Difference between revisions of "Frequency maps angular power spectra"

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== HFI maps power spectra ==
 
== HFI maps power spectra ==
−
-----------------------------
 
  
−
Angular power spectra of cut sky CMB dominated maps are provided to allow independent cosmological analysis at high $\ell$.
+
 
 +
Angular power spectra of cut sky CMB dominated maps are provided to allow independent cosmological analysis at high <math>\ell</math>.
  
 
===Product description===
 
===Product description===
Line 12: Line 12:
 
The mask used is apodized to reduce the power leakage from large scale to small scale (see input section). Except for the removal of the most contaminated pixels through masking, no attempt at astrophysical components separation has been performed.  
 
The mask used is apodized to reduce the power leakage from large scale to small scale (see input section). Except for the removal of the most contaminated pixels through masking, no attempt at astrophysical components separation has been performed.  
  
−
For each pair of detectors $X$ and $Y$,  are provided,
+
For each pair of detectors <math>X</math> and <math>Y</math>,  are provided,
−
* the unbinned ''estimated'' power spectrum $\hat{C}^{XY}_\ell$ for all $\ell$ from 0 to 3508 (see [[#all_dscl|Figure 1]] below), as well as
+
* the unbinned ''estimated'' power spectrum <math>\hat{C}^{XY}_\ell</math> for all <math>\ell</math> from 0 to 3508 (see [[#all_dscl|Figure 1]] below), as well as
 
* the unbinned symmetric covariance matrix
 
* the unbinned symmetric covariance matrix
 
\begin{align}
 
\begin{align}
Line 19: Line 19:
 
   \label{eq:covmatCl}
 
   \label{eq:covmatCl}
 
\end{align}
 
\end{align}
−
for all $\ell$ on the same range. At the price of some extra hypotheses, that information can be used to build the likelihood of a given theoretical power spectrum $C_{\ell}$ given the data, and therefore determine the best cosmological models fitting the data. Several examples of such high-$\ell$ likelihoods are described, discussed and compared in <cite>#planck2013-p08</cite> {{p2013|8}}.
+
for all <math>\ell</math> on the same range. At the price of some extra hypotheses, that information can be used to build the likelihood of a given theoretical power spectrum <math>C_{\ell}</math> given the data, and therefore determine the best cosmological models fitting the data. Several examples of such high-<math>\ell</math> likelihoods are described, discussed and compared in {{PlanckPapers|planck2013-p08}}.
  
 
$
 
$
Line 27: Line 27:
 
\newcommand{\lmax}{\ell_{\mathrm{max}}}
 
\newcommand{\lmax}{\ell_{\mathrm{max}}}
 
$
 
$
−
Note that $\hat{\bfM}$ only describes the statistical covariance of the power spectrum
+
Note that <math>\hat{\bfM}</math> only describes the statistical covariance of the power spectrum
 
induced by the signal and noise of the input map on the cut sky begin analyzed.  
 
induced by the signal and noise of the input map on the cut sky begin analyzed.  
−
Most sources of systematic effects (such as uncertainty on the beam modeling) as well as post-processing steps (such as foreground subtraction) will increase the covariance. In the particular case of the uncertainty on the beam window functions $B(l)$,
+
Most sources of systematic effects (such as uncertainty on the beam modeling) as well as post-processing steps (such as foreground subtraction) will increase the covariance. In the particular case of the uncertainty on the beam window functions <math>B(l)</math>,
−
the [[The RIMO|RIMO]] provides for each pair $XY$ a set of eigen-vectors $E_{p}^{XY}(\ell)$ of the relative error on $B^{XY}_{\ell}$ (see "HFI time response and beams paper"<cite>planck2013-p03c</cite> {{P2013|7}}), defined for $p$ in $[1,5]$ and $\ell$ in $[0, \lmax]$ (with $\lmax$ being 2500, 3000 or 4000 when the lowest of the nominal frequencies of the detectors $X$ and $Y$ is respectively 100, 143 or 217GHz). The extra contribution to the covariance of $C^{XY}_\ell$ is then  
+
the [[The RIMO|RIMO]] provides for each pair <math>XY</math> a set of eigen-vectors <math>E_{p}^{XY}(\ell)</math> of the relative error on <math>B^{XY}_{\ell}</math> (see {{PlanckPapers|planck2013-p03c|1|HFI time response and beams paper}}), defined for <math>p</math> in <math>[1,5]</math> and <math>\ell</math> in <math>[0, \lmax]</math> (with <math>\lmax</math> being 2500, 3000 or 4000 when the lowest of the nominal frequencies of the detectors <math>X</math> and <math>Y</math> is respectively 100, 143 or 217GHz). The extra contribution to the covariance of <math>C^{XY}_\ell</math> is then  
 
\begin{align}
 
\begin{align}
 
   \hat{M}^{XY, \mathrm{beam}}_{\ell_1 \ell_2} = 4 \hat{C}^{XY}_{\ell_1} \hat{C}^{XY}_{\ell_2} \sum_{p=1}^{5} E^{XY}_p(\ell_1) E^{XY}_p(\ell_2).
 
   \hat{M}^{XY, \mathrm{beam}}_{\ell_1 \ell_2} = 4 \hat{C}^{XY}_{\ell_1} \hat{C}^{XY}_{\ell_2} \sum_{p=1}^{5} E^{XY}_p(\ell_1) E^{XY}_p(\ell_2).
Line 38: Line 38:
 
<!--          ===================================================    -->
 
<!--          ===================================================    -->
 
<div id="all_dscl">
 
<div id="all_dscl">
−
[[File:all_dscl.png  | 500px  | center | thumb | '''Figure 1:''' The 91 auto- (dotted lines) and cross- (solid lines) angular power spectra $\hat{C}_\ell$, shown here after a binning of $\Delta \ell = 31$, grouped by frequencies. For instance the top left panel, tagged ''100x100 (3)'', contains the three spectra 100-ds1x100-ds2, 100-ds1x100-ds1 and 100-ds1x100-ds2. The auto spectra are contaminated at high $\ell$ by the instrumental noise, and all of them may be affected by foreground contamination. The grey circles show the best Planck CMB high-$\ell$ power spectrum described in the [[CMB spectrum & Likelihood Code | CMB spectrum & Likelihood Code section]]  ]]
+
[[File:all_dscl.png  | 500px  | center | thumb | '''Figure 1:''' The 91 auto- (dotted lines) and cross- (solid lines) angular power spectra <math>\hat{C}_\ell</math>, shown here after a binning of <math>\Delta \ell = 31</math>, grouped by frequencies. For instance the top left panel, tagged ''100x100 (3)'', contains the three spectra 100-ds1x100-ds2, 100-ds1x100-ds1 and 100-ds1x100-ds2. The auto spectra are contaminated at high <math>\ell</math> by the instrumental noise, and all of them may be affected by foreground contamination. The grey circles show the best Planck CMB high-<math>\ell</math> power spectrum described in the [[CMB spectrum & Likelihood Code | CMB spectrum & Likelihood Code section]]  ]]
 
</div>
 
</div>
 
<!--          ===================================================    -->
 
<!--          ===================================================    -->
Line 45: Line 45:
 
====Auto and Cross Power Spectra====
 
====Auto and Cross Power Spectra====
  
−
The spectra computed up to$l=3508$ using [http://prof.planck.fr/article141.html PolSpice] (<cite>Szapudi2001</cite>, <cite>Chon2004</cite>)
+
The spectra computed up to <math>l=3508</math> using [http://prof.planck.fr/article141.html PolSpice]{{BibCite|Szapudi2001}}{{BibCite|Chon2004}}
 
are corrected from the effect of the cut sky, and from the nominal beam window function and average pixel function. The different steps of the calculation are
 
are corrected from the effect of the cut sky, and from the nominal beam window function and average pixel function. The different steps of the calculation are
−
* computation of the Spherical Harmonics coefficients of the masked input maps $\Delta T^X(p)$ and of the input mask $w(p)$,
+
* computation of the Spherical Harmonics coefficients of the masked input maps <math>\Delta T^X(p)</math> and of the input mask <math>w(p)</math>,
 
\begin{align}
 
\begin{align}
 
   \tilde{a}^X_{\ell m} = \sum_p \Omega_p\, \Delta T^X(p)\, w(p)\, Y^*_{\ell m}(p), \label{eq:almdef}
 
   \tilde{a}^X_{\ell m} = \sum_p \Omega_p\, \Delta T^X(p)\, w(p)\, Y^*_{\ell m}(p), \label{eq:almdef}
Line 54: Line 54:
 
   \tilde{w}^{(n)}_{\ell m} = \sum_p \Omega_p\ w^n(p)\, Y^*_{\ell m}(p); \label{eq:wlmdef}
 
   \tilde{w}^{(n)}_{\ell m} = \sum_p \Omega_p\ w^n(p)\, Y^*_{\ell m}(p); \label{eq:wlmdef}
 
\end{align}
 
\end{align}
−
where the sum is done over all sky pixels $p$, $\Omega_p$ is the pixel area, and $n$ is either 1 or 2;
+
where the sum is done over all sky pixels <math>p</math>, <math>\Omega_p</math> is the pixel area, and <math>n</math> is either 1 or 2;
−
* the sky (cross or auto) pseudo-power spectrum and mask power spectrum are computed from the $\tilde{a}_{\ell m}$ and $\tilde{w}_{\ell m}$,
+
* the sky (cross or auto) pseudo-power spectrum and mask power spectrum are computed from the <math>\tilde{a}_{\ell m}</math> and <math>\tilde{w}_{\ell m}</math>,
 
\begin{align}
 
\begin{align}
 
   \tilde{C}^{XY}_\ell =  \sum_{\ell m} \tilde{a}^X_{ m} \tilde{a}^{Y^*}_{\ell m}  / (2 \ell + 1), \label{eq:alm2cl}
 
   \tilde{C}^{XY}_\ell =  \sum_{\ell m} \tilde{a}^X_{ m} \tilde{a}^{Y^*}_{\ell m}  / (2 \ell + 1), \label{eq:alm2cl}
Line 69: Line 69:
 
   \tilde{\xi}_W(\theta) = \sum_\ell \frac{2\ell+1}{4\pi} \tilde{W}^{(1)}_{\ell} P_\ell(\theta),
 
   \tilde{\xi}_W(\theta) = \sum_\ell \frac{2\ell+1}{4\pi} \tilde{W}^{(1)}_{\ell} P_\ell(\theta),
 
\end{align}
 
\end{align}
−
where $P_\ell$ is the Legendre Polynomial of order $\ell$;
+
where <math>P_\ell</math> is the Legendre Polynomial of order <math>\ell</math>;
 
* the ratio of the sky angular correlation by the mask correlation provides the cut sky corrected angular correlation,
 
* the ratio of the sky angular correlation by the mask correlation provides the cut sky corrected angular correlation,
 
\begin{align}
 
\begin{align}
Line 78: Line 78:
 
   {C'}_\ell =  2\pi \sum_i w_i \xi(\theta_i) P_\ell(\theta_i), \label{eq:xi2cl}
 
   {C'}_\ell =  2\pi \sum_i w_i \xi(\theta_i) P_\ell(\theta_i), \label{eq:xi2cl}
 
\end{align}
 
\end{align}
−
where $w_i$ are the weights of the Gauss-Legendre quadrature, for $\theta$ in $[0, \pi]$;
+
where <math>w_i</math> are the weights of the Gauss-Legendre quadrature, for <math>\theta</math> in <math>[0, \pi]</math>;
−
* the resulting power spectrum is corrected from the nominal beam window function $B_\ell$ and average pixel window function $w_{\mathrm{pix}}(\ell)$, to provide the final Spice estimator $\hat{C}_\ell$,
+
* the resulting power spectrum is corrected from the nominal beam window function <math>B_\ell</math> and average pixel window function <math>w_{\mathrm{pix}}(\ell)</math>, to provide the final Spice estimator <math>\hat{C}_\ell</math>,
 
\begin{align}
 
\begin{align}
 
   \hat{C}_\ell = {C'}_\ell / \left( B^2_\ell w^2_{\mathrm{pix}}(\ell) \right). \label{eq:clfinal}
 
   \hat{C}_\ell = {C'}_\ell / \left( B^2_\ell w^2_{\mathrm{pix}}(\ell) \right). \label{eq:clfinal}
Line 85: Line 85:
  
 
====Covariance Matrices====
 
====Covariance Matrices====
−
The covariance matrix for the pair $XY$ is computed by PolSpice
+
The covariance matrix for the pair <math>XY</math> is computed by PolSpice
−
using the formalism described in <cite>Efstathiou2004</cite><!--[http://adsabs.harvard.edu/abs/2004MNRAS.349..603E Efstathiou (2004)]-->, also sketched in the appendix
+
using the formalism described in{{BibCite|Efstathiou2004}}<!--[http://adsabs.harvard.edu/abs/2004MNRAS.349..603E Efstathiou (2004)]-->, also sketched in the appendix
−
of "CMB power spectra and likelihood paper"<cite>planck2013-p08</cite>, assuming the instrumental noise to be white and uniform.
+
of {{PlanckPapers|planck2013-p08|1|CMB power spectra and likelihood paper}}, assuming the instrumental noise to be white and uniform.
  
 
$
 
$
 
\newcommand{\hC}{\hat C}
 
\newcommand{\hC}{\hat C}
 
$
 
$
−
One note that a good approximation of the covariance matrix $\tilde{M}$ of the pseudo $\tilde{C}_{\ell}$ is related to the underlying ''estimated'' auto- and cross-spectra $\hC_{\ell}$ through
+
One note that a good approximation of the covariance matrix <math>\tilde{M}</math> of the pseudo <math>\tilde{C}_{\ell}</math> is related to the underlying ''estimated'' auto- and cross-spectra <math>\hC_{\ell}</math> through
 
\begin{align}
 
\begin{align}
 
\tilde{M}_{\ell_1\ell_2} \equiv \langle\Delta \tilde{C}^{XY}_{\ell_1}\Delta \tilde{C}^{XY}_{\ell_2}\rangle =  
 
\tilde{M}_{\ell_1\ell_2} \equiv \langle\Delta \tilde{C}^{XY}_{\ell_1}\Delta \tilde{C}^{XY}_{\ell_2}\rangle =  
Line 105: Line 105:
 
\label{eq:covpseudo}
 
\label{eq:covpseudo}
 
\end{align}
 
\end{align}
−
where  $\tilde{W}^{(2)}_{\ell}$ is the power spectrum of the square of the pixel mask (Eqs. \ref{eq:wlmdef} and \ref{eq:wlm2wl} for $n=2$). The covariance matrix $\hat{M}$
+
where  <math>\tilde{W}^{(2)}_{\ell}</math> is the power spectrum of the square of the pixel mask (Eqs. \ref{eq:wlmdef} and \ref{eq:wlm2wl} for <math>n=2</math>). The covariance matrix <math>\hat{M}</math>
−
of the Spice estimator is then computed by applying Eqs. \ref{eq:cl2xi}, \ref{eq:xi_deconv}, \ref{eq:xi2cl} and \ref{eq:clfinal} on each row and column of $\tilde{M}$.
+
of the Spice estimator is then computed by applying Eqs. \ref{eq:cl2xi}, \ref{eq:xi_deconv}, \ref{eq:xi2cl} and \ref{eq:clfinal} on each row and column of <math>\tilde{M}</math>.
  
  
−
<!-- The products described here, spectra ${\hat C}_{\ell}$ and covariances ${\hat M}_{\ell \ell'}$ can be used to  
+
<!-- The products described here, spectra <math>{\hat C}_{\ell}</math> and covariances <math>{\hat M}_{\ell \ell'}</math> can be used to  
−
estimate the high-$\ell$ likelihood of a given theoretical model given the data available. -->
+
estimate the high-<math>\ell</math> likelihood of a given theoretical model given the data available. -->
  
 
===Inputs===
 
===Inputs===
Line 117: Line 117:
 
: The input maps are the 13 HFI detset (see [[Frequency_Maps#Types_of_maps | ''Type of maps'']] section for details) maps available at 100, 143 and 217 GHz. These are the same as the ones used for high-ell part of the [[CMB spectrum & Likelihood Code | likelihood code]], but that code applies  different masks for each cross-spectra in order to minimize further the foreground contamination.
 
: The input maps are the 13 HFI detset (see [[Frequency_Maps#Types_of_maps | ''Type of maps'']] section for details) maps available at 100, 143 and 217 GHz. These are the same as the ones used for high-ell part of the [[CMB spectrum & Likelihood Code | likelihood code]], but that code applies  different masks for each cross-spectra in order to minimize further the foreground contamination.
 
; Sky mask
 
; Sky mask
−
: All maps were analyzed on the 42.8% of the sky defined by the apodized mask ''HFI_PowerSpect_Mask_2048_R1.10.fits'', which masks out Galactic Plane and point sources (see <cite>planck2013-p08</cite>), and which is shown in [[#mask_Cl|Figure 2]] below
+
: All maps were analyzed on the 42.8% of the sky defined by the apodized mask ''HFI_PowerSpect_Mask_2048_R1.10.fits'', which masks out Galactic Plane and point sources (see {{PlanckPapers|planck2013-p08}}), and which is shown in [[#mask_Cl|Figure 2]] below
  
 
<div id="mask_Cl">
 
<div id="mask_Cl">
Line 124: Line 124:
  
 
; Beam Window Function
 
; Beam Window Function
−
: The beam window functions $B(l)$, and their uncertainties, are the ones used in the high-ell likelihood analysis, described in section 6.1 "Error Eigenmodes" of <cite>planck2013-p08</cite> and provided in the HFI [[The RIMO|RIMO]].
+
: The beam window functions <math>B(l)</math>, and their uncertainties, are the ones used in the high-ell likelihood analysis, described in section 6.1 "Error Eigenmodes" of {{PlanckPapers|planck2013-p08}} and provided in the HFI [[The RIMO|RIMO]].
  
 
===Related products===
 
===Related products===
Line 146: Line 146:
 
! Column Name || Data Type || Units || Description
 
! Column Name || Data Type || Units || Description
 
|-
 
|-
−
|TEMP_CL || Real*4 || $\mu$K<sub>cmb</sub><sup>2</sup> || the power spectrum
+
|TEMP_CL || Real*4 || <math>\mu</math>K<sub>cmb</sub><sup>2</sup>|| the power spectrum
 
|-
 
|-
−
|TEMP_CL_ERR || Real*4 || $\mu$K<sub>cmb</sub><sup>2</sup> || estimate of the uncertainty in the power spectrum
+
|TEMP_CL_ERR || Real*4 || <math>\mu</math>K<sub>cmb</sub><sup>2</sup> || estimate of the uncertainty in the power spectrum
 
|- bgcolor="ffdead"   
 
|- bgcolor="ffdead"   
 
! Keyword || Data Type || Value || Description
 
! Keyword || Data Type || Value || Description
Line 158: Line 158:
 
!colspan="4" | 2. EXTNAME = 'PSCOVMAT'  (IMAGE)
 
!colspan="4" | 2. EXTNAME = 'PSCOVMAT'  (IMAGE)
 
|-
 
|-
−
|COVMAT || Real*4 || $\mu$K<sub>cmb</sub><sup>4</sup> || the covariance matrix
+
|COVMAT || Real*4 || <math>\mu</math>K<sub>cmb</sub><sup>4</sup> || the covariance matrix
 
|-
 
|-
 
|- bgcolor="ffdead"   
 
|- bgcolor="ffdead"   
Line 364: Line 364:
 
! Sky mask
 
! Sky mask
 
|-
 
|-
−
| HFI_PowerSpect_Mask_2048_R1.10.fits
+
| {{PLASingleFile|fileType=map|name=HFI_PowerSpect_Mask_2048_R1.10.fits|link=HFI_PowerSpect_Mask_2048_R1.10.fits}}
 
|}
 
|}
  
 +
The full list of HFI power spectra with links to the files in the PLA can be found {{PLASpec|inst=HFI|link=here}}.
  
 
== LFI maps power spectra ==
 
== LFI maps power spectra ==
−
---------------------------------
+
 
  
 
===Product description===
 
===Product description===
Line 379: Line 380:
  
 
===Production process===
 
===Production process===
−
Spectra are computed using cROMAster, a implementation of the pseudo-Cl method described in <cite>#master</cite>. In addition to the original approach, our implementation allows for estimation of cross-power spectra from two or more maps (see <cite>#polenta_CrossSpectra</cite> for details). The software package uses [http://healpix.sourceforge.net/ HEALPix] modules for spherical harmonic transform and Cl calculation. The schematic of the estimation process is as follows:
+
Spectra are computed using cROMAster, a implementation of the pseudo-Cl method described in{{BibCite|master}}. In addition to the original approach, our implementation allows for estimation of cross-power spectra from two or more maps (see{{BibCite|polenta_CrossSpectra}} for details). The software package uses [http://healpix.sourceforge.net/ HEALPix] modules for spherical harmonic transform and Cl calculation. The schematic of the estimation process is as follows:
  
 
* computing the a_lm coefficients from the input temperature map after masking Galactic Plane and point sources.
 
* computing the a_lm coefficients from the input temperature map after masking Galactic Plane and point sources.
Line 558: Line 559:
  
 
== References ==
 
== References ==
−
<biblio force=false>
+
<References />
−
#[[References]]
+
−
</biblio>
+
 
  
  
 
[[Category:Mission products|006]]
 
[[Category:Mission products|006]]

Latest revision as of 17:10, 23 July 2014

HFI maps power spectra[edit]

Angular power spectra of cut sky CMB dominated maps are provided to allow independent cosmological analysis at high [math]\ell[/math].

Product description[edit]

The auto and cross-spectra of the 13 detector set (detset) maps at 100, 143 and 217 GHz, all analyzed on the same 42.8% of the sky, are provided. The mask used is apodized to reduce the power leakage from large scale to small scale (see input section). Except for the removal of the most contaminated pixels through masking, no attempt at astrophysical components separation has been performed.

For each pair of detectors [math]X[/math] and [math]Y[/math], are provided,

  • the unbinned estimated power spectrum [math]\hat{C}^{XY}_\ell[/math] for all [math]\ell[/math] from 0 to 3508 (see Figure 1 below), as well as
  • the unbinned symmetric covariance matrix

\begin{align} 

  \hat{M}^{XY}_{\ell \ell'} \equiv \langle\Delta \hat{C}^{XY}_\ell\Delta \hat{C}^{XY}_{\ell'}\rangle
  \label{eq:covmatCl}

\end{align} for all [math]\ell[/math] on the same range. At the price of some extra hypotheses, that information can be used to build the likelihood of a given theoretical power spectrum [math]C_{\ell}[/math] given the data, and therefore determine the best cosmological models fitting the data. Several examples of such high-[math]\ell[/math] likelihoods are described, discussed and compared in Planck-2013-XV[1].

$ \newcommand{\bfE}{\boldsymbol{\mathrm{E}}} \newcommand{\bfM}{\boldsymbol{\mathrm{M}}} \newcommand{\bfx}{\boldsymbol{\mathrm{x}}} \newcommand{\lmax}{\ell_{\mathrm{max}}} $ Note that [math]\hat{\bfM}[/math] only describes the statistical covariance of the power spectrum induced by the signal and noise of the input map on the cut sky begin analyzed. Most sources of systematic effects (such as uncertainty on the beam modeling) as well as post-processing steps (such as foreground subtraction) will increase the covariance. In the particular case of the uncertainty on the beam window functions [math]B(l)[/math], the RIMO provides for each pair [math]XY[/math] a set of eigen-vectors [math]E_{p}^{XY}(\ell)[/math] of the relative error on [math]B^{XY}_{\ell}[/math] (see HFI time response and beams paper[2]), defined for [math]p[/math] in [math][1,5][/math] and [math]\ell[/math] in [math][0, \lmax][/math] (with [math]\lmax[/math] being 2500, 3000 or 4000 when the lowest of the nominal frequencies of the detectors [math]X[/math] and [math]Y[/math] is respectively 100, 143 or 217GHz). The extra contribution to the covariance of [math]C^{XY}_\ell[/math] is then \begin{align} 

  \hat{M}^{XY, \mathrm{beam}}_{\ell_1 \ell_2} = 4 \hat{C}^{XY}_{\ell_1} \hat{C}^{XY}_{\ell_2} \sum_{p=1}^{5} E^{XY}_p(\ell_1) E^{XY}_p(\ell_2).
  \label{eq:covmatBeam}

\end{align}

Figure 1: The 91 auto- (dotted lines) and cross- (solid lines) angular power spectra [math]\hat{C}_\ell[/math], shown here after a binning of [math]\Delta \ell = 31[/math], grouped by frequencies. For instance the top left panel, tagged 100x100 (3), contains the three spectra 100-ds1x100-ds2, 100-ds1x100-ds1 and 100-ds1x100-ds2. The auto spectra are contaminated at high [math]\ell[/math] by the instrumental noise, and all of them may be affected by foreground contamination. The grey circles show the best Planck CMB high-[math]\ell[/math] power spectrum described in the CMB spectrum & Likelihood Code section


Auto and Cross Power Spectra[edit]

The spectra computed up to [math]l=3508[/math] using PolSpice[3][4] are corrected from the effect of the cut sky, and from the nominal beam window function and average pixel function. The different steps of the calculation are

  • computation of the Spherical Harmonics coefficients of the masked input maps [math]\Delta T^X(p)[/math] and of the input mask [math]w(p)[/math],

\begin{align} 

 \tilde{a}^X_{\ell m} = \sum_p \Omega_p\, \Delta T^X(p)\, w(p)\, Y^*_{\ell m}(p), \label{eq:almdef}

\end{align} \begin{align} 

 \tilde{w}^{(n)}_{\ell m} = \sum_p \Omega_p\ w^n(p)\, Y^*_{\ell m}(p); \label{eq:wlmdef}

\end{align} where the sum is done over all sky pixels [math]p[/math], [math]\Omega_p[/math] is the pixel area, and [math]n[/math] is either 1 or 2;

  • the sky (cross or auto) pseudo-power spectrum and mask power spectrum are computed from the [math]\tilde{a}_{\ell m}[/math] and [math]\tilde{w}_{\ell m}[/math],

\begin{align} 

 \tilde{C}^{XY}_\ell =  \sum_{\ell m} \tilde{a}^X_{ m} \tilde{a}^{Y^*}_{\ell m}   / (2 \ell + 1), \label{eq:alm2cl}

\end{align} \begin{align} 

 \tilde{W}^{(n)}_\ell =  \sum_{\ell m} \tilde{w}^{(n)}_{ m} {\tilde{w}^{(n)}}^*_{\ell m}   / (2 \ell + 1); \label{eq:wlm2wl}

\end{align}

  • the sky and mask angular correlation function are computed from the respective power spectra,

\begin{align} 

 \tilde{\xi}(\theta) = \sum_\ell \frac{2\ell+1}{4\pi} \tilde{C}_{\ell} P_\ell(\theta),\label{eq:cl2xi}

\end{align} \begin{align} 

 \tilde{\xi}_W(\theta) = \sum_\ell \frac{2\ell+1}{4\pi} \tilde{W}^{(1)}_{\ell} P_\ell(\theta),

\end{align} where [math]P_\ell[/math] is the Legendre Polynomial of order [math]\ell[/math];

  • the ratio of the sky angular correlation by the mask correlation provides the cut sky corrected angular correlation,

\begin{align} 

 \xi(\theta) = \tilde{\xi}(\theta) / \tilde{\xi}_W(\theta); \label{eq:xi_deconv}

\end{align}

  • the sky angular correlation function which is then turned into a angular power spectrum,

\begin{align} 

  {C'}_\ell =  2\pi \sum_i w_i \xi(\theta_i) P_\ell(\theta_i), \label{eq:xi2cl}

\end{align} where [math]w_i[/math] are the weights of the Gauss-Legendre quadrature, for [math]\theta[/math] in [math][0, \pi][/math];

  • the resulting power spectrum is corrected from the nominal beam window function [math]B_\ell[/math] and average pixel window function [math]w_{\mathrm{pix}}(\ell)[/math], to provide the final Spice estimator [math]\hat{C}_\ell[/math],

\begin{align} 

 \hat{C}_\ell = {C'}_\ell / \left( B^2_\ell w^2_{\mathrm{pix}}(\ell) \right). \label{eq:clfinal}

\end{align}

Covariance Matrices[edit]

The covariance matrix for the pair [math]XY[/math] is computed by PolSpice using the formalism described in[5], also sketched in the appendix of CMB power spectra and likelihood paper[1], assuming the instrumental noise to be white and uniform.

$ \newcommand{\hC}{\hat C} $ One note that a good approximation of the covariance matrix [math]\tilde{M}[/math] of the pseudo [math]\tilde{C}_{\ell}[/math] is related to the underlying estimated auto- and cross-spectra [math]\hC_{\ell}[/math] through \begin{align} \tilde{M}_{\ell_1\ell_2} \equiv \langle\Delta \tilde{C}^{XY}_{\ell_1}\Delta \tilde{C}^{XY}_{\ell_2}\rangle = \left( \left(\hC^{XX}_{\ell_1} \hC^{YY}_{\ell_1} \hC^{XX}_{\ell_2} \hC^{YY}_{\ell_2}\right)^{1/2} 

   + \hC^{XY}_{\ell_1} \hC^{XY}_{\ell_2} \right) 
 \sum_{\ell_3} \frac{2\ell_3+1}{4\pi} \tilde{W}^{(2)}_{\ell_3} \left(     
   \begin{array}{ccc}
       \! \ell_1\! & \ell_2\!  & \ell_3\!  \\
       \! 0     \! & 0     \!  & 0     \!
     \end{array}
 \right)^2,

\label{eq:covpseudo} \end{align} where [math]\tilde{W}^{(2)}_{\ell}[/math] is the power spectrum of the square of the pixel mask (Eqs. \ref{eq:wlmdef} and \ref{eq:wlm2wl} for [math]n=2[/math]). The covariance matrix [math]\hat{M}[/math] of the Spice estimator is then computed by applying Eqs. \ref{eq:cl2xi}, \ref{eq:xi_deconv}, \ref{eq:xi2cl} and \ref{eq:clfinal} on each row and column of [math]\tilde{M}[/math].


Inputs[edit]

Maps
The input maps are the 13 HFI detset (see Type of maps section for details) maps available at 100, 143 and 217 GHz. These are the same as the ones used for high-ell part of the likelihood code, but that code applies different masks for each cross-spectra in order to minimize further the foreground contamination.
Sky mask
All maps were analyzed on the 42.8% of the sky defined by the apodized mask HFI_PowerSpect_Mask_2048_R1.10.fits, which masks out Galactic Plane and point sources (see Planck-2013-XV[1]), and which is shown in Figure 2 below
Figure 2: Apodized pixel mask used for HFI power spectrum estimation
Beam Window Function
The beam window functions [math]B(l)[/math], and their uncertainties, are the ones used in the high-ell likelihood analysis, described in section 6.1 "Error Eigenmodes" of Planck-2013-XV[1] and provided in the HFI RIMO.

Related products[edit]

None

FITS file structure[edit]

Power spectra are provided for the auto and cross products built from the 13 detsets available at 100, 143 and 217 GHz, namely:

  • 100-ds1, 100-ds2,
  • 143-ds1, 143-ds2, 143-5, 143-6, 143-7,
  • 217-ds1, 217-ds2, 217-1, 217-2, 217-3, 217-4

which makes 13*(13+1)/2 = 91 spectra. Filenames for the auto spectra are HFI_PowerSPect_{detset}_Relnum.fits and HFI_PowerSPect_{detset1}x{detset2}_Relnum.fits for the auto- and cross-spectra, respectively. The list of the 91 files is given below. Each files contains 2 BINTABLE extensions:


Power spectrum file data structure
1. EXTNAME = 'POW-SPEC' (BINTABLE)
Column Name Data Type Units Description
TEMP_CL Real*4 [math]\mu[/math]Kcmb2 the power spectrum
TEMP_CL_ERR Real*4 [math]\mu[/math]Kcmb2 estimate of the uncertainty in the power spectrum
Keyword Data Type Value Description
LMIN Integer 0 First monopole
LMAX Integer value Last monopole
2. EXTNAME = 'PSCOVMAT' (IMAGE)
COVMAT Real*4 [math]\mu[/math]Kcmb4 the covariance matrix
Keyword Data Type Value Description
NAXIS1 Integer dim1 matrix first dimension
NAXIS2 Integer dim2 matrix second dimension

where LMAX is the same for both vectors, and dim1 = dim2 = LMAX+1 by construction.


List of filenames[edit]

FITS filenames
Auto power spectra
HFI_PowerSpect_100-ds1_R1.10.fits
HFI_PowerSpect_100-ds2_R1.10.fits
HFI_PowerSpect_143-5_R1.10.fits
HFI_PowerSpect_143-6_R1.10.fits
HFI_PowerSpect_143-7_R1.10.fits
HFI_PowerSpect_143-ds1_R1.10.fits
HFI_PowerSpect_143-ds2_R1.10.fits
HFI_PowerSpect_217-1_R1.10.fits
HFI_PowerSpect_217-2_R1.10.fits
HFI_PowerSpect_217-3_R1.10.fits
HFI_PowerSpect_217-4_R1.10.fits
HFI_PowerSpect_217-ds1_R1.10.fits
HFI_PowerSpect_217-ds2_R1.10.fits
Cross power spectra
HFI_PowerSpect_100-ds1x100-ds2_R1.10.fits
HFI_PowerSpect_100-ds1x143-5_R1.10.fits
HFI_PowerSpect_100-ds1x143-6_R1.10.fits
HFI_PowerSpect_100-ds1x143-7_R1.10.fits
HFI_PowerSpect_100-ds1x143-ds1_R1.10.fits
HFI_PowerSpect_100-ds1x143-ds2_R1.10.fits
HFI_PowerSpect_100-ds1x217-1_R1.10.fits
HFI_PowerSpect_100-ds1x217-2_R1.10.fits
HFI_PowerSpect_100-ds1x217-3_R1.10.fits
HFI_PowerSpect_100-ds1x217-4_R1.10.fits
HFI_PowerSpect_100-ds1x217-ds1_R1.10.fits
HFI_PowerSpect_100-ds1x217-ds2_R1.10.fits
HFI_PowerSpect_100-ds2x143-5_R1.10.fits
HFI_PowerSpect_100-ds2x143-6_R1.10.fits
HFI_PowerSpect_100-ds2x143-7_R1.10.fits
HFI_PowerSpect_100-ds2x143-ds1_R1.10.fits
HFI_PowerSpect_100-ds2x143-ds2_R1.10.fits
HFI_PowerSpect_100-ds2x217-1_R1.10.fits
HFI_PowerSpect_100-ds2x217-2_R1.10.fits
HFI_PowerSpect_100-ds2x217-3_R1.10.fits
HFI_PowerSpect_100-ds2x217-4_R1.10.fits
HFI_PowerSpect_100-ds2x217-ds1_R1.10.fits
HFI_PowerSpect_100-ds2x217-ds2_R1.10.fits
HFI_PowerSpect_143-5x143-6_R1.10.fits
HFI_PowerSpect_143-5x143-7_R1.10.fits
HFI_PowerSpect_143-5x217-1_R1.10.fits
HFI_PowerSpect_143-5x217-2_R1.10.fits
HFI_PowerSpect_143-5x217-3_R1.10.fits
HFI_PowerSpect_143-5x217-4_R1.10.fits
HFI_PowerSpect_143-5x217-ds1_R1.10.fits
HFI_PowerSpect_143-5x217-ds2_R1.10.fits
HFI_PowerSpect_143-6x143-7_R1.10.fits
HFI_PowerSpect_143-6x217-1_R1.10.fits
HFI_PowerSpect_143-6x217-2_R1.10.fits
HFI_PowerSpect_143-6x217-3_R1.10.fits
HFI_PowerSpect_143-6x217-4_R1.10.fits
HFI_PowerSpect_143-6x217-ds1_R1.10.fits
HFI_PowerSpect_143-6x217-ds2_R1.10.fits
HFI_PowerSpect_143-7x217-1_R1.10.fits
HFI_PowerSpect_143-7x217-2_R1.10.fits
HFI_PowerSpect_143-7x217-3_R1.10.fits
HFI_PowerSpect_143-7x217-4_R1.10.fits
HFI_PowerSpect_143-7x217-ds1_R1.10.fits
HFI_PowerSpect_143-7x217-ds2_R1.10.fits
HFI_PowerSpect_143-ds1x143-5_R1.10.fits
HFI_PowerSpect_143-ds1x143-6_R1.10.fits
HFI_PowerSpect_143-ds1x143-7_R1.10.fits
HFI_PowerSpect_143-ds1x143-ds2_R1.10.fits
HFI_PowerSpect_143-ds1x217-1_R1.10.fits
HFI_PowerSpect_143-ds1x217-2_R1.10.fits
HFI_PowerSpect_143-ds1x217-3_R1.10.fits
HFI_PowerSpect_143-ds1x217-4_R1.10.fits
HFI_PowerSpect_143-ds1x217-ds1_R1.10.fits
HFI_PowerSpect_143-ds1x217-ds2_R1.10.fits
HFI_PowerSpect_143-ds2x143-5_R1.10.fits
HFI_PowerSpect_143-ds2x143-6_R1.10.fits
HFI_PowerSpect_143-ds2x143-7_R1.10.fits
HFI_PowerSpect_143-ds2x217-1_R1.10.fits
HFI_PowerSpect_143-ds2x217-2_R1.10.fits
HFI_PowerSpect_143-ds2x217-3_R1.10.fits
HFI_PowerSpect_143-ds2x217-4_R1.10.fits
HFI_PowerSpect_143-ds2x217-ds1_R1.10.fits
HFI_PowerSpect_143-ds2x217-ds2_R1.10.fits
HFI_PowerSpect_217-1x217-2_R1.10.fits
HFI_PowerSpect_217-1x217-3_R1.10.fits
HFI_PowerSpect_217-1x217-4_R1.10.fits
HFI_PowerSpect_217-1x217-ds1_R1.10.fits
HFI_PowerSpect_217-1x217-ds2_R1.10.fits
HFI_PowerSpect_217-2x217-3_R1.10.fits
HFI_PowerSpect_217-2x217-4_R1.10.fits
HFI_PowerSpect_217-2x217-ds1_R1.10.fits
HFI_PowerSpect_217-2x217-ds2_R1.10.fits
HFI_PowerSpect_217-3x217-4_R1.10.fits
HFI_PowerSpect_217-3x217-ds1_R1.10.fits
HFI_PowerSpect_217-3x217-ds2_R1.10.fits
HFI_PowerSpect_217-4x217-ds1_R1.10.fits
HFI_PowerSpect_217-4x217-ds2_R1.10.fits
HFI_PowerSpect_217-ds1x217-ds2_R1.10.fits
Sky mask
HFI_PowerSpect_Mask_2048_R1.10.fits

The full list of HFI power spectra with links to the files in the PLA can be found here.

LFI maps power spectra[edit]

Product description[edit]

The angular power spectrum provides information about the distribution of power on the sky map at the various angular scales. It is especially important for CMB, because it is characterized by a number of features, most notably the acoustic peaks, that encode the dependence from cosmological parameters. Therefore, angular power spectra are the basic inputs for the Planck Likelihood Code, and for estimation of cosmological parameters in general.

For this release we have computed only temperature power spectra. Polarization is not included.

Please note that these spectra come from frequency maps. No component separation has been applied, and we have only masked Galactic Plane and detected point sources. Units are [math] \rm{ \mu K^2_{CMB}} [/math].

Production process[edit]

Spectra are computed using cROMAster, a implementation of the pseudo-Cl method described in[6]. In addition to the original approach, our implementation allows for estimation of cross-power spectra from two or more maps (see[7] for details). The software package uses HEALPix modules for spherical harmonic transform and Cl calculation. The schematic of the estimation process is as follows:

  • computing the a_lm coefficients from the input temperature map after masking Galactic Plane and point sources.
  • computing the pseudo power spectrum from the alms.
  • estimating the bias due to the noise power spectrum of the map from noise-only Monte Carlo simulations based on detector noise properties
  • correcting for the effect of the adopted mask by computing the mode-mode coupling kernel corresponding to that mask
  • deconvolving the effect due to the finite angular resolution of the telescope by using the beam window function
  • deconvolving the effect due to the finite size of the pixel in the map by using a pixel window function
  • binning the power spectrum from individual multipoles into bandpowers
  • estimating error bars on bandpowers from signal plus noise Monte Carlo simulations, where signal simulations include only CMB anisotropies.

Inputs[edit]

The inputs are the following:

  • LFI Frequency Maps
  • Point source and galactic plane masks (the name being specified in the comment keyword in the header, see Note in Meta Data section below):
Point source masks
LFI_MASK_030-ps_2048_R1.00.fits
LFI_MASK_044-ps_2048_R1.00.fits
LFI_MASK_070-ps_2048_R1.00.fits
Galactic plane masks
COM_MASK_gal-06_2048_R1.00.fits
COM_MASK_gal-07_2048_R1.00.fits

File Names[edit]

LFI_PowerSpect_030_R1.10.fits
LFI_PowerSpect_044_R1.10.fits
LFI_PowerSpect_070_R1.10.fits

Meta Data[edit]

The angular power spectra source list in each frequency is structured as a FITS binary table. The Fits extension is composed by the columns described below:


FITS header
Column Name Data Type Units Description
L Integer*4 ell parameter
TEMP_CL Real*8 uK[math]_{CMB}^2[/math] [math]C_l[/math] (temperature)
TEMP_CL_ERR Real*8 uK[math]_{CMB}^2[/math] [math]C_l[/math] error


Note.- in the comment keyword in the header, the galactic and point source maps used to generate the angular spectra are specified (LFI_MASK_030-ps_2048_R1.00.fits and COM_MASK_gal-06_2048_R1.00.fits in the example below). Note also that, due to an oversight, the mask description related to COM_MASK_gal-xxx is wrong and should refer to the galactic mask.

Below an example of the header. 

XTENSION= 'BINTABLE'           /Written by IDL:  Sat Feb 16 00:44:22 2013
BITPIX  =                    8 /
NAXIS   =                    2 /Binary table
NAXIS1  =                   20 /Number of bytes per row
NAXIS2  =                  130 /Number of rows
PCOUNT  =                    0 /Random parameter count
GCOUNT  =                    1 /Group count
TFIELDS =                    3 /Number of columns
TFORM1  = '1J      '           /Integer*4 (long integer)
TTYPE1  = 'L       '           /
TFORM2  = '1D      '           /Real*8 (double precision)
TTYPE2  = 'TEMP_CL '           /
TFORM3  = '1D      '           /Real*8 (double precision)
TTYPE3  = 'TEMP_CL_ERR'        /
EXTNAME = 'POW-SPEC'           / Extension name
EXTVER  =                    1 /Extension version
DATE    = '2013-02-15'         /Creation date
TUNIT2  = 'uK_CMB^2'           /
TUNIT3  = 'uK_CMB^2'           /
FILENAME= 'LFI_PowerSpect_030_R1.00.fits' /
PROCVER = 'Dx9_delta'          /
COMMENT ---------------------------------------------
COMMENT     Original Inputs
COMMENT ---------------------------------------------
COMMENT TT_30GHz_maskCS0.60_PS30GHzdet_febecopWls
COMMENT Used Point source Mask LFI_MASK_030-ps_2048_R1.00.fits
COMMENT Used Point source Mask COM_MASK_gal-06_2048_R1.00.fits
COMMENT Used FebeCoP 30 GHz wls
END

Below an example of the header of two masks used as input: COM_MASK_gal-06_2048_R1.00.fits and LFI_MASK_030-ps_2048_R1.00.fits: 

XTENSION= 'BINTABLE'           / binary table extension
BITPIX  =                    8 / 8-bit bytes
NAXIS   =                    2 / 2-dimensional binary table
NAXIS1  =                    4 / width of table in bytes
NAXIS2  =             50331648 / number of rows in table
PCOUNT  =                    0 / size of special data area
GCOUNT  =                    1 / one data group (required keyword)
TFIELDS =                    1 / number of fields in each row
TTYPE1  = 'Mask    '           / label for field   1
TFORM1  = 'E       '           / data format of field: 4-byte REAL
TUNIT1  = 'none    '           / physical unit of field
EXTNAME = '06-GalMask'
DATE    = '2013-02-16T11:07:42' / file creation date (YYYY-MM-DDThh:mm:ss UT)
CHECKSUM= 'NaGVNZGUNaGUNYGU'   / HDU checksum updated 2013-02-16T11:07:43
DATASUM = '2540860986'         / data unit checksum updated 2013-02-16T11:07:43
COMMENT
COMMENT *** Planck params ***
COMMENT
PIXTYPE = 'HEALPIX '           / HEALPIX pixelisation
ORDERING= 'NESTED  '           / Pixel ordering scheme, either RING or NESTED
NSIDE   =                 2048 / Resolution parameter for HEALPIX
FIRSTPIX=                    0 / First pixel # (0 based)
LASTPIX =             50331647 / Last pixel # (0 based)
INDXSCHM= 'IMPLICIT'           / Indexing: IMPLICIT or EXPLICIT
OBJECT  = 'FULLSKY '           / Sky coverage, either FULLSKY or PARTIAL
BAD_DATA=          -1.6375E+30
COORDSYS= 'GALACTIC'
FILENAME= 'COM_MASK_gal-06_2048_R1.00.fits'
COMMENT ---------------------------------------------------------------------
COMMENT Combined galactic mask 0.6 sky fraction
COMMENT Objects used:
COMMENT /sci_planck/lfi_dpc_test/ashdown/repository/masks/component_separation/d
COMMENT x9/combined_mask_0.60_sky_fraction.fits
COMMENT ---------------------------------------------------------------------
END

XTENSION= 'BINTABLE'           / binary table extension
BITPIX  =                    8 / 8-bit bytes
NAXIS   =                    2 / 2-dimensional binary table
NAXIS1  =                    4 / width of table in bytes
NAXIS2  =             50331648 / number of rows in table
PCOUNT  =                    0 / size of special data area
GCOUNT  =                    1 / one data group (required keyword)
TFIELDS =                    1 / number of fields in each row
TTYPE1  = 'Mask    '           / label for field   1
TFORM1  = 'E       '           / data format of field: 4-byte REAL
TUNIT1  = 'none    '           / physical unit of field
EXTNAME = '030-PSMask'
DATE    = '2013-02-16T11:03:20' / file creation date (YYYY-MM-DDThh:mm:ss UT)
CHECKSUM= 'fR7ThO7RfO7RfO7R'   / HDU checksum updated 2013-02-16T11:03:21
DATASUM = '3828742620'         / data unit checksum updated 2013-02-16T11:03:21
COMMENT
COMMENT *** Planck params ***
COMMENT
PIXTYPE = 'HEALPIX '           / HEALPIX pixelisation
ORDERING= 'NESTED  '           / Pixel ordering scheme, either RING or NESTED
NSIDE   =                 2048 / Resolution parameter for HEALPIX
FIRSTPIX=                    0 / First pixel # (0 based)
LASTPIX =             50331647 / Last pixel # (0 based)
INDXSCHM= 'IMPLICIT'           / Indexing: IMPLICIT or EXPLICIT
OBJECT  = 'FULLSKY '           / Sky coverage, either FULLSKY or PARTIAL
BAD_DATA=          -1.6375E+30
COORDSYS= 'GALACTIC'
FILENAME= 'LFI_MASK_030-ps_2048_R1.00.fits'
COMMENT ---------------------------------------------------------------------
COMMENT The radius of the holes is 3 times the sigma of the beam at the correspo
COMMENT nding frequency and sigma is FWHM/(2*sqrt(2ln2))
COMMENT FWHM at 30GHz used = 33.158 arcmin
COMMENT Objects used:
COMMENT /planck/sci_ops1/LFI_MAPs/DX9_Delta/MASKs/mask_ps_30GHz_beam33amin_nside
COMMENT 2048.00_DX9_nonblind_holesize3.fits
COMMENT ---------------------------------------------------------------------
END

References[edit]

  1. ↑ 1.01.11.21.3 Planck 2013 results: CMB power spectra and likelihood, Planck Collaboration XV, A&A, in press, (2014).
  2. ↑ Planck 2013 results: HFI time response and beams, Planck Collaboration 2013 VII, A&A, in press, (2014).
  3. ↑ Fast Cosmic Microwave Background Analyses via Correlation Functions, I. Szapudi, S. Prunet, D. Pogosyan, A. S. Szalay, J. R. Bond, ApJS, 548, L115-L118, (2001).
  4. ↑ Fast estimation of polarization power spectra using correlation functions, G. Chon, A. Challinor, S. Prunet, E. Hivon, I. Szapudi, MNRAS, 350, 914-926, (2004).
  5. ↑ Myths and truths concerning estimation of power spectra: the case for a hybrid estimator, G. Efstathiou, MNRAS, 349, 603-626, (2004).
  6. ↑ MASTER of the Cosmic Microwave Background Anisotropy Power Spectrum: A Fast Method for Statistical Analysis of Large and Complex Cosmic Microwave Background Data Sets, E. Hivon, K. M. Górski, C. B. Netterfield, B. P. Crill, S. Prunet, F. Hansen, ApJ, 567, 2-17, (2002).
  7. ↑ Unbiased estimation of an angular power spectrum, G. Polenta, D. Marinucci, A. Balbi, P. de Bernardis, E. Hivon, S. Masi, P. Natoli, N. Vittorio, J. Cosmology Astropart. Phys., 11, 1, (2005).

(Planck) High Frequency Instrument

Cosmic Microwave background

Flexible Image Transfer Specification

Planck Legacy Archive

(Planck) Low Frequency Instrument

Full-Width-at-Half-Maximum

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