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# The Galactic Faraday rotation sky 2020

Sebastian Hutschenreuter The Galactic Faraday rotation sky 2020The Galactic
Faraday rotation sky 2020The Galactic Faraday rotation sky 2020The Galactic
Faraday rotation sky 2020The Galactic Faraday rotation sky 2020The Galactic
Faraday rotation sky 2020 Craig S. Anderson The Galactic Faraday rotation sky
2020The Galactic Faraday rotation sky 2020 Sarah Betti The Galactic Faraday
rotation sky 2020The Galactic Faraday rotation sky 2020 Geoffrey C. Bower The
Galactic Faraday rotation sky 2020The Galactic Faraday rotation sky 2020 Jo-
Anne Brown The Galactic Faraday rotation sky 2020The Galactic Faraday rotation
sky 2020 Marcus Brüggen The Galactic Faraday rotation sky 2020The Galactic
Faraday rotation sky 2020 Ettore Carretti The Galactic Faraday rotation sky
2020The Galactic Faraday rotation sky 2020 Tracy Clarke The Galactic Faraday
rotation sky 2020The Galactic Faraday rotation sky 2020 Andrew Clegg The
Galactic Faraday rotation sky 2020The Galactic Faraday rotation sky 2020
Allison Costa The Galactic Faraday rotation sky 2020The Galactic Faraday
rotation sky 2020 Steve Croft The Galactic Faraday rotation sky 2020The
Galactic Faraday rotation sky 2020The Galactic Faraday rotation sky 2020The
Galactic Faraday rotation sky 2020 Cameron Van Eck The Galactic Faraday
rotation sky 2020The Galactic Faraday rotation sky 2020 B. M. Gaensler The
Galactic Faraday rotation sky 2020The Galactic Faraday rotation sky 2020
Francesco de Gasperin The Galactic Faraday rotation sky 2020The Galactic
Faraday rotation sky 2020The Galactic Faraday rotation sky 2020The Galactic
Faraday rotation sky 2020 Marijke Haverkorn The Galactic Faraday rotation sky
2020The Galactic Faraday rotation sky 2020 George Heald The Galactic Faraday
rotation sky 2020The Galactic Faraday rotation sky 2020 Charles L.H. Hull The
Galactic Faraday rotation sky 2020The Galactic Faraday rotation sky 2020The
Galactic Faraday rotation sky 2020The Galactic Faraday rotation sky 2020The
Galactic Faraday rotation sky 2020The Galactic Faraday rotation sky 2020
Makoto Inoue The Galactic Faraday rotation sky 2020The Galactic Faraday
rotation sky 2020 Melanie Johnston-Hollitt The Galactic Faraday rotation sky
2020The Galactic Faraday rotation sky 2020 Jane Kaczmarek The Galactic Faraday
rotation sky 2020The Galactic Faraday rotation sky 2020 Casey Law The Galactic
Faraday rotation sky 2020The Galactic Faraday rotation sky 2020 Yik Ki Ma The
Galactic Faraday rotation sky 2020The Galactic Faraday rotation sky 2020The
Galactic Faraday rotation sky 2020The Galactic Faraday rotation sky 2020 David
MacMahon The Galactic Faraday rotation sky 2020The Galactic Faraday rotation
sky 2020 Sui Ann Mao The Galactic Faraday rotation sky 2020The Galactic
Faraday rotation sky 2020 Christopher Riseley The Galactic Faraday rotation
sky 2020The Galactic Faraday rotation sky 2020The Galactic Faraday rotation
sky 2020The Galactic Faraday rotation sky 2020 Subhashis Roy The Galactic
Faraday rotation sky 2020The Galactic Faraday rotation sky 2020 Russell
Shanahan The Galactic Faraday rotation sky 2020The Galactic Faraday rotation
sky 2020 Timothy Shimwell The Galactic Faraday rotation sky 2020The Galactic
Faraday rotation sky 2020The Galactic Faraday rotation sky 2020The Galactic
Faraday rotation sky 2020 Jeroen Stil The Galactic Faraday rotation sky
2020The Galactic Faraday rotation sky 2020 Charlotte Sobey The Galactic
Faraday rotation sky 2020The Galactic Faraday rotation sky 2020 Shane P.
O’Sullivan The Galactic Faraday rotation sky 2020The Galactic Faraday rotation
sky 2020 Cyril Tasse The Galactic Faraday rotation sky 2020The Galactic
Faraday rotation sky 2020The Galactic Faraday rotation sky 2020The Galactic
Faraday rotation sky 2020 Valentina Vacca The Galactic Faraday rotation sky
2020The Galactic Faraday rotation sky 2020 Tessa Vernstrom The Galactic
Faraday rotation sky 2020The Galactic Faraday rotation sky 2020 Peter K.G.
Williams The Galactic Faraday rotation sky 2020The Galactic Faraday rotation
sky 2020 Melvyn Wright The Galactic Faraday rotation sky 2020The Galactic
Faraday rotation sky 2020 Torsten A. Enßlin The Galactic Faraday rotation sky
2020The Galactic Faraday rotation sky 2020The Galactic Faraday rotation sky
2020The Galactic Faraday rotation sky 2020

###### Abstract

Aims. This work gives an update to existing reconstructions of the Galactic
Faraday rotation sky by processing almost all Faraday rotation data sets
available at the end of the year 2020. Observations of extra-Galactic sources
in recent years have, among other regions, further illuminated the previously
under-constrained southern celestial sky, as well as parts of the inner disc
of the Milky Way. This has culminated in an all-sky data set of 55,190 data
points, which is a significant expansion on the 41,330 used in previous works,
hence making an updated separation of the Galactic component a promising
venture. The increased source density allows us to present our results in a
resolution of about $1.3\cdot 10^{-2}\,\mathrm{deg}^{2}$
($46.8\,\mathrm{arcmin}^{2}$), which is a twofold increase compared to
previous works.

Methods.  As for previous Faraday rotation sky reconstructions, this work is
based on information field theory, a Bayesian inference scheme for field-like
quantities which handles noisy and incomplete data.

Results.  In contrast to previous reconstructions, we find a significantly
thinner and pronounced Galactic disc with small-scale structures exceeding
values of several thousand $\mathrm{rad}\,\mathrm{m}^{-2}$. The improvements
can mainly be attributed to the new catalog of Faraday data, but are also
supported by advances in correlation structure modeling within numerical
information field theory. We furthermore give a detailed discussion on
statistical properties of the Faraday rotation sky and investigate
correlations to other data sets.

##  1 Introduction

The Faraday effect describes the rotation of the polarization position angle
propagating through a magnetized plasma, and provides information on the line-
of-sight (LOS) component of magnetic fields weighted by the thermal electron
density. These quantities are key puzzle pieces for the characterization and
modeling of the structure of the Milky Way’s magnetic field and of many extra-
Galactic astrophysical objects, such as galaxies (Beck 2015) and galaxy
clusters (Durrer & Neronov 2013). A better understanding of these structures
is not only an interesting topic by its own merit but may also provide a
pathway in understanding the origin of the magnetic field in the Universe as a
whole (Subramanian 2016). A primary objective of research on cosmic magnetism
has therefore been the investigation of the polarimetric properties of
peculiar extra-Galactic objects, in the hope to obtain insights on their
morphology and formation history.

Over the years, this pursuit has led to a dense coverage of Faraday rotation
data across the whole sky, which have been cataloged in exhaustive
compilations (Oppermann et al. 2012; Van Eck et al. in prep.). The
characteristics of this data not only depend on the properties of the objects
in question but on the integral of all environments through which the light
has passed upon arriving at the telescope. This leads to a potential
intertwining of information of completely different astrophysical
environments, such as the intergalactic and interstellar media. In particular,
the plasma of our Galaxy is responsible for a significant amount of Faraday
rotation, see Cooper & Price (1962) or Gardner & Davies (1966) for early
detections of the Galactic Faraday rotation contribution and Haverkorn (2015)
and Beck (2015) for reviews.

In order to disentangle the contributions to Faraday rotation from multiple
sources, reconstruction procedures have been developed. Specifically in the
case of the Galactic Faraday rotation sky, a useful measure for discerning
Galactic and extra-Galactic components is the similarity of RM values of
nearby sources on the sky. One expects that large angular correlations in
Faraday data more likely result from local effects, implying that data
resulting from extra-Galactic processes should be mostly uncorrelated on
angular scales of arc-minutes or larger (Akahori & Ryu 2010). Exceptions to
this might be caused by neighboring parts of the cosmic large-scale structure
or large-scale magnetic fields in the inter-galactic medium in proximity to
the Milky Way Xu et al. (2006). Past all-sky methods to reconstruct the
Galactic Faraday sky were developed for instance by Frick et al. (2001);
Johnston-Hollitt et al. (2004); Dineen & Coles (2005); Xu et al. (2006); Short
et al. (2007); Pshirkov et al. (2011); Oppermann et al. (2012); Xu & Han
(2014).

In this paper, we are continuing the work of Oppermann et al. (2012),
Oppermann et al. (2015) and Hutschenreuter & Enßlin (2020). In the latter
reference, henceforth abbreviated with HE20, the authors introduce two
inference models. The simpler one is a generalization of the model used by
Oppermann et al. (2012), while the more complicated one additionally restricts
the amplitude of the Faraday sky with data on the emission measure (EM) of
thermal electrons, as obtained from the Planck satellite (Planck Collaboration
et al. 2016). These works use Bayesian inference schemes, which utilize the
whole-sky correlation structure of the Galactic part of the Faraday rotation
sky, in order to (a) perform the aforementioned component separation and (b)
to interpolate between data points on the sky. As the precise form of the
correlation structure (which is necessary for the interpolation) is unknown,
it is also inferred jointly with the sky map.

The aim of this paper is to provide the community with the most recent version
of the Galactic Faraday sky with a minimal set of physical assumptions in
unprecedented resolution and to draw attention to previously under-constrained
sky regions, such as the Magellanic Clouds or peculiar substructures of the
Galactic disc. The more complicated inferences using the algorithms developed
in HE20 have also been repeated. However, discussing these results, in
particular in light of the many options that exist for the inclusion of new
data sets and modeling assumptions, exceeds the scope of this paper, which is
to provide a methodologically clean update on the Faraday rotation sky with a
specific focus on high resolution, specific Galactic structures and
statistical properties of the Faraday rotation sky. Thus, we perform the same
analysis as in HE20, but are restricting ourselves to the model which uses
Faraday data only. An accompanying publication is in preparation
(Hutschenreuter & Ensslin in prep.), which focuses on the aforementioned more
complicated models and focuses on the analysis of the component maps. This
present publication should therefore be regarded as the primary reference for
the updated Faraday rotation sky, whereas the other as providing a physical
interpretation of the component maps, which depends more on assumptions. The
resulting posterior mean and uncertainty maps for the Faraday rotation sky are
available for download either as full-sky maps111https://wwwmpa.mpa-
garching.mpg.de/~ensslin/research/data/faraday2020.html or via a cutout
server222CIRADA cutout server.

We structure the paper as following: Section 2 summarizes the relevant physics
and the modeling. Section 3 describes the data used in this work. Section 4
discusses the results and Section 5 gives a conclusive summary.

-250250$\displaystyle d_{\phi}\left[\mathrm{rad}\,\mathrm{m}^{-2}\right]$ (a)

030$\displaystyle\sigma_{d_{\phi}}\left[\mathrm{rad}\,\mathrm{m}^{-2}\right]$
(b)

015counts (c)

Figure 1:  Sky projections of the data set used in this work, the associated
observational errors and the source density. Figure 1(a) shows the Faraday
data, Fig. 1(b) the observed error bars. Data points falling into the same
pixel have been averaged. These plot has been made with a four-times decreased
resolution (corresponding to an pixel area of $0.21\,\mathrm{deg}^{2}$ or
$7.56\cdot 10^{2}\,\mathrm{arcmin}^{2}$ ) compared to that of the inference in
order to enable better visualization. Figure 1(c) shows the source density of
the data set used in this work. The pixels in this map have an area of about
$3.36\,\mathrm{deg}^{2}$ or $1.2\cdot 10^{4}\,\mathrm{arcmin}^{2}$, again for
visualization purposes. These, and all subsequent maps, are presented in
Galactic coordinates centered at $(l,b)=\left($$,$$\right)$.

-250250$\displaystyle\phi_{\mathrm{gal}}[\mathrm{rad}\,\mathrm{m}^{-2}]$ (a)

080$\displaystyle\sigma_{\phi_{\mathrm{gal}}}[\mathrm{rad}\,\mathrm{m}^{-2}]$
(b)

Figure 2:  Inference results for the posterior mean (Fig. 2(a)) and
uncertainties (Fig. 2(b)) of the Galactic Faraday rotation sky using the
updated data catalog. The color scale is saturated at $\pm
250\,\mathrm{rad}\,\mathrm{m}^{-2}$ for the mean and at
$80\,\mathrm{rad}\,\mathrm{m}^{-2}$ for the uncertainties to highlight mid
latitude features.

-15001500$\displaystyle\phi_{\mathrm{gal}}[\mathrm{rad}\,\mathrm{m}^{-2}]$ (a)

0300$\displaystyle\sigma_{\phi_{\mathrm{gal}}}[\mathrm{rad}\,\mathrm{m}^{-2}]$
(b)

Figure 3:  As for Fig. 2, but the color scale is saturated at $\pm
1500\,\mathrm{rad}\,\mathrm{m}^{-2}$ in the mean and
$300\,\mathrm{rad}\,\mathrm{m}^{-2}$ in the uncertainty to highlight disc
features of the Galactic Faraday rotation sky.

-250250$\displaystyle\phi_{\mathrm{HE20}}\left[\mathrm{rad}\,\mathrm{m}^{-2}\right]$ (a)

-250250$\displaystyle\phi_{\mathrm{gal}}-\phi_{\mathrm{HE20}}[\mathrm{rad}\,\mathrm{m}^{-2}]$ (b)

Figure 4:  Figure 4(a) shows the mean of the Galactic Faraday rotation sky as
inferred by HE20 using the the same saturation scale as in Fig. 2(a). Figure
4(b) shows the difference between Fig. 2(a) and Fig. 4(a).

010$\displaystyle e^{\rho}$ (a)

-100100$\displaystyle\chi$ (b)

Figure 5:  Posterior means of the components defined in Eq. (7). Figure 5(a)
shows the amplitude field $e^{\rho}$ and Fig. 5(b) shows the sign field
$\chi$.  90.070.050.030.010.0350.0330.0310.0290.0270.0$\displaystyle
l\,\left[{}^{\circ}\right]$$\displaystyle{-1500}$$\displaystyle{-1000}$$\displaystyle{-500}$$\displaystyle{0}$$\displaystyle{500}$$\displaystyle{1000}$$\displaystyle{1500}$$\displaystyle{2000}$$\displaystyle{2500}$$\displaystyle{3000}$$\displaystyle\phi_{\mathrm{rad}}[\mathrm{rad}\,\mathrm{m}^{-2}]$LocNorNorSctSctSgrSgr$\displaystyle{-1}$$\displaystyle{1}$$\displaystyle
b\left[{}^{\circ}\right]$ Figure 6:  A slice in longitude ($l\in($$,$$)$ )
through the inner part of the Galactic disc from the Galactic Faraday rotation
sky. The colored strips indicate equal latitude strips of the inferred Faraday
sky within $b\in(-$$,$$)$. The plot also contains the data points falling in
the same region indicated by grey crosses. Data points with
$\phi>3000\,\mathrm{rad}\,\mathrm{m}^{-2}$ are indicated with black triangles
at the upper end of the plot. The approximate locations of several spiral arm
tangents (Hou et al. 2009; Vallée 2017) are indicated as grey lines,
specifically these are the Local (Loc), Sagittarius (Sgr), Scrutum (Sct) and
Norma (Nor) arms. Note that these positions are inferred from different
tracers such as HII regions and hence do not necessarily coincide exactly with
the tangent points traced by a potential excess in Faraday rotation. The
region towards $l\approx$$$ is also partly shown in Shanahan et al. (2019),
where the strong excess in Faraday rotation in direction to the tangent of the
Sagittarius arm was first noted.
$\displaystyle{0}$$\displaystyle{20}$$\displaystyle{40}$$\displaystyle{60}$$\displaystyle{80}$$\displaystyle|b|\,\left[{}^{\circ}\right]$$\displaystyle{0}$$\displaystyle{250}$$\displaystyle{500}$$\displaystyle{750}$$\displaystyle{1000}$counts
Figure 7:  Histogram of absolute values of the Galactic latitude positions of
all data points used in this work

##  2 Modeling the Faraday rotation sky

###  2.1 Physics

To provide context for the modeling of the Faraday sky, we first give a short
summary on important physical concepts and general data aquisition methods for
Faraday data.

The differential angle of rotation, $\Delta_{\psi_{\lambda}}$, that the
polarization plane of linearly polarized light experiences between emission
and detection can be described by

$\psi_{\lambda}-\psi_{0}=\Delta_{\psi_{\lambda}}=\mathrm{RM}\,\lambda^{2},$
(1)

where $\lambda$ is the observational wavelength, $\psi_{0}$ is the intrinsic
polarization angle, $\psi_{\lambda}$ the angle observed at wavelength
$\lambda$ and RM is the rotation measure (Burn 1966). Determining RMs has
traditionally been done by observing $\psi_{\lambda}$ at different
wavelengths, and then determining the slope of $\Delta_{\psi_{\lambda}}$ in
$\lambda^{2}$ space. In the ideal case of a thin non-emitting plasma screen
being the only source for the rotation effect, the RM at redshift zero is
equal to the Faraday depth,

$\phi(r)=\frac{e^{3}}{2\pi
m_{e}^{2}c^{4}}\int_{r}^{0}\mathrm{d}l\,n_{\mathrm{th}}(r)B_{\mathrm{\parallel}}(r),$
(2)

where $n_{\mathrm{th}}$ is the thermal electron density and
$B_{\mathrm{\parallel}}$ is the component of the magnetic field parallel to
the LOS. The physical constants $e,m_{e}$ and $c$ describe the elementary
charge, the electron mass and the speed of light, respectively. We conform
with the general sign convention that RM and $\phi$ are positive for
$B_{\mathrm{parallel}}$ pointing toward the observer. Unfortunately, the
relation RM $=\phi$ is generally not correct, for example in case of multiple
components along the LOS. Measurements of $\phi$ are therefore often obtained
via more elaborate reconstruction techniques, such as RM synthesis (Brentjens
& de Bruyn 2005; Bell & Enßlin 2012). In this work we will use data for $\phi$
that were determined either with a simple slope fit in $\lambda^{2}$ space or
via RM synthesis. The sampling in $\lambda^{2}$ space is often sparse, and
this sparsity is generally understood to produce artificial feature in the
Faraday spectrum (Farnsworth et al. 2011). The likelihood of these being
mistaken as true $\phi$-values can be minimized by careful analysis, but not
completely eradicated and hence has to be considered when using the data.

In this work, we are interested in the Galactic component of the Faraday
rotation sky $\phi_{\mathrm{gal}}$. Due to the additivity of $\phi$, one can
write the equation connecting the data $d_{\phi}$ with $\phi_{\mathrm{gal}}$
as

$d_{\phi}=\mathcal{R}\phi+n_{\phi}=\mathcal{R}\left(\phi_{\mathrm{gal}}+\phi_{\mathrm{etc}}\right)+n_{\phi}=\mathcal{R}\phi_{\mathrm{gal}}+\widetilde{n}_{\phi},$
(3)

where the $\mathcal{R}$ is a projection operator connecting the sky with data
space and the (Gaussian) noise term $n_{\phi}$ contains the known
observational error. The non-Galactic component $\phi_{\mathrm{etc}}$ contains
for example extra-Galactic, ionospheric contributions or unresolvable small
scale structure and is absorbed together with the observational noise into an
adapted noise term $\widetilde{n}_{\phi}$, whose covariance needs to be
estimated. It should be emphasized at this point that, for the lack of
accurate radial information connected to the RM data, our decision criterion
on the discrimination between Galactic and extra-Galactic solely relies on
angular correlations. This implies that relatively close-by extra-Galactic
objects such as the Magellanic clouds can end up in the Galactic map, while
unresolvable small scale Galactic structures might be erased by the noise
estimation. A more accurate separation most likely would rely on detailed
three dimensional modeling of the ISM and the inclusion of distance
information, which by far exceeds the scope of this work. The noise estimation
is done using the same noise estimation technique as first described by
Oppermann et al. (2012), i.e. by modeling the adapted noise standard
deviations $\widetilde{\sigma}_{d_{\phi}}$ via

$\widetilde{\sigma}^{2}_{d_{\phi}}=\eta_{\phi}\,\sigma^{2}_{d_{\phi}},$ (4)

where $\eta_{\phi}$ is a parameter to be inferred via an inverse gamma model,

$\mathcal{P}\left(\eta_{\phi}|\alpha_{\phi},\beta_{\phi}\right)=\frac{\beta_{\phi}^{\alpha_{\phi}}}{\Gamma(\alpha_{\phi})}\eta_{\phi}^{-\alpha_{\phi}-1}e^{-\frac{\beta_{\phi}}{\eta_{\phi}}}$
(5)

as detailed by Oppermann et al. (2012) and HE20. The hyper-parameters
$\alpha_{\phi}$ and $\beta_{\phi}$ steer the ability of the noise estimation
to increase the noise and hence down-weights data points with large
$\phi_{\mathrm{etc}}$ in the likelihood. This is equivalent to setting a prior
for our expectation that a data excess should be explained by the sky map or
an increase of the noise. In Oppermann et al. (2012) and HE20 $\alpha_{\phi}$
was set to a single number for the whole sky ($1$ and 2.5, respectively). As
discussed in HE20 we need to adapt the hyper-parameters for the noise
estimation when increasing the resolution to make up for the increase in
degrees of freedom, i.e. to allow for the representation of previously
unresolvable small scale structure. With the increase of the resolution to
about $1.3\cdot 10^{-2}\mathrm{deg}^{2}$ or $46.8\,\mathrm{arcmin}^{2}$ in
this work, we found that setting the parameter to a restrictive value leads to
over-fitting in high latitude regions (i.e. data points clearly dominated by
the extra-Galactic component appearing as point sources in the sky map), while
a looser choice makes it hard to represent the small scale structure in the
Galactic disc. This indicates that we have reached a resolution where a noise
model with a universally chosen $\alpha_{\phi}$ is not able to accurately
represent the non-homogeneous statistics of the Faraday sky anymore. We hence
decided to use a position dependent noise estimation, which gets more
restrictive in regions where we expect small scale structures to be mostly
Galactic, e.g. towards the Galactic disc or the Magellanic clouds. We chose to
use the logarithmic DM map of Yao et al. (2017) as a proxy for this prior, as
it provides a natural scaling with Galactic latitude and includes at least
some important structures such as the Magellanic clouds. This template is then
rescaled such that it reaches $1$ towards the poles (hence complying with the
global value for $\alpha_{\phi}$ chosen in Oppermann et al. (2012)), but
reaches up to $3.5$ towards the disc, implying a very restrictive prior for
$\eta_{\phi}$ in these regions. In accordance with the previous studies,
$\beta_{\phi}$ is then set such that the prior mode of $\eta_{\phi}$ is $1$
for all data points.

13$\displaystyle\alpha$ Figure 8:  Template for the noise estimation parameter
$\alpha_{\phi}$ introduced in Eq. (5), based on the Galactic electron model
derived by Yao et al. (2017).

This noise estimation procedure also takes care of uncorrelated systematic
errors as for example the $n\pi$-ambiguity, as demonstrated in HE20 for data
points identified by Ma et al. (2019). A, potentially correlated, systematic
effect may come from the limited range in Faraday depth that most Faraday
rotation surveys probe. Sources with a Faraday depth out of range are either
ignored or misrepresented in the data sets, depending on the technique used to
analyze the polarimetric data. Such effects are hard to model and cannot
generally be remedied by the noise estimation. Hence, we do not correct for
such systematics.

###  2.2 Nomenclature

Before proceeding with the modeling discussion, we would like to comment on
the terminology used in this paper and its predecessor HE20. There, the
authors use the term “Faraday data” for the data and refer to $\phi$ as
Faraday depth. Accordingly the reconstructed sky is referred to as the
“Galactic Faraday depth sky”. This reflects the common Bayesian ontology that
clearly separates data and signal and relies on the assumption that the data
are a noisy measurement of $\phi$ defined in Eq. 2. This nomenclature is
independent of the way the data were obtained or the physical geometry of the
LOS observed. The term “RM” is omitted by HE20 and implicitly reserved for the
sub set of the data set which was obtained using slope fitting techniques.
This unfortunately does not completely match the general way these terms are
used in the astrophysical literature, where sometimes the naming convention is
more determined by the LOS structure. The label “Faraday depth” for Eq. 2 is
sometimes reserved for cases where rotation and emission are mixed and one has
to deal with complex Faraday spectra (e.g. see Van Eck 2018), while “RM” is
often used in situations that can be explained with a single non-emitting
Faraday screen (e.g. see Mao et al. 2010). This terminology is hard to
maintain in our work as we are mixing these data sets and are agnostic to the
geometry along the LOS, but on the other hand the term “Faraday depth sky” as
used by HE20 might give a false impression of what is actually inferred. We
hence decided to again label Eq. 2 as “Faraday depth”, but refer to the
reconstructed sky simply as the “Faraday rotation sky”.

###  2.3 Sky model

The sky model for $\phi_{\mathrm{gal}}$ has been well motivated by HE20. Here,
we give a very short alternative motivation, which provides connection points
for additional data sets (e.g. on the Galactic dispersion measure (DM)). An
illustrative picture arises if we assume no correlation between
$n_{\mathrm{th}}$ and $B_{\mathrm{\parallel}}$, which is reasonable for the
warm ionized medium of the Galaxy (Passot & Vázquez-Semadeni 2003; Harvey-
Smith et al. 2011; Wu et al. 2015), Eq. 2 can be written for the Galactic
component as

$\displaystyle\phi_{\mathrm{gal}}\propto\mathrm{DM}\,\left\langle
B_{\mathrm{\parallel}}\right\rangle_{\mathrm{LOS}}$ (6)

(Draine 2010), where
$\mathrm{DM}=\int_{\mathrm{LOS}}\mathrm{d}l\,n_{\mathrm{th}}$ and
$\left\langle B_{\mathrm{\parallel}}\right\rangle_{\mathrm{LOS}}$ is the LOS-
parallel component of the magnetic field vector averaged along the LOS. This
equation is used to motivate the simplest model for the Faraday sky,

$\displaystyle\phi_{\mathrm{gal}}=e^{\rho}\chi\,\mathrm{rad}\;\mathrm{m}^{-2},$
(7)

implemented for the first time by HE20. Here, $\rho$ and $\chi$ are Gaussian
fields on the sky with unknown correlation structure that needs to be
determined. The log-normal field $e^{\rho}$ is supposed to take over the role
of the DM, while the sign field $\chi$ models the magnetic field average. As
already noted by HE20, one cannot break the degeneracy between $e^{\rho}$ and
$\chi$ in this model and relate the component fields to the respective
physical quantities without further information and/or assumptions. This is
attempted in a separate work (Hutschenreuter & Ensslin in prep.). Note that
the assumption of no correlation between $n_{\mathrm{th}}$ and
$B_{\mathrm{\parallel}}$ is by no means necessary to motivate the model above
as a separation of the Faraday sky into an amplitude and a sign field is of
course always possible, but is only used to give an illustrative image.

The inference of the correlation functions has been updated in accordance with
recent developments in numerical information field theory, for details see
Arras et al. (2020). The new model is equally flexible in representing
different power spectra, but has the advantage of more intuitive and better
decoupled hyper-parameters. This inference was implemented in the newest
version (v.7) of the NIFTy package (The NIFTy5 team et al. 2019). Nifty is
based on Information Field Theory (IFT) and provides the user with a library
of Bayesian signal inference techniques, mostly aimed at the evaluation of
noisy and very high dimensional problems. Specifically, it makes use of the
MGVI algorithm (Knollmüller & Enßlin 2020) to approximate posterior
distributions via variational inference. It furthermore contains a toolkit to
implement complicated likelihoods and signal models. For a generic reference
and introduction to IFT see Enßlin (2019).

##  3 Data

We have used a newly compiled master catalog of published Faraday data from
extra-Galactic sources such as AGNs 333Tabara & Inoue (1980); Simard-Normandin
et al. (1981); Broten et al. (1988); Clegg et al. (1992); Wrobel (1993); Oren
& Wolfe (1995); Minter & Spangler (1996); Gaensler et al. (2001); Clarke et
al. (2001); Klein et al. (2003); Brown et al. (2003); Roy et al. (2005); Brown
et al. (2007); Rossetti et al. (2008); Mao et al. (2008); Heald et al. (2009);
Feain et al. (2009); Taylor et al. (2009); Mao et al. (2010); Law et al.
(2011); Van Eck et al. (2011); Battye et al. (2011); Mao et al. (2012a, b);
Anderson et al. (2015); Kim et al. (2016); Costa et al. (2016); Kaczmarek et
al. (2017); O’Sullivan et al. (2017); Van Eck et al. (2018); Costa & Spangler
(2018); Vernstrom et al. (2018); Riseley et al. (2018); Shanahan et al.
(2019); Betti et al. (2019); Schnitzeler et al. (2019); Ma et al. (2019,
2020); Riseley et al. (2020) , to be published by Van Eck et al. (in
prep.)444This master catalog can be found online at
https://github.com/Cameron-Van-Eck/RMTable; we used used version 0.1.8 of the
catalog in this paper.. This catalog was assembled independently of the
Oppermann et al. (2012) catalog, but contains most of the same sources plus
many published since then. A list of the individual catalogs included can be
found online with the master catalog.

This new master catalog is not yet complete, with papers being prioritized for
inclusion based on combination of catalog size, recency, ease of data access
(i.e., Vizier or machine readable tables were preferred to LaTeX or PDF
tables, which were preferred to images of tables). The version of the catalog
used here includes data points from 38 papers, which encompasses nearly all of
the catalogs published in the past 30 years that contain more than 30 data
points. Some papers with fewer data points were also included. Sources
reported with multiple data points (from RM synthesis or QU-fitting) had all
components included as separate entries in the catalog. This results in a
catalog of 50,207 data points, which is a significant expansion on the 41,330
used by Hutschenreuter & Enßlin (2020); Oppermann et al. (2012). In accordance
with Stil et al. (2011), the error bars of Taylor et al. (2009) where
multiplied by $1.22$.

Additionally, we were provided with yet unpublished catalogs, mostly including
data compiled in 2020 by the LOFAR Two-metre Sky Survey (LoTSS) (O’Sullivan et
al. (in prep.); 2461 data points), by the Canadian Galactic Plane Survey
(CGPS) (Van Eck et al. (submitted); 2493 data points) and a small data set
provided by Johnston-Hollit et al. (in prep.) (68 data points) which was
already present in the Oppermann catalog. These data sets will become part of
the catalog once published. We have decided in favor of including this data,
as we would like our results to represent the most up to date state of the
Galactic Faraday sky as of the end of the year 2020.

In summary, we therefore have 55190 data points from 41 surveys available.
This number excludes 39 pulsars which were removed from the catalog, as they
in general do not probe the full Galactic LOS (Sobey et al. 2019).

We have not attempted to identify duplicate sources in the catalog; if a
source appears in multiple catalogs, all measurements are kept. While multiple
measurements of the same source provide valuable additional information, the
noise estimation technique of Oppermann et al. (2012) is strictly speaking not
optimal in these cases, as these sources probe the same part of the IGM and
the assumption of independence for the error estimates, which includes the
extra-Galactic signal, does not hold anymore. The severity of this issue
depends on the number of duplicates, which is hard to determine without
extensive source matching. We can, however, give an estimate on this number by
counting the number of data points located in a sky pixel containing more than
one data point. This results in a fraction of $\approx 0.6\%$ of the data set
being identified as potential duplicates at a resolution of about $None$. Only
a small fraction of those will be actual duplicates, on the other hand close
by sources might be correlated by the IGM on such scales even if they are not
duplicates, as demonstrated by Akahori & Ryu (2010). We hence assume the
number of data points with extra-Galactic correlations to be on the order of
$1\%$ and therefore deem the approximation of uncorrelated noise acceptable.
In future studies of the Faraday sky, this might have to be revised. We
assumed an a-priori noise level of $50\%$ of $|d_{\phi}|$ for the 72 sources
without or with zero-valued error bars, which were then subject to the same
noise estimation procedure as all other data points. This is a rather cautious
choice, but we deem the potential downsides of systematic effects introduced
by ‘overconfident’ data higher than of a potentially ignored data point due to
too large error bars. In any case, this choice is automatically corrected to
some degree by the noise estimation. A projection of all data points, their
error bars and the source density on the sky is shown in Fig. 1.

$\displaystyle{305}$$\displaystyle{298}$$\displaystyle{290}$$\displaystyle{282}$$\displaystyle{275}$$\displaystyle
l\left[{}^{\circ}\right]$$\displaystyle{-55}$$\displaystyle{-48}$$\displaystyle{-40}$$\displaystyle{-32}$$\displaystyle{-25}$$\displaystyle
b\left[{}^{\circ}\right]$-8080$\displaystyle\phi_{\mathrm{gal}}-20[\mathrm{rad}\,\mathrm{m}^{-2}]$
(a)

$\displaystyle{305}$$\displaystyle{298}$$\displaystyle{290}$$\displaystyle{282}$$\displaystyle{275}$$\displaystyle
l\left[{}^{\circ}\right]$$\displaystyle{-55}$$\displaystyle{-48}$$\displaystyle{-40}$$\displaystyle{-32}$$\displaystyle{-25}$$\displaystyle
b\left[{}^{\circ}\right]$03$\displaystyle e^{\rho}$ (b)

$\displaystyle{305}$$\displaystyle{298}$$\displaystyle{290}$$\displaystyle{282}$$\displaystyle{275}$$\displaystyle
l\left[{}^{\circ}\right]$$\displaystyle{-55}$$\displaystyle{-48}$$\displaystyle{-40}$$\displaystyle{-32}$$\displaystyle{-25}$$\displaystyle
b\left[{}^{\circ}\right]$LMCSMC0100$\displaystyle\mathrm{H}_{\alpha}[\mathrm{R}]$
(c)

$\displaystyle{305}$$\displaystyle{298}$$\displaystyle{290}$$\displaystyle{282}$$\displaystyle{275}$$\displaystyle
l\left[{}^{\circ}\right]$$\displaystyle{-55}$$\displaystyle{-48}$$\displaystyle{-40}$$\displaystyle{-32}$$\displaystyle{-25}$$\displaystyle
b\left[{}^{\circ}\right]$020$\displaystyle\sqrt{\mathrm{EM}}[\sqrt{\mathrm{pc}\,\mathrm{cm}^{-6}}]$
(d)

Figure 9:  The Magellanic Clouds in our results and as seen with different
observables, respectively. Fig. 9(a) and Fig. 9(b) show the Faraday rotation
(with a $20\,\mathrm{rad}\,\mathrm{m}^{-2}$ offset) and the underlying
amplitude field as inferred in this work. Figure 9(c) and Fig. 9(d) show the
clouds in $\mathrm{H}\alpha$ emission (Finkbeiner 2003; Gaustad et al. 2001),
and in emission measure (EM) as obtained by the Planck satellite (Planck
Collaboration et al. 2016), respectively. Fig. 9(c) indicates the locations of
the clouds.

##  4 Results and Discussion

In the following we present the results of the inference using the model
introduced in Section 2 and the data presented in Section 3.

$\displaystyle{10^{0}}$$\displaystyle{10^{1}}$$\displaystyle{10^{2}}$$\displaystyle{10^{3}}$$\displaystyle\ell$$\displaystyle{10^{7}}$$\displaystyle{10^{9}}$$\displaystyle{10^{11}}$$\displaystyle{10^{13}}$$\displaystyle
C^{\phi}_{\ell}$This workparametric fit (Eq. 8)Oppermann (2012)HE20 Figure 10:
Power spectrum for the Faraday rotation sky derived in this work, compared
with results from HE20 and Oppermann et al. (2012). The red dashed line
indicates a parametric fit to the results of this work using Eq. 8. The
uncertainties of the spectrum derived in this work are indicated by lightblue
lines, which are only slightly offset from the darkblue-lined mean. Figure 11:
Two dimensional histogram of logarithmic noise weighted residuals and absolute
Galactic latitude.

###  4.1 The sky

We show the resulting Faraday rotation sky and uncertainties in Fig. 2 and
Fig. 3 with different saturation scales to highlight the sky morphology at
different latitude scales. For comparison, the previous result of HE20 is
shown in Fig. 4(a) and a difference map between this figure and Fig. 2(a) is
shown in Fig. 4(b). On large scales, the results are in good agreement with
each other, while on smaller scales many more structures are discernible in
the new reconstruction. At higher latitudes, the largest deviations are
discernible in the Southern sky, i.e. the region south of $-$$$ in celestial
declination, which was severely under-constrained in previous inferences but
has since been filled mostly by Faraday data from Schnitzeler et al. (2019),
although the region still has not reached a number density comparable to the
Northern Hemisphere. Moreover, Fig. 4(b) demonstrates considerable differences
in the Galactic disc, in parts in morphology but also in amplitude. This can
be well observed in Fig. 3(a), which reveals a pronounced disc with absolute
values often exceeding $1500\,\mathrm{rad\,m^{-2}}$. The maximum is close to
$3000\,\mathrm{rad\,m^{-2}}$, which stands in strong contrast to the previous
inference results of HE20, which had found a maximum magnitude slightly above
$1000\,\mathrm{rad\,m^{-2}}$. The disc is further investigated in Section
4.2.1.

We further show the component fields $e^{\rho}$ and $\chi$ in Fig. 5. Both
show strong similarity to the analog component fields found in HE20. If we
assume that our model in Eq. 7 is correct and that both maps are a proxy for
the physical components of the Faraday sky, then, at least in terms of
morphology, Fig. 5(a) should trace the Galactic DM map, while Fig. 5(b) should
trace the LOS-averaged magnetic field strength. We give some evidence for this
claim in Section 4.2.2 and Section 4.3.3, while a further more detailed
investigation is to be presented by Hutschenreuter & Ensslin (in prep.). The
$\chi$ map shows some evidence for point-like structures with opposite sign to
their diffuse surroundings, which at high latitudes we cannot exclude to be
remaining extra-Galactic residuals. This indicates that our noise estimation
routine was only partially successful, and that some precautions have to be
taken if the maps are used as foreground reduction template.

###  4.2 Substructures

####  4.2.1 The inner disc

The inner Galactic disc is highlighted in more detail in Fig. 6. Here, we
demonstrate the strong variability of the disc by plotting a fine slice along
Galactic longitude together with data points falling in the same region. The
plot also contains some color coded latitude information, in order to
demonstrate that within the disc the same variability is observed when moving
along latitudes, including frequent sign flips. The plot contains the
approximate directions of the Galactic spiral arm tangents. We note two strong
excesses in amplitude of the Faraday sky, one towards the Galactic center and
another one towards the tangent point of the Sagittarius arm at around
$l\approx$$$ reaching almost $3000\,\mathrm{rad\,m^{-2}}$. The latter excess
has been first reported by Shanahan et al. (2019), with the highest observed
data point above $4000\,\mathrm{rad\,m^{-2}}$. Other arm locations also seem
to be correlated with stronger values in $\phi$, albeit in a much less clear
way. The Sagittarius region has also been discussed extensively by Reissl et
al. (2020), who investigate rotation measures as spiral tracers using
simulations. In their work, the authors confirm the shape of the Sagittarius
excess and attribute its sharp morphology to the geometry of the spiral arm
relative to the observer. They furthermore predict somewhat more extended but
nonetheless relatively strong Faraday rotation features for the other arms and
an overall increase in Faraday depth towards the Galactic center and note that
these features are not visible in the results of Oppermann et al. (2012) or
Hutschenreuter & Enßlin (2020). In this work, we do find stronger disc
amplitudes of the Faraday sky, but cannot confirm the above mentioned overall
morphology, as our results are dominated by much smaller structures. Assuming
the consistency of our inference and that the simulations of Reissl et al.
(2020) produce a correct representation of the Milky Way, the observations can
only be explained by systematic effects in the data. For one, the longitude
region $l\in\left($$,$$\right)$ with latitudes $|b|\in\left($$,$$\right)$
still has under-dense data coverage, as can be seen in Fig. 1(c) and Fig. 7.
This might imply that several strong excess regions have not been noted yet as
they have simply not been probed. Furthermore, as already noted by Reissl et
al. (2020) for the Taylor et al. (2009) data set and previously discussed in
Section 2.1,many Faraday rotation data sets have range limitations in
$d_{\phi}$ and hence might misrepresent regions with exceedingly large Faraday
rotation. Modeling such systematics on the inference side, for example as an
extension of Eq. 7 have not been attempted and potentially would require
strong prior assumptions on the disc, which would be opposed to our general
approach of using generic sky models in order to maintain a high flexibility.
It hence cannot be excluded that systematic effects are still present in the
data, especially within the disc, and future Faraday rotation surveys are
likely necessary to arrive at a complete picture.

####  4.2.2 The Magellanic clouds

To investigate the correspondence of the Faraday sky with other tracers, we
single out the Magellanic clouds as objects of study, due to their relatively
well defined over-all morphology and small-scale structure. The clouds are
shown in Fig. 9 as excerpts of the Faraday rotation sky, the amplitude field
$e^{\rho}$, $\mathrm{H}\alpha$ data (Finkbeiner 2003; Gaustad et al. 2001) and
the EM sky as obtained by Planck (Planck Collaboration et al. 2016), where the
latter is related to the thermal electron density as
$\mathrm{EM}=\int_{\mathrm{LOS}}\mathrm{d}l\,n^{2}_{\mathrm{th}}$ in analogy
to the DM defined in Sec. 2.3. A comparison between Fig. 9(b) and Fig. 9(d)
reveals that the amplitude field indeed seems to trace the morphology of dense
Galactic structures also on intermediate scales, as both clouds are not only
present in Fig. 9(b), but also are weighted correctly. The excerpt on the
Faraday sky has been offset by $20\,\mathrm{rad}\,\mathrm{m}^{2}$, in order to
highlight the structures of the clouds. A significant small scale
correspondence of the Faraday rotation excerpt and the Hα is visible, which
seems to indicate that even the very small scale structures are not residuals
of extra-Galactic contamination, but resemble existing structures.
Furthermore, the region between the clouds seems to indicate a coherent region
of Faraday rotation between the Magellanic clouds. This is again consistent
with an additional coherent magnetic field structure between the clouds as
first reported by Mao et al. (2008), see also Kaczmarek et al. (2017).

###  4.3 Statistical properties

####  4.3.1 Power spectra

The inference method employed in this work has seen a significant upgrade in
correlation structure modeling. The resulting power spectrum is compared to
previous results by plotting the power spectra in Fig. 10. The plot also
contains the corresponding statistical uncertainties of our results. The plot
shows a significant offset of the new spectrum towards small scales compared
to its predecessors. In order to illustrate the result, Fig. 10 also contains
a parametric maximum a-posteriori fit to the new spectrum using the analytic
formula

$C^{\phi}_{\ell,\mathrm{fit}}=\frac{A}{1+\left(\frac{\ell}{\ell_{0}}\right)^{-s}},$
(8)

which results in $s=-2.4$, $\ell_{0}=34$ and $A=1.1\cdot\,10^{14}$. While a
similar power law slope of $\approx\,-2.17$ was found by Oppermann et al.
(2012), the power spectra on small scales were at least somewhat suppressed in
HE20. This is consistent with the new small-scale structures visible in Fig.
2(a) as compared to Fig. 4(a).

$\displaystyle{-1}$$\displaystyle{0}$$\displaystyle{1}$$\displaystyle\rho$$\displaystyle{1}$$\displaystyle{2}$$\displaystyle{3}$$\displaystyle\log(\mathrm{DM}\left[\mathrm{pc}\,\mathrm{cm}^{-3}\right])$00.6251.251.8752.5$\displaystyle\log(1+\mathrm{counts})$
(a)

$\displaystyle{-1}$$\displaystyle{0}$$\displaystyle{1}$$\displaystyle\rho$$\displaystyle{-1}$$\displaystyle{0}$$\displaystyle{1}$$\displaystyle{2}$$\displaystyle\log(\sqrt{\mathrm{EM}}\left[\mathrm{pc^{0.5}}\,\mathrm{cm}^{-3}\right])$00.6251.251.8752.5$\displaystyle\log(1+\mathrm{counts})$
(b)

Figure 12:  Histogram of sky pixels illustrating the correlation between the
log amplitude field $\rho$ and the logarithmic DM (Fig. 12(a)) and logarithmic
square root EM (Fig. 12(b)) skies, respectively. The former is calculated from
a Galactic thermal electron model (Yao et al. 2017), while the latter was
inferred by the Planck survey from extinction-free microwave data (Planck
Collaboration et al. 2016)..

####  4.3.2 Latitude correlations

An important consistency test for the convergence of our inference and a
successful separation of the Galactic component of the Faraday sky is to check
if there are any latitude correlations remaining in the residuals
$d_{\phi}-\mathcal{R}\phi_{\mathrm{gal}}$. Since a data point can be either
fitted by adjusting the sky or by increasing the noise estimate, it is best to
compare the noise weighted residual
$\widetilde{\sigma}_{d_{\phi}}^{-1}|d_{\phi}-\mathcal{R}\phi_{\mathrm{gal}}|$
as a dimensionless tracer of such correlations. The noise
$\widetilde{\sigma}_{d_{\phi}}$ in this case is the standard deviation as
defined in Eq. 4. This is investigated in Fig. 11, which shows a two-
dimensional histogram between the logarithmic noise-weighted residuals and the
absolute value of the Galactic latitude. For latitudes exceeding $None$, no
notable correlation with latitude is discernible. This is not true for the
scatter of the residual, which is strong in the disc and smaller at higher
latitudes. Such an increase is however easily explained when viewing the
distribution of data points along Galactic latitude as depicted in the
histogram Fig. 7. Another peculiar feature of Fig. 11 is the relatively sharp
decrease of the count density for log residuals of $0$. This is an imprint of
the distribution of our uncertainties in our Faraday data catalog, which sees
a relatively hard cut towards large error-bars.

####  4.3.3 Amplitude correlations

The inferences conducted in HE20 and (Hutschenreuter & Ensslin in prep.) make
use of the EM sky as a proxy for the amplitude of the Faraday sky. In order to
test the extent to which this assumption is viable, we investigate the
correlation between the amplitude field (shown in Fig. 5(a)) and both the DM
sky and the square root of the EM sky, Fig. 12. The square root is motivated
by the quadratic dependence of the EM on the thermal electron density, in
contrast to the linear dependence of DM and RM. The latter was inferred from
cosmic microwave data by the Planck survey (Planck Collaboration et al. 2016),
while the former is calculated from a parametric model of the Galactic thermal
electron density (Yao et al. 2017) based on pulsar DM data. Although the EM
map of Planck is much more detailed and the DM map relies on very little data
and strong modeling assumptions, we note much clearer correlations of the
amplitude field with the DM map than with the $\sqrt{\mathrm{EM}}$ map,
aligning with both the model and the physical discussion presented in Section
2.3. This is confirmed by Spearman-rank cross-correlation coefficients of
$0.88$ for the log DM map and $0.72$ for the log square root EM map. In the
case of the DM map some of the missing correlation to the amplitude map can be
explained by the fact that the Yao et al. (2017) electron model lacks small-
scale structure as it relies on sparse data. The differences to the square
root EM map, which itself contains a lot of small scale structure, are best
explained by the missing volume filling factor that needs to be considered in
order to equate the EM and DM skies. Gaensler et al. (2008) show that this
factor is variable over the sky and ranges between 0.04 in the mid-plane to
about 0.3 at larger distances from the Galactic disc. As demonstrated by HE20,
the EM sky is nonetheless a valuable source of information on the Galactic
electron density profile. It is however necessary to model these systematic
volume effects by introducing additional degrees of freedom, as implemented
phenomenologically by HE20. An improved version of this model which takes the
above discussion into account will be given by Hutschenreuter & Ensslin (in
prep.).

##  5 Summary and Conclusion

We have reconstructed the Galactic Faraday rotation sky using a newly-
available Faraday rotation data set, comprising the catalog compiled by (Van
Eck et al. in prep.) and including additional catalogs provided by O’Sullivan
et al. (in prep.); Van Eck et al. (submitted); Johnston-Hollit et al. (in
prep.). This work is an update to the reconstructions by Oppermann et al.
(2012) and HE20. Additionally to the new data set, we have made technical
advancements by employing a new correlation structure model originally
developed by Arras et al. (2020) and providing our results with an increased
angular resolution. We have found significant updates in contrast to older
results, such as an enhanced Galactic disc with Faraday rotation amplitudes
near $3000\,\mathrm{rad\,m^{-2}}$ and drastically improved small-scale
structures in the sky map. Apart from a morphological examination of the full
sky and certain excerpts such as the Galactic disc and the Magellanic clouds,
we have further examined the statistical properties of our results by
investigating the power spectrum of the resulting map and cross correlations
of the map and its components with other ISM tracers. We have found further
motivation to the attempt begun by HE20 to introduce additional tracers for
more holistic Galactic sky inferences. Our results can be used as a foreground
reduction template for extra-Galactic Faraday data, however one should note
that some extra-Galactic residuals might still be present in the posterior
mean maps. An ideal foreground reduction pipeline should hence mirror the
analysis of Oppermann et al. (2015), i.e. by performing a joint analysis of
the extra-Galactic, systematic and Galactic components to the variance. If
such a procedure is deemed too expensive to implement, at least some smoothing
should be applied to the maps, to lessen the effect of potential extra-
Galactic residuals, especially at higher latitudes. Our analysis has
furthermore demonstrated that future Faraday rotation sky inferences might
strongly benefit from future surveys that remedy certain shortcomings of the
existing data sets such as

  * •

the still sparse data in parts of the Galactic disc, the regions above and
below the disc and the southern celestial, sky (although the situation there
has improved considerably in recent years, mainly due to the results of
Schnitzeler et al. (2019)) and

  * •

the systematic biases introduced by several surveys due to their limited
Faraday depth range, which might mean that several strong Faraday rotating
regions are still undetected.

These issues indicate a strong potential for the upcoming polarimetric surveys
of ASKAP, MeerKAT, LOFAR, MWA, VLASS, and SKA in increasing our knowledge on
the Galactic Faraday rotation sky, as they will provide a vast catalog of new
Faraday data (Heald et al. 2020). This information will, in turn, provide a
pivotal role in constraining the Galactic magnetic field and is an important
input for upcoming reconstructions of the magnetized interstellar medium.

## Acknowledgment

The results in this publication have been derived using the NIFTy package
(https://gitlab.mpcdf.mpg.de/ift/NIFTy, The NIFTy5 team et al. (2019)). Some
of the images were produced using the CMasher library
(https://github.com/1313e/CMasher, van der Velden (2020)). We would like to
thank Jennifer West and the CIRADA team providing the cutout server. SH would
like to thank Philipp Frank, Philipp Arras, Martin Reinecke, Jakob
Knollmüller, Reimar Leike, and the rest of the IFT team for valuable
discussions and their continuous work on the NIFTy package. The noise
estimation template in Fig. 8 was produced using the pygedm package
(https://pypi.org/project/pygedm/, (Yao et al. 2017)). We thank Rainer Beck
for his careful reading and valuable comments as the Max Planck Institute for
Radio Astronomy (MPIfR) internal referee.
SH and MH acknowledge funding from the European Research Council (ERC) under
the European Union’s Horizon 2020 research and innovation programme (grant
agreement No 772663). The Australia Telescope is funded by the Commonwealth of
Australia for operation as a National Facility managed by CSIRO. JMS
acknowledges the support of the Natural Sciences and Engineering Research
Council of Canada (NSERC), 2019-04848. JB acknowledges the support of the
Natural Sciences and Engineering Research Council of Canada and the National
Research Council of Canada. Basic research in radio astronomy at the US Naval
Research Laboratory is supported by 6.1 Base funding. CJR acknowledges
financial support from the ERC Starting Grant “DRANOEL”, number 714245.
C.L.H.H. acknowledges the support of the NAOJ Fellowship and JSPS KAKENHI
grants 18K13586 and 20K14527. The Paul G. Allen Family Foundation, the US
Naval Observatory and the US National Science Foundation grants AST-0321309,
AST-0540690 and AST-0838268 have contributed to the ATA project. This paper
has made use of the S-PASS/ATCA RM catalog Schnitzeler et al. (2019). The
Dunlap Institute is funded through an endowment established by the David
Dunlap family and the University of Toronto. C.V.E. and B.M.G. acknowledge the
support of the Natural Sciences and Engineering Research Council of Canada
(NSERC) through grant RGPIN-2015-05948, of the Canada Research Chairs program,
and of the Canada Foundation for Innovation 2017 Innovation Fund through
Project 35999. LOFAR (van Haarlem et al. 2013) is the Low Frequency Array
designed and constructed by ASTRON. It has observing, data processing, and
data storage facilities in several countries, which are owned by various
parties (each with their own funding sources), and which are collectively
operated by the ILT foundation under a joint scientific policy. The ILT
resources have benefited from the following recent major funding sources:
CNRS-INSU, Observatoire de Paris and Université d’Orléans, France; BMBF, MIWF-
NRW, MPG, Germany; Science Foundation Ireland (SFI), Department of Business,
Enterprise and Innovation (DBEI), Ireland; NWO, The Netherlands; The Science
and Technology Facilities Council, UK; Ministry of Science and Higher
Education, Poland; The Istituto Nazionale di Astrofisica (INAF), Italy. This
research made use of the Dutch national e-infrastructure with support of the
SURF Cooperative (e-infra 180169) and the LOFAR e-infra group. The Jülich
LOFAR Long Term Archive and the German LOFAR network are both coordinated and
operated by the Jülich Supercomputing Centre (JSC), and computing resources on
the supercomputer JUWELS at JSC were provided by the Gauss Centre for
Supercomputing e.V. (grant CHTB00) through the John von Neumann Institute for
Computing (NIC). This research made use of the University of Hertfordshire
high-performance computing facility and the LOFAR-UK computing facility
located at the University of Hertfordshire and supported by STFC
[ST/P000096/1], and of the Italian LOFAR IT computing infrastructure supported
and operated by INAF, and by the Physics Department of Turin university (under
an agreement with Consorzio Interuniversitario per la Fisica Spaziale) at the
C3S Supercomputing Centre, Italy.

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