... | ... | @@ -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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![image1](https://gitlab.in2p3.fr/tristram/Xpol/uploads/ee26c49f028906fbae336d882c506e91/image1.png)
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<img name="image1" src="https://gitlab.in2p3.fr/tristram/Xpol/uploads/ee26c49f028906fbae336d882c506e91/image1.png" width=500px">
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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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... | ... | @@ -20,7 +20,7 @@ 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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![image2](https://gitlab.in2p3.fr/tristram/Xpol/uploads/2c65b8ab65d909584c7ee2b31d9bf01a/image2.png)
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<img name="image2" src="https://gitlab.in2p3.fr/tristram/Xpol/uploads/2c65b8ab65d909584c7ee2b31d9bf01a/image2.png" width="500px">
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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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... | ... | @@ -28,15 +28,15 @@ 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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![image3](https://gitlab.in2p3.fr/tristram/Xpol/uploads/eae00a3a178a59cb1e247f23f16e6406/image3.png)
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<img name="image3" src="https://gitlab.in2p3.fr/tristram/Xpol/uploads/eae00a3a178a59cb1e247f23f16e6406/image3.png" width="500px">
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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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<img name="image4" src="https://gitlab.in2p3.fr/tristram/Xpol/uploads/51cb31ce9c2de17c3d8b061b7aa540e8/image4.png" width="500px">
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with
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![image5](https://gitlab.in2p3.fr/tristram/Xpol/uploads/13b08d4211b8c26415f0711eb4ff227e/image5.png)
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<img name="image5" src="https://gitlab.in2p3.fr/tristram/Xpol/uploads/13b08d4211b8c26415f0711eb4ff227e/image5.png" width="500px">
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where M^(2)^ is the quadratic coupling kernel matrix and ![image6](https://gitlab.in2p3.fr/tristram/Xpol/uploads/d874dfcad1e84570f4d8a69741d7d483/image6.png).
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... | ... | @@ -45,22 +45,22 @@ where M^(2)^ is the quadratic coupling kernel matrix and ![image6](https://gitla |
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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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![image7](https://gitlab.in2p3.fr/tristram/Xpol/uploads/d33dcf860da69980eebe863bb322f393/image7.png)
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<img name="image7" src="https://gitlab.in2p3.fr/tristram/Xpol/uploads/d33dcf860da69980eebe863bb322f393/image7.png" width="500px">
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where ![image8](https://gitlab.in2p3.fr/tristram/Xpol/uploads/80598e5129b8616fecb7f68e922f82ef/image8.png)
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where <img name="image8" src="https://gitlab.in2p3.fr/tristram/Xpol/uploads/80598e5129b8616fecb7f68e922f82ef/image8.png" width="100px">
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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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![image9](https://gitlab.in2p3.fr/tristram/Xpol/uploads/2ff0793a35225f82b143a3551c631fda/image9.png)
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<img name="image9" src="https://gitlab.in2p3.fr/tristram/Xpol/uploads/2ff0793a35225f82b143a3551c631fda/image9.png" width="300px">
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the final covariance matrix can be obtained from [Eq.](#likelihood),
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![image10](https://gitlab.in2p3.fr/tristram/Xpol/uploads/0fb05cdfe527c621b7d5f183fe104206/image10.png)
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<img name="image10" src="https://gitlab.in2p3.fr/tristram/Xpol/uploads/0fb05cdfe527c621b7d5f183fe104206/image10.png" width="300px">
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and the final error bars are given by
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![image11](https://gitlab.in2p3.fr/tristram/Xpol/uploads/d71bfea141795d45154a5dc8f009afa1/image11.png)
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<img name="image11" src="https://gitlab.in2p3.fr/tristram/Xpol/uploads/d71bfea141795d45154a5dc8f009afa1/image11.png" width="300px">
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... | ... | |