Cosmic microwave background power spectrum estimation and map reconstruction with the expectation-maximization algorithm

Abstract

We apply the iterative expectation-maximization algorithm (EM) to estimate the power spectrum of the cosmic microwave background (CMB) from multifrequency microwave maps. In addition, we are also able to provide a reconstruction of the CMB map. By assuming that the combined emission of the foregrounds plus the instrumental noise is Gaussian distributed in Fourier space, we have simplified the EM procedure, finding an analytical expression for the maximization step. By using the simplified expression, the CPU time can be greatly reduced. We test the stability of our power spectrum estimator with realistic simulations of Planck data, including point sources and allowing for spatial variation of the frequency dependence of the Galactic emissions. Without prior information about any of the components, our new estimator can recover the CMB power spectrum up to scales l ≈ 1500 with less than 10 per cent error. This result is significantly improved if the brightest point sources are removed before applying our estimator. In this way, the CMB power spectrum can be recovered up to l ≈ 1700 with 10 per cent error and up to l ≈ 2100 with 50 per cent error. This result is very close to the one that would be obtained in the ideal case of only CMB plus white noise, for which all our assumptions are satisfied. Moreover, the EM algorithm also provides a straightforward mechanism for reconstructing the CMB map. The recovered cosmological signal shows a high degree of correlation (r = 0.98) with the input map and low residuals.