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<!-- DOI: 10.1051/0004-6361/200911624 -->

<h2 class="sec">Online Material</h2>



<h2 class="sec"><a name="SECTION000100000000000000000"></a>
<A NAME="sec:d"></A>
Appendix A: Parameterisation the foreground component and choice of a mask
</h2>

<p>
In this appendix, we discuss in more detail the dimension <I>D</I> of
matrix used to represent the covariance of the total galactic
emission, and the choice of a mask to hide regions of strong galactic
emission for the estimation of&nbsp;<I>r</I> with S<SMALL>MICA</SMALL>.

<p>

<h3 class="sec2"><a name="SECTION000101000000000000000"></a>A.1 Dimension <I>D</I> of the foreground component</A>
</h3>

<p>
First, we explain on a few examples the mechanisms which set the rank
of the foreground covariance matrix, to give an intuitive
understanding of how the dimension&nbsp;<I>D</I> of the foregrounds component
used in S<SMALL>MICA</SMALL> to obtain a good model of the data.  Let us consider
the case of a ``perfectly coherent'' physical process, for which the
total emission, as a function of sky direction&nbsp; <IMG
 WIDTH="10" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img25.png"
 ALT="$\xi$">
and frequency&nbsp;<IMG
 WIDTH="9" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img16.png"
 ALT="$\nu$">,
is well described by a spatial template multiplied by a
pixel-independent power law frequency scaling:

<p></p>
<DIV ALIGN="CENTER">

<!-- MATH: \begin{equation}
S_\nu(\xi) = S_0(\xi) \left( \frac{\nu}{\nu_0} \right)^{\beta}\cdot
\end{equation} -->

<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:monodim"></A><IMG
 WIDTH="130" HEIGHT="73"
 SRC="img139.png"
 ALT="\begin{displaymath}S_\nu(\xi) = S_0(\xi) \left( \frac{\nu}{\nu_0} \right)^{\beta}\cdot
\end{displaymath}"></td>
<TD WIDTH=10 ALIGN="RIGHT">
(A.1)</td></tr>
</TABLE>
</DIV><BR CLEAR="ALL"><p></p>
The covariance matrix of this foreground will be of rank one and

<!-- MATH: $\ensuremath{\tens{R}} [S] = [ \mathbf{A} \mathbf{A}^\dag {\rm var}(S_0)]$ -->
<IMG
 WIDTH="124" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img140.png"
 ALT="$\ensuremath{\tens{R}} [S] = [ \mathbf{A} \mathbf{A}^\dag {\rm var}(S_0)] $">,
with 
<!-- MATH: $A_f =
\left( \frac{\nu_f}{\nu_0} \right)^{\beta}$ -->
<IMG
 WIDTH="64" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="img141.png"
 ALT="$A_f =
\left( \frac{\nu_f}{\nu_0} \right)^{\beta} $">.
Now, if the spectral
index <IMG
 WIDTH="10" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img142.png"
 ALT="$\beta$">
fluctuates on the sky, 
<!-- MATH: $\beta(\xi) = \beta +
\delta\beta(\xi)$ -->
<IMG
 WIDTH="95" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img143.png"
 ALT="$\beta(\xi) = \beta +
\delta\beta(\xi)$">,
to first order, the emission at frequency <IMG
 WIDTH="9" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img16.png"
 ALT="$\nu$">around <IMG
 WIDTH="16" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img15.png"
 ALT="$\nu_0$">
can be written:
<p></p>
<DIV ALIGN="CENTER">

<!-- MATH: \begin{equation}
S_\nu(\xi) \approx S_0(\xi) \left( \frac{\nu}{\nu_0} \right)^{\beta} + S_0(\xi) \left( \frac{\nu}{\nu_0} \right)^{\beta} \delta \beta(\xi) \left( \frac{\nu - \nu_0}{\nu_0}\right)\cdot
\end{equation} -->

<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:bidim"></A><IMG
 WIDTH="307" HEIGHT="73"
 SRC="img144.png"
 ALT="\begin{displaymath}S_\nu(\xi) \approx S_0(\xi) \left( \frac{\nu}{\nu_0} \right)^...
...\delta \beta(\xi) \left( \frac{\nu - \nu_0}{\nu_0}\right)\cdot
\end{displaymath}"></td>
<TD WIDTH=10 ALIGN="RIGHT">
(A.2)</td></tr>
</TABLE>
</DIV><BR CLEAR="ALL"><p></p>
This is not necessarily the best linear approximation of the emission,
but supposing it holds, the covariance matrix of the foreground will
be of rank two (as the sum of two correlated rank 1 processes). If the
noise level is sufficiently low, the variation introduced by the first
order term of Eq.&nbsp;(<a href="/articles/aa/full_html/2009/33/aa11624-09/aa11624-09.html#eq:bidim">A.2</a>) becomes truly significant, we can't
model the emission by a mono-dimensional component as in
Eq.&nbsp;(<a href="/articles/aa/full_html/2009/33/aa11624-09/aa11624-09.html#eq:monodim">A.1</a>).

<p>
In this work, we consider two processes, synchrotron and dust, which
are expected to be correlated (at least by the galactic magnetic field
and the general shape of the galaxy). Moreover, significant spatial
variation of their emission law arises (due to cosmic aging, dust
temperature variation ...), which makes their emission only partially
coherent from one channel to another. Consequently, we expect that the
required dimension <I>D</I> of the galactic foreground component will be
at least&nbsp;4 as soon as the noise level of the instrument is low enough.

<p>
The selection of the model can also be made on the basis of a
statistical criterion. For example, Table&nbsp;<a href="/articles/aa/full_html/2009/33/aa11624-09/aa11624-09.html#tab:bic">A.1</a> shows the
Bayesian information criterion (BIC) in the case of the EPIC-2m
experiment (<I>r</I> = 0.01) for 3&nbsp;consecutive values of&nbsp;<I>D</I>. The BIC is a
decreasing function of the likelihood and of the number of
parameter. Hence, lower BIC implies either fewer explanatory
variables, better fit, or both. In our case the criterion reads:
<p></p>
<DIV ALIGN="CENTER">
<!-- MATH: \begin{displaymath}
BIC = -2 \ln \mathcal{L}+ k \ln\sum_q w_q
\end{displaymath} -->


<IMG
 WIDTH="172" HEIGHT="70"
 SRC="img145.png"
 ALT="\begin{displaymath}BIC = -2 \ln \mathcal{L}+ k \ln\sum_q w_q
\end{displaymath}">
</DIV><BR CLEAR="ALL">
<p></p>
where <I>k</I> is the number of estimated parameters and <I>w</I><SUB><I>q</I></SUB> the
effective number of modes in bin&nbsp;<I>q</I>. Taking into account the
redundancy in the parameterisation, the actual number of free
parameters in the model is 
<!-- MATH: $1 + F\times D + Q{D(D+1)}/ 2 - D^2$ -->
<IMG
 WIDTH="182" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img146.png"
 ALT="$ 1 + F\times D + Q{D(D+1)}/ 2 - D^2$">.
However, we usually prefer to rely on the inspection of the mismatch
in every bin of&nbsp;<IMG
 WIDTH="9" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img4.png"
 ALT="$\ell $">,
as some frequency specific features may be
diluted in the global mismatch.

<p>
<A NAME="tab:bic"></A><p class="inset-old"><a href="/articles/aa/full_html/2009/33/aa11624-09/tableA.1.html"><span class="bold">Table A.1:</span></a>&#160;&#160;
Bayesian information criterion of 3 models with increasing dimension of the galactic component for the EPIC-2m mission.</p>
<p>

<h3 class="sec2"><a name="SECTION000102000000000000000"></a>
<A NAME="sec:maskstud"></A>
A.2 Masking influence
</h3>

<p>
The noise level and the scanning strategy remaining fixed in the
full-sky experiments, a larger coverage gives more information and
should result in tighter constraints on both foreground and CMB. In
practice, it is only the case up to a certain point, due to the non
stationarity of the foreground emission. In the galactic plane, the
emission is too strong and too complex to fit in the proposed model,
and this region must be discarded to avoid contamination of the
results.
The main points governing the choice of an appropriate mask are the
following:

<p>
<UL>
<LI>the covariance of the total galactic emission (synchrotron and
  dust polarised emissions), because of the variation of emission laws
  as a function of the direction on the sky, is never <I>exactly</I>
  modelled by a rank <I>D</I>&nbsp;matrix. However it is 
  <I>satisfactorily</I> modelled in this way if the difference between
  the actual second order statistics of the foregrounds, and those of
  the rank <I>D</I>&nbsp;matrix model, are indistinguishable because of the
  noise level (or because of cosmic variance in the empirical
  statistics).  The deviation from the model is more obvious in
  regions of strong galactic emission, hence the need for a galactic
  mask. The higher the noise, the smaller the required mask;
<LI>S<SMALL>MICA</SMALL> provides a built-in measure of the adequacy of the model, which is the value of the spectral mismatch. If too high, the model under-fits the data, and the dimension of the foreground model (or the size of the mask) should be increased. If too low, the model over-fits the data, and&nbsp;<I>D</I> should be decreased;
<LI>near full sky coverage is better for measuring adequately the reionisation bump;
<LI>the dimension of the foreground component must be smaller than
  the number of channels.
</UL>If the error variance is always dominated by noise and cosmic
variance, the issue is solved: one should select the smaller mask that
gives a good fit between the model and the data to minimise the mean
squared error and keep the estimator unbiased.

<p>
If, on the other hand, the error seems dominated by the contribution
of foregrounds, which is, for example, the case of the EPIC-2m
experiment for <I>r</I> = 0.001, the tradeoff is unclear and it may happen
that a better estimator is obtained with a stronger masking of the
foreground contamination. We found that it is not the case. 
Table&nbsp;<a href="/articles/aa/full_html/2009/33/aa11624-09/aa11624-09.html#tab:mask">A.2</a> illustrates the case of the EPIC-2m experiment with the
galactic cut used in Sect.&nbsp;<a href="/articles/aa/full_html/2009/33/aa11624-09/aa11624-09.html#sec:results">4</a> and a bigger
cut. Although the reduction of sensitivity is slower in the presence of
foreground than for the noise dominated case, the smaller mask still
give the better results.
<A NAME="tab:mask"></A><p class="inset-old"><a href="/articles/aa/full_html/2009/33/aa11624-09/tableA.2.html"><span class="bold">Table A.2:</span></a>&#160;&#160;
Estimation of the tensor to scalar ratio with two different galactic cuts in the EPIC-2m experiment.</p>We may also recall that the expression&nbsp;(<a href="/articles/aa/full_html/2009/33/aa11624-09/aa11624-09.html#eq:loglikesingle">7</a>) of the
likelihood is an approximation for partial sky coverage. The scheme
presented here thus may not give fully reliable results when masking
effects become important.

<p>

<h2 class="sec"><a name="SECTION000110000000000000000"></a>
<A NAME="sec:mm"></A>
Appendix B: Spectral mismatch
</h2>

<p>
Computed for each bin <I>q</I> of <IMG
 WIDTH="9" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img4.png"
 ALT="$\ell $">,
the mismatch criterion, 
<!-- MATH: $w_q
\ensuremath{K \left( \ensuremath{\widehat{\tens{R}}}_q, \R_q(\theta^*) \right) }$ -->
<IMG
 WIDTH="98" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img149.png"
 ALT="$w_q
\ensuremath{K \left( \ensuremath{\widehat{\tens{R}}}_q, \R_q(\theta^*) \right) } $">,
between the best-fit model

<!-- MATH: $\R_q(\theta^*)$ -->
<IMG
 WIDTH="39" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img150.png"
 ALT="$\R_q(\theta^*)$">
at the point of convergence <IMG
 WIDTH="15" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img151.png"
 ALT="$\theta^*$">,
and the data

<!-- MATH: $\ensuremath{\widehat{\tens{R}}} _q$ -->
<IMG
 WIDTH="18" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img152.png"
 ALT="$\ensuremath{\widehat{\tens{R}}} _q$">,
gives a picture of the goodness of fit as a function of the
scale. Black curves in Figs.&nbsp;<a href="/articles/aa/full_html/2009/33/aa11624-09/aa11624-09.html#fig:planckmismatch">B.1</a> and&nbsp;<a href="/articles/aa/full_html/2009/33/aa11624-09/aa11624-09.html#fig:epicmm">B.2</a> show the mismatch criterion of the best fits for
Planck and EPIC designs respectively. When the model holds, the value
of the mismatch is expected to be around the number of degrees of
freedom (horizontal black lines in the figures). We can also compute
the mismatch for a model in which we discard the CMB contribution 
<!-- MATH: $w_q
\ensuremath{K \left( \ensuremath{\widehat{\tens{R}}}_q, \R_q(\theta^*) - \ensuremath{\tens{R}}[CMB]_q(r^*) \right) }$ -->
<IMG
 WIDTH="189" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img153.png"
 ALT="$w_q
\ensuremath{K \left( \ensuremath{\widehat{\tens{R}}}_q, \R_q(\theta^*) - \ensuremath{\tens{R}}[CMB]_q(r^*) \right) } $">.
Gray curves in
Figs.&nbsp;<a href="/articles/aa/full_html/2009/33/aa11624-09/aa11624-09.html#fig:planckmismatch">B.1</a> and&nbsp;<a href="/articles/aa/full_html/2009/33/aa11624-09/aa11624-09.html#fig:epicmm">B.2</a> show the mismatch
for this modified model. The difference between the two curves
illustrates the ``weight'' of the CMB component in the fit, as a
function of the scale.

<p>
<div class="inset-old">
<table>
<tr><td><!-- init Label --><A NAME="fig:planckmismatch">&#160;</A><!-- end Label--><A NAME="2427"></A><A NAME="figure2213"
 HREF="img154.png"><IMG
 WIDTH="90" HEIGHT="138" SRC="Timg154.png"
 ALT="\begin{figure}
\par\includegraphics{11624f10}
\end{figure}"></A><!-- HTML Figure number: 8 --></td>
<td class="img-txt"><span class="bold">Figure B.1:</span><p>
Those plots present the distribution in <IMG
 WIDTH="9" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img4.png"
 ALT="$\ell $">
of the
    mismatch criterion between the model and the data for two values
    of <I>r</I> for P<SMALL>LANCK</SMALL>. On the grey curve, the mismatch has
    been computed discarding the CMB contribution from the S<SMALL>MICA</SMALL>     model. The difference between the two curves, plotted in
    inclusion, illustrates somehow the importance of the CMB
    contribution to the signal.</p></td>
</tr><tr><td colspan="2"><a href="http://dexter.edpsciences.org/applet.php?pdf_id=8&DOI=10.1051/0004-6361/200911624" target="DEXTER">Open with DEXTER</a></td></tr>

</table></div>
<p>
Figure&nbsp;<a href="/articles/aa/full_html/2009/33/aa11624-09/aa11624-09.html#fig:planckmismatch">B.1</a> shows the results for Planck for
<I>r</I>=0.3 and&nbsp;0.1. The curves of the difference plotted in inclusion
illustrate the predominance of the reionisation bump. In
Fig.&nbsp;<a href="/articles/aa/full_html/2009/33/aa11624-09/aa11624-09.html#fig:epicmm">B.2</a>, we plot the difference curve on the bottom
panels for the three experiments for <I>r</I>=0.01 and <I>r</I>=0.001. They
illustrate clearly the difference of sensitivity to the peak between
the EPIC-LC design and the higher resolution experiments. In general
it can be seen that no significant contribution to the CMB is coming
from scales smaller than 
<!-- MATH: $\ell = 150$ -->
<IMG
 WIDTH="47" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img155.png"
 ALT="$\ell = 150$">.

<p>
<div class="inset-old">
<table>
<tr><td><!-- init Label --><A NAME="fig:epicmm">&#160;</A><!-- end Label--><A NAME="2428"></A><A NAME="figure2221"
 HREF="img156.png"><IMG
 WIDTH="196" HEIGHT="230" SRC="Timg156.png"
 ALT="\begin{figure}
\par\includegraphics{11624f11} \includegraphics{11624f12}
\end{figure}"><!-- HTML Figure number: 9 --></td>
<td class="img-txt"><span class="bold">Figure B.2:</span><p>
Mismatch criterion for <I>r</I> = 0.01 (<I> top</I>) and <I>r</I> = 0.001 
  (<I> bottom</I>). In each plot, the top panel shows the mismatch criterion between the best fit model and the data (black curve) and the best fit model deprived from the CMB contribution and the data (gray curve). Solid and dashed horizontal lines show respectively the mismatch expectation and 2 times the mismatch expectation. The difference between the gray and the black curve is plotted in the bottom panel and gives an idea of the significance of the CMB signal in each bin of&nbsp;<IMG
 WIDTH="9" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img4.png"
 ALT="$\ell $">.</p></td>
</tr><tr><td colspan="2"><a href="http://dexter.edpsciences.org/applet.php?pdf_id=9&DOI=10.1051/0004-6361/200911624" target="DEXTER">Open with DEXTER</a></td></tr>

</table></div>
<p>
<br>

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