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Three-dimensional dust, grain-size physics, and the sky-projected magnetic field as constraints on polarized-dust frequency decorrelation for CMB $B$-modes

“Decorrelation of the spectral index” is used here in its operational CMB sense: frequency decorrelation of polarized thermal dust. Stokes $Q$ and $U$ at one microwave frequency are not a single scalar multiple of the maps at another frequency, because the dust spectral parameters $\beta_d$ and $T_d$, and the polarization angle $\psi$, vary both across the sky and along each line of sight (LOS). The standard statistic is the BB correlation ratio

which BICEP/Keck denote $\Delta_d$. A single modified blackbody (MBB) template at 353 GHz, scaled by one $(\beta_d,T_d)$ pair, assumes $R=1$. Any LOS or spatial mixture that makes $R<1$ leaves a residual that is not a CMB $B$-mode and is not removed by that scaling.

The “3D maps of the thermal dust emission” that actually exist are 3D extinction-density cubes from stellar photometry and Gaia parallaxes, together with 3D dust-temperature cubes obtained by fitting Planck and IRAS intensity on top of those cubes. No direct 3D millimetre emission map exists. Converting extinction density into an emission model is itself a physical step: it requires the grain size distribution and composition, which set the millimetre opacity per $A_V$, the equilibrium temperature, and which grains are aligned. The sky-projected (plane-of-sky) magnetic field, traced by starlight polarization tomography and, with caveats, by H I morphology, supplies the missing per-cloud angles. The three ingredients do not remove dust. They constrain the shape of the frequency extrapolation and turn an unmodeled bias into a marginalized, prior-informed uncertainty.

1. The problem in numbers

Current $B$-mode pipelines treat polarized dust as a single MBB per pixel or per region,

and use the Planck 353 GHz $Q,U$ maps as a spatial template that is extrapolated to 90–220 GHz. BICEP/Keck’s baseline model is of this form, with $T_d$ fixed at 19.6 K and $\beta_d$ fitted or given a Gaussian prior (BICEP/Keck XIII 2021, hereafter BK18). In the BICEP/Keck field the dust BB amplitude at 353 GHz and $\ell=80$ is $A_d\simeq 4.4\,\mu\mathrm{K}^2$ (BK18; $4.4^{+0.8}_{-0.7}\,\mu\mathrm{K}^2$), historically in the 4–5 $\mu\mathrm{K}^2$ range. For $\beta_d=1.6$ and $T_d=19.6\,\mathrm{K}$, the MBB conversion from 353 to 150 GHz in thermodynamic units is an amplitude factor $\simeq 0.045$, so the dust BB power at 150 GHz is $\simeq 9\times 10^{-3}\,\mu\mathrm{K}^2$. A primordial tensor signal with $r=10^{-3}$ has $D_\ell^{BB}\simeq 7\times 10^{-5}\,\mu\mathrm{K}^2$ at $\ell=80$. Dust at 150 GHz therefore exceeds the target by two orders of magnitude; the cleaning must be accurate at the percent-of-a-percent level in power.

Planck 2018 XI found no evidence for frequency decorrelation of polarized dust and quoted $\beta_d^P=1.53\pm 0.02$ at $T_d=19.6\,\mathrm{K}$, with $\beta_d^P-\beta_d^I=0.05\pm 0.03$. Their limit on $R_\ell^{BB}(217\times 353)$ was stated to suffice for $r\simeq 0.01$ at the recombination peak if it applies to the small patches of sub-orbital experiments, but not for the next decade’s target. Planck Int. L (2017) had reported decorrelation that Sheehy & Slosar (2018) and Planck 2018 XI showed to be noise-biased. BK18 find $\Delta_d$ consistent with 1 ($\Delta_d>0.98$ at 68% C.L. in their field). SPIDER (Ade et al. 2025) likewise reports no significant LOS polarization decorrelation over their full region, while finding spatial variation of $\beta_d$ between sub-regions at $\simeq 3.9\sigma$. What is detected is LOS angle rotation between 217 and 353 GHz in sightlines selected for magnetically misaligned H I clouds (Pelgrims et al. 2021), and hints of SED complexity in Ritacco et al. (2023). Decorrelation is therefore a demonstrated physical mechanism and a demonstrated Planck-scale effect in targeted sightlines; it is not an established, sky-averaged detection at CMB frequencies.

2. Polarized emission along a line of sight

A single cloud with uniform $\beta_d$, $T_d$ and field orientation emits

The optically thin Stokes integrals, in the notation of Lee & Draine (1985) and Wardle & Königl (1990) as used by Planck Collaboration Int. XIX (2015), are \begin{align} I_\nu&=\int S_\nu(s)\bigl[1-p_0\bigl(\cos^2\gamma(s)-\tfrac{2}{3}\bigr)\bigr],ds,\ Q_\nu&=p_0\int S_\nu(s),\cos^2\gamma(s),\cos 2\psi(s),ds,\ U_\nu&=p_0\int S_\nu(s),\cos^2\gamma(s),\sin 2\psi(s),ds. \end{align} Here $\gamma$ is the angle between $\mathbf{B}$ and the plane of the sky, so $\cos\gamma=B_\mathrm{POS}/B$; $\psi$ is the position angle of the emission polarization, which is perpendicular to the sky-projected field $\mathbf{B}_\mathrm{POS}$ (starlight polarization is parallel to $\mathbf{B}_\mathrm{POS}$); and $p_0$ is the intrinsic polarization fraction of the grain population, set by grain shape, the aligned mass fraction, and the Rayleigh reduction factor. The observed polarization fraction after LOS averaging satisfies $p\propto p_0\cos^2\gamma$, further reduced by field tangling. Planck 2018 XII report $p_\mathrm{max}\simeq 22\%$ at 353 GHz.

Sign convention: the equations above are written in the IAU Galactic convention, in which $\psi=\frac12\mathrm{arg}(Q+iU)$ increases from Galactic north toward Galactic east. The COSMO convention used in HEALPix CMB maps flips the sign of $U$ and therefore of $\psi$. All angles quoted below are IAU.

A sum of MBBs is not an MBB. Expanding the intensity about a pivot $(\bar\beta,\nu_0)$ gives the moment series of Chluba, Hill & Abitbol (2017),

with $\omega_n$ the emission-weighted moments of $\beta-\bar\beta$. The spin-2 generalization (Vacher et al. 2023a, A&A 669, A5) is

where the complex moments are

To first order the polarization angle rotates as

Temperature mixing produces an analogous distortion. The logarithmic slope of the Planck function is

At 150 GHz this slope is 1.741 at 15 K and 1.849 at 25 K, a contrast equivalent to $\Delta\beta\simeq 0.11$. Between 150 and 353 GHz, a 1 K shift around $T_d\simeq 20\,\mathrm{K}$ is worth $\Delta\beta\simeq 0.017$–0.019. Temperature variation is therefore not a higher-order correction in the Rayleigh–Jeans tail (Kogut & Fixsen 2016). Polarized emission is weighted toward larger, cooler, aligned grains, so $T_P\neq T_I$ and $\beta_P\neq\beta_I$ even for a single cloud.

3. Line-of-sight angle rotation: a worked example

Two clouds with complex polarized amplitudes $\mathcal{P}_i(\nu)=p_i S_i(\nu)\cos^2\gamma_i\,e^{2i\psi_i}$ combine to a frequency-dependent angle. With $r(\nu)=|\mathcal{P}_2|/|\mathcal{P}_1|$ and $\Delta\psi=\psi_2-\psi_1$,

and

Take $\beta_1=1.45$, $\beta_2=1.65$, $T_1=T_2=20\,\mathrm{K}$, $r(353\,\mathrm{GHz})=0.5$, and $\Delta\psi=45^\circ$. Equal temperatures drop the Planck-function term, so $r(\nu)=r(\nu_0)(\nu/\nu_0)^{\Delta\beta}$ and

Because $2\Delta\psi=90^\circ$, $\tan[2(\psi-\psi_1)]=r$, hence $\psi(353)-\psi_1=\frac12\arctan(0.5)\simeq 13.28^\circ$ and $\psi(150)-\psi_1\simeq 11.42^\circ$. The angle rotates by $\delta\psi\simeq 1.86^\circ$ between 353 and 150 GHz. The corresponding correlation loss on that sightline is

Geometry alone also makes the polarized SED differ from the intensity SED: $|1+r\,e^{2i\Delta\psi}|$ is frequency dependent, so $\beta_P\neq\beta_I$ does not by itself prove different grains (Tassis & Pavlidou 2015; Poh & Dodelson 2017). Multiple clouds with aligned fields produce no angle rotation. Decorrelation requires both SED differences and differences in $\psi$ or $p$; the number of clouds $N_c$ is not a predictor by itself.

4. Quantifying $R$ and the requirement on $1-R$

If $\beta$ varies spatially as a Gaussian of width $\sigma_\beta$, uncorrelated with amplitude, the correlation ratio follows in three lines. Write the scaling as $(\nu/\nu_0)^\beta=(\nu/\nu_0)^{\bar\beta}\exp[(\beta-\bar\beta)\ln(\nu/\nu_0)]$. The Gaussian characteristic function then gives

This is the origin of the BICEP/Keck ansatz

(Vansyngel et al. 2017; BK15; BK18). For a coherent angle rotation the analogous expansion is $R\approx\langle\cos 2\delta\psi\rangle\approx 1-2\langle\delta\psi^2\rangle$ with $\delta\psi$ in radians.

Table 1. Decorrelation $1-R$ from Gaussian $\sigma_\beta$, uncorrelated with amplitude.

Cleaning with a template of correlation $R$ leaves a residual power fraction $1-R^2\approx 2(1-R)$ if the amplitude is known. With $A_d(150)\simeq 9\times 10^{-3}\,\mu\mathrm{K}^2$ and a tensor signal $7\times 10^{-5}\,\mu\mathrm{K}^2$, the residual equals the $r=10^{-3}$ signal when $1-R\simeq 4\times 10^{-3}$. Thus $1-R$ must be known to a few $10^{-3}$ to reach the signal level, and to a few $10^{-4}$ to make the residual a small correction. Table 1 shows that $\sigma_\beta\simeq 0.1$, a plausible cloud-to-cloud scatter, already produces $1-R\simeq 3.7\times 10^{-3}$ between 150 and 353 GHz — comparable to the signal. Pelgrims et al. (2021) found that a $\simeq 15\%$ cloud-to-cloud spread of the 353/217 polarized intensity ratio reproduces the LOS angle rotation they detect; that spread is a $\Delta\beta$ of order 0.1–0.2 depending on temperature, i.e. the middle-to-lower rows of Table 1, but only on the selected, misaligned sightlines.

Two caveats on this Gaussian estimate. First, $\beta$ statistics are not Gaussian and not independent of amplitude. Liu et al. (2025), in a Simons Observatory forecast, find that at the highest complexities considered ($\sigma(\beta_d)\simeq 0.3$) the bias on $r$ can reach $\simeq 0.03$ and is driven by spatial correlations between $\beta_d$ and dust amplitude, particularly at higher orders, rather than by $\sigma_\beta$ alone. Second, Planck itself is systematics-limited for this purpose: Guillet et al. (2026) find that SRoll2 residual covariances are compatible with temperature-only variations, while PR4 is compatible with neither a temperature-only nor a $\beta$-only model.

5. Where the variations come from

Pixel-based MBB fits absorb plane-of-sky variations of $\beta_d$ and $T_d$ into a per-pixel SED. They cannot see LOS variation: two clouds in one pixel with different $(\beta,T,\psi)$ are fitted by a single effective MBB whose residuals are coherent in frequency and mimic CMB.

Temperature. Grains above $\simeq 0.01\,\mu\mathrm{m}$ in radiative equilibrium dominate millimetre emission. Energy balance gives

Draine & Li (2007) quote roughly $T_d\simeq 17\,\mathrm{K}\,U^{1/6}$. Larger grains are cooler, so the size distribution sets the temperature distribution and the $T$-mixing. The radiation field $U$ depends on 3D position (attenuation, local stars). Zelko et al. (2022) reconstruct 3D $T_d$ at 110′, 55′, and 27′ over 3$\pi$ steradians by fitting Planck 217–857 GHz and IRAS 100 $\mu\mathrm{m}$ on Bayestar19 voxels.

Opacity index. In the Rayleigh limit $\beta$ depends on material and structure, not on size. Amorphous silicates, carbonaceous grains, and the astrodust dielectric function (Hensley & Draine 2021, 2023) give different millimetre $\beta$; temperature-dependent two-level-system absorption (Meny et al. 2007; Paradis et al. 2011; Demyk et al. 2017) and coagulation or ice accretion (Köhler, Ysard & Jones 2015; THEMIS: Jones et al. 2013; Ysard et al. 2024) raise millimetre opacity and change $\beta$. Empirically $\beta_\mathrm{mm}$ rises from $\simeq 1.54$ in diffuse gas to $\simeq 1.66$ in dense gas (Planck Int. XIV 2014). One must not write “bigger grains have lower $\beta$”: size sets $T$ and alignment; material and porosity set $\beta$.

Alignment. Radiative torque alignment (Lazarian & Hoang 2007; Hoang & Lazarian 2014, 2016; Andersson, Lazarian & Vaillancourt 2015) aligns grains above $a_\mathrm{align}\simeq 0.05$–$0.1\,\mu\mathrm{m}$, with $a_\mathrm{align}$ growing with $n_\mathrm{H}T_\mathrm{gas}/U$ to roughly the $2/7$ power. The Hensley & Draine (2023) fit has no alignment below $\simeq 0.01\,\mu\mathrm{m}$ and full alignment above $\simeq 0.1\,\mu\mathrm{m}$. Polarized emission is therefore weighted to larger, cooler grains. Planck Int. XXII (2015) found $\beta_P\simeq 1.59$ versus $\beta_I\simeq 1.51$; Planck 2018 XI give $\beta_P=1.53\pm 0.02$ at 19.6 K.

Polarization spectrum $p(\nu)$. A single-composition astrodust model predicts a nearly flat $p(\nu)$ through the millimetre (Hensley & Draine 2021, 2023). Silicate plus carbon mixtures can rise toward long wavelengths (Draine & Fraisse 2009). BLASTPol found $p(\nu)$ consistent with flat in Vela C (Gandilo et al. 2016; Ashton et al. 2018; Shariff et al. 2019). Flatness is what makes a 353 GHz polarized template extrapolable at all. Any frequency dependence of $p_0$ is an additional decorrelation channel.

Size distribution from extinction. $R(V)\equiv A_V/E(B-V)$ traces size and composition together. Zelko & Finkbeiner (2020) show that larger grains give higher $R(V)$, but the observed $R(V)$–$\beta$ anticorrelation requires composition changes; the $A_V/\tau_{353}$ ratio varies by up to a factor $\simeq 2$ with $R(V)$, and ISRF changes of $\times 0.5$ to 2 move $\beta$ by up to 0.15. Zhang & Green (2025) map 3D $R(V)$ from 130 million Gaia XP stars; Green et al. (2024) show that extinction curves have additional, spatially coherent components beyond $R(V)$.

Field geometry. After LOS averaging, $p\propto p_0\cos^2\gamma$, reduced by tangling and by multiple clouds. Halal, Clark & Tahani (2024) find $\simeq 1.5\%$ lower $p$ at fixed column when dust is spread over several distance components within 1.25 kpc (the effect vanishes if only dust within 270 pc is counted). $E/B\simeq 2$ and $TE>0$ arise from filaments aligned with $\mathbf{B}_\mathrm{POS}$ (Clark et al. 2015; Planck Int. XXXVIII 2016; Huffenberger et al. 2020; Konstantinou et al. 2022). $TB$ and $EB$ arise from misalignment of intensity and polarization structures (Clark et al. 2021; Cukierman, Clark & Halal 2023); Bracco et al. (2025) attribute part of the signal to the projected geometry of cold and warm gas. Under LOS mixing, $E/B$ and $EB$ become frequency dependent (Vacher et al. 2023b; Ritacco et al. 2023). At $|b|\gtrsim 60^\circ$ the polarized sky is often dominated by the Local Bubble shell at $\sim 100$–300 pc (Alves et al. 2018; Skalidis & Pelgrims 2019; Pelgrims et al. 2020; O’Neill et al. 2024b), which is favorable for 3D coverage. Halal et al. (2024) find no simple imprint of Local Bubble geometry on the polarization fraction, so dominance of the shell in polarized intensity does not imply a purely tangential field on the bubble wall.

6. What the three ingredients measure today

3D extinction-density maps. Edenhofer et al. (2024) reach 1.25 kpc at up to 14′ angular resolution ($N_\mathrm{side}=256$) and parsec-scale radial resolution, with 12 variational posterior samples. Earlier maps (Green et al. 2019; Leike, Glatzle & Enßlin 2020; Lallement et al. 2022) have complementary volume and resolution. Failure modes: these are extinction, not emission; radial smearing and resolution that degrade with distance; a $\tau_{353}/A_V$ ratio that varies by up to a factor $\simeq 2$; missing nearby volume (Edenhofer et al. start at $\simeq 69\,\mathrm{pc}$); and distant intermediate-velocity clouds (IVCs) that exist at high latitude. Pelgrims et al. (2024) see polarizing clouds to $\simeq 2\,\mathrm{kpc}$. Cheng et al. (2025) find lower FIR emissivity per H in IVCs than in low-velocity gas and, assuming constant $T_d$ along each LOS, $\sigma_T<1.28\,\mathrm{K}$ at high latitude. H I velocity channels give kinematic, not distance, separation and should not be called 3D without qualification; the appropriate joint product is a combination of H I velocities with a 3D extinction cube.

3D temperature. Zelko et al. (2022) fit Planck and IRAS emission on Bayestar19 voxels over 3$\pi$ of sky, matching observations on a 10′ scale, and recover distinct temperatures for two Cepheus components along one LOS. The fit cannot uniquely separate $\beta$ and $T$ per voxel; the conversion factor between emission optical depth and reddening varies by a factor of two, consistent with Zelko & Finkbeiner (2020).

Grain size distribution. Per-star $R(V)$ and higher-order extinction-curve components (Zhang & Green 2025; Green et al. 2024) constrain the size–composition mix in 3D. They do not determine millimetre $\beta$ from optical data alone. What they do provide is a spatially varying $\tau_{353}/A_V$ prior, a prior on $T_d$ through the size–ISRF–temperature relation, and a prior on $p_0(\nu)$ through the aligned-size cutoff.

Sky-projected field. Starlight polarization tomography (Pelgrims et al. 2023, BISP-1; Pelgrims et al. 2024, first degree-scale map over 4 deg$^2$; TRIShUL, Uppal et al. 2026) measures per-cloud $\psi_i$, with emission polarization at $\psi_i+90^\circ$. PASIPHAE (Tassis et al. 2018) aims at 0.1% polarimetric accuracy over more than 10,000 deg$^2$ of the polar caps; degree-scale tomography exists only in small fields. Recovery requires the cloud to induce $p\gtrsim 0.1\%$ (Pelgrims et al. 2023). LOS structure biases Serkowski-based grain constraints by $\simeq 10\%$ (Skalidis 2024). H I velocity-channel orientation is a useful proxy where stars are sparse (Clark & Hensley 2019) but is kinematic. The Planck 353 GHz polarization angle is the emission-weighted average along the full LOS; using it to predict per-cloud angles is circular. Neither starlight nor dust polarization sees $B_\parallel$; Faraday rotation traces ionized gas and does not substitute. Synchrotron polarization traces $\mathbf{B}_\mathrm{POS}$ with cosmic-ray weighting and Faraday rotation and is not a proxy for dust $\psi$.

7. Combined method

Table 2. Parameters of the LOS integral and the ingredient that constrains each.

The forward model has five steps.

1. Segment each high-latitude LOS into distance components using the 3D extinction map, retaining posterior samples (Edenhofer et al. provide 12) so that radial smearing and component confusion propagate.

2. Assign per-cloud emission properties $(\tau,T,\beta\ \mathrm{prior},p_0)$ from the grain-physics layer, and per-cloud $(\psi,\gamma)$ from the field layer. Millimetre amplitudes are not taken from optical data at the percent level; they remain free parameters to be fitted to the microwave maps, with the optical layer supplying the mixing geometry and SED-shape priors.

3. Forward-model $Q_\nu$ and $U_\nu$ per pixel from Eqs. (2)–(4). Three derived products are the scientifically useful output: maps of predicted angle rotation $\delta\psi(\nu)$; maps of predicted $R(\nu_1,\nu_2)$; and maps of the spin-moments $W_1,W_2$. The last is the bridge to existing pipelines: moments computed from 3D data are physically informed priors, not additional free spectral parameters.

4. Use in component separation, in four distinct ways.

   • Moment priors. Moment-expansion fits and constrained-moment ILC deprojection (Remazeilles, Rotti & Chluba 2021; Vacher et al. 2022; Azzoni et al. 2021) deproject $W_1$ (and $W_2$ where warranted). 3D-informed priors decide which moments to deproject where, limiting the noise penalty of deprojecting moments that are consistent with zero.

   • Cloud-basis parametrization. In pixel $p$,

 where the mixing weights $M_{p,i}$ come from the 3D map and the per-cloud SED $f_i$ is shared across all pixels that cloud spans. Relative to a per-pixel $(A,\beta,T,\psi)$ fit this reduces the number of free parameters, makes LOS mixing explicit, and breaks $\beta$–$T$–noise degeneracies. The millimetre amplitudes remain fitted from multi-frequency CMB data.

• Region selection and weighting. Predicted complexity — $N_c$, misalignment of $\psi_i$, $\sigma_\beta$ and $\sigma_T$ among clouds, and $|W_1|$ — is a per-patch score for the BICEP Southern Hole and for Simons Observatory and LiteBIRD deep fields. Weighting or masking by this score is a conservative use that does not require the forward model to be unbiased, only rank-ordered.

• Validated 3D-consistent simulations. Replace ad hoc layered models such as PySM d12 (Martínez-Solaeche, Karakci & Delabrouille 2018) with skies whose distance components, $R(V)$, $T_d$, and $\psi_i$ are those of the real 3D data, complemented by statistical complexity models (McBride, Bull & Hensley 2023; Hervías-Caimapo & Huffenberger 2022; Vacher et al. 2025) and MHD synthetic skies (Kritsuk et al. 2026). Bias on $r$ is then estimated on simulations that have the same LOS structure as the data.

5. Validate against Planck 353 GHz $Q/U$ amplitude and angle; against 217$\times$353 angle differences in sightlines selected by H I or 3D complexity (Pelgrims et al. 2021); against $E/B$, $TE$, and $TB$; and, later, against 280–400 GHz polarization from Simons Observatory and LiteBIRD. Extinction-based amplitude templates and star-based angle templates contain no CMB, no CIB, and no millimetre-instrument systematics, so cross-correlations with them are clean estimators of dust amplitude and $\beta$ at each frequency, as BICEP/Keck XVI demonstrated with H I.

8. Expected gains, limits, and roadmap

Gains are conditional. If a high-latitude LOS is dominated by one or two distance components that the 3D map resolves, and if starlight tomography measures $\Delta\psi$ to a few degrees, the forward model predicts $\delta\psi(\nu)$ and $W_1$ at the level that Table 1 and the worked example show to matter ($1-R\sim 10^{-3}$). In the Local Bubble–dominated caps this condition is the most plausible (O’Neill et al. 2024b). If many misaligned, SED-different clouds contribute comparably, or if the millimetre-emitting grains are a different population from the optical extinctors, the 3D layer still provides a prior and a complexity score but does not determine the residual.

Limits that will not be removed by more of the same data:

• Optical extinction never yields the millimetre amplitude per cloud to 1%. The $\tau/A_V$ factor-of-two variation is a physical floor unless millimetre data themselves calibrate it.
• $\beta_i$ is not an optical observable. Grain models and FIR/submillimetre intensity constrain it; optical $R(V)$ is a prior, not a measurement of $\beta_\mathrm{mm}$.
• $B_\parallel$ is invisible to both starlight and dust polarization.
• Distant IVCs and clouds beyond $\simeq 1.25\,\mathrm{kpc}$ are under-resolved or missing. High-latitude polarized emission is mostly local, not entirely local.
• Starlight tomography currently exists at degree scale only in a few square degrees. Until PASIPHAE-class coverage exists, $\psi_i$ over the BICEP patch must come from sparse polarimetry plus H I orientation, with a large systematic floor.
• Planck 353 GHz cannot be used as an independent per-cloud angle prior.
• Moment deprojection has a noise cost; deprojecting moments that 3D data say are small is a net loss.

Roadmap. Gaia DR4 will tighten stellar distances and XP extinction curves, improving both the Edenhofer-class density cubes and the Zhang–Green $R(V)$ cube. PASIPHAE’s WALOP polarimeters target 0.1% accuracy over the polar caps; that is the data set that makes per-cloud $\psi_i$ a sky-wide product rather than a pathfinder. Simons Observatory’s 280 GHz channel and LiteBIRD’s 280–400 GHz bands measure polarized dust on the Wien side of the CMB, where $T_d$ mixing is larger and the moment expansion is better conditioned. Cross-correlation of those channels with CMB-free optical templates is the most direct validation of predicted $R(\nu_1,\nu_2)$. None of this replaces millimetre component separation; it changes the prior and the simulation suite against which that separation is biased.

9. Conclusion

Polarized thermal dust is the leading foreground to primordial $B$-modes because, even in the BICEP field, its 150 GHz BB power is $\simeq 9\times 10^{-3}\,\mu\mathrm{K}^2$ against $7\times 10^{-5}\,\mu\mathrm{K}^2$ for $r=10^{-3}$. Frequency decorrelation — $R_\ell^{BB}<1$ — is the statement that a 353 GHz template is not a scalar multiple of the dust at CMB frequencies, because $\beta_d$, $T_d$, and $\psi$ vary along the LOS. A Gaussian $\sigma_\beta\simeq 0.1$, or a two-cloud angle rotation of $\simeq 2^\circ$ between 353 and 150 GHz, each produce $1-R\sim$ a few $10^{-3}$, which is the level at which the residual equals the target signal. Planck-scale analyses do not detect this as a sky-averaged effect at CMB frequencies; targeted 217$\times$353 angle differences do detect the LOS mechanism. Three-dimensional extinction and temperature maps segment the LOS; the grain size distribution converts extinction into $\tau_{353}$, $T_d$, and an alignment-weighted $p_0$; starlight tomography supplies per-cloud $\psi$. Combined, they predict spin-moments, $R(\nu_1,\nu_2)$, and $\delta\psi(\nu)$ that can be used as priors, as a cloud-basis mixing model, as a complexity weight, and as a 3D-consistent simulation. The millimetre amplitude remains a microwave fit. If the high-latitude sky is as locally dominated and as coarsely layered as the present 3D maps suggest, this program can turn an unmodeled $r\sim 10^{-3}$ systematic into a quantified, marginalized uncertainty; if the millimetre-emitting, aligned grain population is not the optically extincting one, or if many misaligned clouds hide beyond the maps’ reach, the same program still ranks patches and bounds the bias, but it does not claim to have cleaned the dust.

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Deviations from brief

• Stokes integrals are cited to Lee & Draine (1985), Wardle & Königl (1990), and Planck Collaboration Int. XIX (2015), which is the polarized-dust overview paper that uses this formalism. Planck Int. XX in the 2015 A&A 576 sequence is not the polarized-dust methods paper; pinning the integrals to “Int. XX” would be a mis-citation.
• Zelko et al. (2022) cover 3$\pi$ steradians (Bayestar19), not the full sky; the essay states this.
• Local Bubble dominance of high-latitude polarized intensity (O’Neill et al. 2024b; Pelgrims et al. 2020) is distinguished from an imprint of bubble-wall geometry on the polarization fraction, which Halal et al. (2024) do not find. The brief’s statement is retained for intensity, with that qualification.
• Table 1 entries are the exact Gaussian formula evaluated at 217, 150, and 95 GHz versus 353 GHz; they agree with the brief to the quoted one-to-two significant figures. The 15 K versus 25 K contrast at 150 GHz is $\Delta\beta\simeq 0.11$ rather than 0.1; a 1 K shift between 150 and 353 GHz is $\Delta\beta\simeq 0.017$–0.019 rather than 0.02. The worked example gives $\delta\psi\simeq 1.86^\circ$ and $1-\cos 2\delta\psi\simeq 2.1\times 10^{-3}$.
• No other scientific corrections. The brief’s reading of “3D emission maps” as extinction plus fitted temperature, and of “spectral-index decorrelation” as frequency decorrelation of $Q,U$, is followed throughout.