile, it will be necessary to quantify the impact of the assumptions made on the dust EE/BB ratio and EB SEDs on cosmology. Map-based component separation is sensitive to the variation of the foregrounds polarization angle with frequency, intertwined with the power-spectrum effects discussed in the present work. However, current B-mode only analyses at the power spectrum level are in principle immune to the potential variation of the dust EE/BB, as they model the BB SED independently of that of EE. Still, next-generation experiments that will probe the CMB reionization bump might consider simultaneously EE and BB to tackle the correlations between the tensor-to-scalar ratio r and the reionization optical depth τ and are therefore sensitive to the assumptions made about the dust EE/BB ratio variations. Regarding the dust EB correlation, as already stressed in Diego-Palazuelos et al. (2022b), distortions of the MBB SED could impact cosmic birefringence studies at some level, but as long as this signal remains undetected, using higher signal-to-noise spectra as proxies to EB has to be done with even more caution.
In any case, the 3D variation of the dust composition, physical conditions and the orientation of the Galactic magnetic field produce complex polarization effects in map space and at the angular power-spectrum level and they have to be ruled out or taken into account for precise B-mode and birefringence measurements.  Left: modulus of the total signal at 100 GHz normalized by the pivot MBB (red) and modulus of the analytical derivation from the spinmoment expansion up to second order (black dashed). The modulus of each term is displayed: order 0: |W 0 | (blue), order 1: |W β 1 ln (100/400) | (orange) and order 2: |0.5W β 2 ln (100/400) 2 | (green). Right: Difference of the polarization angles between the two frequencies (red), prediction from the complex ∆β correction 0.5 Im(W β 1 /W 0 ) ln (100/400) (blue) and from the spin-moment expansion up to second order (black dashed).