Analysis of cosmic microwave background data on an incomplete sky

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 l(max) less than or similar to 50, as they required all the modes have appreciable support on the cut-sky, whereas for large 1,,,a., 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 O(l(max)(6)) operations and O(l(max)(4)) storage and so is impractical for l(max) greater than or similar to200 at present. These problems are circumvented for the special case of constant (Galactic) latitude cuts, for which the storage requirements scale as O(l(rmax)(2)) and the operations count scales as O(l(max)(4)). 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.