11institutetext: Department of Physics, University of Crete, Voutes, 70013
Heraklion, Greece 22institutetext: Institute of Astrophysics, Foundation for
Research and Technology-Hellas, Voutes, 70013 Heraklion, Greece
33institutetext: Department of Computer Science, University of Applied
Sciences Trier, 54208 Trier, Germany

# Polarization power spectra and dust cloud morphology

A. Konstantinou<EMAIL_ADDRESS>V. Pelgrims ,
<EMAIL_ADDRESS>F. Fuchs 33 K. Tassis 1122

(Received November 4, 2021; accepted April 19, 2022)

###### Abstract

Context. In the framework of cosmic microwave background polarization studies
and the characterization of its Galactic foregrounds, the angular power
spectrum analysis of the thermal dust polarization map has led to intriguing
evidence of an $E/B$ asymmetry and a positive $TE$ correlation. The
interpretation of these observations are the subject of theoretical and
simulation-driven studies in which the correlation between the density
structure of the interstellar medium (ISM) and the magnetic field appear to be
a key aspect. In this context, and when the magnetized ISM structures are
modeled in three dimensions, dust clouds are generally considered to be
filamentary structures only, while both filamentary and sheet-like shapes are
supported by observational and theoretical evidence.

Aims. In this work, we aim to study the influence of cloud shape and its
connection to the local magnetic field, as well as the influence from the
viewing angle, on the angular power spectra measured on thermal dust
polarization maps; we specifically focus on the dependence of the $E/B$ power
asymmetry and $TE$ correlation.

Methods.  To that end, we simulate realistic interstellar clouds with both
filament-like and sheet-like shapes using the software Asterion which also
allows us to generate synthetic maps of thermal dust polarized emission with
an area of 400 square degrees. Then, we compute their polarization power
spectra in multipole range $\ell\in[100,\,500]$ and focus on the $E/B$ power
asymmetry, quantified through the $\mathcal{R}_{EB}$ ratio, and the
correlation coefficient $r^{TE}$ between $T$ and $E$ modes. We quantify the
dependence of $\mathcal{R}_{EB}$ and $r^{TE}$ values on the offset angle
(between the longest cloud axis and local magnetic field lines) and
inclination angle (between the line of sight and the magnetic field) for both
types of cloud shapes either embedded in a regular magnetic field or coupled
to a non-regular field to mimic turbulence.

Results. We find that both types of cloud shapes cover the same regions of the
($\mathcal{R}_{EB},\,r^{TE}$) parameter space. The dependence on the
inclination and offset angles are similar for both shapes although sheet-like
structures generally show larger scatter as compared to filamentary
structures. In addition to the known dependence on the offset angle, we find a
strong dependence of $\mathcal{R}_{EB}$ and $r^{TE}$ on the inclination angle.

Conclusions. The very fact that filament-like and sheet-like structures may
lead to polarization power spectra with similar ($\mathcal{R}_{EB},\,r^{TE}$)
values complicates their interpretation. We argue that interpreting them
solely in terms of filament characteristics is risky and that in future
analyses, this degeneracy should be accounted for as well as the connection to
the magnetic field geometry. Indeed, our results from 400 square degrees maps
make it clear that the overall geometrical arrangement of the magnetized ISM
surrounding the observer leaves its marks on polarization power spectra.

###### Key Words.:

ISM: dust, magnetic fields – submillimeter: ISM – polarization – (cosmology)
cosmic background radiation

##  1 Introduction

The study of polarization of the cosmic microwave background (CMB) is very
important for the field of cosmology as it is the only foreseeable way to test
observationally whether the Universe has undergone an early period of
inflationary expansion. Indeed, gravitational waves generated during the
inflation era would have left unambiguous imprints on CMB polarization. These
imprints are the so-called $B$-modes in the polarization power spectra which,
although difficult to detect, are in principle observable (Polnarev 1985).

Most of the cosmological information can be obtained from the CMB radiation
through the analysis of its $T$, $E$, and $B$ angular power spectra (intensity
and linear polarization) and their correlation (e.g., Kamionkowski et al.
1997; Seljak & Zaldarriaga 1997; Hu & White 1997) in which the polarization is
decomposed into a scalar, curl-free field, the $E$ mode, and a pseudo-scalar,
curl field, the $B$ mode.

Galactic dust, through which we observe the CMB, contaminates the cosmological
signal as the grain thermal emission is also polarized and contains $B$-mode
components as well. Non-spherical dust grains align their short axis with the
magnetic field of the Galaxy which is present everywhere. As a result, they
emit thermal radiation with their electric vector preferentially aligned with
their long axis (e.g., Andersson et al. 2015). The polarized emission from
Galactic dust grains, with polarization fraction about 20% (e.g., Benoît et
al. 2004; Planck Collaboration XII 2020), higher than that of the CMB, which
is about 5% (Bond & Efstathiou 1987; Seljak 1997; Hinshaw et al. 2003), always
dominates the measured signal at frequency higher than $\approx$ 80 GHz (e.g.,
Planck Collaboration Int. XXX 2016). The $B$-mode component of Galactic
thermal dust emission can be confused with the primordial signal of inflation
(e.g., BICEP2 Collaboration et al. 2014; Planck Collaboration Int. XXX 2016;
Planck Collaboration XI 2020). It is therefore essential to understand the
emission and absorption properties of interstellar dust so as to remove its
contribution to the map of the polarized primordial microwave sky.

In this context, the dust polarization is characterized by its $T$, $E$, and
$B$ representations rather than through its representation in Stokes
parameters $I$, $Q$, and $U$. $E$ and $B$ are invariant under rotations making
the $E$-$B$ decomposition more convenient since it is unnecessary to define a
reference frame.

The characterization of the dust polarization auto- and cross-angular power
spectra has led to the intriguing evidences, first reported in (Planck
Collaboration Int. XXXVIII 2016), for an $E/B$ asymmetry
($\mathcal{R}_{EB}\sim 2$) and a positive $TE$ correlation ($r^{TE}\sim 0.36$)
that were not anticipated, at least for sky regions at intermediate and high
Galactic latitudes in which a weak signal with random polarization
orientations was expected from the dust. Dedicated analysis of the data has
revealed that this -a priori unexpected- signal may likely originate from the
observed correlation between the orientation of ridges seen in intensity maps
and the sky-projected orientation of the integrated magnetic field (Planck
Collaboration Int. XXXII 2016; Planck Collaboration Int. XXXV 2016; Planck
Collaboration Int. XXXVIII 2016; Rotti & Huffenberger 2019; Soler 2019). Such
a correlation between ISM clouds and magnetic field was known for some time
for molecular clouds (e.g., Goodman et al. (1990)) and subsequently observed
for clouds of the diffuse ISM seen in HI emission (McClure-Griffiths et al.
2006; Clark et al. 2014). This observed correlation is likely the result of
the coupling of interstellar matter and the ambient magnetic field, which is
expected in the framework of the magneto-hydrodynamic (MHD) description of ISM
and it has been extensively discussed in the literature (e.g., Hennebelle
2013; Soler et al. 2013; Caldwell et al. 2017; Kandel et al. 2018; Kim et al.
2019).

To make progress in the characterization of Galactic foregrounds to the CMB
polarization, it is necessary to consider the sky and the ISM structures as
three-dimensional (3D) systems projected onto the sky (e.g., Tassis & Pavlidou
2015; Pelgrims et al. 2021a). In this context, dedicated studies have been
carried out to estimate and characterize the result of projection effects of
ISM structures to the measured polarization power spectra (e.g., Rotti &
Huffenberger 2019; Huffenberger et al. 2020; Hervías-Caimapo et al. 2021). In
these studies, the shape of ISM structures have been routinely, if not
exclusively, assumed to be filamentary. The authors have indeed considered and
modeled ISM clouds as being cylinders or prolate (elongated) spheroids.
However, to date, there is no consensus that all ISM clouds have a filament-
like morphology. On the contrary, numerical simulations show that turbulent
flows tend to stretch and compress the ISM gas into sheet-like and filament-
like structures, which appear elongated in column density maps because of
projection effects (e.g., Hennebelle 2013). This shape ambiguity is not new
and has been at the heart of a long lasting debate in the ISM literature
(e.g., Kainulainen et al. 2016 vs. Tritsis & Tassis 2018). Dense, molecular,
self-gravitating clouds generally appear filamentary on the sky, giving rise
to the picture of a cosmic web in which clouds fragment to give rise to star-
forming regions at its nodes (e.g., Myers 2009). However, there are several
observational pieces of evidence indicating the existence of clouds with
sheet-like morphology that are necessary to explain light echoes of supernovae
explosions and pulsar scattering data (e.g., Williamson 1972; Suntzeff et al.
1988; Spyromilio et al. 1995; Brisken et al. 2010; Yang et al. 2016). Such
sheet-like clouds may result from large scale planar shocks when supernovae
induced bubbles interact (Hartmann et al., 2001). In addition, the very fact
that filamentary patterns are not all straight but may show large-scale
curvature, as suggested in (Planck Collaboration Int. XXXVIII 2016), might
also be indicative of sheet-like structures that would, for example, be part
of expanding bubbles, and would appear brighter where viewed tangentially
(e.g., Tahani et al. 2022). Hence, while diverse observations of the diffuse
ISM -from relatively low-resolution submillimeter observations by Planck to
high-resolution infrared observations by Herschel, and of HI observation in
the radio waves (e.g., Kalberla & Kerp 2009; Miville-Deschênes et al. 2010;
André et al. 2014; Clark et al. 2014)- reveal its filamentary appearance, it
might very well be that it is made of clouds having both filament-like and
sheet-like shapes.

The preferred shape of ISM clouds in the diffuse ISM, along with their
specific relation to the magnetic field orientation, is expected to depend on
the exact balance of several factors: magnetic field strength, gravitational
energy, and turbulent energy (e.g., Heiles & Crutcher 2005; Crutcher et al.
2010; Soler et al. 2013). However, it is very challenging to determine
observationally the regime corresponding to any given ISM region. The
characterization of relative orientations between apparent structures and
projected magnetic field is an equally difficult task, with results that may
also depend on the details of the specific method used in the analysis (e.g.,
Micelotta et al. 2021).

In this paper, motivated by the fact that the possibility for sheet-like
clouds has been disregarded so far in CMB-foreground characterization studies
that want to account for 3D effects, we explore the possible effects from ISM-
cloud morphology on the characterization of polarization power spectra.

Our main goal is to study how the shape of interstellar clouds affects the
angular power spectra of emitted polarized radiation from dust. We set up a
simulation-based experiment to investigate the degeneracy that cloud shapes
might lead to in the properties of observed polarization power spectra. Our
toy-models allow us to quantify the effect of the angle between the structure-
major axes and the local magnetic field orientation, as well as the effect of
the viewing angle, i.e. the angle that the sight-lines make with the magnetic
field lines.

We simulate interstellar clouds with both filamentary and sheet-like structure
using the software Asterion that we developed and present in Sect. 2, where we
also produce synthetic polarization maps and compute their angular power
spectra. In Sect. 3 we present the results of our exploration of the effect
the two angles have on the power asymmetry between $E$ and $B$ modes, and on
the correlation between $T$ and $E$ modes. We discuss our results in Sect. 4
and provide our conclusions in Sect. 5.

##  2 Numerics & synthetic data

###  2.1 Asterion

####  2.1.1 The software

Asterion is a scientific tool to simulate the magnetized ISM of our Galaxy in
3D, including dust clouds and the magnetic field that permeates them, to
directly visualize the data in 3D and interact with it into an immersed
virtual environment.

Ultimately, we hope that Asterion will assist in the reconstruction of the 3D
structure of the magnetized ISM of our Galaxy, a long-standing problem in the
quest for the origin of the magnetic field in our Galaxy. Asterion relies on
real time 3D visualization techniques with virtual reality capabilities that
reach a high degree of performance. It allows us to render the details of the
magnetized ISM and enables the user to interact with the simulated environment
as it is done in video games.

Asterion is not meant to be another visualization and ray-tracing tool, with
or without radiative transfer implemented, as already existing ones such as
Polaris (Reissl et al. 2016). Asterion enables real-time reprocessing of
portions of the visualized data at a higher resolution, allowing sub-grid
details to be computed and added to the visualized data in real-time. Asterion
is implemented in the Real-Time-Engine ‘Unreal Engine 4’ (UE4)111UE4 is a
complete suite of development tools that allows for cutting edge visualization
and immersive virtual worlds, multi-platform deployment, asset and plugin
marketplace, among other features (https://www.unrealengine.com/en-
US/unreal)..

The core features of creating and visualizing 3D structures use the parallel-
computation paradigm and utilize the benefits of modern Graphics Processing
Units (GPU). A significant amount of hardware threads (Shaders) performs
calculations independently and presents solutions at once, as 3D textures,
available for further computation and visualization.

Within Asterion (currently at the prototype level), the user is immersed in a
3D environment that emulates the Galaxy. The user can fly through the Galactic
space and visualize dust density distribution and magnetic field. For the
purpose of this work the user can specify, through a set of parameters, the
large-scale dust-density distribution model, the large-scale Galactic magnetic
field model and, clouds of dust with non-trivial morphology following some
orientation relation with the ambient magnetic field lines. The structures
become accessible using interactive controls and volume-rendering techniques.
The user can also specify a virtual-telescope (Observer) in the environment
and output two-dimensional (2D) projections (polarization and column density
maps) with given angular size. These polarization maps, intended to simulate
observations of the polarized thermal dust emission in the sub-millimeter,
encode information on the density and the magnetic field properties of the
observed portion of the Galaxy. These outputs can then be analyzed externally
as if they were actual observations. To allow for statistical analyses and to
make Asterion produce massive amounts of selected outputs, input
specifications in the form of setup files (CSV format) can be passed into
Asterion, which runs over each and produces the corresponding output maps in
an automated way.

In its current implementation, Asterion simulates both the Galaxy at the large
scales and dust clouds with higher spatial resolution in a smaller volume.
Both volumes are sampled by cubic grids made of $256^{3}$ voxels. The low-
resolution, large-scale, grid is 40 kpc on a side and the high-resolution grid
has a size that can be adjusted by the user. Both grids are represented as
Volume-Texture on the GPUs and, for each cell, local density and magnetic
field as defined by model settings is rendered.

Within Asterion, both the location of the high-resolution observed box (OB)
and the observer can be determined by the user who is free to move
independently through space. The position of the observer is set in a
heliocentric Cartesian coordinate system and the OB location is fixed through
specification of the longitude, latitude and distance as seen from the
observer. The outputs that are generated by the software and that are of
interest in this study (see Sect. 2.1.2) are related to the polarized thermal
dust emission and correspond to the view of the OB (containing simulated
clouds at high-resolution) as seen by the observer. Only the volume spanned by
the OB is mapped into the outputs. We make this choice to make it possible, in
the future, to simulate large volumes with high-resolution capabilities by the
production of mosaic in 3D.

####  2.1.2 Polarized thermal dust emission model

For a given setup of the simulated magnetized ISM, synthetic observation of
the thermal dust polarization and column density can be produced by the
software. These come in the form of four maps: $I$, $Q$, $U$ for the three
Stokes parameters of the linear polarization of the thermal dust emission, and
$K$ for the total column density. These quantities are obtained through
integration along the lines of sight through the OB according to the following
equations:

$\displaystyle I(\mathbf{n})\propto$
$\displaystyle\,\int_{0}^{+\infty}dr\,n_{\rm{d}}(r,\mathbf{n})\,\left\\{1+\mathfrak{p_{0}}\left(\frac{2}{3}-\sin^{2}\alpha(r,\mathbf{n})\right)\right\\}$
(1) $\displaystyle Q(\mathbf{n})\propto$
$\displaystyle\,\mathfrak{p_{0}}\int_{0}^{+\infty}{dr\,n_{\rm{d}}(r,\mathbf{n})\,\sin^{2}\alpha(r,\mathbf{n})\,\cos[2\,\psi(r,\mathbf{n})]}$
(2) $\displaystyle U(\mathbf{n})\propto$
$\displaystyle\,\mathfrak{p_{0}}\int_{0}^{+\infty}{dr\,n_{\rm{d}}(r,\mathbf{n})\,\sin^{2}\alpha(r,\mathbf{n})\,\sin[2\,\psi(r,\mathbf{n})]}$
(3) $\displaystyle K(\mathbf{n})\propto$
$\displaystyle\,\int_{0}^{+\infty}dr\,n_{d}(r,\mathbf{n})$ (4)

where $r$ is the radial distance from the observer along a given line of sight
with direction specified by $\mathbf{n}$. The different terms in the equation
are:

  * •

$\mathfrak{p_{0}}$ is a parameter dependent on dust polarization properties
(grain cross sections and the degree of alignment with the magnetic field),
taking a value of $0.25$ in this work 222 This is a rather high value compared
to the value of $0.2$ used in Planck Collaboration Int. XX (2015). The choice
of this value is motivated by the fact that it leads to average degree of
polarization of $\approx 21\%$ in the maps that we generate in Sect. 2.2 when
the magnetic field is in the plane of the sky. We note that the exact value of
this parameter does not affect our analysis nor our main conclusion. A purely
multiplicative rescaling of $\mathfrak{p}$ affects the scaling of the power
spectra but not the parameters $\mathcal{R}_{EB}$ and $r^{TE}$ (Huffenberger
et al. 2020).

  * •

$n_{\rm{d}}(r,\,\mathbf{n})$ is the 3D dust grain density at position
$(r,\,\mathbf{n})$

  * •

$\alpha(r,\,\mathbf{n})$ is the inclination angle between the magnetic field
and the line of sight at $(r,\,\mathbf{n})$

  * •

$\psi(r,\,\mathbf{n})$ is the local plane-of-sky polarization angle expressed
as

$\psi(r,\,\mathbf{n})=\frac{1}{2}\arctan\left(\frac{-2\,B_{\theta}(r,\,\mathbf{n})\,B_{\phi}(r,\,\mathbf{n})}{B_{\phi}(r,\,\mathbf{n})^{2}-B_{\theta}(r,\,\mathbf{n})^{2}}\right),$
(5)

where $B_{\theta}$ and $B_{\phi}$ are the local transverse components of the
magnetic field in the local spherical coordinate basis
($\mathbf{e}_{r},\,\mathbf{e}_{\theta},\,\mathbf{e}_{\phi}$) with
$\mathbf{e}_{\theta}$ pointing towards the South pole. $\psi(r,\,\mathbf{n})$
is rotated 90∘ from the position angle of the plane-of-sky component of the
local magnetic field.

This formulation assumes that the emissivity of the dust grains is constant
through the Galaxy despite known evidences for variations of the dust spectral
energy distribution (e.g., Finkbeiner et al. 1999; Planck Collaboration XI
2014; Pelgrims et al. 2021a).

As we do not consider multi-frequency analysis in this study, we are not
affected by such variations which, in any case, should be small since we only
produce maps corresponding to small 3D volumes to achieve high spatial
resolution.

filaments |  sheets
---|---
|
Figure 1:  Column density ($K$) maps obtained for filament-like (left) and
sheet-like (right) density structures with different wiggle parameters. The
wiggle intensity ($w_{i}$) is constant in columns and increases from left to
right and the wiggle correlation length ($w_{c}$) is constant in rows and
increases from top to bottom. In our settings, $w_{i}$ and $w_{c}$ have units
of [pc/(100 $\times$ 200)]. The shape parameters of filaments and sheets are
the same in each panel and read as follows (in pc). Filaments: $L\in[7,\,25]$
and $R\in[1.5,\,4.5]$; Sheets: $L\in[5,\,20]$, $R_{12}\in[10,\,20]$ and
$R_{13}\in[1,\,3]$ (see Sect. 2.1.4 and Table 1 for parameter definitions and
labels.) The circular regions displayed on those maps have an angular radius
of 12.53∘. All maps share the same color scale, which is expressed in
$10^{21}\,\mathrm{cm}^{-2}$.

####  2.1.3 Populating the Galactic space with dust

Within Asterion the Galaxy is populated by dust grains according to a smooth
large-scale density distribution on top of which dust-cloud structures are
added. The large-scale density distribution follows a parametric model that
implements either the exponential disk model (ED) or the four logarithmic-
spiral arms model (ARM4) discussed and adjusted to a full-sky Planck map in
Pelgrims et al. (2021b). Their ARM4 model with best-fit parameter values is
implemented as default in Asterion and we use that model in this work. The
large-scale model fills the low-resolution grid which is then integrated to
generate a dust column density map as seen from the Sun’s location. A latitude
profile is subsequently built and a normalization factor is computed such that
it fits the well-known relation (Kulkarni & Heiles 1987; Heiles & Troland
2003):

$N_{\mathrm{H}_{\mathrm{I}}}(b)=\frac{3.7}{\sin(|b|)}\times
10^{20}\;[\textrm{cm}^{-2}],$ (6)

where $N_{\mathrm{H}_{\mathrm{I}}}$ is the column density of neutral hydrogen
atoms and $b$ is Galactic latitude.

The normalization factor fixes the proportionality in Eq. 4 leading the large-
scale density distribution model to determine the number density of hydrogen
atoms in any place in the Galaxy. The propagation of this normalization factor
to Eqs. 1 to 3 implies that our polarization maps ($I$, $Q$, and $U$) are
given in units of dust grain emissivity, as stated above.

The particle density assigned to the high-resolution box (i.e., the OB) is
calculated from this large-scale model which is evaluated at the location of
the center of the OB. Inside the OB, the total number of particles is
determined by assuming that the matter in the OB is constant and therefore,
scales with the OB volume. The total number of particles is then divided into
two phases of the ISM which are the Warm Neutral Medium (WNM) and the Cold
Neutral Medium (CNM) (e.g., Wolfire et al. 2003). In Asterion we assume that
the WNM corresponds to a very diffuse component that follows the large-scale
density distribution model, whereas the CNM forms the ISM structures (the
clouds) which will appear brighter on the maps. The modeling of the clouds
which we consider as being either filament-like or sheet-like structures is
discussed in Sect. 2.1.4. The relative contribution of CNM and WNM to build
the column density is a free parameter in Asterion. We currently do not
consider the possibility for a contribution from an unstable neutral medium
phase (Ghosh et al. 2017; Adak et al. 2020).

Assuming a given number density of dust particles in clouds and given the
volume of the dust clouds that we model, the number of clouds required to
account for the mass of the CNM in the OB is automatically determined. The
clouds are then placed randomly within the OB according to a uniform
distribution. Both the particle number density and the morphology of the
clouds are parameters that the user can tune.

The orientation of clouds in the 3D space are not random. It is now well
established observationally that the main axes of the projected shapes of ISM
structures show orientations that are not independent of the orientation of
the ambient plane-of-sky component of the magnetic field. Both orientations
appear preferentially parallel or perpendicular, depending on their column
density (e.g., Planck Collaboration Int. XXXII 2016; Planck Collaboration Int.
XXXV 2016; Clark 2017; Soler 2019) and this alignment and misalignment may
lead to differences in characteristics of the polarization power spectra
(Planck Collaboration Int. XXXVIII 2016; Huffenberger et al. 2020; Clark et
al. 2021). To control this correlation, Asterion allows the user to vary the
angle of each cloud in 3D with the local magnetic field. We refer to this
angle as the offset angle which we denote $\omega$. The implementation of
magnetic field is discussed in Sect. 2.1.5. The correlation between dust
density structures and local magnetic field is further discussed in Sect.
2.1.6.

filaments | sheets
---|---
|
Figure 2:  Stokes $Q$ polarization maps obtained for filaments (left) and
sheets (right) with different parameter values for the stochastic component of
the magnetic field $B_{\rm{sto}}$. The component amplitude ($n_{i}$) is
constant in column and increases from left to right and the correlation length
($n_{c}$) is constant in rows and increases from top to bottom. The circular
regions, here centered on the North Galactic pole, have an angular radius of
12.53∘. Shape parameters, including those controlling the wiggles, are the
same as in Fig. 3 and are kept fixed between panels. All maps share the same
color scale.

####  2.1.4 Dust cloud structures

Asterion allows for the creation and visualization of dust clouds of the ISM
that are either filament-like or sheet-like. The basic regular geometric
shapes, which are then perturbed, are cylinders and flat rectangular
parallelepipeds for filaments and sheets, respectively. The user can control
the shape parameters (size, axis ratio, etc) and the density of dust grains
within the structure. Generally, input parameters control the ranges of
possible values and, internally, the software seeks random numbers from a
uniform distribution to create any given implementation. This feature ensures
intrinsic scatter which account for the fact that all dust clouds in the ISM
are not identical.

These basic shapes are then perturbed in order to make their appearance more
realistic, producing ‘wiggles’. For this purpose, Asterion relies on the use
of Perlin noise. Perlin noise is a type of gradient noise used to increase
appearance of realism in computer graphics simulating textures (Perlin 1985).
This technology is extensively used in video games and by the imaging
industry. Two parameters control the properties of the noise, an amplitude
($w_{i}$) and a correlation length ($w_{c}$), that the user can change.

Finally, the density distribution within the OB is smoothed in 3D through a
distance-weighted averaging of the densities from direct neighboring voxels.
This smoothing is such that the total mass is conserved. It is applied to
avoid sharp transitions between structures and their surroundings.

When the emission is integrated along the lines of sight (see Eq. 1), dust
clouds can appear as illustrated in Fig. 1 where we show the intensity maps
(Eq. 1) of a population of filaments (left panel) and sheets (right panel)
obtained for different combinations of the wiggle parameters. In these sky
maps, the major axes of all the structures are horizontal. In the case of
sheet-like clouds, the orientation of the minor axes are not bounded, which
makes it possible to observe a cloud either edge-on or face-on. Among maps of
filaments (sheets), only the wiggle parameters change, the 3D positioning of
the clouds and their main orientation remain unchanged. In Fig. 1, the wiggle
amplitude is constant through columns and increases from left to right while
the wiggle correlation length is constant through rows and increases from top
to bottom. The larger $w_{i}$ and the smaller $w_{c}$, the noisier the
structures.

####  2.1.5 Magnetic field

A parametric model for the regular large-scale Galactic magnetic field is
assumed in Asterion. Two models of the magnetic field are implemented: the
Logarithmic Spiral Arm model of Page et al. (2007) with the best-fit
parameters obtained by Pelgrims et al. (2021b) for a fit to dust polarization
maps, and a four logarithmic-spiral arms model with plane-parallel field lines
but with an azimuthal modulation of the strength that follow the large-scale
dust density. The latter is the model that we use in this work despite the
fact that it has never been tested against data. Our choice is only motivated
by the fact that this model allows for an easier control of the relative
orientation of the field lines with respect to the lines of sight of
observation (see Sect. 2.2).

####  2.1.6 Mimicking MHD turbulence

Following the description above, our simplistic approach to model the
magnetized ISM does not couple matter density distribution and magnetic field
as MHD equations do. To implement such a coupling in our toy-model and allow
for the magnetic field lines to depart from the large-scale regular model, we
proceed as follows.

First, we add a stochastic component to the magnetic field in the OB. The
stochastic component is a random realization of a Perlin-noise vector field.
This choice has the advantage of creating anisotropic fluctuations in the
magnetic field directions. Two parameters control the stochastic component of
the magnetic field; the amplitude of the fluctuations ($n_{i}$) measured in
$\mu$G and a correlation length ($n_{c}$). Then, we add the density structures
(clouds). They are first placed and oriented as before and then modified
(perturbed). To account for coupling we rely on the flux-freezing
approximation (expected to be very accurate at the low density clouds we
consider here for the high Galactic latitudes of interest), which dictates
that matter can move freely along magnetic field lines. Therefore, a shift
along the local (total) magnetic field line is imposed to each voxel of each
cloud. The amplitude of the shifts and their spatial correlation is obtained
from a scalar Perlin noise (the one that is used to generate the wiggles in
the previous section) which is again controlled by the wiggle intensity
($w_{i}$) and wiggle correlation ($w_{c}$) parameters. Given our
implementation, the structures can be squeezed or stretched along the magnetic
field lines and compression and rarefaction can happen if the field lines
converge or diverge. As a result, within individual structures, overdensities
generally appear where magnetic field lines are more compressed as observed in
the magnetized ISM (e.g., Heyvaerts & Priest 1983; Heiles & Troland 2003).

The effect of the stochastic component in the magnetic field is demonstrated
in Fig. 2 where we show maps of the Stokes $Q$ (Eq. 2) of a population of
filaments (left) and sheets (right) located in a circular region towards the
North Galactic pole. The orientation of the regular magnetic field crosses the
sky regions making an angle of 28∘ with the horizontal which corresponds to
the value of the pitch angle of the spiral arms in our model. In those panels,
only the parameters of the stochastic magnetic-field component vary; $n_{i}$
is constant in columns and increases from left to right with values of $n_{i}$
= 0.02, 0.05 and 0.08, and $n_{c}$ is constant in rows and increases from top
to bottom with values of $n_{c}$ = 0.005, 0.015, 0.035. The parameters of the
wiggles are fixed to $(w_{i},\,w_{c})=(0.2,\,0.001)$.

The effects of varying the wiggle parameters to perturb the density structures
in this coupling scheme are illustrated in Fig. 3. To produce those synthetic
maps we proceed as for Fig. 1 (with
$(\omega,\,\alpha)=(0^{\circ},\,90^{\circ})$) and fix the realization of the
stochastic component in the magnetic field (including random seed value) to be
the same with parameters $(n_{i},\,n_{c})=(0.05,\,0.0015)$. The amplitude of
the perturbation ($w_{i}$) is constant in a column and increases from left to
right with values of $n_{i}$ = 0.1, 0.2 and 0.3 and the correlation length
($n_{c}$) is constant in a row and increases from top to bottom with values of
$n_{c}$ = 0.0005, 0.0010, 0.0015. Comparison of Figs. 1 and 3 reveals, as
expected, that the coupling of structure perturbations to magnetic field lead
to significant difference in structures’ morphology as the wiggles are not
isotropic anymore. As a side effect of our implementation of the coupling
between magnetic field and matter density, structures may generally appear
more aligned with the magnetic field than as imposed by the offset angle
$\omega$, and may appear more elongated than when turbulence is switched off.
Both effects are in qualitative agreement with what MHD theory predicts (e.g.,
Brandenburg & Lazarian 2013).

filaments | sheets
---|---
|
Figure 3:  Same as for Fig. 1 but for the case of non-regular magnetic field
and density perturbation coupled to the local magnetic field 3D orientation.
The realization of the magnetic-field stochastic component is the same for
each panel and corresponds to $(n_{i},\,n_{c})=(0.05,\,0.0015)$.

###  2.2 Synthetic data: Input parameters

In this work, we want to study how the shape of ISM clouds and their alignment
with the ambient magnetic field impact the measured polarization power spectra
as well as to gauge the importance of the viewing angle. This goal, together
with the requirement to observe sky patches covering at least one per cent of
the full sky (see next section), leads us to favour certain observer-OB
configurations within Asterion.

First, we want to study a portion of the ISM typical of the intermediate-to-
high Galactic latitude sky. Therefore, we select a region placed at
$(l_{\odot},\,b_{\odot})\approx(180^{\circ},\,44^{\circ})$ so that the entire
observed region, as seen from the Sun, is above latitude 30∘. We want to
resolve density structures at the pc scale and therefore decide for an OB size
of 200 pc (the voxels are 0.78 pc on a side). To achieve an observed sky area
of 1% of the full sky, the circle defined by the intersection of the
observation cone and the back side of the OB has to have an angular size of
$\approx$ 25∘ and, therefore, the center of the OB has to be placed at 350 pc
from the observer. The distance between the Sun and the OB is
$d_{\odot}\approx$ 370 pc and the observer will not be at Sun position.

Second, the fact that the observables returned by Asterion reflect only the
portion of the ISM that is encompassed by the OB makes it possible to study
the same ISM configuration from different viewpoints. We take advantage of
this feature to control values of the inclination angle between the (regular)
magnetic field orientation at the center of the OB and the line of sight
toward the center of the OB (the angle $\alpha$ in Eqs. 1 to 3). We notice
that in general $\alpha$ may vary across the outputs. The large-scale regular
magnetic field model implements magnetic field lines following a logarithmic-
spiral pattern with a constant pitch angle of 28∘ and with zero out-of-plane
component. Therefore, we choose to place the observer at a height $z=$ 257.1
pc ($=d_{\odot}\,\sin(b_{\odot})$) above the Galactic disk. We move the
observer in this plane while keeping the same distance to the OB center in
order to explore the effect of the inclination angle on the observations. In
this case, all observations made by the observer are at latitude 0∘ and moving
the observer around the OB changes the longitude only. We sample the
inclination angles from 90∘ (magnetic field in the plane of the sky) to 0∘
(magnetic field along the line of sight pointing to the center of the OB) with
step of 15∘. The angle that the main axes of the clouds make with the
orientation of the magnetic field is the other parameter that we want to
explore. We vary this angle from 0 to 90∘ in 15∘ step. To allow us to focus
only on the effects from the shape, the viewing angle and the misalignment
between structures and magnetic field, we start our study focusing on
simulating data without the addition of the stochastic component in the
magnetic field and the correlations between density perturbations and magnetic
field. Another set of synthetic maps is generated in Sect. 3.4 where we study
the effects from those variables.

To decide on a default setting for the large number of free parameters allowed
by Asterion, we rely on observational constraints and on visual inspection of
the resulting column density maps so that the latter appear as realistic as
possible.

First, according to Heiles & Troland (2003), we consider that the CNM (the
cloud structures) must account for 39% of the gas mass and that the remaining
61% is attributed to the WNM (the large-scale density distribution). Second,
based on Spitzer (1978), we consider that the particle number density in
clouds must be chosen randomly for each cloud in the range of 10 to 70
$\mathrm{cm}^{-3}$.

Following Heiles (1976), the typical length of clouds is expected to be around
20 pc with a width in an approximate ratio of 10:1. We explored Asterion’s
parameter space around those values also varying the parameters describing the
wiggles (the fluctuations in the density distribution within the clouds), and
the values or range of values of all parameters we choose are listed in Table
1. The wiggle parameters are fixed to the same values for filaments and
sheets. For filaments, the length is drawn from a uniform random distribution
in the range 7 to 25 pc while the thickness varies uniformly from 1.5 to 4.5
pc. For sheets, the length is drawn from a uniform random distribution in the
range from 5 to 20 pc, the width of the sheets takes random values in the
range from 10 to 20, and the thickness (the height) in the range 1 to 3.

We adopt the observer-OB configurations presented above together with the
chosen settings for the cloud morphology and generate 10 realizations of the
magnetized ISM for each view point by changing the random seed in Asterion. To
summarize, for both filaments and sheets, we explore seven values for the
inclination angle, seven values for the offset angle (between the structures’
main axis and the magnetic field) and we generate 10 realizations for each
couple of angles in order to infer the variance induced by the positioning of
the clouds and the scatter from the specific choices of cloud size and shape
and small-scale fluctuations. Therefore, we generate a set of 490 observations
for each. An example of $I$, $Q$ and $U$ polarization maps is shown in Fig. 4
for clouds being filaments (top) or sheets (bottom) with major axis aligned
with the ambient magnetic field lines; the latter making an angle of
$90^{\circ}$ with the line of sight for the filament output and $60^{\circ}$
for the sheet output. Since the magnetic field runs horizontally, most of the
polarization signal is observed in positive Stokes $Q$.

However, given the large area of our output maps, some signal is also observed
in Stokes $U$ far from the center of the maps because of projection effects.
Even for the case of a uniform magnetic field in the OB, the effective
inclination and position angles of the magnetic field with respect to the line
of sight and in the plane of the sky vary in the image. The smaller the
inclination angle, the larger the variation in the position angle.

Table 1: Parameters and values kept constant in our toy-models for filaments and sheets. shape | parameters | label | value (range)
---|---|---|---
Filaments | Length [pc] | $L$ | $[7.0-25.0]$
| Thickness [pc] | $R$ | $[1.5-4.5]$
Sheets | Length [pc] | $L$ | $[5.0-20.0]$
| Width [pc] | $R_{12}$ | $[10.0-20.0]$
| Thickness [pc] | $R_{13}$ | $[1.0-3.0]$
Both | Background Density |  | $0.61$
|  Particle density [cm-3] | $n_{\rm{d}}$ | $[10.0-70.0]$
| Wiggle Intensity | $w_{i}$ | $0.003$
| Wiggle Correlation | $w_{c}$ | $0.001$

###  2.3 Polarization power spectra

####  2.3.1 Formalism and main characteristics

The analyses of polarization power spectra are generally carried out using
pseudo angular power spectra defined as:
$\mathcal{D}_{\ell}^{XY}=\ell(\ell+1)\,C_{\ell}^{XY}/(2\pi)$ where
$C_{\ell}^{XY}$ are the angular (auto- or cross-) power spectra and where $X$
and $Y$ refer either to $T$, $E$ or $B$ (e.g., Bracco et al. 2019b).

The $E/B$ power asymmetry is measured through the $\mathcal{R}_{EB}$ ratio,
which is obtained by averaging the ratio of the auto-power spectra
$\mathcal{D}_{\ell}^{EE}$ and $\mathcal{D}_{\ell}^{BB}$ over a specified
multipole range

$\displaystyle\mathcal{R}_{EB}\equiv\left<\frac{\mathcal{D}_{\ell}^{EE}}{\mathcal{D}_{\ell}^{BB}}\right>,$
(7)

where $\left\langle\cdot\right\rangle$ stands for the mean over multipole
bins.

To quantify the correlation between the $T$ and $E$ power spectra we use the
normalized parameter $r^{TE}_{\ell}$ introduced by Caldwell et al. (2017).
$r^{TE}_{\ell}$ takes values $1$, $-1$ and $0$ in case of perfect correlation,
perfect anti-correlation and absence of correlation, respectively, and is
defined as

$r^{TE}=\left\langle
r^{TE}_{\ell}\right\rangle=\left\langle\frac{C_{\ell}^{TE}}{\sqrt{C_{\ell}^{TT}\,C_{\ell}^{EE}}}\right\rangle\;.$
(8)

The correlation between the power spectra $T$ and $B$ discussed in Sect. 4 is
also computed through the correlation coefficient $r^{TB}$ obtained by
substituting $E$ for $B$ in Eq. 8. Additionally, visual inspection of the
power spectra informs us that most of the $TT$, $EE$ and $BB$ auto-power
spectra present an apparent power-law dependence in $\ell$ in the range
$\ell\in[100,\,500]$. Consequently, we decide to adjust a power-law power
spectrum model of the form

$D_{\ell}^{XX}=A^{XX}_{80}(\ell/80)^{a_{XX}+2}$, where $X\in\\{E,B\\}$, and to
characterize the spectra through the values of $A^{XX}_{80}$ and $a_{XX}$.

####  2.3.2 Computation from synthetic maps

$Filaments$

$Sheets$

Figure 4: From left to right: Intensity ($I$), $Q$ and $U$ polarization maps
and polarization power spectra obtained for filament-like (top) and sheet-like
(bottom) clouds. The values of quantities extracted from the power spectra are
$\mathcal{R}_{EB}=1.69$, $r^{TE}=0.65$, $a_{TT}=-2.47$, $a_{EE}=-2.45$,
$a_{BB}=-2.47$ for filaments and $\mathcal{R}_{EB}=1.36$, $r^{TE}=0.61$,
$a_{TT}=-2.19$, $a_{EE}=-2.40$, $a_{BB}=-1.91$ for sheets.

We use the Xpol code333https://gitlab.in2p3.fr/tristram/Xpol (Tristram et al.
2005) to compute the polarization power spectra of the dust polarized sky and
account for incomplete sky coverage. The reliability of Xpol at estimating
polarization power spectra has been tested for sky area larger or equal to one
per cent of the full-sky444Tristram 2020, private communication. This limit
fixes the minimum angular size of the output that we generate using Asterion.
This choice, together with the spatial resolution that we want to achieve in
the OB, further sets the size of the OB and its distance to the observer (see
previous section).

In practice, we project Asterion’s outputs onto a full-sky HEALPix map with
resolution parameter $N_{\rm{side}}=2048$. Most of the map is empty. We then
define a circular mask of radius 11.5∘ centered on the outputs’ center. This
choice allows us to retain only the sky area that is free of artifacts from
ray-tracing the volume-limited cubic OB. The sightlines that are kept pass
through both the front and back faces of the OB, as seen by the observer. To
avoid power leakage we further apodize (smooth) the mask using a 0.5 degree
FWHM beam. This mask is applied to Asterion’s outputs when projected on
HEALPix map. The effective sky fraction of the mask ($f_{\rm{sky}}$) is 1% of
the full-sky. Using Monte Carlo simulations we checked that our mask does not
introduce bias or leakage in the power spectrum estimation following the same
validation procedure as in (Pelgrims et al. 2022).

For the computation of the power spectra there are some constraints imposed
for the multipole moments. The maximum value of $\ell$ is defined by the
effective angular resolution achieved in Asterion’s outputs. Visual inspection
of several power spectra informs us that above $\ell=500$ there is a sudden
drop of power, reflecting the effects of the limited spatial resolution
achieved in the OB and the smoothing of density structure.

The threshold at low multipole values comes from uncertainties in power
spectrum estimation.

Since our synthetic maps do not contain observational noise, we consider only
the sampling variance as a source of uncertainty in our power spectrum
estimates. The latter is linked to the number of unmasked pixels and their
spatial arrangement on the sky. The analytical estimate from the sampling
variance is given by (Planck Collaboration Int. XXX 2016):

$\sigma^{XX}_{C_{\ell}}=\sqrt{\frac{2}{(2\ell+1)\,f_{\mathrm{sky}}\,\Delta\ell_{\rm{bin}}}}\,C^{XX}_{\ell}\;,$
(9)

where $f_{\rm{sky}}$ and $\Delta\ell_{\rm{bin}}$ are the sky fraction and the
width of the multipole bins used to estimate the power spectra, respectively.
We choose to reject power spectrum estimates for multipole $\ell$ below 100 so
as to ensure that $C_{\ell}/\sigma_{C_{\ell}}\geq 3$ always.

To summarize, we use Xpol to estimate the polarization power spectra from
Asterion outputs in the multipole range $\ell\in[100,\,500]$ adopting bins of
width 20. We compute the $TT$, $EE$, $BB$ and $TE$ spectra for every output of
our sample and characterize them through their $\mathcal{R}_{EB}$ and $r^{TE}$
values. An example of such a set of polarization power spectra is shown in
Fig. 4 along with the corresponding polarization maps.

##  3 Analysis

In this section, we explore the dependence of the values of $\mathcal{R}_{EB}$
and $r^{TE}$ with respect to (i) the offset angle between the clouds’ long
axes and the magnetic field ($\omega$) and (ii) the inclination angle of the
magnetic field with the line of sight, for both filament-like and sheet-like
clouds. We vary both angles from 0∘ to 90∘ with a step of 15∘. Since the exact
values of $\mathcal{R}_{EB}$ and $r^{TE}$ depend sensitively on the specific
choice of parameters used to produce the synthetic maps, we are primarily
interested in the trends that they exhibit as a function with $\omega$ and
$\alpha$.

###  3.1 $\mathcal{R}_{EB}$ and $r^{TE}$ dependence on $\omega$

Here we infer the dependence of $\mathcal{R}_{EB}$ and $r^{TE}$ on the angle
that the major axes of the structure makes in three dimensions with the local
magnetic field orientation.

We first select the synthetic maps created with the magnetic field in the
plane of the sky ($\alpha=90^{\circ}$) and we sort them according to the
offset angle that the major axes of the structures make with the local
magnetic field. In the top row of Fig. 5, we show the dependence of
$\mathcal{R}_{EB}$ (left) and $r^{TE}$ (right) as a function of $\omega$, for
both shape families of clouds.

For filament-like clouds, we observe that $\mathcal{R}_{EB}$ as a function of
$\omega$ follows a parabolic trend starting from a maximum when clouds’ major
axes are perfectly aligned with the magnetic field ($\omega=0^{\circ}$), then
decreases to reach a minimum toward $\omega=45^{\circ}$ and increases back for
structures perpendicular to the magnetic field ($\omega=90^{\circ}$). This
trend reflects well the generally accepted picture according to which $E$
modes are maximized for structures perfectly parallel or perpendicular to the
magnetic field whereas $B$ modes dominate when (projected) linear structures
make an angle of $45^{\circ}$ with it (e.g. Zaldarriaga 2001; Huffenberger et
al. 2020; Clark et al. 2021). Obtaining this result, for filaments, validates
our modeling and analysis pipeline. Still for filament-like clouds, we observe
that $r^{TE}$ as a function of $\omega$ follows a decreasing squared-cosine
trend, even showing negative values for large offset angles ($\omega\geq
60^{\circ}$). This trend is reminiscent of the result obtained for individual
straight cylinders (Huffenberger et al. 2020) and is understood by the same
argument as above. When $E$ modes are reduced due to misalignment, the
corresponding $E$ map systematically looses its correlation with the $T$ map.
The two even get anti-correlated when the structures are perpendicular to the
field since $E$ modes become negative in such a configuration.

For sheet-like clouds, the same parabolic trend is recovered for
$\mathcal{R}_{EB}$ as a function of $\omega$ but with a smaller absolute
difference between the extrema than in the case of filament-like structures.
This is also true for the squared-cosine trend of $r^{TE}$ versus $\omega$. We
understand these observations by the fact that sheets have one additional
degree of freedom in their orientation. The fact that they could be seen
either face-on, edge-on or with any intermediate angles, even though the
orientation of their principal axes is fixed as compared to the magnetic field
orientation (here parallel to it), has the effect of reducing the imbalance
between $E$ and $B$ modes. Even in the case the sheets’ major axes are
parallel to the magnetic field, when projected on the sky, wiggles of the
sheet-like structures may show appreciable angles with the magnetic field
lines, contributing to the production of $B$ modes rather than $E$ modes. This
reduces both the $\mathcal{R}_{EB}$ and $r^{TE}$ maximum values.

For both sheet- and filament-like structures, we observe that the absolute
difference between extrema in the trend of $\mathcal{R}_{EB}$ versus $\omega$
is reduced when the inclination angle decreases, for example when the field
lines depart from the plane of the sky. This is again understood by projection
effects and is further explored in Sect. 3.2.

In the bottom row of Fig. 5, we show the scatter plots of $r^{TE}$ versus
$\mathcal{R}_{EB}$ obtained for filament-like clouds (left) and sheet-like
clouds (right), for all offset and inclination angles. The scatter plots are
color-coded according to the offset angles. These plots offer a different and
generalized view of the trends observed in the top panels which were
restricted to the case $\alpha=90^{\circ}$.

$\omega$ dependence of $\mathcal{R}_{EB}$ and $r^{TE}$

Figure 5: Dependence of $\mathcal{R}_{EB}$ and $r^{TE}$ values on $\omega$,
the misalignment angle between magnetic field’s orientation and structures’
main axes for both filament and sheet structures’ shapes. Top row shows
$\mathcal{R}_{EB}$ (left) and $r^{TE}$ (right) versus $\omega$ for
$\alpha=90^{\circ}$ for both filaments (blue) and sheets (orange). The shaded
areas include 68% of the data around the median (thick lines) obtained through
10 random realizations of the same setup. Bottom row shows scatter plots of
$(\mathcal{R}_{EB},\,r^{TE})$ pairs for filaments (left) and sheets (right)
color-coded according to the $\omega$ value and obtained for the complete set
of synthetic map. For visual reference, the black star in the bottom row
indicate the fiducial Planck values $(\mathcal{R}_{EB}=1.89,\,r^{TE}=0.36)$.

Particularly, for the case of filament-like clouds (left), we notice a tail in
high $\mathcal{R}_{EB}$ and $r^{TE}$ values which corresponds to small offset
angles ($\omega\lesssim 30^{\circ}$). Then, as $\omega$ increases up to
$\approx 45^{\circ}$, both the values of $\mathcal{R}_{EB}$ and $r^{TE}$
decrease forming a roundish cluster centered on
$(\mathcal{R}_{EB},\,r^{TE})\approx(1,\,0.1)$. As structures’ major axes
become perpendicular to the field, another smaller tail appears with negative
$r^{TE}$ and large $\mathcal{R}_{EB}$ values. This general picture is also
observed for sheet-like clouds (Fig. 5, bottom right panel) although the
scatter of the data points is larger; as expected due to the increase of the
degree of freedom. The comparison between the 2D distributions of
$(\mathcal{R}_{EB},\,r^{TE})$ pairs obtained from filament- and sheet-like
clouds is further developed below.

###  3.2 $\mathcal{R}_{EB}$ and $r^{TE}$ dependence on $\alpha$

Here we infer the dependence of the measured $\mathcal{R}_{EB}$ and $r^{TE}$
as a function of the angle between the magnetic field lines and the line of
sight.

We first select the synthetic maps created with all the clouds having their
major axis parallel to the magnetic field ($\omega=0^{\circ}$) and we sort
them according to the inclination angle of the field with respect to the line
of sight. As in the previous subsection, we have for each $\alpha$ value 10
realizations with filament-like and sheet-like clouds of the same ISM volume
that is observed from seven different observer positions (see Sect. 2.2). A
different set of polarization power spectra corresponds to each realization.
In the top row of Fig. 6, we show the dependence of the corresponding values
of $\mathcal{R}_{EB}$ (left) and $r^{TE}$ (right) as a function of $\alpha$.

For filament-like clouds, the viewing-angle and projection effects on
$\mathcal{R}_{EB}$ and $r^{TE}$ are clearly shown. $\mathcal{R}_{EB}$
decreases following a squared-sine trend from its maximum (at $\approx 1.8$)
obtained for lines of sight perpendicular to the field lines
($\alpha=90^{\circ}$) to its minimum (at $\approx 1$) obtained for lines of
sight parallel to the field ($\alpha=0^{\circ}$). As for $r^{TE}$, it also
decreases from its maximum when $\alpha\approx 90^{\circ}$ to zero when
$\alpha\approx 0^{\circ}$ but following a sine trend. For lines of sight and
field lines nearly parallel, the degree of polarization is low and any small
variations generate polarization with random orientations. This generates
equal amounts of $E$ and $B$ modes (thus $\mathcal{R}_{EB}\approx 1$) with no
particular correlation with intensity (thus $r^{TE}\approx 0$).

Similar trends are observed for sheet-like clouds with, however, much reduced
difference between extrema. As before, we interpret this reduction by the
presence of wiggles and the extra freedom in orientation that sheet-like
structures allow for. The maxima of both $\mathcal{R}_{EB}$ and $r^{TE}$ also
do not appear for $\alpha=90^{\circ}$ but at lower values ($\gtrsim
60^{\circ}$). We speculate that this peculiarity may be related to projection
effects of the 3D structures.

$\alpha$ dependence of $\mathcal{R}_{EB}$ and $r^{TE}$

Figure 6: Dependence of $\mathcal{R}_{EB}$ and $r^{TE}$ values on $\alpha$,
the angle that makes the magnetic field lines with the line of sight. Top row
shows $\mathcal{R}_{EB}$ (left) and $r^{TE}$ (right) versus $\alpha$ for
$\omega=0^{\circ}$ for both filaments (blue) and sheets (orange). Plotting
conventions are the same as in Fig. 5 except that the color-coded in the
bottom row is according to $\alpha$.

In the bottom row of Fig. 6, we show the same scatter plots as shown in the
bottom row of Fig. 5 but color-coded by their corresponding inclination angle
values. We observe that the data points corresponding to $\alpha=90^{\circ}$
draw a croissant-like pattern (already outlined in the bottom panels of Fig.
5) in which $\mathcal{R}_{EB}$ is maximized for $\omega=0^{\circ}$ and
$90^{\circ}$ and minimized for $\omega=45^{\circ}$, and that is tighter for
filament-like clouds than for sheet-like clouds. Then, as $\alpha$ decreases,
the distributions of $(\mathcal{R}_{EB},\,r^{TE})$ pairs depart from the
croissant shape and become rounder with a center at about $(1,\,0.1)$. The
extreme cases are for $\alpha=0^{\circ}$ for which the data points (all
$\omega$ values included) show a very large scatter for both filament-like and
sheet-like clouds.

###  3.3 Ambiguity from the shapes of dust clouds

It is obvious that sheet-like structures may appear filament-like if they are
seen nearly edge-on, that is, if their minor axes is close to the plane of the
sky. Such an example is shown in Fig. 4. However, if dust clouds of the ISM
are sheet-like rather than filament-like structures, and given that generally
the 3D orientation of structures does not depend on the observer position
(us), there is no reason why all the minor axes of ISM clouds would lie in the
plane of the sky555We note that even in the scenario where the Sun is embedded
in the Local Bubble and the magnetic field lines follow the surface of the
shell of the Local Bubble (as discussed in Sect. 4), it is difficult to find a
physical argument that would make the minor axes of all the sheets to lie in
the plane of the sky.. As a consequence, due to projection effects, ripples in
the sheets may produce filament-like patterns (ridges) uncorrelated with the
projected orientation of the principal axes of the sheets, or of the magnetic
field. Therefore, this could induce different patterns in the polarization
maps and therefore, different characteristics of the (polarization) power
spectra.

We studied the dependence of the summarizing characteristics of the
polarization power spectra on the offset angle and on the inclination
(viewing) angle for outputs generated out of both, filament-like and sheet-
like clouds. Beside the overall similar trend observed for both shape types,
we showed that sheet-like structures lead to a generally looser correlation of
the $\mathcal{R}_{EB}$ and $r^{TE}$ values than filaments. This reflects the
increase of degree of freedom in the projected orientation of substructures in
sheets than in filaments. In addition, reaching simultaneously large
$\mathcal{R}_{EB}$ and large positive $r^{TE}$ values for sheets appears less
common. This increase of the possible ways to project a sheet on the sky
results in the reduction of the $E$ to $B$ power asymmetry and the correlation
between $T$ and $E$.

However, despite these small differences, we find that the distributions of
$(\mathcal{R}_{EB},\,r^{TE})$ pairs from maps of filament-like and sheet-like
clouds largely overlap. This is best seen in Fig. 7 where we show contour
plots of the distributions of points on the $(\mathcal{R}_{EB},\,r^{TE})$
plane for both shape categories on a same figure. The three embedded contours
enclose about 10, 70 and 90 per cent of the data points. The overlap is
striking and the larger scatter of data points from outputs made out of sheet-
like clouds is evident.

Figure 7 makes it clear that the shape of the cloud structures cannot be
distinguished by the unique consideration of the $\mathcal{R}_{EB}$ and
$r^{TE}$ values. The observation of an $(\mathcal{R}_{EB},\,r^{TE})$ pair
appears almost as likely coming from either family of shapes. This is even
more true if one relaxes the constraints of the morphology (axis ratios) and
size of the clouds, which we have kept fixed in this study (see Sect. 2.2).

$r^{TE}$ |
---|---
|  $\mathcal{R}_{EB}$
Figure 7: Contour plots of the 2D distributions of
$(\mathcal{R}_{EB},\,r^{TE})$ from filament-like (blue) and sheet-like
(orange) clouds. The embedded contours enclose about 10, 70 and 90 per cent of
the data points. To guide the eye, the black star indicates the fiducial
Planck values $(\mathcal{R}_{EB}=1.89,\,r^{TE}=0.36)$.

$Filaments$ $Sheets$

$a_{EE},\,a_{BB}$ $a_{EE},\,a_{BB}$

Figure 8: Histograms of spectral indices of the power-law power spectra fitted
to the $E$ (orange) and $B$ (green) auto-power spectra obtained for filament-
like clouds (left) and sheet-like clouds (right); the full sample is included.

Aside from the $\mathcal{R}_{EB}$ and $r^{TE}$ characteristics, power spectra
of outputs made from filaments and sheet-like structures are similar. This can
already be inferred from the right column of Fig. 4 where, for those examples,
it is seen that amplitudes and slopes of spectra have similar values. This is
better demonstrated in Fig. 8 where we present histograms of the spectral
indices of power-law fits to the auto-power spectrum of $E$ and $B$ modes, for
filament (left) and sheet-like (right) clouds. No substantial difference can
be spotted between cloud shapes.

Based on these considerations, we argue that interpreting polarization power
spectra solely in terms of filament characteristics and their relationship to
the ambient magnetic field, is risky because of this degeneracy on the
intrinsic shape of the clouds.

###  3.4  Turbulent ISM and magnetic field

The toy-model simulations of the magnetized ISM corresponding to the
polarization maps used in the analysis so far are somewhat unrealistic in the
sense that they lack known physics. In our model, the orientations of density
caustics and magnetic field are not correlated at small scales although we
expect them to be, at least to some extent, due to flux-freezing (Heiles &
Crutcher 2005). These simple, but non-trivial simulations have allowed us to
study in detail the dependence of the summary characteristics of the
polarization power spectra on the inclination angle, the offset angle and the
type of dust cloud morphology. Here, we want to study the effects of coupling
density structures to a non-regular magnetic field on our results. To do so,
we exploit the possibility in Asterion to add a stochastic component to the
regular magnetic field and to generate density perturbations along the local
(perturbed) magnetic field lines (see Sect. 2.1.6). Example of outputs
obtained with such coupling were presented in Figs. 2 and 3 for Stokes $Q$
maps and $K$ maps, respectively. For the purpose of our analysis we fix the
parameters for the stochastic component in magnetic field and for the coupled
wiggles such that (i) the outputs are visually similar to the dust
polarization sky at high-Galactic latitudes and (ii) the characteristics of
polarization power spectra of maps made out of filaments are close to observed
values when $\omega=0^{\circ}$ and $\alpha=90^{\circ}$. This choice, leads to
the set of parameters $(n_{i},\,n_{c})=(0.05,\,0.0015)$ and
$(w_{i},\,w_{c})=(0.2,\,0.001)$ which we do not change while exploring the
effect from varying the $\omega$ and $\alpha$ angles as in Sect. 3.

With those values we notice that generally the simulated density structures
coupled to the magnetic field show their effective major axes more aligned
with the local mean magnetic field ($\omega_{\mathrm{eff}}\lesssim
45^{\circ}$) than set by the input $\omega$ values. This is a desirable
feature motivated by observational evidences that clouds of the diffuse ISM,
both HI fibers and dust clouds, tend to be preferentially aligned with the
ambient magnetic field (McClure-Griffiths et al. 2006; Clark et al. 2014;
Clark et al. 2015; Martin et al. 2015; Planck Collaboration Int. XXXII 2016;
Planck Collaboration Int. XXXV 2016; Planck Collaboration Int. XXXVIII 2016;
Soler 2019). As a result, and as illustrated in Fig. 11, we find that
$\mathcal{R}_{EB}$ remains globally larger than unity and $r^{TE}$ remains
positive while varying $\omega$. This is the case for both cloud shapes.
Comparing Figs. 11 and 12 with Figs. 5 and 6, we find that the different
scalings present for the range $\omega\gtrsim 30^{\circ}$ are severely
attenuated, if not suppressed, when density structures are coupled to the
magnetic field and that $\mathcal{R}_{EB}$ and $r^{TE}$ values of outputs from
sheet-like structures are in general closer to outputs from filament-like
structures. The main effects of coupling the density perturbations and the
magnetic field is that 2D distributions of $\mathcal{R}_{EB}$ and $r^{TE}$
from the two types of shapes do overlap even more than in the absence of such
coupling and that negative $r^{TE}$ values are much less frequent. This is
clearly seen in Fig. 9.

$r^{TE}$ |
---|---
|  $\mathcal{R}_{EB}$
Figure 9:  Same as Fig. 7 but with noisy magnetic field and correlated
perturbations in density structures.

##  4 Discussion

In this paper we do not attempt to model the actual dust polarization sky
observed at submillimeter wavelengths, for example by Planck at 353 GHz.
Rather, our focus is to explore whether the assumed shape of the ISM clouds
may lead to erroneous conclusions in the interpretation of dust polarization
power spectra.

In order to model the dust polarization sky and in particular the measured
polarization power spectra, it would be necessary to sample a large volume of
the Galactic space with our OB, generating a 3D mosaic, possibly in a way
similar to what has been proposed by Hervías-Caimapo et al. (2021). However,
our analysis shows that the viewing angle -with which the portion of the ISM
is observed- is significant in determining the ratio and correlation between
power spectrum modes. We find an extra dependence on $\alpha$ than that
proposed by Huffenberger et al. (2020) for $r^{TE}$ only. Our results make it
clear that the overall geometrical arrangement of the magnetized ISM
surrounding the observer (us) leaves its marks on the power spectra and in
particular on the $\mathcal{R}_{EB}$ and $r^{TE}$ parameters. We see it
unlikely that the geometry dependence averages out when considering large
swaths of the sky. This is reinforced by the fact that all viewing angles are
not equally likely, especially at high Galactic latitudes. The reason is that
the Sun sits in the Local Bubble, a cavity of hot plasma that was presumably
created by supernova explosions, which is surrounded by a shell of cold
neutral gas and dust with a radius between 100 to 300 pc, depending on
observation direction. The polarized signal observed in the polar caps
($|b|>60^{\circ}$) is dominated by the emission from the shell of the Local
Bubble (Skalidis & Pelgrims 2019). The formation of this structure has shaped
the local, magnetized ISM surrounding the Sun (Alves et al. 2018; Pelgrims et
al. 2020) and has left the magnetic field to lie mostly in the plane of the
sky at high $|b|$ but may show strong departures in some places (Pelgrims et
al. 2020). In light of the highlighted $\alpha$-dependence that we find, we
caution that the exact morphology of the magnetic field in regions dominating
the polarization signal needs to be accounted for to model large areas of the
sky. This would involve the production of 3D mosaics, and the computation of
polarization power spectra to be compared to actual observations. This is a
highly complex task, which is beyond the scope of this paper.

Recently, Pelgrims et al. (2022) showed from maps synthesized from Galaxy-
sized MHD simulations that the specific observer location has a strong effect
on the characterization of the polarization power spectra. In light of our
results, it would be interesting to figure out how much of the reported
variance can be attributed to the peculiar 3D geometry of the part of the
magnetic field that is imprinted in maps and thus, in power spectra.

On the other hand, Bracco et al. (2019b) have studied polarization power
spectra from Planck data at intermediate-to-low Galactic latitudes in sky
patches of the same size as ours. They found that the $\mathcal{R}_{EB}$ and
$r^{TE}$ parameters span large ranges of values. With our toy-model
simulation-based study we show that a significant source of scatter in those
quantities may actually come from the geometrical arrangement of the
magnetized ISM, in addition to the scatter that may arise from the fact that
the offset angle in different clouds is probably not constant but rather
follows some distribution (see e.g., Huffenberger et al. 2020). Our toy-
models, however, cannot be simply compared to the results of their
observational study since our synthetic maps do not account for the full cone
of observation, which becomes important at low latitudes, and also because
they pre-processed the polarization sky maps in order to get rid of the large-
scale Galactic gradient. We do not know how this processing affects their
results.

Furthermore, in an attempt to model the actual sky, it would probably be
relevant to consider that dust clouds can be described by a mixed population
of sheet-like and filament-like structures with varying axis ratios. In this
study, we did not explore such a possibility because we want to quantify the
possible cloud-shape degeneracy of the $\mathcal{R}_{EB}$ and $r^{TE}$ values.

Currently, our results are limited by the achieved spatial resolution that is
fixed by the minimum angular size of the sky patch that Xpol can handle, and
the size of Asterion grid ($256^{3}$). We achieve a spatial resolution of
$\sim 0.78$ pc which is more than twice the thickness obtained for actual
sheet clouds of the neutral ISM ($\lesssim 0.3$ pc according to Kalberla et
al. 2016).

Another source of limitation of our study comes from the limited portion along
distance that is mapped. This reduces the number of structures that appear on
our maps as compared to real observations. This does not affect our analysis
since our main results relate to the (relative) comparison of power spectrum
characteristics between (i) different cloud shapes, (ii) different offset
angles, and (iii) different inclination angles.

Besides the special attention paid to the $E/B$ power asymmetry and the
correlation between $T$ and $E$ modes which were detected with high
significance in dust polarization maps from Planck, a more marginal signal has
been reported between the $T$ and $B$ modes (Planck Collaboration Int. XXX
2016; Planck Collaboration XI 2020). Other recent studies (Weiland et al.
2020; Clark et al. 2021) relied on a more accurate version of the Planck data
and on external data sets to prove that the weak positive $TB$ signal (a mark
of parity violation in the dust polarization sky) is a real property of the
Galactic emission and infer its origin.

A detailed study of the effect of cloud morphology on the $TB$ signal (e.g.,
Bracco et al. 2019a) and a careful analysis of the dependence on $\omega$ and
$\alpha$ along with the scale dependence of the signal is not the focus of
this paper and we postpone this to future work. Meanwhile, we show in Fig. 10,
that the correlation coefficient $r^{TB}$ between $T$ and $B$ modes measured
from filament-like or sheet-like structures does not strongly differ, at least
when the scatter from varying $\omega$ and or $\alpha$ is taken into account.
All distributions are centered on zero and, as with $\mathcal{R}_{EB}$ and
$r^{TE}$, we find that sheet-like structures lead to broader distributions
than filament-like structures do.

Regular B field Turbulent ISM

$r^{TB}$ $r^{TB}$

Figure 10: Histograms of $r^{TB}$ obtained for filament-like clouds (blue) and
sheet-like clouds (orange) for regular magnetic field (left) and with
”turbulence” switched on (right); the full sample is included.

##  5 Conclusions

The search for the primordial $B$ modes in the polarization of CMB radiation
calls for the understanding and characterization of the Galactic foregrounds
in order to disentangle their contribution to the maps of the polarized sky.

In this project, we investigated whether the morphology of the dust clouds has
an impact on the statistical characterization of the polarized emission. We
simulated interstellar clouds with filament-like and sheet-like shapes and
produced corresponding synthetic polarization maps, using the software
Asterion. Then, we computed their polarization power spectra in multipole
range $\ell\in[100,\,500]$ and we focused our study on the power asymmetry
between $E$ and $B$ modes, and on the cross correlation between $T$ and $E$
modes, through the parameters $\mathcal{R}_{EB}$ and $r^{TE}$, respectively.
We explored the dependence of $\mathcal{R}_{EB}$ and $r^{TE}$ on the
inclination angle ($\alpha$, between magnetic field and the line of sight),
and the offset angle ($\omega$, between the longest cloud axis and the
magnetic field) for both types of shape. Our study provides important insights
on the interpretation of the polarization power spectra of dust polarization
in terms of ISM properties and could help in the characterization of Galactic
foregrounds to the CMB polarization.

Our results for filaments are consistent with theoretical predictions and
observations. For filamentary structures aligned with the magnetic field
($\omega\approx 0^{\circ}$), something that seems to be true in nature for the
diffuse ISM, the power ratio $\mathcal{R}_{EB}$ can be as large as $\sim$ 2,
in agreement with Planck measurements (e.g., Planck Collaboration Int. XXX
2016). Moreover, for regular magnetic field and large inclination angles,
where polarization is maximized, $E$-mode power of structures is maximized for
offset angles $\omega$ = 0∘ and 90∘ and $B$-mode power is maximized for
$\omega$ = 45∘. This behaviour is in agreement with theoretical expectations
Zaldarriaga (2001) and previous studies (Rotti & Huffenberger 2019;
Huffenberger et al. 2020). As a result, $\mathcal{R}_{EB}$ as a function of
$\omega$ draws a parabola with minimum at $\omega\approx 45^{\circ}$, where
$B$ modes slightly dominate $E$ modes. In parallel, the correlation
coefficient $r^{TE}$ goes from its maximum at $\omega=0^{\circ}$ to its
minimum at $\omega=90^{\circ}$. These trends (with little variance from
specific ISM realizations) are similarly observed for other inclination angle
values, though with lower difference between extrema as $\alpha$ decreases.

To the best of our knowledge, this is the first time that an analysis of
polarization power spectra obtained from sheet-like interstellar clouds has
been performed. We found that the parameters $\mathcal{R}_{EB}$ and $r^{TE}$
measured from maps made from sheet-like structures show similar trends as a
function of $\omega$ and $\alpha$ angles to those observed for filaments.
However, the amplitude between extrema is generally smaller and the variance
with respect to the specific ISM realizations is larger. We understood these
characteristics in terms of the extra degree of freedom that sheet-like
structures possess related to the orientation of their corrugations with the
long axis.

Varying only the inclination and offset angles, we found that sheet-like
structures and filament-like structures lead to polarization power spectra
that cannot be distinguished in the plane of $(\mathcal{R}_{EB},\,r^{TE})$
parameters.

Consequently, we argued that measured $\mathcal{R}_{EB}$ and $r^{TE}$ values
cannot be used alone to discriminate among characteristics of ISM structures
and that this degeneracy should be accounted for in the interpretation and
modeling of observational data. We checked that our main conclusions remain
valid when a stochastic component is added to the large-scale magnetic field
and, most importantly, when density structure perturbations are added such
that the orientations of density caustics and magnetic field are correlated at
small scales as we expect them to be due to MHD physics.

Our analysis also highlights and quantifies the dependence of the $E/B$
asymmetry and $TE$ correlation on the inclination angle, a dependence that is
changed by the inclusion of fluctuations in the magnetic field lines and their
coupling to density perturbations but that preserves its significance. This
strongly suggests that the geometrical arrangement of the large scale magnetic
field as viewed by the observer is a critical factor shaping polarization
power spectra.

This was already suggested by Bracco et al. (2019a) and could explain the
large scatter on $\mathcal{R}_{EB}$ and $r^{TE}$ values observed from
different sky patches (Bracco et al. 2019b) and the strong cosmic variance
reported from MHD simulation-based studies (Kim et al. 2019; Pelgrims et al.
2022).

Finally, and for completeness, we also examined the correlation coefficient
between the $T$ and $B$ modes ($r^{TB}$) and observe that this feature of the
power spectrum cannot be used to discriminate cloud shape types either.
Instead, we find that variation in $\omega$ and $\alpha$ generates significant
dispersion in the values of $r^{TB}$. Further work will be needed to
investigate the possible scale dependence of the $TB$ signal as reported by
real sky observation of dust polarization.

###### Acknowledgements.

We would like to thank G. Panopoulou, R. Skalidis, T. Ghosh, and V. Pavlidou
for insightful discussions related to this project. We warmly thank our
referee, Andrea Bracco, for his thorough review and for providing us with
sound comments which help us improve the quality of this paper. This project
has received funding from the European Research Council (ERC) under the
European Unions Horizon 2020 research and innovation programme under grant
agreement No. 771282.

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##  Appendix A Power spectrum characteristics and turbulent ISM

Figures 11 and 12 show the main results obtained when we consider our simple
implementation to account for density structures correlated to non-regular
magnetic field, that is, to mimic basic effects expected from MHD physics.
These figures are obtained following the very same procedure as Figs. 5 and 6
obtained with a uniform magnetic field permeating the OB.

$\omega$ dependence of $\mathcal{R}_{EB}$ and $r^{TE}$

Figure 11:  Same as for Fig. 5 but for when a 3D vector noise is added to the
regular magnetic field and the wiggles of density structures are constrained
to follow the local magnetic field lines.

$\alpha$ dependence of $\mathcal{R}_{EB}$ and $r^{TE}$

Figure 12:  Same as for Fig. 6 but for when a 3D vector noise is added to the
regular magnetic field and the wiggles of density structures are constrained
to follow the local magnetic field lines.