Model of the polarized foreground diffuse Galactic emissions from 33 to 353 GHz
Abstract
We present 3D models of the Galactic magnetic field including regular and turbulent components, and of the distribution of matter in the Galaxy including relativistic electrons and dust grains. By integrating along the line of sight, we construct maps of the polarized Galactic synchrotron and thermal dust emissions for each of these models. We perform a likelihood analysis to compare the maps of the Ka, Q, V and W bands of the Wilkinson Microwave Anisotropy Probe (Wmap) and the 353 GHz Archeops data to the models obtained by varying the pitch angle of the regular magnetic field, the relative amplitude of the turbulent magnetic field and the extrapolation spectral indices of the synchrotron and thermal dust emissions. The bestfit parameters obtained for the different frequency bands are very similar and globally the data seem to favor a negligible isotropic turbulent magnetic field component at large angular scales (an anisotropic lineofsight ordered component can not be studied using these data). From this study, we conclude that we are able to propose a consistent model of the polarized diffuse Galactic synchrotron and thermal dust emissions in the frequency range from 33 to 353 GHz, where most of the CMB studies are performed and where we expect a mixture of these two main foreground emissions. This model can be very helpful to estimate the contamination by foregrounds of the polarized CMB anisotropies, for experiments like the Planck satellite.
keywords:
ISM: general – Methods: data analysis – Cosmology: observations – millimeter1 Introduction
The Planck
As summarized in fauvet the main polarized foreground
contributions come from the diffuse Galactic synchrotron emission (page2007)
and from thermal dust emission (benoit2004a; ponthieu2005). The polarized
synchrotron emission have already been modelled by page2007; sun; jaffe; jansson; fauvet based on
models of the Galactic magnetic field (han2004; han2006) and of
the relativistic electrons in the Galaxy. Concerning the thermal dust
emission, modeling based on the physical origins of this polarized emission
has already been discussed in ponthieu2005; page2007; fauvet.
We propose here an extended analysis of the 3D joint model of
the Galactic polarized diffuse
emissions discussed in fauvet. In the previous analysis we
focused on the Wmap data at 23 GHz and Archeops at 353 GHz data
where synchrotron and thermal dust emissions dominate, respectively.
Here, we use complementary data : the other frequency bands of Wmap from 33
to 94 GHz, where a mixing of those
emissions is expected, and the Archeops data at 353 GHz. Furthermore
we apply here a
pixeltopixel likelihood based comparison instead of a Galactic
profilebased method as discussed in fauvet.
The paper is structured as follows: Sect. 2 describes the fiveyear Wmap and Archeops data set used in the analysis. In Sect.3 we describe in detail models for the polarized foreground emissions. Section 4 discusses the 3D model of the Galaxy using to construct the polarized Galactic emissions. The models are statistically compared to the data in Section 5 and we discuss the results in Section 6. We finally conclude in Section 7.
2 Observations
2.1 Diffuse Galactic synchrotron emission
The synchrotron emission is an important contributor to the
diffuse sky emission at both radio and microwave observation
frequencies.
In intensity, the 408MHz allsky continuum survey (haslam), at a resolution of of 0.85 degrees, is a good tracer of the synchrotron emission and it will be used in the following
as a template. In particular, we use the 408MHz allsky map available on the LAMBDA website in the
HEALpix pixelisation scheme (gorski). We correct this map from the contribution from freefree emission which estimated to be about 30 %. The corrected maps was then
downgraded to the resolution of the WMAP and Archeops maps discussed below.
In polarization, Faraday rotation introduces complications into the interpretation of the radio data
since strong depolarization is observed for frequencies lower than
10 GHz, particularly concerning the inner part of the Galaxy
(burn1966; sun; jaffe; jansson; 2006A&A...455L...9L). A detailed discussion on the depolarization process, based
on data from the Leiden survey, can be found in porta2006. For this reason the best polarized Galactic diffuse synchrotron tracers
are at high frequency such as the Wmap survey (page2007).
We used here the fiveyear Wmap Q and U low resolution maps for the frequency channels centered at 33 (Ka band), 41 (Q band), 61 (V band) and 94 GHz (W band) (page2003; gold). These data are available on the LAMBDA website in the HEALpix pixelisation scheme at . The associated noise is estimated using the full noise correlation matrix also available on the LAMBDA website in the same resolution.
2.2 Thermal dust
The thermal dust emission is significant in the Wmap data only for
frequencies above 70 GHz, then we also used here the Archeops 353 GHz Q and U maps as tracers of the polarized thermal dust
emission. Those maps cover about 20 % of the sky (macias) and were filtered and downgraded to to make them
comparable to the Wmap ones.
In intensity, the most accurate measurements of the thermal dust emission are those of the IRAS satellite (neugebauer) and in particular at 100 m. We use here predicted fullsky maps of sub millimeter and microwave emission from the diffuse interstellar dust in the Galaxy from finkbeiner which were produced combining the IRAS data at 6.1 arcmin and the COBE DIRBE data at 40 arcmin (schlegel). These maps were downgraded to the resolution of the WMAP and Archeops maps presented above.
3 Emissivity model in polarization
We present in this section a realistic model of the diffuse
polarized synchrotron and dust emissions
using a 3D model of the Galactic magnetic field and of the matter
density in the Galaxy. We will
consider the distribution of relativistic cosmicray electrons (CREs),
, for the synchrotron emission and the distribution of
dust grains, , for the thermal dust
emission. Following fauvet we calculate the Stokes parameters
I, Q and U for the Galactic polarized emission along the line of sight
as follows.
For the synchrotron emission (ribicki) we write :
(1)  
(3)  
(5)  
where is the magnetic component along the lineofsight n, and and the magnetic field components on a plane perpendicular to the lineofsight. Notice that the 3 vectors n,l,t form an orthonormal basis being l and t oriented to the north and to the east respectively in a plane perpendicular to n. The polarization fraction is set to 75% (ribicki). The polarization angle is given by :
(6) 
The distribution of relativistic electrons,
, is described in detail in
section 4. is the reference map in
intensity
constructed from the 408 MHz all sky continuum
survey haslam after subtraction of the bremsstrahlung
(freefree) emission
and is the frequency of observation. To subtract the freefree
contribution
we used the Wmap Kband freefree foreground map generated from the
maximum entropy method (MEM) (hinshaw; bennett2003a). Notice that
we do not use the synchrotron MEM intensity map at 23 GHz
(hinshaw) as a synchrotron template to avoid any possible
Anomalous Microwave Emission (AME) contamination (the Wmap team made no attempt to fit for
the latter). The spectral index used to extrapolate maps at various
frequencies is a free parameter of the model. The SED of the synchrotron emission in the radio and microwave domain,
and in particular in the 33 to 353 GHz range, can be well approximated by a power law in antenna temperature units (ribicki).
This is due to the fact that the energy spectrum of the Galactic relativistic electrons producing the radio and microwave synchrotron emission is also well approximated by a power law
(2004ApJ...601L..13K). .
For the thermal dust emission we write
(7)  
(9)  
(11)  
where the dust polarization fraction is set to 10
% (ponthieu2005) based on the Archeops data, and is the dust
grain distribution discussed in section 4. The
term accounts for
the geometrical suppression and is an
empirical factor which accounts for the misalignment between dust
grains and the magnetic field lines (see (fauvet) for details).
The reference map, was taken to be model 8 in finkbeiner at 545 GHz. The spectral index used to
extrapolate maps at various frequencies is a free parameter of the model.
In the following we work on antenna temperature, RayleighJeans units. Assuming nearly constant dust temperature
across the sky then a powerlaw approximation for the thermal dust emission in antenna temperature
units can be used (planck).
4 A 3D modeling of the Galaxy
We describe here the 3D model of the Galaxy as used in the previous Stokes parameter definitions both for synchrotron and dust.
4.1 Matter density model
In galactocentric cylindrical coordinates we consider an exponential distribution of relativistic electrons on the Galactic disk motivated by drimmel:
(12) 
where defines the width of the distribution
vertically and it is set to kpc in the
following. defines the distribution radially and
it is set to kpc (see sun; jaffe; fauvet). The value of
is set to .cm (sun).
The density distribution of dust grains in the Galaxy is poorly known and we therefore choose to describe it in the same way as for relativistic electrons:
(13) 
where and are the radial and vertical widths of the distribution. In the following we set them to 3 and 1 kpc respectively. Notice that we expect these two parameters to be strongly correlated for both dust grain and electron distributions, hence we decided to fix one of them as in previous analyses (sun; jaffe). We have tested different values of these two parameters and found no impact on the final results.
4.2 Galactic magnetic field model
The Galactic magnetic field model consists of a regular component and a turbulent component such that
(14) 
where is the amplitude of the turbulent component and it is a free parameter of the model. In the following we will express it as a relative amplitude with respect to the amplitude of the regular component.
Regular component
The regular part is a Modified Logarithmic Spiral model, discussed in detail in fauvet. In galactocentric cylindrical coordinates it reads
(15)  
where is the pitch angle, and is the radial scale set to 7.1 kpc. is the vertical scale, with degrees and kpc. Following taylor we restrict our model to the range kpc. The lower limit is set to avoid the center of the Galaxy for which the physics is poorly constrained and the model diverges. The intensity of the regular field is fixed using pulsar Faraday rotation measurements by han2006
(16) 
where the largescale field intensity at the Sun position is
and the physical cut kpc. The distance
between the Sun and the Galactic center, is set to 8 kpc
(eisenhauer; reid).
Turbulent component
In addition to the largescale Galactic magnetic field, Faraday rotation measurements on pulsars in our vicinity have revealed a turbulent component on scales smaller than a few hundred pc (lyne). Moreover it seems to be present on large angular scales (han2004) with an amplitude estimated to be of the same order of magnitude as that of the regular one (han2006). The magnetic energy in Fourier space, , associated with the turbulent component is well described by a power spectrum of the form (han2004; han2006)
(17) 
where and .
To obtain the 3D spatial distribution of the turbulent magnetic field we produced independent Gaussian simulations
from the above power spectrum in the , and directions on boxes of of 512 pixels at a resolution of 56 pc.
We renormalize the simulated boxes so that the total amplitude of the turbulent component is .
Notice that we do not include here an anisotropic/ordered
component as discussed in jaffe. As in (fauvet), our regular
component is then equivalent to the sum of what jaffe call the
coherent and ordered fields. The latter, also called ordered random
component, will be not considered in this paper because it can not be
distinguished from the largescale magnetic field when studying
polarization intensity only.
5 Method
We compute I, Q and U maps for the synchrotron and thermal dust emissions with
a modified version of
the Hammurabi code (waelkens). Each map is generated by
integrating in 100 steps along each lineofsight defined
by the HEALpix(gorski) pixel centres. The integration
continues out to 25 kpc from the observer situated
8.5 kpc from the Galactic centre. These fullsky maps are computed for a
grid of models obtained by varying the pitch angle, ,
the turbulent component amplitude, and the spectral indices
of the synchrotron and thermal dust emissions and .
The latter are assumed to be spatially constant on the sky. Dealing with
a more realistic
varying spectral index (see kogut; porta2008 for
detailed studies) is beyond the scope of this paper. However, we ensured
that this hypothesis does not impact the results for the other free
parameters in the model. Indeed, we produced
simulated Wmap observations with spatially varying synchrotron
spectral index and analyzed them assuming a constant one.
No significant bias was observed for any of the other parameters and the
error bars were equivalent to those in the case of a constant spectral index.
The range and binning step considered for each of the above
parameters are given in Table 1.
All the other parameters of the models of the Galactic magnetic field
and matter density are fixed to values proposed
in Section 4. Notice that to be able to compare the dust
models to the Archeops 353 GHz data, the simulated maps
are multiplied by a mask to account for the Archeops partial sky
coverage of 30% (macias).
Parameter  Range  Binning 

(deg)  
5.1 Likelihood based analysis
To compare the Wmap data sets and the model of Galactic polarized emissions we used a maximum likelihood analysis where the the loglikelihood function is given by
(18) 
where and
correspond respectively to the data set and model for the frequency of observation
. and index the polarization states Q and U and the pixel
number in the maps respectively. is the
pixeltopixel inverse covariance matrices for the frequency band
. These matrices allow us to estimate the noise for a given pixel and
the correlation between pixels. They are composed of
61446144 elements allocated in 4 blocks of 30723072
elements each. Each element represents the autocorrelated noise
associated to a
pixel or the correlation between 2 pixels. The elements linked to the
pixel located outer the processing mask are set to
0 (hinshaw09; limon). These matrices are available on the LAMBDA website.
We used the processing mask of the Wmap team. This processing mask is built from intensity cuts on the Wmap polarized maps at 23 GHz and the thermal dust emission model of the
Wmap team (kogut; page2007). The pointsources have also been
subtracted. More details concerning this mask can be found in hinshaw09; limon.
We also used a maximum likelihood analysis to compare the thermal dust emission model to the Archeops data. The loglikelihood function is defined by
(19) 
where and correspond to the Archeops data and to the thermal dust emission model at 353 GHz. and index the Stokes parameters Q and U and the pixel number respectively. The error bars associated to the Archeops data are estimated from 600 simulations of the time ordered data (TOD) (see ponthieu2005; macias). The instrumental noise at the TOD level was estimated following benoit2003a. From these simulations we conclude that the noise in the Archeops maps can be well approximated by anisotropic white noise on the maps. We thus compute the variance per pixel.
6 Results
Frequency band  zone  
in P06  
out of P06  
allsky  
in P06  
out of P06  
allsky  
in P06  
out of P06  
allsky  
in P06  
out of P06  
allsky  
all Wmap bands  allsky  
Archeops 353 GHz  30 % sky  

First of all, we consider a null turbulent magnetic field component.
For this case, the constraints on the parameters of the polarized Galactic
emission model, , and , using the Wmap and Archeops data are presented in Table 2.
For illustration, we also present on Figure 1 the combined marginalized likelihood
at 1 and 2D using all the Wmap data. We present the 68.8% (black), 95.4% (dark grey) and 98% (grey) confidence level contours.
The pitch angle, , of the regular magnetic field component does not seem to be correlated neither with nor .
Similar results are found in the case of the Archeops data. The bestfit values obtained for magnetic field pitch angle, , are compatible for the Wmap and Archeops data,
indicating that the same magnetic field can describe both emissions. For , as it might be expected, constraints are only reliable at high frequency (94 and
353 GHz) and are compatible within the 1 error bars. These results are also compatible at
1 level with results presented in boulanger; gold. Notice that the value obtained is for a fix degree of polarization.
An independent determination of the degree of polarization and of the dust spectral index is not possible given
the available data. Indeed, both parameters act as multiplicative factors. In order to discriminate between them we would
need a larger frequency coverage and sampling.The parameter is well constrained at the Wmap frequencies, with a bestfit value of , using all
the frequency bands and it is consistent across the frequency range. This result is consistent at the 1 with the results obtained by fauvet; gold; gold2011; page2007; sun; jansson.
For the Wmap data we computed the likelihood for the full sky, inside and outside the Wmap processing mask. In the case of , the results obtained inner and outer the mask are consistent with those from the full sky analysis within the error bars.
Thus, we can conclude that the spectral variations are smaller than the error bars in the determination of .
This seems to be consistent with the results from gold; gold2011 which found that the synchrotron spectral index is relatively
constant (with respect to our error bars) across the sky but for the Galactic plane where they found a significant increased.
Although the uncertainties on the dust spectral index are much larger our results seem to be consistent with those in early24 that favor a
larger dust spectral index at high Galactic latitudes.
channel  

Ka  
Q  
V  (95.4 % CL)  
W  
all  (95.4 % CL)  
Archeops 353 GHz  (95.4 % CL) 
The constraints on the parameters of the polarized Galactic emission model including a turbulent magnetic field component are presented in Table 3 for the Wmap and Archeops data. As above, the bestfit parameters of the model are consistent from 30 to 353 GHz but the constraints are much more loosy. In particular, the relative amplitude of the turbulent component, , is poorly constrained although the data do not seem to favour a strong turbulent component. We present on Figure 2 the combined marginalized likelihood at 1 and 2D using all the Wmap data. We present the 68.8% (black), 95.4% (dark grey) and 98% (grey) confidence level contours.
Figures 3 and 4 compare the Q (top) and U (bottom) Stokes parameter maps obtained from the bestfit parameters (right) to the Wmap Ka and W band maps (left), respectively. The maps are presented in units. At 33 GHz the synchrotron emission dominates the signal and the data is well represented by the model. However, at 94 GHz thermal dust emission dominates the signal. This can be observed both on the Q and U maps, although the latter is too noisy for a clear detection of thermal dust emission.
7 Conclusions
We proposed in this paper an extended study of the polarized
Galactic diffuse
emissions presented in fauvet. We have constructed coherent
models of these two
foreground emissions based on a 3D representation of the Galactic magnetic
field and of the distributions of relativistic electrons and dust
grains in the Galaxy. For the Galactic magnetic field we considered a
modified logarithmic spiral model for the largescale regular
component, plus a turbulent one. The relativistic
electrons and dust grains distributions have been modeled with
exponentials peaking at the Galactic center.
We performed a likelihood analysis to compare the available
Wmap and Archeops data to a set of models obtained by varying the
pitch angle of the regular magnetic field, the relative amplitude of
the turbulent magnetic field as well as the extrapolation spectral
indices for the synchrotron and thermal dust emissions.
From this analysis, we observe that the bestfit parameters are compatible
across the frequency range explored, indicating that the
polarized sky emission at the different frequencies are of the same nature.
Using the full data set, we have been able to set constraints on the
pitch angle, degrees. The bestfit value for the
spectral index of the synchrotron emission, ,
is lower but compatible with other values found in the literature (see for example kogut; gold).
This low spectral index indicates that the synchrotron emission observed at microwave frequency
is produced by relativistic electrons with an steep energy spectrum (spectral index ).
An upper limit on the relative amplitude of the turbulent component is obtained although it
seems that this turbulent part is not required to reproduce the microwave data at large angular scales.
However, we only accounted here for a statistically isotropic turbulent component and we did not for the ordered turbulent component which
in our case can not be distinguished from the regular component.
From above we can conclude that a simple model of the polarized Galactic diffuse foreground emission including synchrotron and thermal dust can account for the observed sky emission in the frequency range from 30 to 353 GHz at large angular scales. This is of great interest for the analysis of Planck satellite mission data, as Planck will measure the polarized CMB anisotropies in the frequency range from 70 to 217 GHz and we expect the foreground emissions to be dominant at large angular scales (gold2011; fauvet2012).
References
Footnotes
 journal: Astroparticle Physics
 http://www.rssd.esa.int/index.php?project=Planck
 http://lambda.gsfc.nasa.gov/product/map/current/