a Power Spectrum estimator based on cross correlation between maps.
Xpol is a method to estimate the polarized angular power spectra Cls by computing the cross-power spectra between a collection of input maps coming either from multiple detectors of the same experiment or from different instruments. The `pseudo' cross-power spectra are explicitly corrected for incomplete sky coverage, beam smoothing, filtering and pixelization. Assuming no correlation between the noise contribution from two different maps, each of the corrected cross-power spectra is an unbiased estimate of the Cls. Analytical error bars are derived for each of them. The cross-power spectra, that do not include the classical ''auto''-power spectra, are then combined using a Gaussian approximation of the likelihood function. This result can be compared to the auto-spectra for which the noise bias in multipole domain is requested as input.
The method computes an analytical estimate of the cross-correlation matrix directly from the data avoiding any Monte Carlo simulations. This allows to include naturally the mode coupling in this matrix. Further, this permits the computation of analytical error bars which are very compatible with those obtained from simulations (see Tristram et al. 2005a for the temperature case). Because of the above, this method can be applied without modification to the estimation of the power spectrum of the correlated signal between a set of maps of the sky coming from multiple instruments with potentially different sky coverages. This method has been used on Archeops data to estimate the CMB angular power spectrum Tristram et al. 2005b and the polarized foreground emission at the sub-millimeter and millimeter wavelength in Ponthieu et al. 2005.
Generalizing MASTER equation for classical auto-spectra, we can compute the `pseudo' cross-power spectrum Cl(X1xY2) between any two detectors
- Bl 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;
- pl is the transfer function of the pixelization scheme of the map describing the effect of smoothing due to the finite pixel size and geometry;
- Fl is an effective function that represents any filtering applied to the time ordered data that can also be computed via Monte Carlo;
- <Nl> is the noise power spectrum;
- Mll' the coupling kernel matrix computed analytically from the weighting function as intensively described in Kogut et al. 2003.
The main advantage of using cross-power spectra is that the noise is generally uncorrelated between different detectors so that
(note that for any common systematic residual inside maps, cross-spectra will exhibit a bias correlation term). The system can thus be solved for large sky fraction, otherwise one has to bin multipoles into bandpowers.
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 :
To combine the cross-power spectra and obtain the best estimate of the power spectrum Cl, we maximize the Gaussian approximated likelihood function
Neglecting the correlation between adjacent multipoles, the estimate of the angular power spectrum reads
the final covariance matrix can be obtained from Eq.,
and the final error bars are given by
- [Efstathiou 2004 & 2005] Efstathiou G., 2004, MNRAS, 349, 603
- [Efstathiou 2004 & 2005] Efstathiou G., 2005, MNRAS, 370, 343
- [Hivon et al. 2002] Hivon E., Gorski K. M., Netterfield C. B., Crill B. P., Prunet S., Hansen F., 2002, ApJ, 567, 2
- [Kogut et al. 2003] Kogut A. et al., 2003 ApJS 148 161
- [Ponthieu et al. 2005] Ponthieu N. et al., 2005 A&A 444 327
- [Tristram et al. 2005a] Tristram M. et al., 2005 MNRAS 358 833
- [Tristram et al. 2005b] Tristram M. et al., 2005 A&A 436 785
- [Varshalovich et al. 1988] Varshalovich D. A., Moskalev A. N., Khersonoskii V. K., 1988, ''Quantum theory of Angular Momentum'', World Scientific, Singapore.