The Power Spectra of Polarized, Dusty Filaments

We next discuss the projection of a three-dimensional filament onto the plane of the sky. Many important quantities depend on the angle to the line of sight of (1) the long axis of the filament (θ_L) and (2) the magnetic field vector (θ_H).³ Another important quantity is the plane-of-sky projection of the angle between these vectors (ψ_LH), which controls the polarization angle and the amounts of E/B polarization present. We depict these angles in Figure 3. If the magnetic field direction aligns somewhat with the filament direction, as is the case in strong-field MHD, all these angles will be correlated.

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We assume that, on average, the filaments align with the local magnetic field. In the Appendix, we use simple geometry to compute the distribution of the magnetic field projection angle ${\theta }_{H}$ and relative orientation angle ψ_LH as functions of θ_L. We base the distribution on the assumption of a Gaussian distribution for the angle (θ_LH) between the filament and the magnetic field in three dimensions, characterized by the dispersion rms(θ_LH).

The field angle θ_H is correlated with ψ_LH, so our numerical procedure yields the tabulated joint distribution,

$$\begin{eqnarray}&&p({\psi }_{{LH}},{\theta }_{H}| {\theta }_{L}).\end{eqnarray} \tag{ 4 }$$

This distribution centers on aligned filaments (ψ_LH = 0°, θ_H = θ_L), and the distribution for ψ_LH broadens for filaments nearly along the line of sight. Its precise form is not vital for this discussion and is plotted in the Appendix in Figure A1.

On the other hand, the probability distribution for the line-of-sight angle of randomly oriented filaments is determined purely by geometry,

$$\begin{eqnarray}&&p({\theta }_{L})=\sin {\theta }_{L},\end{eqnarray} \tag{ 5 }$$

for ${\theta }_{L}\in [0^\circ ,180^\circ ]$.

These quantities relate immediately to the polarization. Although the dust polarization fraction depends on the microphysical details of the emission, it has a geometric dependence similar to ${f}_{\mathrm{pol}}\propto {\sin }^{2}{\theta }_{H}$ (Fiege & Pudritz 2000). Meanwhile, the polarization angle for a filament projected along the x-axis is ${\psi }_{\mathrm{pol}}={\psi }_{{LH}}+90^\circ$.

We model the filament as a prolate spheroid, and label the major axis as L_a and the minor axis as L_b. The axis ratio is thus $\epsilon ={L}_{b}/{L}_{a}\lt 1$. The column density (and therefore the surface brightness and ultimately the observed temperature perturbation) is proportional to the density and the line-of-sight distance through the filament, and so (approximately)

$$\begin{eqnarray}\begin{array}{rcl}{T}_{0} & \propto & {\rho }_{0}{\left({L}_{a}^{2}{\cos }^{2}{\theta }_{L}+{L}_{b}^{2}{\sin }^{2}{\theta }_{L}\right)}^{1/2}\\ & \propto & {\rho }_{0}{L}_{a}{\left({\cos }^{2}{\theta }_{L}+{\epsilon }^{2}{\sin }^{2}{\theta }_{L}\right)}^{1/2},\end{array}\end{eqnarray} \tag{ 6 }$$

where ρ₀ is a characteristic density for the filament. So all else being equal, a filament that lies along the line of sight will have the greatest column density and the brightest temperature signal. On the other hand, ${f}_{\mathrm{pol}}\propto {\sin }^{2}{\theta }_{H}$, so if the magnetic field lies along the line of sight, there is no polarization. The polarization fraction is maximum when the magnetic field is perpendicular to the line of sight.

Figure 4 describes the relationship between the line-of-sight angle, column density, and polarization fraction. It also shows that since the polarized amplitude depends on the product of these two, the filaments with the brightest polarization are inclined, but not perpendicular, to the line of sight. The polarization maximum depends on axis ratio through its impact on the column density. Nearly round filaments have the polarization maximum when oriented near 90° to the line of sight, while in the limit of thin filaments ($\epsilon \to 0$) the orientation of polarization maximum approaches θ_L = 45° for perfect magnetic field alignment. If there is significant misalignment of the magnetic field and filament directions, the situation can become more complicated, depending on the particular combination of axis ratio and misalignment dispersion. In such cases, filaments along the line of sight can have significant polarization. Still, the typical line-of-sight orientation angle for maximum polarization, averaging over the magnetic field directions, is around θ_L = 45°.

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We compute the filament's projected angular sizes along its two axes as if it were a cylinder. These depend on its distance R and are

$$\begin{eqnarray}\begin{array}{rcl}{{\rm{\Theta }}}_{a} & = & {\left({L}_{a}^{2}{\sin }^{2}{\theta }_{L}+{L}_{b}^{2}{\cos }^{2}{\theta }_{L}\right)}^{1/2}/R\\ & = & {\left({\sin }^{2}{\theta }_{L}+{\epsilon }^{2}{\cos }^{2}{\theta }_{L}\right)}^{1/2}{L}_{a}/R\\ {{\rm{\Theta }}}_{b} & = & {L}_{b}/R=\epsilon {L}_{a}/R.\end{array}\end{eqnarray} \tag{ 7 }$$

Thus the projected axis ratio is

$$\begin{eqnarray}&&{\epsilon }_{{\rm{\Theta }}}={{\rm{\Theta }}}_{b}{/{\rm{\Theta }}}_{a}=\displaystyle \frac{\epsilon }{{\left({\sin }^{2}{\theta }_{L}+{\epsilon }^{2}{\cos }^{2}{\theta }_{L}\right)}^{1/2}}\end{eqnarray} \tag{ 8 }$$

which goes to unity for filaments along the line of sight, and to the true value () for filaments perpendicular to the line of sight.

For several terms in our power spectrum calculation, we need the Fourier transform of the projected filament profile:

$$\begin{eqnarray}&&f({\boldsymbol{\ell }})=\int {d}^{2}x\ f({\boldsymbol{x}})\exp (-i{\boldsymbol{\ell }}\cdot {\boldsymbol{x}}).\end{eqnarray} \tag{ 9 }$$

Rather than project rays through a three-dimensional model to obtain the filament profile, we make a simplifying assumption for computational efficiency. From the size and orientation of a filament, we take the angular dimensions and compute under the assumption that the profile is a distortion from an axisymmetric function g:

$$\begin{eqnarray}\begin{array}{rcl}f(x,y) & = & g(x/{{\rm{\Theta }}}_{a},y/{{\rm{\Theta }}}_{b})\\ & = & g({x}^{* },{y}^{* })\quad =g(r)\end{array}\end{eqnarray} \tag{ 10 }$$

where Θ_a, Θ_b are the semimajor and semiminor axes of the elliptical distortion. As we stated, by convention and without loss of generality, we orient the long axis of the filament along the x-axis.

Then the transform of f is simply related to the transform of g:

$$\begin{eqnarray}\begin{array}{rcl}f({\boldsymbol{\ell }}) & = & {{\rm{\Theta }}}_{a}{{\rm{\Theta }}}_{b}\displaystyle \int {d}^{2}{x}^{* }\ g({{\boldsymbol{x}}}^{* })\exp (-i({{\rm{\Theta }}}_{a}{{\ell }}_{x}{x}^{* }+{{\rm{\Theta }}}_{b}{{\ell }}_{y}{y}^{* }))\\ & = & {{\rm{\Theta }}}_{a}{{\rm{\Theta }}}_{b}\,g({{\ell }}^{* })\end{array}\end{eqnarray} \tag{ 11 }$$

where ${{\ell }}^{* }({\boldsymbol{\ell }})={\left({{\rm{\Theta }}}_{a}^{2}{{\ell }}_{x}^{2}+{{\rm{\Theta }}}_{b}^{2}{{\ell }}_{y}^{2}\right)}^{1/2}$ and

$$\begin{eqnarray}\begin{array}{rcl}g({{\ell }}^{* }) & = & \displaystyle \int {d}^{2}{x}^{* }\ g({{\boldsymbol{x}}}^{* })\exp (i{{\boldsymbol{\ell }}}^{* }\cdot {{\boldsymbol{x}}}^{* })\\ & = & 2\pi \displaystyle \int {dr}\ r\ g(r){J}_{0}({{\ell }}^{* }r).\end{array}\end{eqnarray} \tag{ 12 }$$

The input profile $g({{\boldsymbol{x}}}^{* })$ is real and even, and so the Fourier transform is too. The power spectra we find are not very sensitive to the profile that we use. In this work we have used an exponential for the basic filament profile ($g(r)=\exp (-r)$), but we have checked our best-fitting power spectrum model with a Gaussian profile ($g(r)=\exp (-{r}^{2}/2)$) and a Plummer profile ($g{(r)=(1+{r}^{2})}^{-5/2}$), and find the same results.

For a parameterized set of filament properties,

$$\begin{eqnarray*}&&\alpha =({L}_{a},{L}_{b},{\psi }_{{LH}},{\theta }_{L},{\theta }_{H},R,...),\end{eqnarray*}$$

we can write the number density distribution n(α), so that the average number of filaments in a realization of the sky is

$$\begin{eqnarray}&&\langle N\rangle =\int d{\rm{\Omega }}\ d\alpha \ n(\alpha )\end{eqnarray} \tag{ 13 }$$

where the integral is over

$$\begin{eqnarray}&&d\alpha ={{dL}}_{a}{{dL}}_{b}d{\psi }_{{LH}}d{\theta }_{H}d{\theta }_{L}{dR}.\end{eqnarray} \tag{ 14 }$$

Expressed another way, $n(\alpha )=\langle N\rangle p(\alpha )$, where the normalized probability distribution of the filament population is

$$\begin{eqnarray}&&p(\alpha )=p({L}_{a},{L}_{b})p({\psi }_{{LH}},{\theta }_{H}| {\theta }_{L})p({\theta }_{L})p(R).\end{eqnarray} \tag{ 15 }$$

This integral over the population is at least six-dimensional. For a screen at a distance R, it is a five-dimensional integral. Since the angular power spectrum for foregrounds is a power law, if we can reproduce it on a single screen, putting that screen at different distances will maintain the same power spectrum. If we further fix the physical aspect ratio of the filaments, it is a four-dimensional integral, over L_a, ψ_LH, θ_L, θ_H. (The projected aspect ratio will still vary with the line-of-sight angle θ_L.)

The contributions to the power spectrum from filaments correlated with themselves are

$$\begin{eqnarray}\begin{array}{rcl}{C}_{{\ell }}^{{TT}} & = & \displaystyle \frac{1}{2\pi }\displaystyle \int d{\phi }_{{\ell }}\displaystyle \int d\alpha \ n(\alpha )\ | T({\boldsymbol{\ell }},\alpha ){| }^{2},\\ {C}_{{\ell }}^{{EE}} & = & \displaystyle \frac{1}{2\pi }\displaystyle \int d{\phi }_{{\ell }}\displaystyle \int d\alpha \ n(\alpha )\ | E({\boldsymbol{\ell }},\alpha ){| }^{2},\\ {C}_{{\ell }}^{{BB}} & = & \displaystyle \frac{1}{2\pi }\displaystyle \int d{\phi }_{{\ell }}\displaystyle \int d\alpha \ n(\alpha )\ | B({\boldsymbol{\ell }},\alpha ){| }^{2},\\ {C}_{{\ell }}^{{TE}} & = & \displaystyle \frac{1}{2\pi }\displaystyle \int d{\phi }_{{\ell }}\displaystyle \int d\alpha \ n(\alpha )\ T({\boldsymbol{\ell }},\alpha )E{\left({\boldsymbol{\ell }},\alpha \right)}^{* }.\end{array}\end{eqnarray} \tag{ 16 }$$

Similar expressions hold for the other cross-correlations, but these vanish if the orientations of the filaments are random. These computations of power spectra are directly analogous to the one-halo term in the cosmological halo model (Seljak 2000).

We can relate the slope of a power-law spectrum to scaling relations for parameters in the filament profiles. This allows us to place constraints on the distribution of filament sizes and the scaling of other parameters. For a generic parameter α₀, if the filament's contribution to the power spectrum scales as

$$\begin{eqnarray}&&{C}_{{\ell }}\propto \int d{\alpha }_{0}\ n({\alpha }_{0})\times {\alpha }_{0}^{q}\,F({\alpha }_{0}^{r}\,{\ell })\end{eqnarray} \tag{ 17 }$$

for any function F, and furthermore if the weighting distribution for the parameter is a power law, $n({\alpha }_{0})\propto {\alpha }_{0}^{p}$, then we can rescale the integration with a straightforward substitution, $u={\alpha }_{0}{{\ell }}^{1/r}$:

$$\begin{eqnarray}&&\begin{array}{l}{C}_{{\ell }}\propto {{\ell }}^{-(p+q+1)/r}\\ \times \,\displaystyle \int d({\alpha }_{0}{{\ell }}^{1/r})\ {\left({\alpha }_{0}{{\ell }}^{1/r}\right)}^{p}\times {\left({\alpha }_{0}{{\ell }}^{1/r}\right)}^{q}\,F({\left({\alpha }_{0}{{\ell }}^{1/r}\right)}^{r})\\ \propto \,{{\ell }}^{-(p+q+1)/r}\times \displaystyle \int {du}\ {u}^{p}{u}^{q}\,F({u}^{r}).\end{array}\end{eqnarray} \tag{ 18 }$$

The integral no longer has any multipole dependence and evaluates to some constant value, whatever the details of F. Thus we are left with a power-law power spectrum with

$$\begin{eqnarray}&&{C}_{{\ell }}\propto {{\ell }}^{-(p+q+1)/r}.\end{eqnarray} \tag{ 19 }$$

This argument holds not just for filaments, but for any signal with a power-law power spectrum that is built from a set of objects that are similarly related to each other, so long as they are weighted by power-law scalings and distributions. So if we observe a power-law spectrum with ${C}_{{\ell }}\propto {{\ell }}^{s}$, it implies that the parameter distribution's index is $p=-{rs}-q-1$, regardless of the objects' profiles.

We walk through this scaling argument for a simple (and unrealistic) case—with plane-of-sky filaments with identical surface brightnesses (T₀ is the same for all filaments) and a constant projected axis ratio (Θ_b = Θ_a)—and analyze the distribution for Θ_a, the angular size of filaments. For the filament Fourier transform, we have $f({\boldsymbol{\ell }})\propto {{\rm{\Theta }}}_{a}^{2}g({{\boldsymbol{\ell }}}^{* })$ with ${{\boldsymbol{\ell }}}^{* }\propto {{\rm{\Theta }}}_{a}$. The contribution to the power spectrum is proportional to f², so comparing the scaling for angular size parameter Θ_a to Equation (17), we find q = 4 and r = 1. In the polarization case, to reproduce ${C}_{{\ell }}\propto {{\ell }}^{-2.4}$ (meaning $s=-2.4$), the number density distribution of such objects on the sky must approximately scale like $n({{\rm{\Theta }}}_{a})\propto {{\rm{\Theta }}}_{a}^{p}$ where $p=2.4-4-1=-2.6$. Indeed this yields the desired slope of the power spectrum when calculated in our model.

In a more realistic case, with three-dimensional filaments, we can make a similar argument to deduce the distribution of filament lengths. Surface brightness depends on column density, which is proportional to length (after integrating out any distribution of axis ratios—compare Equation (6)—and assuming a density normalization independent of length). The solid angle scales like length squared. After squaring those three powers during the computation of the power spectrum, the overall scaling is q = 6. The multipole ℓ scaling should also go like length, so r = 1, the same as the plane-of-sky case above. So with no other dependence on length, we should have distribution of lengths $n{({L}_{a})\propto ({L}_{a})}^{p}$ where $p=2.4-6-1=-4.6$.⁴

We have verified that this distribution produces the proper slope in Figure 5. All the temperature and polarization spectra have the specified slope in common. The complications of the modeling of the three-dimensional orientation are not important to the slope, only the weighting and distribution of filament size.

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Other than the slope, there are no clear features in the Planck-measured spectra. We note that features in the distribution of filament sizes would break the power-law behavior of the resulting spectra. For example, if we impose a maximum filament size (semi-transparent lines in Figure 5), it causes the low-ℓ behavior of C_ℓ to deviate: at scales much larger than the filament, the temperature spectrum adopts the flat, white, Poisson spectrum of point sources. The EE and BB spectra flatten the same way, and on scales large compared to the filaments, the aspect ratios of the filaments become unimportant and the powers in EE and BB equalize. The TE cross-correlation falls off at large scales, possibly because the positive and negative contributions to E are being averaged over. If, on the other hand, we impose a minimum filament size, it causes the high-ℓ behavior to deviate: the spectrum will drop off with increasing ℓ, where there is no more contribution to the power.

In yet more realistic cases, we can make the slopes of all the temperature and polarization spectra differ. For example, this can happen if there is another effect that changes the size scaling of the polarization relative to the temperature. For example, to make polarization slope shallower than the temperature slope, one could make smaller objects more polarized than large ones, or the aspect ratio of filaments could change as a function of size.

Considering the Planck data, it is not immediately clear what conclusion to draw. Planck Collaboration et al. (2018) quote EE, BB, TE slopes for the dust foreground, but not the TT slope. Using our own tools, we have computed the TT power spectrum based on the Planck data, and find a TT slope that is about −2.6, somewhat steeper than the polarization spectra at −2.4 to −2.5. Like Planck Collaboration et al. (2018), for the mask we used the LR71 polarization mask supplemented with a point-source mask (based on intensity maps), resulting in a mask with effective f_sky ≃ 0.6. We can approximately reproduce a temperature slope of −2.6 and a polarization slope of −2.4 with $n({L}_{a})\propto {L}_{a}^{-4.4}$ and ${f}_{\mathrm{pol}}\propto {L}_{a}^{-0.08}$. This argues that in the unmasked region, the polarization is higher in smaller filaments.

However, this conclusion may not be correct because it depends sensitively on the mask. We reasoned that small filaments, oriented along the line of sight, might look like a point source and be included in the masked area. These end-on filaments could also have low polarization (note Figure 4), and excluding them might not have much effect on the polarization results. Thus for comparison, we recomputed the TT spectrum with different masks. When we use only the polarization LR71 mask without removing the additional point sources from the intensity map, we get a shallower TT spectrum with slope −2.5. When we use a mask that keeps the same large-scale features but does not mask any point sources (Planck's publicly available GAL70 mask) we find a TT slope of −2.1, notably shallower than the polarization spectra. The polarization spectra change somewhat between these masks, but the changes in the polarization slopes are small compared to the change in the TT slopes. Some of the masked sources are extragalactic, so this slope with all point sources unmasked is probably too shallow to describe the ISM component, but can serve as a bound. The upshot is that we are not certain whether the spectrum for all filaments is steeper or shallower in temperature than polarization, and so it is difficult to draw conclusions on the size dependence of the polarization fraction.

Another feature of the Planck data is the differing slopes in E and B. We can reproduce this feature by varying the aspect ratio as a function of filament size. For the LR71 mask in Planck Collaboration et al. (2018), the slopes for (BB, EE, TE) are roughly $(-2.5,-2.4,-2.5)$ and we found a TT slope of −2.6. So BB is steeper than EE, which should happen if smaller filaments are proportionally thinner than longer ones. Modifying the aspect ratio in this way also affects the TT slope, breaking the simple relation that we saw earlier in this section. By trial and error, we found that this set of slopes are approximately reproduced with the following parameter dependence: $\epsilon \propto {L}_{a}^{0.1}$, $n({L}_{a})\propto {L}_{a}^{-4.45}$, and ${f}_{\mathrm{pol}}\propto {L}_{a}^{-0.1}$. Here we are simply exploring what is possible, but the relationship between the measured slopes and these filament parameters should be made more systematic and quantitative.

In light of these complications and uncertainties, in what follows we keep a common slope of −2.4 for all the temperature and polarization components while we explore their other parameter dependences.

The aspect ratio of the filaments is the major factor determining the ratio of B-mode power to E-mode power. In Figure 6, we plot the power ratio against the aspect ratio for varying degrees of filament–magnetic field misalignment. To reproduce the Planck-observed ratio of ∼0.5, filaments need to have an aspect ratio slightly less than 0.26, so filaments must be slightly less than four times longer than they are wide. If the model deviates too much from this ratio, the required magnetic field misalignment is made so large that the model has trouble fitting the TE correlation.

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It is difficult to compare this result quantitatively to the aspect ratios of observed filaments from the literature without making a detailed accounting of the filament selection function. Projection effects will also tend to lower observed aspect ratios. The stacked filaments in Figure 7 of Planck Collaboration et al. (2016b) appear to have axis ratios not so far from what we are finding here. The filaments identified by the rolling Hough transformation in Clark et al. (2014) on H i maps tend to be longer and thinner than this.

In the context of the filament model, we find that the level of the TE cross-correlation implies that filaments cannot be precisely aligned to their local magnetic field direction. The correlation coefficient is defined as

$$\begin{eqnarray}&&{r}_{{\ell }}^{{TE}}={C}_{{\ell }}^{{TE}}/\sqrt{{C}_{{\ell }}^{{EE}}{C}_{{\ell }}^{{TT}}},\end{eqnarray} \tag{ 20 }$$

and perfect alignment of the filaments and the fields causes far too much TE correlation compared to the Planck observations.

The Planck dust data show ${r}_{{\ell }}^{{TE}}\approx 0.35$ with little scale dependence (Planck Collaboration et al. 2018). Figure 7 shows that to match this, the field misalignment angle θ_LH must have an rms dispersion of nearly 50°, while maintaining the axis ratio  ≈ 0.26 needed to reproduce the BB/EE power ratio. If the misalignment dispersion is independent of filament size, as in our modeling, it causes no scale dependence: ${r}_{{\ell }}^{{TE}}$ is constant.

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Projection effects cause the distribution of the projected angle ψ_LH to have a positive kurtosis (see Appendix, Figure A3), and so we can describe its dispersion in a few ways. For the case with rms(θ_LH) = 50°, 68% of the probability is bounded by $| {\psi }_{{LH}}| \lt 45^\circ$. Alternatively, ${[\mathrm{Var}({\psi }_{{LH}})]}^{1/2}=48^\circ$. For comparison, Planck Collaboration et al. (2016b) fit a Gaussian with 19° dispersion (1σ) to the projected field–projected filament histogram of relative orientations for the filaments they found. Again, this comparison is not direct because of selection effects. Their Hessian-based selection of filaments would disfavor those with small projected aspect ratios (close to the line of sight), and such filaments can have the largest differences in the projected orientation.

The overall level of ${C}_{{\ell }}^{{TE}}$ (and the polarization spectra) depends on the polarization fraction. The relative power ratio has a dependence according to

$$\begin{eqnarray}&&{C}_{{\ell }}^{{TE}}/{C}_{{\ell }}^{{EE}}\propto \langle {f}_{\mathrm{pol}}\rangle /\langle {f}_{\mathrm{pol}}^{2}\rangle ,\end{eqnarray} \tag{ 21 }$$

while for the cross-correlation it is

$$\begin{eqnarray}&&{r}_{{\ell }}^{{TE}}\propto \langle {f}_{\mathrm{pol}}\rangle /\langle {f}_{\mathrm{pol}}^{2}{\rangle }^{1/2}.\end{eqnarray} \tag{ 22 }$$

Thus a purely multiplicative rescaling of the polarization fraction affects the ratios of the powers in ${TT}/{TE}/{EE}$ but not the correlation coefficient r^TE.

To reproduce the Planck-measured ratio ${C}_{{\ell }}^{{TE}}/{C}_{{\ell }}^{{EE}}\sim 2.7$ (in our case that already fits ${C}_{{\ell }}^{{BB}}/{C}_{{\ell }}^{{EE}}$ and ${r}_{{\ell }}^{{TE}}$) requires ${f}_{\mathrm{pol}}=0.15\,{\sin }^{2}{\theta }_{H}$. We have only modeled the amplitude of polarization fraction and the geometric dependence on the magnetic field orientation, but in addition, the polarization fraction depends on grain geometry and small-scale turbulence (Fiege & Pudritz 2000), and filaments need not in reality all have the same intrinsic polarization fraction.

Our other tests have shown that the TE correlations differ in their sensitivity to intrinsic dispersion in the polarization fraction. For example, the correlation r^TE is not very sensitive to the maximum polarization fraction, but the power ratio is very sensitive to it: decreasing the maximum polarization fraction decreases ${C}_{{\ell }}^{{TE}}$ but decreases the denominator ${C}_{{\ell }}^{{EE}}$ more, and so raises the ratio.

One surprising finding in the dust spectra of Planck Collaboration et al. (2018) is a non-zero TB correlation, with ${r}_{{\ell }}^{{TB}}\approx 0.05$. Like r^TE, the observed r^TB correlation has little scale dependence (up to multipoles of several hundred).

Because non-zero TB and EB are parity-violating correlations, our model cannot reproduce them for randomly oriented filaments. To get a positive TB correlation we would need to favor polarization angles in the range ${\psi }_{\mathrm{pol}}\in [90^\circ ,180^\circ ]$ relative to the filament direction (compare Figure 2). Equivalently, this corresponds to projected field angles in the range ${\psi }_{{LH}}\in [0^\circ ,90^\circ ]$. Such an effect may be due to some large-scale feature in the Galaxy's magnetic field or differential gas flow (e.g., Planck Collaboration et al. 2018; Bracco et al. 2019).

We can determine how far away from random this correlation is by artificially weighting the distribution of the projected misalignment angles, favoring the ψ_LH > 0 portion of the distribution of $p({\psi }_{{LH}},{\theta }_{H}| {\theta }_{L})$ over the ψ_LH < 0 portion, while keeping the same functional form. We find that we can approximately reproduce the Planck-measured TB correlation by giving the preferred ψ_LH directions about 55% of the total weight, rather than the 50% that comes naturally from randomly oriented filaments. Similar to the r^TE correlation (also set by field–filament misalignment), this effect is not scale-dependent, and so r^TB is constant to high ℓ in this model.

Our modeling comes from the one-halo term only, and shows that the TB correlation can be explained if filament orientations in projection are slightly twisted counterclockwise from the projected local magnetic field. Our model does not address the structure of that underlying field, but we may speculate that some differential, shearing hydrodynamic forcing could preferentially twist the filaments, according to our point of view, from the global mean field direction of the Milky Way.

Bracco et al. (2019) argue that the observed TE and TB correlations may be features of the large-scale structure of the Galactic magnetic field. They show that a helical component can create such correlations, but in their modeling the correlations show a strong scale dependence, with ${r}_{{\ell }}^{{TE},{TB}}$ falling substantially already by ${\ell }=22$. The Planck spectra have much flatter ${r}_{{\ell }}^{{TE},{TB}}$ correlations, consistent with the filament modeling here.

Planck did not detect an EB correlation, but since E and B both have a factor of the polarization fraction, we would naturally expect this correlation to be smaller. It may be there, hidden in the noise. In the presence of a positive TB correlation, in the context of the filament modeling, we would expect a positive EB correlation too (including at high ℓ), and it should be a target for future experiments. Both TB and EB dust correlations can potentially interfere with sky-calibration of the polarization angles of CMB instruments (Abitbol et al. 2016) or with CMB lensing reconstruction (e.g., Fantaye et al. 2012; Challinor et al. 2018).