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 mΓm 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 Cl power spectrum once the Γm coefficients have been measured. This recovery is performed numerically below the 1 per cent level for colatitudes Θ > 80°. In addition, taking advantage of the smoothness of Cl and of Γm, we provide analytical expressions that allow the recovery of one of the spectra at the 1 per cent level, the other one being known.