Instability of reconstruction of the low CMB multipoles

We present an analysis of the cross-correlation for the WMAP ILC(III) low multipoles and foregrounds and show that these correlations are very high. By analysis of the 10 Monte Carlo simulations of the CMB-foreground separation presented by Eriksen et al. 2004a, we show that most likely the Lagrangian Internal Linear Combination method (LILC) produced negative shift of the distribution function P (K) with the mean value of the coefficient of cross-correlation about 〈K(l = 2)〉 = −0.254 for the quadrupole component, and 〈K(l = 3)〉 = −0.171 for the octupole. Moreover, practically for 40% realizations of the CMB quadrupole having the phase of the l = 2,m=0 mode equivalent to the phase of the foreground component, the reconstructed phase has a shift by factor π after implementation of the LILC method, increasing the coefficient of cross-correlation of the output CMB and the foregrounds. By implementation of the perturbation technique for the weighting coefficients Wj of LILC method, we show that the shape of the CMB–foreground cross-correlation coefficient in the vicinity of the optimal values Wj = W op j is very sharp. Small (∼ 1 − 5%) fluctuation of Wj around W op j can change K(l = 2) from −0.5 to 0.5 both for the ILC(III) and the WMAP foregrounds, and for the ILC(III) and Haslam et al. (1982) synchrotron map quadrupoles. We analyze the dependence of the multipole vectors for ILC(III) quadrupole on the perturbations of weighting coefficients around W op j and show that these vectors reflects directly the instability of reconstruction of l = 2,m=0 mode of the CMB quadrupole.