Difference between revisions of "CMB spectrum & Likelihood Code"

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<span style="color:red"> UPDATE COMMANDER: Over the multipole range <math> \ell </math> = 2–29, the power spectrum is derived from a component-separation algorithm, ''Commander'':  applied to maps in the frequency range 30–353 GHz over 91% of the sky {{PlanckPapers|planck2013-p06}} . The asymmetric error bars associated to this spectrum are the 68% confidence limits and include the uncertainties due to foreground subtraction. </span>
 
<span style="color:red"> UPDATE COMMANDER: Over the multipole range <math> \ell </math> = 2–29, the power spectrum is derived from a component-separation algorithm, ''Commander'':  applied to maps in the frequency range 30–353 GHz over 91% of the sky {{PlanckPapers|planck2013-p06}} . The asymmetric error bars associated to this spectrum are the 68% confidence limits and include the uncertainties due to foreground subtraction. </span>
  
For multipoles equal or greater than <math>\ell=30</math>, instead, the spectrum is derived from the ''Plik'' likelihood {{PlanckPapers|planck2014-a13}} by optimally combining the spectra in the frequency range 100-217 GHz, and correcting them for unresolved foregrounds using the best-fit foreground solution from Planck TT+lowP run. . Associated 1-sigma errors include beam  uncertainties. Both ''Commander'' and ''Plik'' are described in more details in the sections below.
+
For multipoles equal or greater than <math>\ell=30</math>, instead, the spectrum is derived from the ''Plik'' likelihood {{PlanckPapers|planck2014-a13}} by optimally combining the spectra in the frequency range 100-217 GHz, and correcting them for unresolved foregrounds using the best-fit foreground solution from Planck TT+lowP LCDM run. Associated 1-sigma errors include beam  uncertainties. Both ''Commander'' and ''Plik'' are described in more details in the sections below.
  
 
[[File: Planck2014 TT Dl NORES bin30 w180mm.jpeg|thumb|center|700px|'''CMB TT spectrum. Logarithmic x-scale up to <math>\ell=30</math>, linear at higher <math>\ell</math>; all points with error bars. The red line is the Planck best-fit primordial power spectrum (cf Planck TT+lowP in Table 3 of {{PlanckPapers|planck2014-a15}}). The blue shaded area shows the uncertainties due to cosmic variance alone.''']]
 
[[File: Planck2014 TT Dl NORES bin30 w180mm.jpeg|thumb|center|700px|'''CMB TT spectrum. Logarithmic x-scale up to <math>\ell=30</math>, linear at higher <math>\ell</math>; all points with error bars. The red line is the Planck best-fit primordial power spectrum (cf Planck TT+lowP in Table 3 of {{PlanckPapers|planck2014-a15}}). The blue shaded area shows the uncertainties due to cosmic variance alone.''']]
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====TE and EE====
 
====TE and EE====
 
The Planck best-fit CMB polarization and temperature-polarization cross-correlation power spectra, shown in the figure below, cover the multipole range <math> \ell </math> = 30-1996. The data points relative to the multipole range <math> \ell </math> = 2-29 will be released in a second moment.
 
The Planck best-fit CMB polarization and temperature-polarization cross-correlation power spectra, shown in the figure below, cover the multipole range <math> \ell </math> = 30-1996. The data points relative to the multipole range <math> \ell </math> = 2-29 will be released in a second moment.
Analogously to the TT case, the <math> \ell\ge 30 </math> spectrum is derived from the ''Plik'' likelihood {{PlanckPapers|planck2014-a13}} by optimally combining the spectra in the frequency range 100-217 GHz, and correcting them for unresolved foregrounds using the best-fit foreground solution from Planck TT,TE,EE+lowP run.  
+
Analogously to the TT case, the <math> \ell\ge 30 </math> spectrum is derived from the ''Plik'' likelihood {{PlanckPapers|planck2014-a13}} by optimally combining the spectra in the frequency range 100-217 GHz, and correcting them for unresolved foregrounds using the best-fit foreground solution from a Planck TT,TE,EE+lowP LCDM run.  
  
 
{|style="margin: 0 auto;"
 
{|style="margin: 0 auto;"
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The <math>\ell \ge 30</math>  part of the TT, TE and EE power spectra have been derived by the Plik likelihood, a code that implements a pseudo-Cl based technique, extensively described in Sec. 2 and the Appendix of {{PlanckPapers|planck2013-p08}} and {{PlanckPapers|planck2014-a13}}. Frequency spectra are computed as cross-spectra between half-mission maps. Mask and multipole range choices for each frequency spectrum are summarized in Section 3.3 of {{PlanckPapers|planck2014-a15}} and in {{PlanckPapers|planck2014-a13}}. The final power spectrum is an optimal combination of the 100, 143, 143x217 and 217 GHz spectra, corrected for the best-fit unresolved foregrounds and inter-frequency calibration factors, as derived from the full likelihood analysis (for TT we use the best-fit solutions for the nuisance parameters from the Planck+TT+lowP data combination, while for TE and EE we use the best fit from Planck+TT+lowP, cf Table 3 of {{PlanckPapers|planck2014-a15}}). A thorough description of the models of unresolved foregrounds is given in {{PlanckPapers|planck2014-a13}}. The spectrum covariance matrix accounts for cosmic variance and noise contributions, together with beam uncertainties. The <math>\ell  \ge 30</math>  CMB TT spectrum and associated covariance matrix are available in two formats:
 
The <math>\ell \ge 30</math>  part of the TT, TE and EE power spectra have been derived by the Plik likelihood, a code that implements a pseudo-Cl based technique, extensively described in Sec. 2 and the Appendix of {{PlanckPapers|planck2013-p08}} and {{PlanckPapers|planck2014-a13}}. Frequency spectra are computed as cross-spectra between half-mission maps. Mask and multipole range choices for each frequency spectrum are summarized in Section 3.3 of {{PlanckPapers|planck2014-a15}} and in {{PlanckPapers|planck2014-a13}}. The final power spectrum is an optimal combination of the 100, 143, 143x217 and 217 GHz spectra, corrected for the best-fit unresolved foregrounds and inter-frequency calibration factors, as derived from the full likelihood analysis (for TT we use the best-fit solutions for the nuisance parameters from the Planck+TT+lowP data combination, while for TE and EE we use the best fit from Planck+TT+lowP, cf Table 3 of {{PlanckPapers|planck2014-a15}}). A thorough description of the models of unresolved foregrounds is given in {{PlanckPapers|planck2014-a13}}. The spectrum covariance matrix accounts for cosmic variance and noise contributions, together with beam uncertainties. The <math>\ell  \ge 30</math>  CMB TT spectrum and associated covariance matrix are available in two formats:
#Unbinned. TT: 2479 bandpowers (<math>\ell=30-2508</math>); TE or EE:1697 bandpowers (<math>\ell=30-1996</math>).
+
#Unbinned. TT: 2479 bandpowers (<math>\ell=30-2508</math>); TE or EE: 1697 bandpowers (<math>\ell=30-1996</math>).
 
#Binned, in bins of <math> \Delta\ell=30 </math>. TT: 83 bandpowers. TE or EE: 66 bandpowers. We bin the <math> C_\ell </math> power spectrum with a weight proportional to <math> \ell (\ell+1) </math>, so that the <math> C_{\ell_b} </math> binned bandpower centered in <math> \ell_b </math> is: <math> \\ C_{\ell_b}=\Sigma_{\ell \in b}  w_{\ell_b\ell} C_\ell \quad \text{with} \quad w_{\ell_b\ell}=\frac{\ell (\ell+1)}{\Sigma_{\ell \in b} \ell (\ell+1)}. \\</math> Equivalently, using the matrix formalism, we can construct the binning matrix B as: <math>\\ B_{\ell_b \ell}=w_{\ell_b\ell} \\ </math> where B is a <math> n_b\times n_\ell</math> matrix, with <math>n_b=83</math> the number of bins and <math>n_\ell=2479</math> the number of unbinned multipoles. Thus:  <math> \\ \vec{C}_\mathrm{binned}=B \, \vec{C} \\ \mathrm{cov_\mathrm{binned}}= B\, \mathrm{cov}\, B^T \\ \ell_b=B\, \ell \\ </math> Here, <math> \vec{C}_{binned}\, (\vec{C}) </math>  is the vector containing all the binned (unbinned) <math>C_\ell</math> bandpowers, <math>\mathrm{cov} </math> is the covariance matrix and <math>\ell_b</math> is  the weighted average multipole  in each bin. The binned <math>D_{\ell_B}</math> power spectrum is then calculated as: <math> \\ D_{\ell_b}=\frac{\ell_b (\ell_b+1)}{2\pi} C_{\ell_b} </math>.
 
#Binned, in bins of <math> \Delta\ell=30 </math>. TT: 83 bandpowers. TE or EE: 66 bandpowers. We bin the <math> C_\ell </math> power spectrum with a weight proportional to <math> \ell (\ell+1) </math>, so that the <math> C_{\ell_b} </math> binned bandpower centered in <math> \ell_b </math> is: <math> \\ C_{\ell_b}=\Sigma_{\ell \in b}  w_{\ell_b\ell} C_\ell \quad \text{with} \quad w_{\ell_b\ell}=\frac{\ell (\ell+1)}{\Sigma_{\ell \in b} \ell (\ell+1)}. \\</math> Equivalently, using the matrix formalism, we can construct the binning matrix B as: <math>\\ B_{\ell_b \ell}=w_{\ell_b\ell} \\ </math> where B is a <math> n_b\times n_\ell</math> matrix, with <math>n_b=83</math> the number of bins and <math>n_\ell=2479</math> the number of unbinned multipoles. Thus:  <math> \\ \vec{C}_\mathrm{binned}=B \, \vec{C} \\ \mathrm{cov_\mathrm{binned}}= B\, \mathrm{cov}\, B^T \\ \ell_b=B\, \ell \\ </math> Here, <math> \vec{C}_{binned}\, (\vec{C}) </math>  is the vector containing all the binned (unbinned) <math>C_\ell</math> bandpowers, <math>\mathrm{cov} </math> is the covariance matrix and <math>\ell_b</math> is  the weighted average multipole  in each bin. The binned <math>D_{\ell_B}</math> power spectrum is then calculated as: <math> \\ D_{\ell_b}=\frac{\ell_b (\ell_b+1)}{2\pi} C_{\ell_b} </math>.
  
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* compact sources catalog
 
* compact sources catalog
  
; High-l spectrum (<math>30  \ge 30 \ell \lessim 2500</math>):  
+
; High-l spectrum (<math>30  \ge \ell \lesssim 2500</math>):  
 
   
 
   
 
* 100, 143, 143x217 and 217 GHz spectra and their covariance matrix (Sec. 3.3 {{PlanckPapers|planck2014-a15}})
 
* 100, 143, 143x217 and 217 GHz spectra and their covariance matrix (Sec. 3.3 {{PlanckPapers|planck2014-a15}})
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: with the low ell part of the spectrum, not binned, and for l=2-49.  The table columns are
 
: with the low ell part of the spectrum, not binned, and for l=2-49.  The table columns are
 
# ''ELL'' (integer): multipole number
 
# ''ELL'' (integer): multipole number
# ''D_ELL'' (float): <math>D_l</math> as described below
+
# ''D_ELL'' (float): <math>D_l</math> as described above
 
# ''ERRUP'' (float): the upward uncertainty
 
# ''ERRUP'' (float): the upward uncertainty
 
# ''ERRDOWN'' (float): the downward uncertainty
 
# ''ERRDOWN'' (float): the downward uncertainty
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# ''L_MIN'' (integer): lowest multipole of bin
 
# ''L_MIN'' (integer): lowest multipole of bin
 
# ''L_MAX'' (integer): highest multipole of bin
 
# ''L_MAX'' (integer): highest multipole of bin
# ''D_ELL'' (float): <math>D_l<\math> as described below
+
# ''D_ELL'' (float): <math>D_\ell<\math> as described above
 
# ''ERR'' (float): the uncertainty
 
# ''ERR'' (float): the uncertainty
 
;BINNED COV-MAT (IMAGE)
 
: with the covariance matrix of the high-ell part of the spectrum in a 83x83 pixel image, i.e., covering the same bins as the ''HIGH-ELL'' table. Note that this is the covariance matrix of the <math>C_\ell</math>, not of the <math>D_\ell</math>.
 
  
 
; UNBINNED HIGH-ELL (BINTABLE)  
 
; UNBINNED HIGH-ELL (BINTABLE)  
 
: with the high-ell part of the spectrum, unbinned, in 2979 bins covering <math>\langle l \rangle = 30-2508\ </math>. The table columns are as follows:
 
: with the high-ell part of the spectrum, unbinned, in 2979 bins covering <math>\langle l \rangle = 30-2508\ </math>. The table columns are as follows:
 
# ''ELL'' (integer):  multipole  
 
# ''ELL'' (integer):  multipole  
# ''D_ELL'' (float): $D_l$ as described below
+
# ''D_ELL'' (float): $D_l$ as described above
 
# ''ERR'' (float): the uncertainty
 
# ''ERR'' (float): the uncertainty
  
; UNBINNED COV-MAT (IMAGE)
 
: with the covariance matrix of the high-ell part of the spectrum in a 2979x2979 pixel image, i.e., covering the same bins as the ''HIGH-ELL'' table. Note that this is the covariance matrix of the <math>C_\ell</math>, not of the <math>D_\ell</math>.
 
  
  
The spectra give $D_\ell = \ell(\ell+1)C_\ell / 2\pi$ in units of $\mu\, K^2$, and the covariance matrix is in units of $\mu\, K^4$ (NOTE that the covariance matrix is for the <math>C_\ell</math>, not for the <math>D_\ell</math>). 
+
The spectra give <math>D_\ell = \ell(\ell+1)C_\ell / 2\pi </math> in units of <math>\mu\, K^2</math>.
 +
<!--
  
 
The CMB spectrum is also given in a simple text comma-separated file:
 
The CMB spectrum is also given in a simple text comma-separated file:
 
* ''{{PLASingleFile | fileType=cosmo | name=COM_PowerSpect_CMB_R1.10.txt |link=COM_PowerSpect_CMB_R1.10.txt}}''
 
* ''{{PLASingleFile | fileType=cosmo | name=COM_PowerSpect_CMB_R1.10.txt |link=COM_PowerSpect_CMB_R1.10.txt}}''
  
 +
-->
 
==Likelihood==
 
==Likelihood==
 
The likelihood will soon be released with an accompanying paper and an Explanatory Supplement update.  
 
The likelihood will soon be released with an accompanying paper and an Explanatory Supplement update.  

Revision as of 17:07, 4 February 2015


CMB spectra[edit]

General description[edit]

TT[edit]

The Planck best-fit CMB temperature power spectrum, shown in figure below, covers the wide range of multipoles [math] \ell [/math] = 2-2508. UPDATE COMMANDER: Over the multipole range [math] \ell [/math] = 2–29, the power spectrum is derived from a component-separation algorithm, Commander: applied to maps in the frequency range 30–353 GHz over 91% of the sky Planck-2013-XII[1] . The asymmetric error bars associated to this spectrum are the 68% confidence limits and include the uncertainties due to foreground subtraction.

For multipoles equal or greater than [math]\ell=30[/math], instead, the spectrum is derived from the Plik likelihood Planck-2015-A11[2] by optimally combining the spectra in the frequency range 100-217 GHz, and correcting them for unresolved foregrounds using the best-fit foreground solution from Planck TT+lowP LCDM run. Associated 1-sigma errors include beam uncertainties. Both Commander and Plik are described in more details in the sections below.

CMB TT spectrum. Logarithmic x-scale up to [math]\ell=30[/math], linear at higher [math]\ell[/math]; all points with error bars. The red line is the Planck best-fit primordial power spectrum (cf Planck TT+lowP in Table 3 of Planck-2015-A13[3]). The blue shaded area shows the uncertainties due to cosmic variance alone.

TE and EE[edit]

The Planck best-fit CMB polarization and temperature-polarization cross-correlation power spectra, shown in the figure below, cover the multipole range [math] \ell [/math] = 30-1996. The data points relative to the multipole range [math] \ell [/math] = 2-29 will be released in a second moment. Analogously to the TT case, the [math] \ell\ge 30 [/math] spectrum is derived from the Plik likelihood Planck-2015-A11[2] by optimally combining the spectra in the frequency range 100-217 GHz, and correcting them for unresolved foregrounds using the best-fit foreground solution from a Planck TT,TE,EE+lowP LCDM run.

CMB EE spectrum. The red line is the Planck best-fit primordial power spectrum (cf Planck TT+lowP in Table 3 of Planck-2015-A13[3]). The blue shaded area shows the uncertainties due to cosmic variance alone.
CMB TE spectrum. The red line is the Planck best-fit primordial power spectrum (cf Planck TT+lowP in Table 3 of Planck-2015-A13[3]). The blue shaded area shows the uncertainties due to cosmic variance alone.

Production process[edit]

UPDATE COMMANDER The [math]\ell[/math] < 50 part of the Planck TT power spectrum is derived from the Commander approach, which implements Bayesian component separation in pixel space, fitting a parametric model to the data by sampling the posterior distribution for the model parameters Planck-2013-XII[1]. The power spectrum at any multipole [math]\ell[/math] is given as the maximum probability point for the posterior [math]C_\ell[/math] distribution, marginalized over the other multipoles, and the error bars are 68% confidence level Planck-2013-XV[4].

The [math]\ell \ge 30[/math] part of the TT, TE and EE power spectra have been derived by the Plik likelihood, a code that implements a pseudo-Cl based technique, extensively described in Sec. 2 and the Appendix of Planck-2013-XV[4] and Planck-2015-A11[2]. Frequency spectra are computed as cross-spectra between half-mission maps. Mask and multipole range choices for each frequency spectrum are summarized in Section 3.3 of Planck-2015-A13[3] and in Planck-2015-A11[2]. The final power spectrum is an optimal combination of the 100, 143, 143x217 and 217 GHz spectra, corrected for the best-fit unresolved foregrounds and inter-frequency calibration factors, as derived from the full likelihood analysis (for TT we use the best-fit solutions for the nuisance parameters from the Planck+TT+lowP data combination, while for TE and EE we use the best fit from Planck+TT+lowP, cf Table 3 of Planck-2015-A13[3]). A thorough description of the models of unresolved foregrounds is given in Planck-2015-A11[2]. The spectrum covariance matrix accounts for cosmic variance and noise contributions, together with beam uncertainties. The [math]\ell \ge 30[/math] CMB TT spectrum and associated covariance matrix are available in two formats:

  1. Unbinned. TT: 2479 bandpowers ([math]\ell=30-2508[/math]); TE or EE: 1697 bandpowers ([math]\ell=30-1996[/math]).
  2. Binned, in bins of [math] \Delta\ell=30 [/math]. TT: 83 bandpowers. TE or EE: 66 bandpowers. We bin the [math] C_\ell [/math] power spectrum with a weight proportional to [math] \ell (\ell+1) [/math], so that the [math] C_{\ell_b} [/math] binned bandpower centered in [math] \ell_b [/math] is: [math] \\ C_{\ell_b}=\Sigma_{\ell \in b} w_{\ell_b\ell} C_\ell \quad \text{with} \quad w_{\ell_b\ell}=\frac{\ell (\ell+1)}{\Sigma_{\ell \in b} \ell (\ell+1)}. \\[/math] Equivalently, using the matrix formalism, we can construct the binning matrix B as: [math]\\ B_{\ell_b \ell}=w_{\ell_b\ell} \\ [/math] where B is a [math] n_b\times n_\ell[/math] matrix, with [math]n_b=83[/math] the number of bins and [math]n_\ell=2479[/math] the number of unbinned multipoles. Thus: [math] \\ \vec{C}_\mathrm{binned}=B \, \vec{C} \\ \mathrm{cov_\mathrm{binned}}= B\, \mathrm{cov}\, B^T \\ \ell_b=B\, \ell \\ [/math] Here, [math] \vec{C}_{binned}\, (\vec{C}) [/math] is the vector containing all the binned (unbinned) [math]C_\ell[/math] bandpowers, [math]\mathrm{cov} [/math] is the covariance matrix and [math]\ell_b[/math] is the weighted average multipole in each bin. The binned [math]D_{\ell_B}[/math] power spectrum is then calculated as: [math] \\ D_{\ell_b}=\frac{\ell_b (\ell_b+1)}{2\pi} C_{\ell_b} [/math].

Inputs[edit]

UPDATE COMMANDER

Low-l spectrum ([math]\ell \lt 30[/math])
High-l spectrum ([math]30 \ge \ell \lesssim 2500[/math])

File names and Meta data[edit]

CHECK EXTENSION NAMES

The CMB spectrum and its covariance matrix are distributed in a single FITS file named

  • COM_PowerSpect_CMB_R2.nn.fits

which contains 5 extensions

LOW-ELL (BINTABLE)
with the low ell part of the spectrum, not binned, and for l=2-49. The table columns are
  1. ELL (integer): multipole number
  2. D_ELL (float): [math]D_l[/math] as described above
  3. ERRUP (float): the upward uncertainty
  4. ERRDOWN (float): the downward uncertainty
BINNED HIGH-ELL (BINTABLE)
with the high-ell part of the spectrum, binned into 83 bins covering [math]\langle l \rangle = 47-2499\ [/math] in bins of width [math]l=30[/math] (with the exception of the last bin that is smaller). The table columns are as follows:
  1. ELL (integer): mean multipole number of bin
  2. L_MIN (integer): lowest multipole of bin
  3. L_MAX (integer): highest multipole of bin
  4. D_ELL (float): [math]D_\ell\lt \math\gt as described above # ''ERR'' (float): the uncertainty ; UNBINNED HIGH-ELL (BINTABLE) : with the high-ell part of the spectrum, unbinned, in 2979 bins covering \lt math\gt \langle l \rangle = 30-2508\ [/math]. The table columns are as follows:
  5. ELL (integer): multipole
  6. D_ELL (float): $D_l$ as described above
  7. ERR (float): the uncertainty


The spectra give [math]D_\ell = \ell(\ell+1)C_\ell / 2\pi [/math] in units of [math]\mu\, K^2[/math].

Likelihood[edit]

The likelihood will soon be released with an accompanying paper and an Explanatory Supplement update.

References[edit]

Cosmic Microwave background

Flexible Image Transfer Specification