| ... | @@ -8,7 +8,7 @@ The method computes an analytical estimate of the cross-correlation matrix direc |
... | @@ -8,7 +8,7 @@ The method computes an analytical estimate of the cross-correlation matrix direc |
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Generalizing MASTER equation for classical auto-spectra, we can compute the `pseudo' cross-power spectrum C<sub>l</sub>^X1xY2 between any two detectors
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Generalizing MASTER equation for classical auto-spectra, we can compute the `pseudo' cross-power spectrum C<sub>l</sub>^X1xY2 between any two detectors
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where
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where
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* B<sub>l</sub> is the beam transfer function describing the beam smoothing effect which can be computed, for example via models and/or Monte Carlo signal-only simulations;
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* B<sub>l</sub> is the beam transfer function describing the beam smoothing effect which can be computed, for example via models and/or Monte Carlo signal-only simulations;
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| ... | @@ -21,7 +21,7 @@ where |
... | @@ -21,7 +21,7 @@ where |
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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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The main advantage of using cross-power spectra is that the noise is generally uncorrelated between different detectors so that
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%height=30px%Attach:image2.png
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(note that for any common systematic residual inside maps, cross-spectra will exhibit a bias correlation term).
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(note that for any common systematic residual inside maps, cross-spectra will exhibit a bias correlation term).
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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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The system can thus be solved for large sky fraction, otherwise one has to bin multipoles into bandpowers.
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| ... | @@ -29,38 +29,39 @@ The system can thus be solved for large sky fraction, otherwise one has to bin m |
... | @@ -29,38 +29,39 @@ The system can thus be solved for large sky fraction, otherwise one has to bin m |
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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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%height=30px%Attach: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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%height=30px%Attach:image4.png
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with
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with
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%height=30px%Attach:image5.png
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where M^(2)^ is the quadratic coupling kernel matrix and %height=15px%Attach:image6.png .
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where M^(2)^ is the quadratic coupling kernel matrix and .
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\\
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\\
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To combine the cross-power spectra and obtain the best estimate of the power spectrum C<sub>l</sub>, we maximize the Gaussian approximated likelihood function
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To combine the cross-power spectra and obtain the best estimate of the power spectrum C<sub>l</sub>, we maximize the Gaussian approximated likelihood function
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[[#likelihood]]%height=30px%Attach:image7.png
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where %height=15px%Attach:image8.png 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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where 
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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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Neglecting the correlation between adjacent multipoles, the estimate of the angular power spectrum reads
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%height=30px%Attach:image9.png
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the final covariance matrix can be obtained from [Eq.](#likelihood),
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the final covariance matrix can be obtained from [Eq.](#likelihood),
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%height=30px%Attach:image10.png
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and the final error bars are given by
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and the final error bars are given by
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%height=30px%Attach:image11.png
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\\
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| ... | @@ -82,3 +83,4 @@ and the final error bars are given by |
... | @@ -82,3 +83,4 @@ and the final error bars are given by |
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* [[#archeops_cl2]] [Tristram et al. 2005b] Tristram M. et al., 2005 A&A 436 785
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* [[#archeops_cl2]] [Tristram et al. 2005b] Tristram M. et al., 2005 A&A 436 785
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* [[#angularmomentum]] [Varshalovich et al. 1988] Varshalovich D. A., Moskalev A. N., Khersonoskii V. K., 1988, ''Quantum theory of Angular Momentum'', World Scientific, Singapore.
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* [[#angularmomentum]] [Varshalovich et al. 1988] Varshalovich D. A., Moskalev A. N., Khersonoskii V. K., 1988, ''Quantum theory of Angular Momentum'', World Scientific, Singapore.
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