... | ... | @@ -17,7 +17,6 @@ where |
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* <N<sub>l</sub>> is the noise power spectrum;
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* M<sub>ll</sub> the coupling kernel matrix computed analytically from the weighting function as intensively described in [Kogut et al. 2003](#wmap_polar).
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\\
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The main advantage of using cross-power spectra is that the noise is generally uncorrelated between different detectors so that
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... | ... | @@ -27,11 +26,11 @@ The main advantage of using cross-power spectra is that the noise is generally u |
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The system can thus be solved for large sky fraction, otherwise one has to bin multipoles into bandpowers.
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From @@N@@ input maps we can obtain @@N(N-1)/2@@ cross-power spectra for each polarized mode (@@TT@@, @@EE@@, @@BB@@, @@TE@@, @@TB@@ and @@EB@@) which are unbiased estimates of the angular power spectrum but which are obviously not independent. For each polarized power spectra independently, *Xpol* can estimate the cross-correlation matrix between cross-spectra and multipoles from which error bars and covariance matrix in multipole space can be deduced for each cross-power spectra. Each element of this matrix reads :
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From *N* input maps we can obtain *N(N-1)/2* cross-power spectra for each polarized mode (*TT*, *EE*, *BB*, *TE*, *TB* and *EB*) which are unbiased estimates of the angular power spectrum but which are obviously not independent. For each polarized power spectra independently, *Xpol* can estimate the cross-correlation matrix between cross-spectra and multipoles from which error bars and covariance matrix in multipole space can be deduced for each cross-power spectra. Each element of this matrix reads :
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![image3](https://gitlab.in2p3.fr/tristram/Xpol/uploads/eae00a3a178a59cb1e247f23f16e6406/image3.png)
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Considering the completeness relation for spherical harmonics [Varshalovich et al.1988](#angularmomentum) and in the limit of large sky coverage [Efstathiou 2004 & 2005](#efstathiou), @@Ξ@@ reads (see [Tristram et al. 2005](#xspect)) :
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Considering the completeness relation for spherical harmonics [Varshalovich et al.1988](#angularmomentum) and in the limit of large sky coverage [Efstathiou 2004 & 2005](#efstathiou), Ξ reads (see [Tristram et al. 2005](#xspect)) :
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![image4](https://gitlab.in2p3.fr/tristram/Xpol/uploads/51cb31ce9c2de17c3d8b061b7aa540e8/image4.png)
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... | ... | @@ -49,7 +48,7 @@ To combine the cross-power spectra and obtain the best estimate of the power spe |
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![image7](https://gitlab.in2p3.fr/tristram/Xpol/uploads/d33dcf860da69980eebe863bb322f393/image7.png)
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where ![image8](https://gitlab.in2p3.fr/tristram/Xpol/uploads/80598e5129b8616fecb7f68e922f82ef/image8.png)
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is the cross-correlation matrix of the cross-power spectra described before (@@i@@ and @@j@@ ∈ {AB, A ≠ B}). The auto-power spectra are not considered.
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is the cross-correlation matrix of the cross-power spectra described before (*i* and *j* ∈ {AB, A ≠ B}). The auto-power spectra are not considered.
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Neglecting the correlation between adjacent multipoles, the estimate of the angular power spectrum reads
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... | ... | |