# Planck constraints on cross-correlations between anisotropic cosmic
birefringence and CMB polarization

M. Bortolami  M. Billi  A. Gruppuso  P. Natoli  and L. Pagano

###### Abstract

Cosmic Birefringence (CB) is the in-vacuo rotation of the linear polarization
direction of photons during propagation, caused by parity-violating extensions
of Maxwell electromagnetism. We build low resolution CB angle maps using
Planck Legacy and NPIPE products and provide for the first time estimates of
the cross-correlation spectra $C_{L}^{\alpha E}$ and $C_{L}^{\alpha B}$
between the CB and the CMB polarization fields. We also provide updated CB
auto-correlation spectra $C_{L}^{\alpha\alpha}$ as well as the cross-
correlation $C_{L}^{\alpha T}$ with the CMB temperature field. We report
constraints by defining the scale-invariant amplitudes $A^{\alpha X}\equiv
L(L+1)C_{L}^{\alpha X}/2\pi$, where $X=\alpha,T,E,B$, finding no evidence of
CB. In particular, we find $A^{\alpha E}=(-7.8\pm 5.6)$ nK deg and $A^{\alpha
B}=(0.3\pm 4.0)$ nK deg at 68% C.L..

##  1 Introduction

The standard model of particle physics predicts that the electromagnetic (EM)
interaction is invariant under the parity operator. Nonetheless, models of
parity-violating electromagnetism have been proposed, reflecting the
phenomenology of weak interactions [1]. A popular model extends Maxwell’s
Lagrangian with a parity-violating Chern-Simons term [2, 3]:

$\mathcal{L}=\mathcal{L}_{EM}+\mathcal{L}_{CS}=-\dfrac{1}{4}F_{\mu\nu}F^{\mu\nu}-\dfrac{1}{2}p_{\mu}A_{\nu}\widetilde{F}^{\mu\nu},$
(1.1)

where $F_{\mu\nu}$ is the EM field tensor, $p_{\mu}$ is a constant kinematic
four-vector with dimensions of mass, $A_{\nu}$ is the four-potential and
$\widetilde{F}^{\mu\nu}=\frac{1}{2}\epsilon^{\mu\nu\alpha\beta}F_{\alpha\beta}$,
$\epsilon^{\mu\nu\alpha\beta}$ being the four-dimensional Levi-Civita symbol.
The additional parity-violating term $\mathcal{L}_{CS}$ is responsible for the
Cosmic Birefringence (CB) effect, the in-vacuo rotation of the linear
polarization plane of photons during propagation.

Hence, to observe such an effect one needs a source of linear polarized light.
The Cosmic Microwave Background (CMB) is linearly polarized at the level of 1%
to 10% due to Thomson scattering off free electrons at the surface of last
scattering and it is therefore a good observable to constrain this phenomenon
[4, 5]. Moreover, the CMB is the farthest EM radiation available in nature and
this increases the chances of detecting the CB effect, as the polarization
plane had more time to rotate, since the effect accumulates during
propagation.

The CB effect induces correlations between the temperature field T and the
B-modes of the polarization field or between the E- and B-modes of the
polarization field, that are null with standard EM as the parity symmetry
prevents these correlations to be active. Thus, we can use TB and/or EB
correlations to detect the CB effect. However, for the Planck data, which are
employed in this work, the signal-to-noise ratio due to TB correlation is
about a factor of two smaller than EB [6]. Hence, in this paper we focus only
on the EB correlation.

If the parity-violating cross-correlations of the CMB fields do not depend on
the direction of observation [7, 8, 9, 10, 11, 12, 13, 14, 15, 16], the CB
effect is isotropic and it can be modelled with a single angle $\alpha$.
Otherwise, the CB effect is anisotropic [17, 18, 19, 20] and it has to be
modelled with a set of angles $\alpha(\hat{n})$. Recent constraints on the CB
effect can be found in [6, 21, 22, 23, 24, 25, 26, 27] for the isotropic
case111For a detailed summary of previous constraints see e.g. references in
[6]. and in [28, 21, 29, 30] for the anisotropic case. Isotropic and
anisotropic CB can be originated by different physical models, see e.g. [31].
In this work, following a phenomenological approach, we focus on the
anisotropic case but we also estimate, as consistency check with previous
results, the CB monopole, i.e. the isotropic signal.

Parity violating electromagnetism Lagrangian also induce a non-null cross-
correlation between the CB field and the CMB temperature and polarization
maps, see e.g. [32] and references therein for a detailed treatment of cross-
spectra and bi-spectra. Furthermore, in recent years, early dark energy models
inspired by ultra-light axion fields [33, 34] have been proposed to alleviate
the $H_{0}$ tension [35]. These models are known to produce CB signal and a
non-zero cross-correlation between CB and the CMB fields, see e.g. [36].

In this paper we analyze CB extending what was done in Ref. [21]. In
particular, using the Planck Public Data Release 3 (PR3) products, we provide
for the first time constraints on the cross-correlation spectra $C_{L}^{\alpha
E}$ and $C_{L}^{\alpha B}$ between the CB angle maps and the CMB polarization
maps at large angular scales. We also provide the CB auto-correlation spectra
$C_{L}^{\alpha\alpha}$ and its cross-correlation spectra $C_{L}^{\alpha T}$
with the CMB temperature map up to $L=24$, exploiting the Planck Public Data
Release 4 (NPIPE) data in addition to the Planck PR3 products.

This paper is organized as follows: in Section 2 we review the features of the
Planck datasets and simulations employed for this study; in Section 3 we
describe the analysis pipeline; in Section 4 we present the results of this
work; in Section 5 we draw the conclusions; in Appendix A we discuss the
purification of E and B modes for the CMB Angular Power Spectra (APS)
estimation; in Appendix B we explain the effective beam calculation used for
the CMB spectra; in Appendix C we report the results of the analysis on
simulations; in Appendix D we perform robustness tests for the analysis
pipeline; in Appendix E we set 2-dimensional (2D) joint constraints on
$A^{\alpha\alpha}$ and $A^{\alpha T}$.

##  2 Datasets and Simulations

In this work we employ CMB maps provided by the Planck collaboration in the
PR3 analysis [37] and in the subsequent NPIPE re-analysis [38]. The PR3 maps
have been cleaned from foregrounds by the four official Planck component
separation methods Commander, NILC, SEVEM and SMICA [39], while NPIPE maps
have been processed only by Commander and SEVEM [38]. The corresponding
simulations, provided by the Planck collaboration, have been also employed.
The resolution of the maps, following the
HEALPix222http://healpix.sourceforge.net [40] pixelization scheme, is
$N_{side}=2048$. The NPIPE products have never been used to estimate the CB
auto-correlation spectra $C_{L}^{\alpha\alpha}$ before this work.

The PR3 simulation set is composed of 1000 CMB Monte Carlo simulations
generated using the Planck $\Lambda$CDM best-fit model, 300 noise simulations
for the full-mission (FM) and 300 noise simulations for each of the two half-
mission (HM) data splits, and for each of the component separation methods. We
sum the first 300 CMB realizations with each set of 300 HM noise simulations
that realistically describe the Planck 2018 data. We employ the two data
splits in the construction of the $\alpha$ map. We also sum the first 300 CMB
realizations with the 300 FM noise simulations in order to obtain 300 FM
simulations. We use these maps for the cross-correlation spectra
$C_{L}^{\alpha E}$ and $C_{L}^{\alpha B}$ estimation. The PR3 simulations
contain beam leakage effects [41], ADC non linearities, thermal fluctuations
and other systematic effects [42].

The NPIPE data and simulation sets are an evolution over PR3 [38]. One of the
effects of these improvements is lower levels of noise and systematics in the
CMB maps at essentially all angular scales [38]. We employ the 400 simulations
available for Commander333The simulations used in the analysis are actually
399 because one turned out corrupted. and the 600 simulations available for
SEVEM. The NPIPE Commander simulations contain the PR3 CMB signal and the
instrumental noise. To extract the noise-only maps, we subtract the signal-
only PR3 simulations to the complete NPIPE simulations. The NPIPE SEVEM CMB
simulations are divided in noise-only and signal-only as the PR3 ones, which
we sum for a set of noisy CMB simulations. The NPIPE data splits, where the
systematics between the splits are expected to be uncorrelated, are called A
and B (see again [38] for a full description). We calculate the NPIPE CMB APS
by cross-correlating those A and B maps. The NPIPE simulations employed in
this work are the “noise aligned” ones, see again [38] for details.

##  3 Analysis pipeline

As first step of the analysis we need to define the processing masks. We
divide the HEALPix sky at $N_{side}=2048$ in small regions following the
pixelization scheme at $N_{side}=8$. In the rest of the paper we refer to
these regions as “patches”. The sky fraction covered by one single patch is
$f_{sky,patch}\simeq 0.13\%$ and the total number of available patches is 768.
Working at resolution $N_{side}=8$ is an improvement with respect to [21],
where they used $N_{side}=4$. We tried to increase the resolution to
$N_{side}=16$, but the patch sky fraction was too low to get reliable CMB
spectra.

We then combine the masks of these patches with a polarization mask that
excludes the regions unobserved or heavily contaminated by foregrounds. For
PR3 we combine the foreground mask (Galactic and point sources) with the HM
missing pixels mask. For NPIPE only the foreground mask is employed. The
polarization masks used are shown in the top panels of Fig. 1 (left for PR3,
right for NPIPE).

Figure 1:  Top: polarization masks used in this work (left for PR3, right for
NPIPE). Bottom: percentage of active pixels in each patch (left for PR3, right
for NPIPE). The grey patches are not used in the analysis. The patches marked
with a red x in the bottom left panel are the ones whose CMB spectra
validation is shown in Fig. 2, i.e. patches with HEALPix indexes 1, 207, 542
from top to bottom.

Some of the aforementioned patches contain masked pixels. A large fraction of
masked pixels in a patch prevents the extraction of reliable CMB spectra. For
this reason, we select and analyze only the patches that have at least 50% of
active (i.e. non masked) pixels. This corresponds to $f_{sky,patch}\gtrsim
0.065\%$. The maps showing the selected patches and the fraction of active
pixels in each of them are shown in the bottom panels of Fig. 1 (left for PR3,
right for NPIPE), where the grey patches are the excluded ones. As expected,
among the selected patches, the ones close to the Galactic mask borders or the
ones containing a large number of point sources have a lower percentage of
active pixels. The number of selected patches is 571 for PR3 and 592 for
NPIPE. The final masks are not apodized because the apodization process
reduces the effective sky fraction covered by each patch and, for the sky
fraction considered, it does not provide any benefit in the CMB computed APS,
see also the discussion about polarization purification in the next paragraph.

For each selected patch, the CMB APS is calculated with the Python package
Pymaster444https://namaster.readthedocs.io [43] by cross-correlating the two
splits both for data and simulations. We recall that the data splits are HM 1
and 2 for PR3, A and B for NPIPE, as explained in Sec. 2. We compute the APS
in cross-mode to reduce systematic effects and noise mismatches [44, 42]. When
analysing a masked CMB map to produce the APS, E to B leakage can be generated
[45]. We have the possibility to purify the E and/or B modes in order to
remove the misinterpreted modes, at a cost of information loss [45, 43]. We
choose not to purify E and B modes: for the noise level of Planck, the size of
the patches considered and for the multipole range of interest, the
purification does not provide any substantial advantage, and does increase the
error bars (see App. A for more details). We calculate the APS for multipoles
from 2 to 2047. We deconvolve a non-Gaussian effective beam obtained from
simulations, as explained in App. B. The APS is binned with $\Delta l=60$
multipoles in order to reduce the errors and the correlations induced by the
cut-sky [46].

We show in the top panels of Fig. 2 the PR3 Commander validation plots for the
EE, BB and EB power spectra for three patches, in terms of the bandpowers
$D_{\ell}=\ell(\ell+1)C_{\ell}/2\pi$.

Figure 2: Top: EE (left), BB (center) and EB (right) CMB power spectra for
three patches and for the Commander component separation method. The black
line represents the fiducial, while the blue, orange and green points are the
simulation average spectra for the patches with HEALPix RING ordering indexes
1, 207 and 542, respectively (highlighted with a red cross in Fig. 1). The
error bars represent the error of the mean. Center: EE, BB and EB difference
between the simulation average and the fiducial spectra, divided by the error
of the mean. Bottom: EE, BB and EB errors of the mean.

The three patches, following the HEALPix RING ordering scheme, have indexes 1
(blue), 207 (orange) and 542 (green) and fractions of patch active pixels
100%, 74.5% and 51.1%, respectively. The dots represent the mean of the
simulations and the error bars show the uncertainty on the mean. The fiducial
power spectra are represented by the black lines in the plots. These plots
show that the simulations CMB spectra are validated. The scatter around the
fiducial at the highest multipoles is large and, for this reason, we do
consider the APS with $\ell\lesssim 1500$. We show in the central panels of
Fig. 2 the residuals plots, i.e. the difference between the simulation average
and the fiducial spectra, divided by the error of the mean. For all the cases
considered the simulation average is less than $3.5\sigma_{\rm mean}$ away
from the fiducial. The residuals do not show any systematic behaviour. We
report in the bottom panels of Fig. 2 the error of the mean. As expected, the
error increases for higher multipoles and for lower numbers of active pixels
in the patch. We verified that the behaviour of other patches and other
component separation methods is similar to the one shown in Fig. 2. For the
NPIPE case the simulations are validated with a similar procedure, the only
difference is that the errors are slightly smaller. This is due to the lower
level of noise with respect to the PR3 simulation set [38].

Given the low number of simulations, the CMB covariance matrices for each
patch and for each component separation method are obtained via the analytical
implementation of Pymaster [43]. The module that produces the matrices needs
the CMB signal APS, the noise APS and the smoothing function. The CMB signal
APS used for the covariance matrix calculation is the Planck FFP10 fiducial
[42]. The noise APS for each of the two data splits (HM1 and HM2 for PR3, A
and B for NPIPE) is obtained by averaging the auto-correlation of the
corresponding noise-only simulations. The smoothing function considered is
obtained as discussed in App. B. We verified that these analytical covariance
matrices are a good approximation of the ones obtained using the simulations.
This is not valid for the first multipole bin, so we exclude the latter from
the analysis and use the CMB spectra starting from $\ell_{min}=62$.

After obtaining the CMB APS and the corresponding covariance matrices, we
estimate the CB angle in each patch assuming that the CB effect is isotropic
inside each patch555Since the scales of the anisotropic birefringence angle we
are interested in, are larger than the size of the aforementioned patches, we
can consider constant the birefringence angle at the scale of the patches or
smaller.. The CMB APS is rotated by the isotropic CB effect in the following
way [16]:

$\displaystyle C_{\ell}^{TT,obs}$ $\displaystyle=C_{\ell}^{TT},$ (3.1)
$\displaystyle C_{\ell}^{TE,obs}$ $\displaystyle=C_{\ell}^{TE}\cos(2\alpha),$
(3.2) $\displaystyle C_{\ell}^{TB,obs}$
$\displaystyle=C_{\ell}^{TE}\sin(2\alpha),$ (3.3) $\displaystyle
C_{\ell}^{EE,obs}$
$\displaystyle=C_{\ell}^{EE}\cos^{2}(2\alpha)+C_{\ell}^{BB}\sin^{2}(2\alpha),$
(3.4) $\displaystyle C_{\ell}^{BB,obs}$
$\displaystyle=C_{\ell}^{BB}\cos^{2}(2\alpha)+C_{\ell}^{EE}\sin^{2}(2\alpha),$
(3.5) $\displaystyle C_{\ell}^{EB,obs}$
$\displaystyle=\frac{1}{2}\left(C_{\ell}^{EE}-C_{\ell}^{BB}\right)\sin(4\alpha),$
(3.6)

where $\alpha$ is the CB angle, the APS with the $obs$ tag is the rotated
(observed) one and the APS without the $obs$ tag is the unrotated one. Without
the CB effect, i.e. $\alpha=0$, the TB and EB spectra are null, as expected.

Harmonic based estimators for $\alpha$ can be built exploiting the observed EB
and/or TB spectra, see e.g. [16]. Since the signal-to-noise ratio is smaller
for the TB-based estimator as compared to EB one, we use the EB-only
D-estimator defined in [16]:

$D_{\ell}^{EB,obs}=C_{\ell}^{EB,obs}\cos(4\alpha)-\frac{1}{2}\left(C_{\ell}^{EE,obs}-C_{\ell}^{BB,obs}\right)\sin(4\alpha)\thinspace.$
(3.7)

We obtain an estimate of the isotropic CB angle $\alpha$ in each patch by
minimizing the following $\chi^{2}_{EB}$:

$\chi^{2}_{EB}=\sum_{\ell,\ell^{\prime}}D_{\ell}^{EB,obs}M_{\ell\ell^{\prime}}^{EB}D_{\ell^{\prime}}^{EB,obs},$
(3.8)

where $D_{\ell}^{EB,obs}$ are the D-estimators defined in Eq. (3.7) and
$M_{\ell\ell^{\prime}}^{EB}=\langle
D_{\ell}^{EB,obs}D_{\ell^{\prime}}^{EB,obs}\rangle^{-1}$ is their covariance
matrix. The $M_{\ell\ell^{\prime}}^{EB}$ matrix reduces to the EB covariance
matrix, i.e. without the CB effect [16], when the noise on the E and B CMB
fields is equivalent. Since the latter is valid for the Planck maps we are
employing [44, 42], we use the EB covariance matrix in the $\chi^{2}_{EB}$
minimization. Only the CMB spectra and covariance matrix related to a specific
patch are used in the $\chi^{2}_{EB}$ to estimate the CB angle in that patch.

For each CMB map we minimize the $\chi^{2}_{EB}$ in each patch obtaining a map
of CB angles. From these maps we firstly estimate the CB monopole averaging
the angle values in the pixels outside the masks shown in the bottom panels of
Fig. 1. Then, we calculate their auto-correlation spectra
$C_{L}^{\alpha\alpha}$ and their cross-correlation spectra $C_{L}^{\alpha T}$,
$C_{L}^{\alpha E}$ and $C_{L}^{\alpha B}$ with the CMB T, Q, U maps employing
a Quadratic Maximum Likelihood (QML) estimator [47, 48, 49, 50], where
$\alpha$ represents an extra scalar field. The maps and the associated
covariance matrices used in the QML estimator are described in the next
paragraphs.

The CB fields for data and simulations are obtained through the
$\chi^{2}_{EB}$ minimization, described above. Following [21], the covariance
matrices associated to the data are assumed to be diagonal, and they are
computed from the variance of the simulations in each pixel. See App. C for
more details about this choice. In the computation of the angular power
spectra, we marginalize over monopole (for all the cases) and dipole (for the
$\alpha T$, $\alpha E$ and $\alpha B$ cases) contributions following the
procedure described in [51, 52]. Hence, note that we retain $L=1$ as the
lowest multipole for $\alpha\alpha$ spectra.

For the CMB temperature field we always use the Commander map since it is the
one used by the Planck collaboration for the large scale temperature
cosmological analysis. We produce 300 signal-only CMB temperature simulations
at HEALPix resolution $N_{side}=8$ smoothed with a Gaussian beam with 880’
FWHM and an analytic pixel window function, to which we sum isotropic white
noise realizations with standard deviation of 500 nK for regularization. The
Commander data map is down-sampled and regularized accordingly. The CMB
temperature covariance matrix is analytically calculated from the FFP10
fiducial spectrum [42] and contains the marginalization of monopole and dipole
and a diagonal term to take into account the 500 nK regularization noise.

For the CMB polarization fields, instead, we use the SMICA polarization maps
because it is the only component separation method providing harmonic
weights666https://wiki.cosmos.esa.int/planck-legacy-
archive/index.php/SMICA_propagation_code [39, 53] that allow to correctly
compute the pixel space noise covariance matrix, from single frequency noise
covariances. We down-sample the 300 PR3 polarization simulations and the data
maps from $N_{side}=2048$ to $N_{side}=8$, applying, in addition to an
analytic pixel window function, a cosine window function [54] defined as:

$b_{\ell}=\left\\{\begin{matrix}1&\mbox{for $\ell<N_{side}$}\\\
\frac{1}{2}\left(1+\sin\left(\frac{\pi}{2}\frac{\ell}{N_{side}}\right)\right)&\hskip
20.0pt\mbox{for $N_{side}\leq\ell<3\ N_{side}$}\\\ 0&\mbox{otherwise}\\\
\end{matrix}\right..$ (3.9)

We then sum a 20 nK regularization noise. The corresponding covariance matrix
contains: the correlated noise term (computed using the SMICA weights and the
FFP8 [41] noise covariance matrices), the signal term based on a fiducial
power spectrum, and the regularization noise term. We limit the estimates of
the $\alpha E$ and $\alpha B$ spectra at $L=23$ by definition of the
considered smoothing window.

We show in Fig. 3 the masks used for the extraction of the spectra with the
QML estimator. The mask employed with the $\alpha$ and $T$ maps has $\approx$
74% of non-masked pixels, while the fraction of active pixels for the $Q$ and
$U$ mask is $\approx$ 47%. The masks are obtained as the product of the PR3
$\alpha$ mask, that excludes the grey patches shown in the bottom left panel
of Fig. 1, and the CMB galactic mask for intensity ($\alpha$ and $T$ maps) or
polarization ($Q$ and $U$ maps).

Figure 3:  Masks used for the CB spectra QML estimator for the $\alpha$ and
$T$ (left), $Q$ and $U$ (right) maps.

After having obtained the $C_{L}^{\alpha X}$ spectra, where X can be $\alpha$,
T, E or B, we set constraints in terms of the scale-invariant amplitude
$A^{\alpha X}$. To do so, we minimize the following $\chi^{2}\left(A^{\alpha
X}\right)$:

$\chi^{2}\left(A^{\alpha
X}\right)=\sum_{L,L^{\prime}}\left(\dfrac{L(L+1)}{2\pi}C_{L}^{\alpha
X}-A^{\alpha
X}\right)M_{LL^{\prime}}^{-1}\left(\dfrac{L^{\prime}(L^{\prime}+1)}{2\pi}C_{L^{\prime}}^{\alpha
X}-A^{\alpha X}\right),$ (3.10)

where $C_{L}^{\alpha X}$ are the data spectra and
$M_{LL^{\prime}}=\left<\dfrac{L(L+1)}{2\pi}C_{L}^{\alpha
X}\dfrac{L^{\prime}(L^{\prime}+1)}{2\pi}C_{L^{\prime}}^{\alpha X}\right>$ is
calculated from simulations.

##  4 Results

In this section we show the results of our analysis, adopting the following
color code: red for Commander, orange for NILC, green for SEVEM, and blue for
SMICA.

In Figure 4 we show the CB maps (left panels) extracted from CMB data, and the
associated error maps computed from simulations (right panels) for the PR3
dataset. In Figure 5 the same quantities are displayed for NPIPE.

Figure 4: Cosmic Birefringence angle maps obtained from the PR3 polarization
maps for the Commander, NILC, SEVEM and SMICA component separation methods
(from top to bottom). Left: data. Right: standard deviation of simulations.

Figure 5: Cosmic Birefringence angle maps obtained from the NPIPE polarization
maps for the Commander and SEVEM component separation methods (from top to
bottom). Left: data. Right: standard deviation of simulations.

By visually comparing PR3 CB maps, we can recognize similar patterns for all
the four component separation methods. The same applies to the NPIPE dataset.
The errors associated to NPIPE are generally lower with respect to the PR3
ones. This is simply due to the fact that the NPIPE CMB maps are slightly less
noisy than the PR3 ones [38]. Furthermore some NPIPE low resolution patches
contain a larger number of observed pixels, with respect to the PR3 dataset,
whose mask contains also the contribution of half mission missing pixels (see
top panels of Fig. 1). Finally, the darker regions in error maps, which
represent patches where the error on the CB angle is lower, are well
correlated with the Planck scanning strategy [55], that observes more deeply
the ecliptic poles.

case | $\bm{\alpha\ \mathrm{[deg]}}$
---|---
PR3 Commander | $\mathrm{0.27\pm 0.05\ (stat)\pm 0.28\ (syst)}$
PR3 NILC | $\mathrm{0.26\pm 0.05\ (stat)\pm 0.28\ (syst)}$
PR3 SEVEM | $\mathrm{0.27\pm 0.05\ (stat)\pm 0.28\ (syst)}$
PR3 SMICA | $\mathrm{0.24\pm 0.05\ (stat)\pm 0.28\ (syst)}$
NPIPE Commander | $\mathrm{0.33\pm 0.04\ (stat)\pm 0.28\ (syst)}$
NPIPE SEVEM | $\mathrm{0.33\pm 0.04\ (stat)\pm 0.28\ (syst)}$
[23] (PR3) | $\mathrm{0.35\pm 0.14\ (stat)}$
[24] (NPIPE) | $\mathrm{0.30\pm 0.11\ (stat)}$
[26] (NPIPE + WMAP) | $\mathrm{0.30^{+0.094}_{-0.091}\ (stat)}$
Table 1: Monopole of the data CB angle maps produced in this work. The values
found in [23], [24], and [26] are also reported for reference.

In Table 1 we report the monopoles of the CB maps. They are obtained as a
weighted average of the data maps with weights corresponding to the inverse
variance of the simulations. All the monopole values are consistent with zero
provided the systematic error is taken into account. Moreover they are well
compatible, within the statistical uncertainty, across the different component
separation methods. The statistical error contribution is computed from the
standard deviation of the simulations, while the systematic error is due to
the uncertainty in the orientation of Planck’s polarization-sensitive
bolometers [56] and is taken from Ref. [6]. Also here we note that, due to the
lower level of the instrumental noise [38] and to the larger number of non-
masked pixels, the NPIPE data provide smaller statistical errors. The monopole
values found in this work agree with the ones reported in literature, see e.g.
[6, 21, 22, 23, 24, 26].

Figure 6: Auto-correlation spectra of the data CB angle maps and their cross-
correlation with CMB. The errors are the standard deviations of the
simulations. The vertical dashed lines highlight a change in the vertical axis
range. Top left panel: PR3 $\alpha\alpha$. Top right panel: NPIPE
$\alpha\alpha$. Central panel: PR3 $\alpha T$. Bottom left panel: PR3 $\alpha
E$. Bottom right panel: PR3 $\alpha B$.

In Figure 6 we show the auto-correlation of the CB maps and their cross-
correlation with CMB777All the spectra shown in this paper are given in terms
of bandpowers, i.e. $D_{L}^{\alpha X}=L(L+1)C_{L}^{\alpha X}/2\pi$, with
$X=\alpha,T,E,B$.. As expected the scatter of the NPIPE $\alpha\alpha$ spectra
around zero and the associated errors are slightly lower with respect to PR3.
Across the spectra, we find a general good compatibility with the null effect.
This is assessed by comparing the harmonic $\chi^{2}$ of data and simulations,
reported in Fig. 7.

Figure 7: Harmonic $\chi^{2}$ plots (histogram for simulations, black line for
data). Top left panel: PR3 $\alpha\alpha$. Top right panel: NPIPE
$\alpha\alpha$. Central panel: PR3 $\alpha T$. Bottom left panel: PR3 $\alpha
E$. Bottom right panel: PR3 $\alpha B$.

The data $\chi^{2}$ for all the spectra lies within the histograms, suggesting
that the data are compatible with the simulations, hence with the null effect.
However, we note that for the $\alpha E$ and $\alpha B$ cases the $\chi^{2}$
of the data are closer to the right tail of the corresponding histograms, with
the following Probability To Exceed (PTE): 0.9–10.6% for $\alpha E$ and
1.4–2.5% for $\alpha B$ (see also Tab. 2 where all the PTEs are reported).

case | Commander | NILC | SEVEM | SMICA
---|---|---|---|---
$\alpha\alpha$ PR3 | 47.6% | 37.1% | 9.0% | 2.8%
$\alpha\alpha$ NPIPE | 47.1% | - | 6.2% | -
$\alpha T$ PR3 | 10.9% | 27.4% | 23.6% | 7.5%
$\alpha E$ PR3 | 1.6% | 10.6% | 4.2% | 0.9%
$\alpha B$ PR3 | 2.5% | 1.4% | 2.1% | 2.1%
Table 2: Probability To Exceed obtained from the harmonic $\chi^{2}$ reported
in Fig. 7.

Despite the overall good compatibility with the null effect, some single
multipole fluctuations deserve a closer look. Considering the $\alpha T$
spectra, $L=5$ shows a PTE between 1% and 2.7% for the four component
separation methods, i.e. $\sim 2.5\sigma$ fluctuation. For the $\alpha E$
spectra, the multipoles $L=5,12,14,19$ have the largest deviation from the
null effect, if compared with the corresponding empirical distributions. These
PTEs are below 1% for $D_{19}^{\alpha E}$ Commander and $D_{12}^{\alpha E}$
SEVEM, but still within $3\sigma$ fluctuation. The same happens for
$D_{2}^{\alpha B}$ NILC, SEVEM and SMICA, all with PTEs lower than 1%, but
within $3\sigma$. The PTE for $D_{13}^{\alpha B}$ is between 2% and 2.7%,
which corresponds to $\sim 2.3\sigma$ fluctuation. All the spectra obtained
with the SMICA component separation method are reported in Tab. 3 along with
the statistical uncertainty at $1\sigma$ level for each multipole888The other
CB spectra, along with the CB angle maps, are made publicly available at
https://github.com/marcobortolami/AnisotropicBirefringence_patches.git.. The
averaged spectra of the simulations are shown in App. C. They all show a nice
compatibility with the null-hypothesis, meaning that the residual systematic
effects included in the simulations do not have a significant impact on the
spectra considered.

$\bm{L}$ | $\bm{D_{L}^{\alpha\alpha}[deg^{2}]}$ | $\bm{D_{L}^{\alpha T}[\mu K\ deg]}$ | $\bm{D_{L}^{\alpha E}[\mu K\ deg]}$ | $\bm{D_{L}^{\alpha B}[\mu K\ deg]}$
---|---|---|---|---
1 | $-0.004\pm 0.006$ | - | - | -
2 | $\phantom{-}0.026\pm 0.019$ | $\phantom{-}1.404\pm 2.683$ | $\phantom{-}0.027\pm 0.021$ | $\phantom{-}0.048\pm 0.016$
3 | $-0.013\pm 0.028$ | $-1.123\pm 2.996$ | $\phantom{-}0.040\pm 0.025$ | $\phantom{-}0.013\pm 0.013$
4 | $-0.063\pm 0.049$ | $-1.678\pm 3.351$ | $\phantom{-}0.013\pm 0.024$ | $-0.032\pm 0.015$
5 | $\phantom{-}0.166\pm 0.060$ | $-11.140\pm 3.668$ | $-0.054\pm 0.026$ | $-0.022\pm 0.014$
6 | $\phantom{-}0.194\pm 0.083$ | $-7.479\pm 4.151$ | $-0.006\pm 0.022$ | $\phantom{-}0.011\pm 0.014$
7 | $-0.074\pm 0.090$ | $\phantom{-}7.492\pm 4.141$ | $-0.013\pm 0.021$ | $-0.001\pm 0.016$
8 | $\phantom{-}0.058\pm 0.112$ | $-1.114\pm 4.663$ | $-0.018\pm 0.022$ | $-0.002\pm 0.018$
9 | $-0.211\pm 0.137$ | $\phantom{-}1.536\pm 4.577$ | $-0.001\pm 0.023$ | $-0.010\pm 0.019$
10 | $\phantom{-}0.206\pm 0.135$ | $-7.389\pm 4.740$ | $-0.035\pm 0.023$ | $\phantom{-}0.020\pm 0.019$
11 | $-0.215\pm 0.169$ | $-3.359\pm 5.139$ | $\phantom{-}0.016\pm 0.025$ | $-0.014\pm 0.023$
12 | $-0.294\pm 0.188$ | $\phantom{-}3.900\pm 4.707$ | $-0.062\pm 0.028$ | $\phantom{-}0.027\pm 0.025$
13 | $\phantom{-}0.097\pm 0.224$ | $\phantom{-}1.340\pm 5.185$ | $-0.041\pm 0.029$ | $\phantom{-}0.057\pm 0.024$
14 | $\phantom{-}0.213\pm 0.253$ | $-4.532\pm 6.245$ | $\phantom{-}0.087\pm 0.034$ | $-0.046\pm 0.031$
15 | $\phantom{-}0.399\pm 0.266$ | $-7.444\pm 5.834$ | $-0.045\pm 0.038$ | $-0.055\pm 0.031$
16 | $-0.373\pm 0.307$ | $-3.625\pm 6.447$ | $-0.034\pm 0.039$ | $\phantom{-}0.002\pm 0.037$
17 | $\phantom{-}0.015\pm 0.298$ | $-9.853\pm 6.462$ | $-0.017\pm 0.045$ | $\phantom{-}0.021\pm 0.039$
18 | $\phantom{-}0.071\pm 0.357$ | $-6.756\pm 7.213$ | $-0.033\pm 0.049$ | $-0.098\pm 0.050$
19 | $-0.729\pm 0.403$ | $\phantom{-}0.848\pm 7.662$ | $\phantom{-}0.095\pm 0.065$ | $\phantom{-}0.042\pm 0.064$
20 | $-0.131\pm 0.406$ | $-4.962\pm 8.737$ | $-0.000\pm 0.095$ | $\phantom{-}0.039\pm 0.093$
21 | $\phantom{-}0.221\pm 0.450$ | $\phantom{-}0.500\pm 10.144$ | $\phantom{-}0.106\pm\phantom{-}0.163$ | $-0.104\pm 0.149$
22 | $-0.218\pm 0.497$ | $-5.240\pm 10.748$ | $-0.241\pm 0.365$ | $\phantom{-}0.024\pm 0.338$
23 | $-0.209\pm 0.476$ | $\phantom{-}11.141\pm 11.989$ | $-0.213\pm 1.477$ | $\phantom{-}0.526\pm 1.429$
24 | $-0.455\pm 0.470$ | $\phantom{-}11.452\pm 14.336$ | - | -
Table 3: CB spectra for the PR3 SMICA case. The $\alpha E$ and $\alpha B$
spectra are calculated up to $L=23$ as the $coswin$ smoothing (and thus the
spectra) are null for higher multipoles, as discussed in Sec. 3.

In Figure 8 we display the posterior distributions for the scale invariant
amplitude, $A^{\alpha X}$, computed using Eq. (3.10). In Table 4 we report the
upper limits at 95% C.L., for $A^{\alpha\alpha}$, and the constraints at 68%
C.L., for $A^{\alpha T}$, $A^{\alpha E}$ and $A^{\alpha B}$. As before, also
for these parameters, we find a good compatibility among the different
component separation methods and with the null effect. Furthermore, for
$A^{\alpha\alpha}$, there is good compatibility between the PR3 and the NPIPE
results. The NPIPE upper limit using the Commander component separation method
is looser with respect to the PR3 one because the posterior peak is slightly
shifted to the right. Even if a small shift is present also for SEVEM, the
lower width of the posterior of NPIPE results in a tighter constraint on
$A^{\alpha\alpha}$, when compared with SEVEM PR3. The $\alpha T$ constraints
are compatible with, but tighter than [21], where they used the same approach
adopted in this work but at the lower resolution $N_{side}=4$. Thus, a higher
resolution provides more constraining power for $\alpha T$.

Figure 8: Posteriors for the scale invariant spectrum fit. Top left panel: PR3 and NPIPE $\alpha\alpha$. Top right panel: PR3 $\alpha T$. Bottom left panel: PR3 $\alpha E$. Bottom right panel: PR3 $\alpha B$. parameter | Commander | NILC | SEVEM | SMICA
---|---|---|---|---
$\mathrm{A^{\alpha\alpha}\ [deg^{2}]\ PR3}$ | $<0.007$ | $<0.007$ | $<0.010$ | $<0.007$
$\mathrm{A^{\alpha\alpha}\ [deg^{2}]\ NPIPE}$ | $<0.010$ | - | $<0.009$ | -
$\mathrm{A^{\alpha T}\ [\mu K\ deg]\ PR3}$ | $-1.827\pm 0.953$ | $-1.229\pm 0.873$ | $-2.037\pm 1.038$ | $-1.916\pm 0.945$
$\mathrm{A^{\alpha E}\ [nK\ deg]\ PR3}$ | $-3.5\pm 6.0$ | $-1.0\pm 5.6$ | $-9.7\pm 6.0$ | $-7.8\pm 5.6$
$\mathrm{A^{\alpha B}\ [nK\ deg]\ PR3}$ | $\hphantom{-}2.4\pm 4.0$ | $-1.2\pm 3.7$ | $\hphantom{-}4.0\pm 4.4$ | $\hphantom{-}0.3\pm 4.0$
Table 4: Constraints on the scale invariant power spectrum $A^{\alpha X}$,
with $X=\alpha,T,E,B$, set in this work.

##  5 Conclusions

In this work we build CB maps at angular scales larger than $\sim 7$ deg,
exploiting Planck CMB component separated maps from both PR3 and NPIPE
releases. From these CB maps we estimate the monopoles (see Tab. 1), i.e. the
isotropic birefringence angle, finding a very good compatibility with previous
results [6, 21, 22, 23, 24]. We also compute their spectra and the cross-
correlation with the CMB temperature map. Moreover, we provide for the first
time the spectra of the cross-correlation of the CB field with the CMB E and B
fields. The data CB angle maps and spectra, for both auto- and cross-
correlations with CMB, are made publicly
available999https://github.com/marcobortolami/AnisotropicBirefringence_patches.git.

We quantify the compatibility of all the aforementioned spectra with null
effect through an harmonic $\chi^{2}$ analysis, see Fig. 7, and through a
scale invariant amplitude $A^{\alpha X}$, with $X=\alpha,T,E,B$, see Fig. 8.
We find no significant evidence of deviation from the null effect. The
constraints on $A^{\alpha X}$ are summarized in Tab. 4. The latter are
obtained through a $\chi^{2}$ minimization assuming null CB, which is
supported by data. We also constrain jointly $A^{\alpha\alpha}$ and $A^{\alpha
T}$ with a pixel-based likelihood which naturally takes into account the
effect of CB-induced cosmic variance, see App. E.

In this work we extend to $L=24$ the multipole range of the $\alpha\alpha$ and
$\alpha T$ spectra covered in [21], which adopts the same technique for the
construction of the CB maps. Note that for the PR3 $\alpha T$ case we obtain
constraints compatible with, but tighter than, [21]. In addition, we provide
for the first time estimates of the Planck NPIPE $\alpha\alpha$ spectra.
However, the main novelty of this work is represented by the estimates of the
Planck PR3 $\alpha E$ and $\alpha B$ spectra, which are presented here for the
first time. The latter might be fruitfully considered to constrain models of
anisotropic birefringence that predict correlations with the CMB fields.

## Acknowledgments

We acknowledge the financial support from the INFN InDark initiative and from
the COSMOS network (www.cosmosnet.it) through the ASI (Italian Space Agency)
Grants 2016-24-H.0 and 2016-24-H.1-2018, as well as 2020-9-HH.0 (participation
in LiteBIRD phase A). We acknowledge the use of the healpy [57, 40], NaMaster
[43], numpy [58], matplotlib [59] software packages, and the use of computing
facilities at CINECA.

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##  Appendix A E and B modes purification for CMB APS extraction

In each patch we estimate the CMB power spectra using Pymaster [43], which is
a pseudo-$C_{\ell}$ estimator. This technique is known to suffer from E/B
leakage when applied on a masked sky. This effect can be alleviated
considering a purification process, which removes the misinterpreted modes, at
a cost of information loss. However, for our case, i.e. the Planck noise and a
sky fraction of the size of the patches, the standard master algorithm
provides unbiased spectra and the purification process just worsens the
uncertainty of the estimated spectra. For this reason, we decide not to apply
any purification. In order to prove the former statement, we compare the
estimated spectra for all the possible purification cases: master (i.e. no
purification), pure E (i.e. purification of the E modes only), pure B (i.e.
purification of the B modes only) and pure EB (i.e. purification of both the E
and B modes). We employ 100 CMB simulations with a Planck-like noise level,
i.e. the polarization noise is $\sigma_{P}$ = 57.7 $\mu$K arcmin, and a sky
fraction of $f_{sky,patch}\simeq 0.13\%$.

In Figure 9 we show the results of this test.

Figure 9: Top: EE (left), BB (center) and EB (right) CMB power spectra for the
four purification cases and for 100 Planck-like simulations. The black line
represents the fiducial, while the colored points are the simulation average
spectra. The error bars represent the error of the mean. Center: EE, BB and EB
difference between the simulation average and the fiducial spectra, divided by
the error of the mean. Bottom: EE, BB and EB errors of the mean.

The spectra are validated for all the four cases. The error on the spectra is
similar for high multipoles, while it is smaller for the master case at low
multipoles. Thus, for the analysis conditions considered in this work, the
purification process increases the uncertainty of the estimated spectra and
turns out to be not useful.

##  Appendix B Effective beam calculation

Strictly speaking, both CMB data and simulations contain a non-Gaussian beam
and a non-ideal pixel window function (see Figure F.1. of [60]). Thus, we
employ an effective “smoothing function” $b_{\ell,eff}^{2}$ obtained by taking
these steps:

  1. 1.

From 999 signal-only Planck PR3 simulated $a_{\ell m}^{i}$, not smoothed, we
compute the corresponding $C_{\ell}^{i}$ for each simulation $i$.

  2. 2.

From 999 signal-only Planck PR3 simulated maps at $N_{side}=2048$, that are
smoothed with a realistic beam, get the corresponding $C_{\ell,sm}^{i}$ in
full sky for each simulation $i$.

  3. 3.

Divide the two power spectra to obtain
$\left(b_{\ell,eff}^{2}\right)^{i}=C_{\ell,sm}^{i}/C_{\ell}^{i}$ for each
simulation $i$.

  4. 4.

Average the smoothing function over the simulation index $i$.

The smoothing function obtained contains both an effective beam window
function and an effective pixel window function. This procedure is followed
for the PR3 simulations of the 4 component separation methods maps separately,
obtaining 4 different effective smoothing functions for Commander, NILC, SEVEM
and SMICA. For NPIPE Commander and SEVEM maps we use the same smoothing
function computed for PR3. For each component separation method, the same
smoothing function is used for each patch.

##  Appendix C CB auto- and cross-correlation results on simulations

We report in Fig. 10 the auto-correlation of the CB angle maps and their
cross-correlation with CMB temperature and polarization fields obtained on
simulations. All the spectra are given in terms of bandpowers, as in Sec. 4.
Figure 10 shows that the systematic effects included in the simulations have a
negligible impact because all the spectra are well compatible with zero, as
the deviations from zero are less than $3\sigma$ for all the multipoles
considered. As for the data spectra, the scatter around zero of the simulation
NPIPE $\alpha\alpha$ spectra and the errors are lower with respect to the PR3
$\alpha\alpha$ case. This is again due to the lower levels of noise present in
the NPIPE maps.

Figure 10: Auto-correlation spectra of the simulation CB angle maps and their
cross-correlation with CMB. The dots represent the average of the simulation
spectra, while the errors are the errors of the average. The vertical dashed
lines highlight a change in the vertical axis range. Top left panel: PR3
$\alpha\alpha$. Top right panel: NPIPE $\alpha\alpha$. Central panel: PR3
$\alpha T$. Bottom left panel: PR3 $\alpha E$. Bottom right panel: PR3 $\alpha
B$.

##  Appendix D Analysis robustness tests

In order to test the robustness of our results, we modify some analysis
settings of our pipeline. Since we find good stability for all the considered
spectra, for the sake of brevity we present here robustness tests only for the
cross-correlation between the CB and the CMB temperature field. This choice is
mainly due to the fact that, even if statistically well compatible, the most
likely value for $A^{\alpha T}$ is found here negative while in [21] it is
positive. Since the analysis in [21] was performed at HEALPix resolution
$N_{side}=4$ (while our analysis is performed at $N_{side}=8$) we rerun our
codes at lower resolution for the Commander component separation method. The
maximum multipole for the $\alpha T$ spectra at this resolution is 12. We find
that data and simulations spectra are compatible with zero, as at
$N_{side}=8$. However, the scale-invariant amplitude moves towards positive
values, as we obtain $\mathrm{A^{\alpha T}=(0.192\pm 1.102)\ \mu K\ deg}$. The
constraint is worse than the one at $N_{side}=8$ due to the lower resolution
of the patch maps. In addition, if we use a more aggressive mask by extending
our mask by the size of one patch as done in [21] ($\approx 15^{\circ}$ at
$N_{side}=4$), we obtain $\mathrm{A^{\alpha T}=(1.890\pm 1.297)\ \mu K\ deg}$.
Our constraints at $N_{side}=4$ are well compatible (even for the sign of the
most likely value) with [21] and with null cross-correlation between the CB
and the CMB temperature fields. To test if the change in resolution can
explain the shift of the data spectra to more negative values, we calculate
the difference between the two data spectra for the first 12 multipoles at the
two different resolutions and we compare it with the same quantity calculated
on simulations. We find that the shift in the bandpowers is compatible with
the scatter of the simulations.

In order to see if the negative sign of the $\mathrm{A^{\alpha T}}$ parameter
is given by a bias in the simulations, we subtract the average of the CB
simulation maps from all the CB simulations and data maps. We then run the QML
estimator for the $D_{L}^{\alpha T}$ spectra and fit with a scale-invariant
spectrum for the $\mathrm{A^{\alpha T}}$ parameter. The posteriors move
towards slightly less negative values of $\mathrm{A^{\alpha T}}$ and the
errors are slightly lower, but this shift cannot explain the change in sign of
$\mathrm{A^{\alpha T}}$. The constraints in the debias case are reported in
Tab. 5.

case | Commander | NILC | SEVEM | SMICA
---|---|---|---|---
reference | $-1.827\pm 0.953$ | $-1.229\pm 0.873$ | $-2.037\pm 1.038$ | $-1.916\pm 0.945$
debias | $-1.766\pm 0.948$ | $-1.192\pm 0.868$ | $-1.997\pm 1.034$ | $-1.886\pm 0.941$
extended | $-1.383\pm 1.098$ | $-0.724\pm 0.993$ | $-1.636\pm 1.164$ | $-1.354\pm 1.066$
$\ell_{min}=302$ | $-1.931\pm 0.995$ | $-1.439\pm 0.899$ | $-2.468\pm 1.092$ | $-2.122\pm 0.987$
Table 5: Constraints on the scale invariant power spectrum $A^{\alpha T}$,
given in $\mu K\ deg$. The reference case is discussed in Sec. 3. The debias,
extended and $\ell_{min}=302$ cases are explained in this Appendix.

To study the effect of possible foregrounds not excluded by the mask applied
to the CB angle and CMB temperature maps, we try to use a more aggressive
mask. We thus extend the mask shown in the left panel of Fig. 3 by the size of
one patch, i.e. $\approx 8^{\circ}$, leaving $\approx$ 60% of the sky non-
masked. The scatter of the data spectra around zero is slightly larger than
for the non extended mask due to the lower sky fraction, but the compatibility
with the null effect is still obtained. The posteriors move towards less
negative values of the scale invariant spectrum, but the errors on the
parameter increase, again due to the reduced sky fraction. The constraints
with the extended mask are reported in Tab. 5.

Finally, we change the CMB multipole range used for the minimization of the
$\chi^{2}$ to obtain the CB angles in each patch. We increase the minimum
multipole considered from 62 to 302 and then produce the CB angle maps and get
the final results. The posteriors move towards more negative values of
$\mathrm{A^{\alpha T}}$, but the errors on this parameter increase and there
is again good compatibility with null cross-correlation between CB and CMB
temperature fields. The constraints with $\ell_{min}=302$ case are reported in
Tab. 5.

##  Appendix E $\alpha\alpha$ and $\alpha T$ joint constraints

We set 2D constraints on $A^{\alpha\alpha}$ and $A^{\alpha T}$ following the
methodology explained in [21]. We report in Fig. 11 the validation of the 2D
analysis for the SMICA case. The posterior shown in Fig. 11 is the product of
the posteriors obtained from the single simulations. Since the simulations do
not contain the CB effect and the posterior is consistent with zero, the
analysis is validated. We obtain similar results for the other component
separation methods.

Figure 11: 2D contour plots for the $\mathrm{A^{\alpha\alpha}}$ and
$\mathrm{A^{\alpha T}}$ parameters obtained with the SMICA simulations. The
darker (lighter) area represents the 68% (95%) confidence region.

In Figure 12 we show the 2D contour plots for the $\mathrm{A^{\alpha\alpha}}$
and $\mathrm{A^{\alpha T}}$ parameters obtained with the Commander, NILC,
SEVEM and SMICA CB data maps for the PR3 case and with the Commander CMB
temperature map. We use two different masks for the CB angles and CMB
temperature maps in order to exploit the largest amount of data: the former
has a sky fraction of 74%, while the latter of 84%. The covariance matrices
are built analytically. For this motivation, the uncertainties in the 2D case
are larger with respect to the 1D analysis described in Sec. 3, as in the
latter the CB cosmic variance is not taken into account. The four component
separation methods provide similar constraints. The SEVEM component separation
method is the one having the lowest compatibility with null CB. However, the
deviation is not larger than 3 standard deviations. Thus, we find no evidence
for the CB effect.

We then marginalize the 2D posterior distribution functions over
$\mathrm{A^{\alpha T}}$, obtaining the posterior distribution functions of
$\mathrm{A^{\alpha\alpha}}$ reported in Fig. 13 (solid curves). We also slice
the 2D probability at $\mathrm{A^{\alpha T}}=0$, obtaining the probability
distribution of $\mathrm{A^{\alpha\alpha}}$ shown in Fig. 13 (dashed curves).
The posterior is closer to zero for the sliced case, providing more stringent
constraints than the marginalized posterior. The four component separation
methods agree among themselves, even if the SEVEM distribution is larger than
the other three, reflecting what discussed for Fig. 12.

If we instead marginalize over $\mathrm{A^{\alpha\alpha}}$, we obtain the
posterior distribution functions for $\mathrm{A^{\alpha T}}$ reported in Fig.
14 (solid curves). Slicing the 2D contours at $\mathrm{A^{\alpha\alpha}}=0$,
we obtain the probability distribution of $\mathrm{A^{\alpha T}}$ shown in
Fig. 14 (dashed curves). Also here the sliced case provides tighter posteriors
than for the marginalized one, as expected. All these results are compatible
with null CB.

Figure 12: 2-dimensional contour plots for the $\mathrm{A^{\alpha\alpha}}$ and
$\mathrm{A^{\alpha T}}$ parameters obtained with the Commander (top left
panel), NILC (top right panel), SEVEM (bottom left panel) and SMICA (bottom
right panel) data. The darker (lighter) area represents the 68% (95%)
confidence region.

Figure 13: Posterior distribution functions for the
$\mathrm{A^{\alpha\alpha}}$ parameter obtained from Commander (top left
panel), NILC (top right panel), SEVEM (bottom left panel) and SMICA (bottom
right panel) data. The solid curves are computed marginalising the 2D
distribution functions over $\mathrm{A^{\alpha T}}$ and the dashed ones
slicing the 2D distribution functions at $\mathrm{A^{\alpha T}}=0$.

Figure 14: Posterior distribution functions for the $\mathrm{A^{\alpha T}}$
parameter obtained from Commander (top left panel), NILC (top right panel),
SEVEM (bottom left panel) and SMICA (bottom right panel) data. The solid
curves are computed marginalising the 2D distribution functions over
$\mathrm{A^{\alpha\alpha}}$ and the dashed ones slicing the 2D distribution
functions at $\mathrm{A^{\alpha\alpha}}=0$.