Concerning the connection between the Cl power spectrum of the cosmic microwave background and the Γm Fourier spectrum of rings on the sky

In this article we present and study a scaling law of the m Gamma (m) cosmic microwave background Fourier spectrum on rings that allows us to (i) combine spectra corresponding to different colatitude angles (e.g. several detectors at the focal plane of a telescope) and (ii) recover the C (l) power spectrum once the Gamma (m) coefficients have been measured. This recovery is performed numerically below the 1 per cent level for colatitudes Theta > 80degrees. In addition, taking advantage of the smoothness of C (l) and of Gamma (m) , we provide analytical expressions that allow the recovery of one of the spectra at the 1 per cent level, the other one being known.