Difference between revisions of "CMB spectrum & Likelihood Code"

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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. Note that following this definition, <math>\ell_b</math> can be a non-integer. 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===
 
===Inputs===
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* compact sources catalog
 
* compact sources catalog
  
; High-l spectrum (<math>30 \le \ell \ge 2500</math>):  
+
; High-l spectrum (<math>30 \ge \ell \le 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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# ''ERRDOWN'' (float): the downward uncertainty
 
# ''ERRDOWN'' (float): the downward uncertainty
  
; BINNED HIGH-ELL (BINTABLE)  
+
; TT 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:
 
: 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:
 
# ''ELL'' (integer): mean multipole number of bin
 
# ''ELL'' (integer): mean multipole number of bin
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# ''ERR'' (float): the uncertainty
 
# ''ERR'' (float): the uncertainty
  
; UNBINNED HIGH-ELL (BINTABLE)  
+
; TT 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  
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# ''ERR'' (float): the uncertainty
 
# ''ERR'' (float): the uncertainty
  
 +
; TE BINNED HIGH-ELL (BINTABLE)
 +
: with the high-ell part of the spectrum, binned into 83 bins covering <math>\langle l \rangle = 47-1988\ </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:
 +
# ''ELL'' (integer): mean multipole number of bin
 +
# ''L_MIN'' (integer): lowest multipole of bin
 +
# ''L_MAX'' (integer): highest multipole of bin
 +
# ''D_ELL'' (float): <math>D_\ell</math> as described above
 +
# ''ERR'' (float): the uncertainty
 +
 +
; TE UNBINNED HIGH-ELL (BINTABLE)
 +
: with the high-ell part of the spectrum, unbinned, in 2979 bins covering <math>\langle l \rangle = 30-1996\ </math>. The table columns are as follows:
 +
# ''ELL'' (integer):  multipole
 +
# ''D_ELL'' (float): <math>D_\ell</math> as described above
 +
# ''ERR'' (float): the uncertainty
 +
 +
; EE BINNED HIGH-ELL (BINTABLE)
 +
: with the high-ell part of the spectrum, binned into 83 bins covering <math>\langle l \rangle = 47-1988\ </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:
 +
# ''ELL'' (integer): mean multipole number of bin
 +
# ''L_MIN'' (integer): lowest multipole of bin
 +
# ''L_MAX'' (integer): highest multipole of bin
 +
# ''D_ELL'' (float): <math>D_\ell</math> as described above
 +
# ''ERR'' (float): the uncertainty
 +
 +
; EE UNBINNED HIGH-ELL (BINTABLE)
 +
: with the high-ell part of the spectrum, unbinned, in 2979 bins covering <math>\langle l \rangle = 30-1996\ </math>. The table columns are as follows:
 +
# ''ELL'' (integer):  multipole
 +
# ''D_ELL'' (float): <math>D_\ell</math> as described above
 +
# ''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>.
+
The spectra give <math>D_\ell = \ell(\ell+1)C_\ell / 2\pi </math> in units of <math>\mu\, K^2</math>. The covariance matrices of the spectra will be released in a second moment.
 
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Revision as of 17:22, 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 a Planck TT+lowP [math]\Lambda[/math]CDM 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 [math]\Lambda[/math]CDM 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. Note that following this definition, [math]\ell_b[/math] can be a non-integer. 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 \le 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_ell[/math] as described above
  3. ERRUP (float): the upward uncertainty
  4. ERRDOWN (float): the downward uncertainty
TT 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[/math] as described above
  5. ERR (float): the uncertainty
TT 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:
  1. ELL (integer): multipole
  2. D_ELL (float): [math]D_\ell[/math] as described above
  3. ERR (float): the uncertainty
TE BINNED HIGH-ELL (BINTABLE)
with the high-ell part of the spectrum, binned into 83 bins covering [math]\langle l \rangle = 47-1988\ [/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[/math] as described above
  5. ERR (float): the uncertainty
TE UNBINNED HIGH-ELL (BINTABLE)
with the high-ell part of the spectrum, unbinned, in 2979 bins covering [math]\langle l \rangle = 30-1996\ [/math]. The table columns are as follows:
  1. ELL (integer): multipole
  2. D_ELL (float): [math]D_\ell[/math] as described above
  3. ERR (float): the uncertainty
EE BINNED HIGH-ELL (BINTABLE)
with the high-ell part of the spectrum, binned into 83 bins covering [math]\langle l \rangle = 47-1988\ [/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[/math] as described above
  5. ERR (float): the uncertainty
EE UNBINNED HIGH-ELL (BINTABLE)
with the high-ell part of the spectrum, unbinned, in 2979 bins covering [math]\langle l \rangle = 30-1996\ [/math]. The table columns are as follows:
  1. ELL (integer): multipole
  2. D_ELL (float): [math]D_\ell[/math] as described above
  3. 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]. The covariance matrices of the spectra will be released in a second moment.

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