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

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Volume 705 (January 2026)

A&A, 705 (2026) A177

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Table A.1

Dictionary of all defined quantities, with complex quantities in bold.

Notation	Meaning	Reference
$P_{i} = Q_{i} + \dot{i} U_{i}$ $\[\mathbf{P}_{i}=Q_{i}+\dot{\mathbb{i}} U_{i}\]$	complex polarized intensity in channel i	Eq. (1), Sect. 2.2
ψ_i	polarization angle in channel i	Sect. 2.2
ν₀	reference frequency	Sect. 2.2
P₀	polarized intensity map at the reference frequency	Sect. 2.5
I₀	total intensity map at the reference frequency	Sect. 3.2
p₀	polarization fraction p₀ = P₀/I₀ at the reference frequency	Sect. 3.2

T, β, ε_ν	temperature, spectral index and emissivity of aligned grains	Eq. (3)
$\bar{T}, \bar{β}$ $\[\overline{T}, \overline{\beta}\]$	pivot temperature and spectral index defining the pivot SED	Eq. (6), Sect. 2.2
${\bar{ε}}_{i} = {\bar{ε}}_{ν_{i}} (\bar{β}, \bar{T})$ $\[\overline{\varepsilon}_{i}=\overline{\varepsilon}_{\nu_{i}}(\overline{\beta}, \overline{T})\]$	value of the pivot SED in channel i of frequency ν_i	Eq. (6)

$a_{i, n}^{s}, a_{i, n}^{T}, a_{i, n}^{β}$ $\[a_{i, n}^{s}, a_{i, n}^{T}, a_{i, n}^{\beta}\]$	order n of spectral gradients, for s ∈ {T, β} and for T and β	Eq. (7)
$W_{n}^{s}, W_{n}^{β}, W_{n}^{T}$ $\[\boldsymbol{\mathcal{W}}_{n}^{s}, \boldsymbol{\mathcal{W}}_{n}^{\beta}, \boldsymbol{\mathcal{W}}_{n}^{T}\]$	polarization-weighted complex spin-moments of order n	Eq. (8)
$ω_{n}^{s}, ω_{n}^{β}, ω_{n}^{T}$ $\[\omega_{n}^{s}, \omega_{n}^{\beta}, \omega_{n}^{T}\]$	intensity-weighted moments of order n	Eq. (11)
$Δ_{n}^{s} = W_{n}^{s} - ω_{n}^{s}$ $\[\boldsymbol{\Delta}_{n}^{s}=\boldsymbol{\mathcal{W}}_{n}^{s}-\omega_{n}^{s}\]$	difference between moments characterizing depolarization effects	Eq. (12)
${\bar{W}}_{n}^{s}, {\bar{ω}}_{n}^{s}, {\bar{Δ}}_{n}^{s}$ $\[\overline{\boldsymbol{\mathcal{W}}}_{n}^{s}, \overline{\omega}_{n}^{s}, \overline{\boldsymbol{\Delta}}_{n}^{s}\]$	mean values over the sky weighted by $P_{0}^{2}$ $\[P_{0}^{2}\]$	Eq. (17), Sects. 2.4, 2.5

${\tilde{ρ}}_{i}$ $\[\tilde{\rho}_{i}\]$	correction to the polarization efficiency in channel i	Sects. 4.1 & 4.3
${\bar{r}}_{i}$ $\[\overline{\mathbf{r}}_{i}\]$	mean complex SED normalized to the reference frequency	Eq. (16) and (42), Sects. 2.4 & 4.3
R_i	complex residual map for channel i	Eq. (18)
$R_{i}^{P}$ $\[R_{i}^{P}\]$	residual map of polarized intensity in channel i	Eq. (21)
$R_{i}^{ψ}$ $\[R_{i}^{\psi}\]$	residual map of polarization angle in channel i	Eq. (22)
$R_{i}^{ε}$ $\[R_{i}^{\varepsilon}\]$	residual map of dust emissivity in polarization	Eq. (23)
Cov(X, Y)	covariance of X and Y maps, weighted by $P_{0}^{2}$ $\[P_{0}^{2}\]$	Eq. (25), Sect. 2.6
$c_{i j}^{P}$ $\[c_{i j}^{P}\]$	covariance of $R_{i}^{P}$ $\[R_{i}^{P}\]$ and $R_{j}^{P}$ $\[R_{j}^{P}\]$	Eq. (26)
$c_{i j}^{ψ}$ $\[c_{i j}^{\psi}\]$	covariance of $R_{i}^{ψ}$ $\[R_{i}^{\psi}\]$ and $R_{j}^{ψ}$ $\[R_{j}^{\psi}\]$	Eq. (27)
$c_{i j}^{ψ P}, c_{i j}^{P ψ}$ $\[c_{i j}^{\psi P}, c_{i j}^{P \psi}\]$	mixed covariance of $R_{i}^{ψ}$ $\[R_{i}^{\psi}\]$ and $R_{j}^{P}$ $\[R_{j}^{P}\]$	Eq. (28)
$c_{i j}^{ε}, {\hat{c}}_{i j}^{ε}$ $\[c_{i j}^{\varepsilon}, \hat{c}_{i j}^{\varepsilon}\]$	covariance of $R_{i}^{ε}$ $\[R_{i}^{\varepsilon}\]$ and $R_{j}^{ε}$ $\[R_{j}^{\varepsilon}\]$ , and its estimate	Eqs.(29) and (39)
$c_{i j}^{P}$ $\[\mathbf{c}_{i j}^{\mathbf{P}}\]$	complex covariance of R_i and R_j	Eq. (30)
$ρ_{i j}^{P}, ρ_{i j}^{P}, ρ_{i j}^{ψ}$ $\[\rho_{i j}^{\mathbf{P}}, \rho_{i j}^{P}, \rho_{i j}^{\psi}\]$	Pearson correlation coefficients between residual maps (i ≠ j)	Table 1, Sect. 4.4

$σ_{s}^{ψ}, σ_{T}^{ψ}, σ_{β}^{ψ}$ $\[\sigma_{s}^{\psi}, \sigma_{T}^{\psi}, \sigma_{\beta}^{\psi}\]$	standard deviations fitting the spectral dependence of $c_{i j}^{ψ}$ $\[c_{i j}^{\psi}\]$	Sect. 5.1
$σ_{s}^{ε}, σ_{T}^{ε}, σ_{β}^{ε}$ $\[\sigma_{s}^{\varepsilon}, \sigma_{T}^{\varepsilon}, \sigma_{\beta}^{\varepsilon}\]$	standard deviations fitting the spectral dependence of $c_{i j}^{ε}$ $\[c_{i j}^{\varepsilon}\]$	Sect. 5.1
$Σ_{T}^{ψ P}$ $\[\Sigma_{T}^{\psi P}\]$	mixed covariance fitting the spectral dependence of $c_{i j}^{ψ P}$ $\[c_{i j}^{\psi P}\]$	Sect 6.1

$\bar{η}, η (p)$ $\[\overline{\eta}, \eta(p)\]$	parameters characteristic of the magnetic field model	Appendix C

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