Variance of dust temperature and spectral index in Planck polarization data using spin-moment expansion

TL;DR

This work develops a spin-moment, complex-residual framework to quantify frequency-dependent variations of the dust polarization SED within the Planck beam, linking residual covariances to dust temperature and spectral-index fluctuations. By constructing complex residual maps and covariances, the authors derive testable predictions, including near-perfect cross-frequency correlation of residuals for temperature fluctuations and distinct scaling relations for emissivity and angle variances. Validation with Planck SRoll2 data shows polarized-intensity residuals are highly cross-correlated across frequencies, while polarization-angle residual correlations are weaker and frequency-dependent, with CO contamination a notable complication at 100 and 217 GHz. Analyses of Planck PR4 data reveal data-version differences and challenges to pure-$T$ or pure-$\beta$ models, underscoring the need for refined dust models; nevertheless, the framework provides robust diagnostics and a path toward more realistic foreground models essential for upcoming CMB polarization experiments and cosmic birefringence studies.

Abstract

Thermal dust is the major polarized foreground hindering the detection of primordial cosmic microwave background (CMB) B-modes. Its signal exhibits complex behavior in frequency space, arising from the combined variation in our Galaxy of the orientation of magnetic fields and the spectral properties of dust grains aligned with magnetic field lines. In this work, we present a new framework for analyzing the thermal dust signal using polarized microwave data. We introduce residual maps, represented as complex quantities, which capture deviations of the local polarized spectral energy distribution (SED) from the mean complex SED averaged over the sky mask. We present simple predictions that relate the values of the statistical correlation and covariances between the residual maps to the physical properties of the emitting aligned grains. Testing these predictions provides valuable information about the nature of the dust signal. We evaluated our predictions using Planck data over a 97% mask excluding the inner Galactic plane. Despite its simplicity, our model captures a significant part of the statistical properties of the data. For the SRoll2 version of the data, the spectral dependence of the covariances between residual maps is compatible with a dust model that includes only temperature variations rather than spectral index variations. In contrast, for the PR4 Planck official release, it is incompatible with both models. Our methodology can be used to analyze future high-precision polarization data and to build more accurate dust models for use by the CMB community.

Figures (18)

• Figure 1: Values of $a_\nu^{\,\beta}$ and $a_\nu^T$ for orders $n=1$ (blue), $n=2$ (orange), and $n=3$ (red), shown for ${\nu_0}=353$ GHz and pivot temperature ${\overline{T}}=20\,$K. The points represent the four Planck HFI channels.
• Figure 2: Rotation of the SED in the complex plane for a sum of seven MBB SEDs with distinct spectral indices drawn from a Gaussian distribution with mean $1.5$ and standard deviation $0.1$ (red), or with temperatures drawn from a Gaussian distribution with mean $20$ K and standard deviation $5$ K (blue). The 3D orientation of the magnetic field is a random realization of the turbulent magnetic field inspired by Planck_XLVIII. The dashed lines indicate frequencies from 40 to 400 GHz, and the four colored points represent the Planck HFI bands.
• Figure 3: HEALPix masks with growing $f_{\rm sky}$ ($N_{\mathrm{side}}=32$): 85% (dark blue), 90% (light blue), 92% (green), 95% (orange), and 97.3% (brown). The 97.3% mask additionally excludes the inner Galactic plane within 3 degrees of latitude and pixels within 4 degrees of the Crab pulsar (gray).
• Figure 4: Mean complex polarized SED $\mathbf{\overline{r}}_i$ in the log complex plane for our 97% mask: debiased SRoll2 data (solid black line), raw SRoll2 data (dashed-dotted blue line), simulations (mean shown by the dashed red line with error bars representing the standard deviation), and the model d1 (dotted line). Errors bars are based on 200 simulations (see Sect. \ref{['subsec:Mean_SED_Planck']}). Planck-HFI frequencies are indicated.
• Figure 5: Test of predictions \ref{['eq:P2']}, \ref{['eq:P3']}, and \ref{['eq:P4']} : dependence on the observed polarization fraction $p_0=P_0/I_0$ at 353 GHz of the variances ${\operatorname{Re}}\,\boldsymbol{\rm c}^\mathbf{P}$, $c^P$, $c^\psi$, and $c^{\varepsilon}$ computed at 217 GHz. Our models for $c^{\varepsilon}$\ref{['eq:P2']} and $c^\psi$\ref{['eq:P3']} are overplotted as dotted orange and red lines, respectively, using the mean value of the slope derived from these equations, multiplied by $\left\langle P_0^2\right\rangle$ and $\left\langle I_0^2\right\rangle$ in each bin of $p_0=P_0/I_0$.