Interpreting map-based $E$/$B$ spectral properties of CMB foregrounds

Abstract

Map-space $E$/$B$ decompositions of linear polarization are attractive for foreground and CMB analyses because they isolate the $B$-family patterns that contaminate primordial tensor searches from $E$-family patterns that trace coherent Galactic structures. However, the $E$/$B$ transform is non-fully-local and induces apparent spectral complexity in projected fields even when the underlying sky is spectrally simple in $\underline{P}=Q+iU$. We quantify this effect for synchrotron emission. We introduce a complex-parameter description of the frequency dependence of $\underline{P}$, its spin-preserving projections $\underline{P}_E$ and $\underline{P}_B$, and the scalar $\underline{S}=E+iB$, using complex log--Taylor and moment expansions (with simple transformation rules under $E$/$B$ projection) and linking their coefficients to spectral-index variations, line-of-sight mixing, synchrotron ageing, and Faraday effects. Using a toy model and a PySM template, we find that scalar combinations, especially $|E|$ and $|B|$, acquire the largest induced complexity, while $\underline{S}$ is less affected but lacks a directly interpretable amplitude and angle. By contrast, $\underline{P}_E$ and $\underline{P}_B$ retain a clear geometric meaning and exhibit only moderate spectral distortions, while satisfying the closure relation $\underline{P}=\underline{P}_E+\underline{P}_B$ (which extends to all spectral orders in the moment formalism). Finally, with three frequency channels, we compare low-order spectral truncations and propose diagnostics to test whether the data favour a single power law in $P$ or independent power laws in $(P_E,P_B)$. This work is intended to be of practical relevance for both Galactic science and CMB $B$-mode analyses and lays the conceptual foundation for a series of papers applying the framework to observational data.

Figures (16)

• Figure 1: Example of the power law superposition effect (Sec. \ref{['sec:superposition']}): Eq. \ref{['eq:superposition_equation']} with two components, $\underline{P}_\nu = \underline{P}_{a, \nu} + \underline{P}_{b, \nu}$ with $\underline{P}_{a, 0} = e^{2i\pi/8}$, $\beta_a = -3.5$, $\underline{P}_{b, 0} = e^{-2i\pi/8}$ and $\beta_b = -2.5$. Upper panel: SED, lower panel: polarization angle, both as functions of frequency. The blue curves describe the baseline polarization spectrum, which is modified into the total black curves by the superimposition effect (whose correction $\underline{C}_\textrm{LoS}(\nu)$ is shown in dashed-red).
• Figure 2: Example of a single-component intrinsic positive curvature effect (Sec. \ref{['sec:curvature']}): Eq. \ref{['eq:curvature_equation']} with $\underline{P}_{0}=e^{2i\pi/8}$, $\overline{\beta}=-3.2$ and $\gamma = 1$. The panels and curves are otherwise similar to the ones in Fig. \ref{['fig:superposition']}.
• Figure 3: Example of the ageing effect (Sec. \ref{['sec:ageing']}): Eq. \ref{['eq:ageing_equation']} with $\overline{\beta}=-3.2$, $\sigma=0.5$, $\Delta\alpha=1$ and $\nu_b = 7\,\mathrm{GHz}$. The panels and curves are otherwise similar to the ones in Fig. \ref{['fig:superposition']}.
• Figure 4: Example of the combination of internal and external Faraday effects (Sec. \ref{['sec:faraday']}): Eq. \ref{['eq:faraday_equation']} with $\underline{P}_{0}=e^{2i\pi/8}$, $\overline\beta=-3.2$, $\sigma_\mathrm{RM,int}=\sigma_\mathrm{RM,ext}=0.5/c^2$, $\mathrm{RM}_\mathrm{int}=-2/c^2$ and $\mathrm{RM}_\mathrm{ext}=4/c^2$, with RMs expressed in units of $\mathrm{GHz}^{2}/c^{2}$. The panels and curves are otherwise similar to the ones in Fig. \ref{['fig:superposition']} (for this particular illustration, (i) polarization angles are left unwrapped along $\nu$ and are between $[-\pi/2, \pi/2]$, (ii) two distinct correction curves are represented, in red the internal Faraday effect and in orange the external Faraday effect).
• Figure 5: Toy-model polarization sky at a single frequency. The numbered structures in the total polarization amplitude panel (top row, centre) correspond to the components defined in Table 1, each chosen to illustrate a distinct $E/B$ balance and morphology. The first row shows the full Stokes fields $(Q,U)$ together with the resulting total polarization amplitude $P$ and angle $\psi$. The second and third rows display the corresponding $E$- and $B$-family fields $(Q_E,U_E,P_E,\psi_E)$ and $(Q_B,U_B,P_B,\psi_B)$ obtained through the map-space E/B decomposition.