11institutetext: CNRS-UCB International Research Laboratory, Centre Pierre
Binétruy, IRL 2007, CPB-IN2P3, Berkeley, CA 94720, USA 22institutetext:
Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720,
USA

# Planck data revisited: low-noise synchrotron polarisation maps from the WMAP
and Planck space missions

Jacques Delabrouille<EMAIL_ADDRESS>Planck data revisited: low-
noise synchrotron polarisation maps from the WMAP and Planck space
missionsPlanck data revisited: low-noise synchrotron polarisation maps from
the WMAP and Planck space missionsPlanck data revisited: low-noise synchrotron
polarisation maps from the WMAP and Planck space missionsPlanck data
revisited: low-noise synchrotron polarisation maps from the WMAP and Planck
space missions

(Received ; accepted )

###### Abstract

Context. Observations of cosmic microwave background polarisation, essential
for probing a potential phase of inflation in the early universe, suffer from
contamination by polarised emission from the Galactic interstellar medium.

Aims. This work combines existing observations from the WMAP and Planck space
missions to make a low-noise map of polarised synchrotron emission that can be
used to clean forthcoming CMB observations.

Methods. We combine WMAP K, Ka and Q maps with Planck LFI 30 GHz and 44 GHz
maps using weights that near-optimally combine the observations as a function
of sky direction, angular scale, and polarisation orientation.

Results. We publish well-characterised maps of synchrotron $Q$ and $U$ Stokes
parameters at $\nu=30$ GHz and 1 degree angular resolution. A statistical
description of uncertainties is provided with Monte-Carlo simulations of
additive and multiplicative errors.

Conclusions. Our maps are the most sensitive full-sky maps of synchrotron
polarisation to date, and are made available to the scientific community on a
dedicated web site.

###### Key Words.:

Cosmology: cosmic background radiation – Cosmology: observations – Cosmology:
inflation – Radio continuum: ISM – ISM: magnetic fields

##  1 Introduction

The cosmic microwave background (CMB), relic radiation emitted when electrons
and nuclei first combined to form neutral atoms, has been transformational for
understanding the origin, evolution, and matter and energy content of the
Universe we live in. Observations from a series of space-borne (Smoot et al.,
1992; Mather et al., 1994; Bennett et al., 2013; Planck Collaboration et al.,
2020a), balloon-borne (e.g., Hanany et al., 2000; Netterfield et al., 2002;
Benoît et al., 2003), and ground-based (e.g., Das et al., 2014; BICEP2
Collaboration et al., 2016; Louis et al., 2017; POLARBEAR Collaboration et
al., 2017; Henning et al., 2018) CMB experiments, complemented by other
cosmological probes (e.g., Perlmutter et al., 1999; Eisenstein et al., 2005;
Alam et al., 2017; Abbott et al., 2022), have led to a standard cosmological
scenario, inflationary $\Lambda$CDM, in which an expanding universe is filled
today with about 30% pressureless matter and 70% dark energy. In this
paradigm, galaxies and clusters of galaxies form from initial density
perturbations generated during an early phase of rapid expansion, dubbed
cosmic inflation. The fast expansion in the inflationary phase expands tiny
local quantum fluctuations of the spacetime metric to supra-horizon scales,
which later-on re-enter the horizon and seed the gravitational collapse of
structures observable today (Bardeen et al., 1983). While appealing, the
inflationary hypothesis still requires observational confirmation, as well as
the identification of the correct model among a plethora of viable
alternatives (Martin et al., 2014).

In many inflationary models, initial perturbations of the spacetime metric
comprise, in addition to a scalar term corresponding to initial density
perturbations responsible for the formation of the large scale structures, a
tensor term corresponding to primordial gravitational waves. These tensor
modes of metric perturbation are expected to imprint, on the CMB, tiny
fluctuations of polarisation of the B-mode type (of odd parity), while density
perturbations generate, at the time of CMB emission, only E-mode type
polarisation (of even parity). For this reason, the detection of primordial
$B$ modes of CMB polarisation is considered an essential next step in
observational cosmology (Kamionkowski & Kovetz, 2016). If detected with an
angular power spectrum matching theoretical prediction, they would be a
“smoking gun” for cosmic inflation. This motivates the deployment of ambitious
next-generation CMB observatories, on the ground and in space, dedicated to
the detection of primordial CMB polarisation B-modes (Hui et al., 2018; Ade et
al., 2019; Abazajian et al., 2019; Delabrouille et al., 2018; Hanany et al.,
2019; LiteBIRD Collaboration et al., 2023).

The amplitude of primordial gravitational waves, parameterised by the tensor-
to-scalar ratio $r$, is already constrained by existing CMB observations to
$r\leq 0.036$ and $r\leq 0.032$ at 95% CL, in two recent independent analyses
combining Planck and BICEP/Keck data (BICEP/Keck Collaboration et al., 2022;
Tristram et al., 2022). At this level and below, over most of the sky, for
most of the harmonic scales, and at all observing frequencies, CMB B-mode
fluctuations are fainter than fluctuations of microwave polarised emission
originating from the interstellar medium (ISM) of our own Milky Way. Two
processes of ISM emission are known to contribute a significant amount of
microwave polarization. Elongated galactic dust grains, aligned preferentially
perpendicularly to the local Galactic magnetic field, polarise the light from
background stars parallel to the local Galactic magnetic field (Davis &
Greenstein, 1951), and emit polarised microwave emission perpendicular to the
magnetic field. Polarised dust emission in the microwave and sub-millimeter
has been detected at 353 GHz by the Archeops stratospheric balloon (Benoît et
al., 2004), and in several frequency bands by the high frequency instrument
aboard the Planck space mission (Planck Collaboration et al., 2020c, d). At
lower frequency, energetic electrons in the ionised ISM spiralling in the
Galactic magnetic field generate synchrotron emission that is highly
polarised. This polarised synchrotron emission has been mapped over the full
sky, although with limited signal-to-noise ratio at high galactic latitude, by
the WMAP space mission (Kogut et al., 2007), and by the low frequency
instrument (LFI) aboard the Planck satellite (Planck Collaboration et al.,
2016, 2020d).

Channel / Frequency | $\beta_{s}=-2.9$ | $\beta_{s}=-3.0$ | $\beta_{s}=-3.1$ | $\beta_{s}=-3.2$ | $\beta_{s}=-3.3$
---|---|---|---|---|---
WMAP K | 2.22 | 2.28 | 2.34 | 2.42 | 2.49
LFI 30 GHz | 1.20 | 1.21 | 1.22 | 1.23 | 1.24
30 GHz | 1.00 | 1.00 | 1.00 | 1.00 | 1.00
WMAP Ka | 0.77 | 0.76 | 0.75 | 0.75 | 0.74
WMAP Q | 0.42 | 0.41 | 0.40 | 0.39 | 0.37
LFI 44 GHz | 0.34 | 0.33 | 0.32 | 0.31 | 0.29
Table 1: Synchrotron frequency scaling (in KCMB units) for various values of
the spectral index $\beta_{s}$. Relative errors on the frequency scalings are
seen to be of the order of a few per cent (10% at most). This will induce
errors of at most a few per cent of the signal in the final maps.

Future sub-orbital CMB observations, such as planned with the BICEP array (Hui
et al., 2018), the Simons observatory (Ade et al., 2019), AliCPT (Salatino et
al., 2020), or with the future CMB-S4 experiment (Abazajian et al., 2019),
must monitor the contamination of CMB observations by Galactic dust and
Galactic synchrotron polarisation. To that effect, one may envisage using
existing data from WMAP and Planck to help model and subtract polarised
emission from the ISM in ground-based CMB observations. Template maps for
polarised dust and synchrotron emission have been produced with a component
separation pipeline from the final Planck space mission data (Planck
Collaboration et al., 2020c), and made available to the scientific community
in the Planck space mission legacy archive at
ESA.111http://www.cosmos.esa.int/web/planck/pla These maps, however, suffer
from limitations, some of which can be addressed making better use of existing
data.

The main limitation of the published WMAP and Planck polarised synchrotron
maps is the level of instrumental noise. In particular, maps are strongly
noise-dominated in regions of sky of lowest Galactic foreground emission,
which would be the main targets for upcoming ground-based observations. This
comes from the limited sensitivity of existing surveys, and from the fact that
published maps do not best combine all existing observations: the latest
Planck published synchrotron maps (Planck Collaboration et al., 2020c), for
instance, use only Planck data.

In this paper, we combine WMAP and Planck LFI data in a simple yet near-
optimal way, to produce well-characterised maps of the $Q$ and $U$ Stokes
parameters of “Planck-Revisited” polarised synchrotron emission at 30 GHz,
that can be used for foreground cleaning in upcoming CMB observations with
ground-based experiments. While our paper was in preparation, Watts et al.
(2023) have combined Planck and WMAP observations to produce 30 GHz maps of
polarised synchrotron emission amplitude with noise levels lower than those of
the WMAP K channel alone, using a conceptually different approach. Data
products are part of the Cosmoglobe first data release (Cosmoglobe DR1). In
that analysis, Watts et al. (2023) use a pipeline that first corrects for a
model of systematic residuals in WMAP data and for calibration uncertainties
in Planck HFI data, and then fit for a multi-component model of sky emission
in multi-frequency maps. However, our focussed pipeline has specific
advantages, which we discuss in this paper. In particular, our data products
have better signal-to-noise ratio in regions of low foreground contamination,
and their effective angular resolution is better characterised. The increased
sensitivity is of specific interest for ground-based experiments targeting
primordial B-mode detection in patches of sky with low foreground emission,
such as the Simons Observatory (Ade et al., 2019), various geneerations of
BICEP/Keck (Hui et al., 2018; Schillaci et al., 2020) and AliCPT (Salatino et
al., 2020).

The rest of the paper is organised as follows. In Sec. 2, we describe the
input WMAP and Planck observations that are used for this work. In Sec. 3, we
describe the data pipeline that is used to combine those observations to make
a single map of polarised synchrotron emission at a frequency of 30 GHz.
Section 4 gives and discusses the main results and data products made
available to the scientific community. We conclude in Sec. 5.

##  2 Input observations and modeling assumptions

WMAP, launched in June 2001 by NASA, has observed the full microwave sky in
five frequency bands, centred from 23 GHz to 94 GHz. The Planck space mission
has observed the sky in 9 frequency bands centred at frequencies ranging from
27 GHz to 857 GHz.

In this paper, we use DR5 (9-year) WMAP data release polarisation maps for the
K, Ka, and Q bands (Bennett et al., 2013), and Planck DR4 LFI 30 GHz and 44
GHz data (Planck Collaboration et al., 2020b). Contamination by polarised dust
emission at these frequencies is strongly sub-dominant as compared to either
synchrotron emission or instrumental noise. We do not use the WMAP V and W
bands, which have low synchrotron signal-to-noise ratio over the largest
fraction of sky, nor the 70 GHz Planck LFI map or any of the Planck HFI maps,
which provide little additional information for mapping polarised synchrotron
specifically, and may contain non-negligible polarised dust emission. However,
we make use of the CMB temperature and polarisation maps obtained with the
SMICA method (Delabrouille et al., 2003; Cardoso et al., 2008) to correct the
low-frequency maps from a sub-dominant CMB polarisation contamination.

To first order, synchrotron emission $s(\nu)$ scales in frequency
proportionally to a power of frequency, i.e. for each sky pixel

$s(\nu)\propto\nu^{\beta_{s}}.$ (1)

The spectral index $\beta_{s}$ depends on the spectral energy distribution of
the relativistic electrons, and is hence theoretically supposed to depend on
the physical properties of the region of emission.

In the frequency range of interest, a simple power law with fixed
$\beta_{s}\simeq-3.1$ (when the emission is expressed in antenna temperature
(KRJ) units) is a good approximation, but there is evidence of slight
variations $\delta\beta_{s}\simeq\pm 0.1$ as a function of the direction in
the sky (Krachmalnicoff et al., 2018; de la Hoz et al., 2023; Watts et al.,
2023). More generally, the power-law scaling approximation can break down at
higher frequency, e.g. because of a depletion of high-energy electrons,
leading to a possible steepening of the frequency dependence. At low
frequency, below $\nu\sim 5$ GHz, Faraday rotation of polarisation and line-
of-sight integration leads to depolarisation that this power law invalid for
the $Q$ and $U$ Stokes parameters. Several authors have attempted to map the
synchrotron spectral index using existing data sets, but results are noisy and
uncertain (see, e.g., Krachmalnicoff et al., 2018; Fuskeland et al., 2021; de
Belsunce et al., 2022; Weiland et al., 2022; de la Hoz et al., 2023; Watts et
al., 2023, for recent results). Spectral index maps obtained using intensity
observations in the frequency domain of interest suffer from contamination by
dust “anomalous microwave emission” (AME) and free-free emission, which are
hard to separate from synchrotron by reason of lack of observing frequencies.
Spectral index maps obtained from polarisation data, negligibly impacted by
AME and free-free which are at most very faintly polarised, are either
contaminated by large scale systematic effects due to intensity to
polarisation leakage, or suffer from Faraday rotation effects and/or poor
signal to noise ratio.

For the work presented here, we assume a fixed synchrotron spectral index
$\beta_{s}=-3.1$. Table 1 gives the frequency scalings of synchrotron for
several values of the spectral index, normalised to unity at 30 GHz. Relative
differences from 23 GHz to 44 GHz are in the few per cent range. Errors made
by using a fixed spectral index are thus of the order of a few per cent of the
total emission. Further discussion of the impact of this choice is deferred to
Sec. 4.1.

##  3 Method

Figure 1: Noise standard deviation maps for the three WMAP channels used in
this work (K, Ka and Q), re-scaled to equivalent 30 GHz noise assuming a
synchrotron spectral index of $-3.1$. Units are in $\mu$Kcmb. Each map
displays the average of the standard deviation of the Q and U Stokes parameter
maps.

Figure 2: Noise standard deviation maps for the two Planck LFI channels used
in this work (30 GHz and 44 GHz), re-scaled to equivalent 30 GHz noise
assuming a synchrotron spectral index of $-3.1$. Units are in $\mu$Kcmb. Each
map displays the average of the standard deviation of the Q and U Stokes
parameter maps. Note, here and in Fig. 1, the difference in color scale for
the different frequency channels, and the variablilty of the noise level
across the sky. The map depth varies by factors of a few across the sky, and
the lowest noise regions are different between the maps. This calls for pixel-
dependent weights for map combination.

The present works aims at combining WMAP and Planck synchrotron observations
for increased signal to noise ratio. For both WMAP and Planck, the noise level
strongly depends on the sky pixel considered (Fig. 1 and 2).

Figure 3: Beam-corrected noise spectra for the WMAP and Planck channels used
in this work, re-scaled to equivalent 30 GHz noise assuming a synchrotron
spectral index of $-3.1$. The relative sensitivity of the different channels
strongly depends on the harmonic scale $\ell$. On the largest scales, the WMAP
K band and the LFI 30 GHz channels have comparable mean synchrotron
sensitivity (with, however, a strong pixel dependence, as seen in Figs. 1 and
2. At $\ell\simeq 800$, however the WMAP Q and LFI 44 GHz are about as
sensitive as the LFI 30 GHz channel, and much more sensitive than the WMAP K
band.

Figure 4: Cosine-shaped needlet bands $h_{J}(\ell)$ used in this work. As
usual, those are normalised so that $\sum_{J}h_{J}(\ell)^{2}=1$ for a useful
range of $\ell$. Here, this normalisation holds up to $\ell=950$. For higher
$\ell$, the sum goes smoothly to zero at $\ell_{\rm max}=1000$ with a cosine-
square half-arch.

By reason of frequency-dependent angular resolution, it also depends on
angular scale (Fig. 3). Finally, it depends on the Stokes parameter being
considered, $Q$ or $U$. We hence propose to combine the observations using
relative weights that depend on sky direction, harmonic mode, and polarisation
orientation. This is done using a decomposition of the maps onto a needlet
frame. The so-called “needlets” are a kind of wavelets defined on the sphere.
For our purpose, needlet decomposition is performed by filtering the original
maps in a set of harmonic windows $h_{J}(\ell)$, where $J$ indexes the scale
and $\ell$ is the harmonic mode of spherical harmonics transforms. These
needlet windows are normalised so that

$\sum_{J}h_{J}(\ell)^{2}=1$ (2)

over a useful range of $\ell$. Maps of needlet coefficients are obtained by an
inverse Spherical Harmonic Transform (SHT) on the filtered harmonic
coefficients. Map reconstruction from the needlets is done by filtering again
each needlet coefficient map by the corresponding spectral window, co-addition
in harmonic space, and an inverse SHT of the recombined $a_{\ell m}$
coefficients.

Stokes parameters $Q$ and $U$ maps are defined with respect to a set of axes.
They are not invariant with a transformation of those local reference axes.
For this reason, they are not scalar quantities, but spin-2 fields. In CMB
data analysis, it is customary to decompose polarisation maps into an
alternate basis, the $E$ and $B$ polarisation modes, respectively scalar (even
parity) and pseudo-scalar (odd parity). In previous work using needlets for
polarisation, needlet windows have been used to filter $E$ and $B$ fields,
which are then transformed into needlet coefficient maps for $E$ and $B$,
processed, and then recombined. While this makes sense for the analysis of CMB
maps, for which $E$ modes and $B$ modes are a natural decomposition, with $E$
modes outshining $B$ modes by orders of magnitude, it is not the case for maps
of Galactic foreground components, for which $E$ and $B$ modes are of the same
order of magnitude and have no special theoretical justification. On the
contrary, the transformation of $Q$ and $U$ into $E$ and $B$ is non-local
(while foreground emission is), and also sub-optimally smoothes the
inhomogeneity of the noise level (the standard deviation is pixel-dependent).
Moreover, the noise level is different in $Q$ and $U$ maps, as it depends on
the orientation of the detectors as they scan through the sky pixels, a
property that should be taken into account for the use of optimal weights for
the production of low noise maps.

In this work, noise-weighted combinations are performed pixel by pixel in
needlet coefficient maps, but instead of working on needlet coefficients for
$E$ and $B$ as done in previous work (Basak & Delabrouille, 2013; Aurlien et
al., 2023; Errard et al., 2022), we transform back $E$ and $B$ maps filtered
by the spectral windows $h_{J}(\ell)$ into maps of $Q$ and $U$, make noise-
weighted combinations of those “special needlets” in pixel space, transform
them back into $E$ and $B$ $a_{\ell m}$s which are then filtered again by the
spectral windows and co-added to produce $Q$ and $U$ synchrotron maps with
near-optimal signal to noise ratio.

The specific data processing pipeline set up in this work is summarised as
follows:

  1. 1.

Planck 30 GHz and 44 GHZ frequency maps, and WMAP K, Ka and Q maps are
smoothed to 60′ angular resolution in harmonic space;

  2. 2.

CMB polarisation is subtracted from all frequency maps to minimise residual
CMB in the maps;

  3. 3.

All maps are rescaled to equivalent synchrotron emission at the 30 GHz
reference frequency, assuming a power law frequency scaling with spectral
index $\beta_{s}=-3.1$;

  4. 4.

We compute needlet coefficients by filtering $E$ and $B$ $a_{\ell m}$s using a
set of 20 bands spanning the $\ell=2$–1000 range of harmonic modes, and
producing corresponding maps of $Q$ and $U$ needlets at nside=512;

  5. 5.

We compute the noise level of each frequency channel in this needlet space, as
a function of scale and sky pixel, and get inverse variance weighted averages
for $Q$ and $U$ in each needlet band;

  6. 6.

Those needlet coefficients are then used to reconstruct a single map of
synchrotron $Q$ and $U$ polarisation at 60′ angular resolution, which can be
used as a template for polarised synchrotron emission at 30 GHz;

  7. 7.

Two hundred maps of simulated noise for the end-product are obtained by co-
adding WMAP and Planck noise simulations with the weights used in the
production of the polarised synchrotron maps.

Each of these steps is further detailed in the following sub-sections.

###  3.1 Step 1: Map smoothing

Original WMAP and Planck maps used in this work have angular resolutions
ranging from about 27.9′ (for the Planck LFI 44 GHz band, with the best
resolution of the five) to about 52.8′ (for the WMAP K band). As a first step,
we transform original maps into spherical harmonics, and filter the map
$a_{\ell m}$s by the ratio of the $B_{\ell}$ of a 60′ beam and the $B_{\ell}$
of the original map. For WMAP we use tabulated beam $B_{\ell}$s provided with
the WMAP DR5. For the Planck LFI, we assume the original beams are 33.1′ and
27.9′ for the 30 GHz and 44 GHz respectively. Small differences between those
assumed values and the actual beams (which are known to be somewhat
elliptical) have little impact on the final products, for which high-$\ell$
power is further suppressed.

The choice of 60′ as the final resolution is motivated by an inspection of the
power spectrum of the initial maps, which is noise dominated above values of
$\ell$ ranging from $\ell=200$ at best (for near-full sky LFI 30 GHz maps) to
$\ell=20$-30 at worst (for high Galactic latitude maps at 44 GHz and in the
WMAP Q band. The whole pipeline has been run also with 50′ and 40′ target
resolution, but the maps produced are noisier and contain little extra signal
information in regions of faint emission.222Those maps can be made available
upon request, but are not part of the default data products made available
with this paper.

Figure 5: Weights used for the reconstruction of the synchrotron $Q$ (top two)
and $U$ (bottom two) synchrotron polarisation maps in the first needlet band.
Only the weights of the WMAP K and the Planck LFI 30 GHz maps are shown, as
only those are significant for that scale (and add-up to almost 1). Note the
differences between $Q$ and $U$ weights.

Figure 6: Stokes Q (top) and U (bottom) 30 GHz polarisation maps obtained with
the pipeline described in this paper, at one degree angular resolution.

Figure 7: Beam-corrected noise spectra for the WMAP and Planck channels used
in this work, re-scaled to equivalent 30 GHz noise assuming a synchrotron
spectral index of $-3.1$, as in Fig. 3. An extra black curve shows the noise
spectrum of the final composite synchrotron map obtained in this work, which
near-optimally combines all observations.

###  3.2 Step 2: CMB subtraction

All of the maps used in this work contain some CMB polarisation. This CMB
“contamination” is sub-dominant as compared to synchrotron emission and to
noise contamination, but for the sake of optimality we subtract from each of
the maps a Wiener-filtered version of the SMICA polarisation maps.
Multivariate Wiener filtering is performed assuming theoretical CMB intensity
and polarisation spectra corresponding to the best-fit Planck cosmological
model. The multivariate spectrum of the noisy CMB maps is computed directly on
the SMICA maps. Subtracting a Wiener-filtered CMB (rather than the SMICA CMB
polarisation maps themselves) ensures that CMB subtraction does not add more
noise than subtracts CMB. The residual noise in the maps is subdominant and,
since it originates predominantly from HFI observations, is not significantly
correlated with original noise in the WMAP and LFI channels. Its contribution,
as well as that of the residual CMB, can safely be neglected in the
computation of the weights to be used for the production of the synchrotron
map.

We do not describe this step in more detail as it has almost no impact on the
final products. Synchrotron $Q$ and $U$ maps have been produced with pipelines
including or omitting CMB subtraction. Differences are imperceptible. Still,
one should be aware that there is low-level residual CMB in the final products
even for the pipeline that includes CMB subtraction (the difference between
the Wiener filtered CMB and the original CMB is present in the maps). This
residual can be characterised precisely if necessary for specific analyses.
333Residuals are small, and for most uses can be safely neglected. In case of
doubt contact the author. .

###  3.3 Step 3: Rescaling to 30 GHz reference frequency

After the two previous steps, we are left with five noisy synchrotron-
dominated sky maps, all of which share the effective angular resolution of a
1-degree Gaussian beam over their common range of used $\ell$ values. For each
of them, the useful $\ell$ range is restricted by the availability or by the
accuracy of the instrumental effective beam window functions. We use
$\ell_{\rm max}=[700,750,1000,850,1000]$ for WMAP K, Ka, Q, and LFI 30 GHz and
44 GHz respectively.

We then use the WMAP and LFI bandpasses to integrate a single power law with
spectral index of -3.1 (in KRJ units) and use the corresponding synchrotron
bandpass-integrated coefficients to rescale each map to equivalent synchrotron
emission at 30 GHz (Table 1). This rescaling, by reason of variation of the
synchrotron spectral index (or more generally, the synchrotron frequency
scaling) across the sky, is slightly inaccurate. We evaluate the amplitude of
the errors by computing the scaling coefficients for spectral indices ranging
from -2.9 to -3.3. Differences of a few per cent at most across frequency
channels guarantee that scaling errors are at most a few per cent of the total
signal (and in practice less, taking averaging between input maps on both
sides of the reference frequency into account). These errors are below the
noise over most of the sky, the only exception being regions of emission with
signal to noise ratio of a few tens or more (in amplitude), for which there is
no need to use the synchrotron maps produced in this paper rather than the
original observations. We evaluate uncertainties more rigorously in Sec. 4.

###  3.4 Step 4: Needlet coefficients

CMB-corrected maps, smoothed to 60′ resolution and scaled to 30 GHz
synchrotron assuming a $\beta_{s}=-3.1$ spectral index are filtered in a set
of cosine-shaped “needlet bands”, shown in Fig. 4. Those are chosen to be wide
enough to allow for localisation on the scale of about a degree, allowing for
adapting weights to map depth inhomogeneities on that scale. They are narrow
enough for the signal to noise ratio of the various channels not to vary too
much in each band.

In practice, $Q$ and $U$ maps in each channel are transformed into $E$ and $B$
$a_{\ell m}$s, which are then multiplied by the windows $h_{J}(\ell)$, and
transformed back using an inverse polarised SHT to produce a number $N_{J}$ of
each of the $Q$ and $U$ needlet coefficient maps.

###  3.5 Step 5: Inverse noise variance weighting

For each “needlet” map of $Q$ and $U$ Stokes parameters, we estimate the local
noise variance integrated in the band. This is done by multiplying the
original noise level in map space by a re-scaling factor that takes into
account noise reduction in the band by reason of map smoothing (i.e. taking
into account the ratio of the beam $B_{\ell}$s squared, integrated in the
needlet band. We then co-add the needlet maps for the various channels with
weights proportional to $1/\sigma^{2}$, normalised for unit response. This
gives us 20 maps of needlet coefficients for each of $Q$ and $U$, which near-
optimally combine the five original WMAP and LFI $Q$ and $U$ needlet
coefficients for that scale into one single map of synchrotron emission
needlet coefficients for each of $Q$ and $U$ at 30 GHz.

The relative weights for the various channels depend on the needlet scale
(WMAP K and LFI 30 GHz dominate the weights on large scale, while other
channels contribute at smaller scale), on the sky pixel, and on the Stokes
parameter (weights are different for $Q$ and $U$, because the noise levels in
each of those depend on the orientation of the scanning with respect to local
polarisation reference axes.

For illustration purposes, weights for the WMAP K band and for the LFI 30 GHz
channel, for the first needlet scale, are shown in Fig. 5. In regions near the
ecliptic poles, most of the weight is given to the LFI 30 GHz map, while over
a large fraction of the sky, the WMAP K channel has a larger weight in the
reconstruction.

###  3.6 Step 6: Coaddition

Maps of needlet coefficients for $Q$ and $U$ are straightforwardly recombined
into co-added maps as usual: polarised SHTs are performed to transform needlet
$Q$ and $U$ maps into $E$ and $B$ $a_{\ell m}$s for each band. Those $a_{\ell
m}$s are multiplied each by the appropriate needlet window $h_{J}(\ell)$.
Those are then coadded before a final inverse SHT is performed to get final
$Q$ and $U$ synchrotron maps with all scales co-added.

Figure 8: Synchrotron and noise polarised intensity as obtained in this work.
Top: Total polarised intensity $P=\sqrt{Q^{2}+U^{2}}$ of the synchrotron map.
Middle: Noise contribution, computed from 200 noise simulations. Increased map
depth is visible both in regions of low Planck HFI noise near the ecliptic
poles, and along the deeper integration rings of the WMAP scanning. Bottom:
Synchrotron polarised intensity after de-biasing from the noise contribution.
In the cleanest sky regions, pixels where
$(Q^{2}+U^{2})-(\sigma_{Q}^{2}+\sigma_{U}^{2})$ is negative appear in grey.

Figure 9: Masks used for computing power spectra shown in Fig. 12. In the top
panel, the low, medium, and high-amplitude regions of emission are shown in
black, dark grey, and light grey respectively. The white region, located in
the inner Galactic ridge, corresponds to the strongest region of polarised
synchrotron emission, and is excluded for all power spectra computations. The
bottom panel shows a mask for compact sources.

Figure 10: Impact of spectral index uncertainty on the polarised intensity
maps. Top: Standard deviation of 50 maps obtained with $\beta_{s}$ drawn at
random with a Gaussian distribution centered on $\beta_{s}=-3.1$ and with a
standard deviation of 0.1. Bottom: Sum in quadrature of the above error and of
the noise standard deviation. Figure 11: Ratio of cosmoglobe to Planck-
Revisited $EE$ and $BB$ power spectra over the region of sky with highest
emission.

Figure 12: Power spectra of polarised synchrotron emission in three different
sky regions, selected using the masks outlined in Fig. 9. The data is plotted
after smoothing the spectra with a $\Delta\ell=3$ window. In the high and
medium emission regions (top and middle panels), spectra are strongly signal-
dominated. There is excellent agreement between our maps and the cosmoglobe
DR1 maps up to $\ell\simeq 30$, above which the cosmoglobe spectra are
somewhat lower. At scales where signal dominates over noise, this discrepancy,
increasing with increasing $\ell$, must be due to different effective beams.
In the low emission region (bottom panel), of specific interest for future
ground-based observations of CMB B-modes, the cosmoglobe polarisation maps
have significantly higher power for $50<\ell<100$ than our maps. This is
indicative of higher noise contamination, while the lower power of the
cosmoglobe maps at higher $\ell$ likely indicates larger effective beam. For
the medium and low contamination regions, we show the noise-debiased power
spectra of our synchrotron polarisation maps with “$+$” data points. Noise
spectra are computed from 200 simulations of WMAP and Planck LFI noise,
propagated through the analysis pipeline.

###  3.7 Step 7: Noise simulations

Synchrotron maps obtained following the previous steps comprise noise from
WMAP and Planck observations, combined in a non-trivial way (non stationary,
scale dependent, polarisation orientation dependent contribution from the
various channels). This noise is not easily characterised analytically, as it
is also correlated between pixels by reason of smoothing of the original
observations. Two hundred simulated noise maps are obtained by projecting WMAP
and Planck LFI simulated noise using the weights used for the various channels
as a function of sky pixel and scale. For each simulation, random noise is
generated using pixel-based covariance matrices for the $I$, $Q$ and $U$ noise
provided by the WMAP and Planck collaborations, produced at the native nside
and angular resolution. These noise maps are then smoothed to 60′ angular
resolution, rescaled to 30 GHz using the synchrotron frequency scalings,
decomposed into $Q$ and $U$ needlets, weighted and recombined into a single
noise realisation. The statistics across the 200 realisations are sufficient
to characterise noise contamination properties with a few per cent error, and
global statistics (i.e. noise rms, spectrum) with sub percent accuracy.

##  4 Results, discussion and data products

The main final data product is a pair of estimated $Q$ and $U$ synchrotron
emission maps at 30 GHz, with a one degree Gaussian beam angular resolution.
Maps of $Q$ and $U$ are shown in Fig. 6. The co-addition of the original WMAP
and Planck LFI observations using weights that depend on sky pixel and on
angular scale for each of $Q$ and $U$ independently allows for significantly
improved signal to noise ratio as compared to any of the original input maps.

Figure 7 compares the $E$ and $B$ noise spectra of the final synchrotron
polarisation obtained in this work to the original noise level of the input
WMAP and Planck LFI channels (all re-scaled to 30 GHz assuming a synchrotron
spectral index $\beta_{s}=-3.1$). As expected, the noise of our final
synchrotron maps is lower than any of the noise spectra of the input maps used
in this analysis.

Figure 8 shows a map of synchrotron polarised amplitude, together with a map
of noise standard deviation obtained from 200 simulated noise maps. Low-noise
regions corresponding to both WMAP and Planck deeper integration zones are
clearly visible in the standard deviation map, showing how the present method
takes the best of the various observations used.

No specific attempt has been made to subtract point source emission in the
initial maps nor in the final polarisation maps. Well known bright radio
sources can be identified in the final map. This is the case of supernova
remnants in our own Galaxy such as Tau A and Cas A; of bright radio-galaxies
such as Cen A, Fornax A and Virgo A; of bright distant quasars such as 3C273
and 3C279. A catalogue of polarised radio sources detected with Planck is
available for identification and masking. Note that since the emissions from
these sources do not necessarily scale with frequency with a power law with a
-3.1 spectral index, their flux in the maps is not expected to correspond to
their true 30 GHz emission. For convenience, a compact source mask (shown in
Fig. 9) is made available as part of the data products.

We anticipate that the present maps can be used as synchrotron polarisation
$Q$ and $U$ templates for foreground cleaning in upcoming CMB observations, in
particular at few degree angular scales for detecting primordial B-modes of
CMB polarisation. They will also be useful for making new models of Galactic
foregrounds to be used in sky simulation tools such as the PSM (Delabrouille
et al., 2013) and the PySM (Thorne et al., 2017; Zonca et al., 2021), to be
used for the development of data analysis pipelines in preparation of future
observations at millimeter to centimeter wavelengths. The good
characterisation of the maps in terms of effective beam and the availability
of 200 noise simulations will make it easy to combine the map with independent
observations for yet-improved polarised synchrotron emission maps.

###  4.1 Impact of the choice of a spectral index

This work produces synchrotron polarisation $Q$ and $U$ maps assuming a fixed
synchrotron spectral index $\beta_{s}=-3.1$ across the sky. We evaluate the
impact of the variability of $\beta_{s}$, and of the uncertainty on its actual
value, by repeating the analysis 50 times, with spectral indices drawn at
random with a Gaussian distribution centered on $\beta_{s}=-3.1$ and with a
standard deviation of 0.1. The standard deviation of the output polarised
intensity map is shown in the top panel of Fig. 10. As expected, this error is
larger in the regions of higher total polarised emission, and is subdominant
over most of the sky, except in the Galactic plane and towards bright compact
sources.

###  4.2 Comparison with other polarised synchrotron maps

Full-sky synchrotron polarization maps from Planck data alone have been
obtained for the Planck 2018 data release using the SMICA (Delabrouille et
al., 2003; Cardoso et al., 2008) and the Commander (Eriksen et al., 2004,
2008; Seljebotn et al., 2019) component separation pipelines (Planck
Collaboration et al., 2020d). In both cases however, only Planck data have
been used, resulting in a noise penalty with respect to the composite
polarised synchrotron polarisation maps presented here. Our “Planck-revisited”
maps are substantially better in terms of noise contamination, and of
characterisation of the effective angular resolution. As all data products are
publicly available, we leave further comparison of our maps with those data
products to the reader.

More interesting is the comparison with maps of synchrotron amplitude and
spectral index obtained from a combined analysis of Planck and WMAP data
recently been published by Watts et al. (2023), which are also available for
download on a dedicated web site.444 cosmoglobe.uio.no/products/cosmoglobe-
dr1.html

Those synchrotron polarisation $Q$ and $U$ maps are produced nominally at 60′
angular resolution, but a smoothing prior of 30′ applied during the fitting,
and an additional one degree smoothing added when taking the mean over the
Gibbs chain, are susceptible to further degrade the effective angular
resolution of the cosmoglobe maps.

We estimate the cosmoglobe map angular resolution by computing the ratio

$B_{\ell}^{2}=\frac{C^{\rm CG}_{\ell,1\times 2}}{C_{\ell}^{\rm
PR}-N_{\ell}^{\rm PR}},$ (3)

where $C^{\rm CG}_{\ell,1\times 2}$ is the cross-spectrum of the HM1 and HM2
cosmoglobe maps, and $C_{\ell}^{\rm PR}-N_{\ell}^{\rm PR}$ is the noise-
debiased auto-spectrum of our Planck-Revisited map. The ratio, which is
roughly compatible with extra $30^{\prime}$ smoothing in the cosmoglobe maps,
is displayed in Fig. 11 for the most interesting $\ell$-range for the
detection of primordial CMB B-modes.

The noise level in the total polarisation amplitude,

$\sigma_{P}=\sqrt{\sigma_{Q}^{2}+\sigma_{U}^{2}},$ (4)

is 3.4 $\mu$KRJ in the cosmoglobe DR1. For comparison, the noise standard
deviation obtained with the present analysis is of about 2.78 $\mu$KRJ at 60′
resolution. When the map is further smoothed with a 30′ beam to match the
cosmoglobe map effective resolution, as detemined above, this noise standard
deviation reduces to 2.32 $\mu$KRJ, 32% lower than that of the cosmoglobe DR1
maps. The low noise contamination we achieve in the Planck-Revisited
synchrotron map is mostly due to efficient co-addition of the input maps
across scales, pixels and as a function of the Stokes parameter being
considered, $Q$ or $U$.

Power spectra for three different sky regions, corresponding to low, medium,
and high polarised synchrotron emission amplitude (using the masks displayed
in Fig. 9), are shown in Fig. 12, together with power spectra computed from
the cosmoglobe DR1 data products. For each region, the mask selecting the
level of contamination is apodised with a 2-degree cosine-squared transition,
and point sources are masked using the mask shown in the bottom panel of Fig.
9, apodised with a 1-degree cosine-squared transition.

Overall, there is good agreement between the two data products at low $\ell$.
For intermediate $\ell$, the cosmoglobe synchrotron maps are noisier. At high
$\ell$, the overall power in the cosmoglobe maps is lower, indicative of extra
smoothing, as already determined above.

###  4.3 Data products

“Planck Revisited” data products are made available on a dedicated
website.555https://portal.nersc.gov/project/cmb/Planck_Revisited Data products
specific to this work comprise:

  1. 1.

Maps of synchrotron $Q$ and $U$ Stokes parameters as obtained with the method
described in Sec. 3, at 60′ resolution (maps at higher resolution can be made
available by the author upon request);

  2. 2.

Masks dividing the sky in three regions of different synchrotron polarised
amplitude (high, medium, and low polarised synchrotron emission);

  3. 3.

A mask for strong polarised compact sources, which can be completed if
necessary using the Planck catalogue of non-thermal source catalogue (Planck
Collaboration et al., 2018);

  4. 4.

Maps of simulated noise for the 60′ resolution synchrotron $Q$ and $U$
polarisation data products (additional simulations can be made available if
necessary).

No attempt has been made to correct for residual systematics in the maps, nor
fit for the scaling of synchrotron emission as a function of sky direction.
For these reasons, caution must be used when exploiting our maps in regions of
strong emission, where the calibration uncertainty due to scaling with
frequency, as well as potential intensity to polarisation leakage, can
generate errors larger than the map noise. At very low $\ell$ (e.g. below
$\ell=20$), systematics in the original data products are expected to project
onto our synchrotron maps. However, in regions where the polarised synchrotron
emission is low and for $\ell$-ranging from 20 to 100, our maps are the most
accurate full-sky tracers of polarised synchrotron emission to date. In
particular, they can be used as polarised synchrotron templates for upcoming
searches, in low-foreground sky regions, of a recombination peak (around
$\ell=80$) in CMB primordial B-modes.

##  5 Conclusion

In this paper, we have applied a simple, yet effective pipeline to combine
WMAP K, Ka and Q maps and Planck LFI 30 GHz and 44 GHz maps into a single set
of $Q$ and $U$ synchrotron polarisation maps at 30 GHz. The effective angular
resolution of the maps is that of a 60′ Gaussian beam, with a smooth cosine-
square-shaped apodisation from $\ell=950$ to $\ell=1000$. Additive errors from
instrumental noise are evaluated with two hundred noise simulations propagated
through the analysis pipeline. Multiplicative errors from the assumption of a
uniform synchrotron emission law with $\beta_{s}=-3.1$, evaluated with a
Monte-Carlo pipeline over a distribution of spectral indices, are evaluated
and shown to be sub-dominant over most of the sky.

The maps and noise simulations are made available to the scientific community
on a dedicated data server, accessible from the web.

###### Acknowledgements.

Some of the results in this paper have been derived using the HEALPix
pixellisation scheme (Górski et al., 2005). The author thanks Duncan Watts for
useful discussions regarding comparison with the cosmoglobe analysis and DR1
data products.

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