Analysis of cosmic microwave background data on an incomplete sky

Abstract

Measurement of the angular power spectrum of the cosmic microwave background is most often based on a spherical harmonic analysis of the observed temperature anisotropies. Even if all-sky maps are obtained, however, it is likely that the region around the Galactic plane will have to be removed as a result of its strong microwave emissions. The spherical harmonics are not orthogonal on the cut sky, but an orthonormal basis set can be constructed from a linear combination of the original functions. Previous implementations of this technique, based on Gram–Schmidt orthogonalization, were limited to maximum Legendre multipoles of lmax & 50, as they required all the modes have appreciable support on the cut-sky, whereas for large lmax the fraction of modes supported is equal to the fractional area of the region retained. This problem is solved by using a singular value decomposition to remove the poorly supported basis functions, although the treatment of the non-cosmological monopole and dipole modes necessarily becomes more complicated. A further difficulty is posed by computational limitations – orthogonalization for a general cut requires Oðl 6 maxÞ operations and Oðl 4 maxÞ storage and so is impractical for lmax * 200 at present. These problems are circumvented for the special case of constant (Galactic) latitude cuts, for which the storage requirements scale as Oðl 2 maxÞ and the operations count scales as Oðl 4 maxÞ. Less clear, however, is the stage of the data analysis at which the cut is best applied. As convolution is ill-defined on the incomplete sphere, beam-deconvolution should not be performed after the cut and, if all-sky component separation is as successful as simulations indicate, the Galactic plane should probably be removed immediately prior to power spectrum estimation.