Estimating the spectral indices of correlated astrophysical foregrounds by a second-order statistical approach

We present the first tests of a new method, the Correlated Component Analysis (CCA) based on second-order statistics, to estimate the mixing matrix, a key ingredient to separate astrophysical foregrounds superimposed to the Cosmic Microwave Background (CMB). In the present application, the mixing matrix is parameterized in terms of the spectral indices of Galactic synchrotron and thermal dust emissions, while the free-free spectral index is prescribed by basic physics, and is thus assumed to be known. We consider simulated observations of the microwave sky with angular resolution and white stationary noise at the nominal levels for the PLANCK satellite, and realistic foreground emissions, with a position dependent synchrotron spectral index. We work with two sets of PLANCK frequency channels: the low frequency set, from 30 to 143 GHz, complemented with the Haslam 408 MHz map, and the high frequency set, from 217 to 545 GHz. The concentration of intense free-free emission on the Galactic plane introduces a steep dependence of the spectral index of the global Galactic emission with Galactic latitude, close to the Galactic equator. This feature makes difficult for the CCA to recover the synchrotron spectral index in this region, given the limited angular resolution of PLANCK, especially at low frequencies. A cut of a narrow strip around the Galactic equator (|b|<3 deg), however, allows us to overcome this problem. We show that, once this strip is removed, the CCA allows an effective foreground subtraction, with residual uncertainties inducing a minor contribution to errors on the recovered CMB power spectrum.