About the connection between the $C_{\mathrm{\ell }}$ power spectrum of the Cosmic Microwave Background and the ${\mathrm{\Gamma }}_m$ Fourier spectrum of rings on the sky

In this article we present and study a scaling law of the $m{\mathrm{\Gamma }}_m$ CMB Fourier spectrum on rings which allows us (i) to combine spectra corresponding to different colatitude angles (e.g. several detectors at the focal plane of a telescope), and (ii) to recover the $C_l$ power spectrum once the ${\mathrm{\Gamma }}_m$ coefficients have been measured. This recovery is performed numerically below the 1% level for colatitudes $\mathrm{\Theta }>{80}^{\circ }$ degrees. In addition, taking advantage of the smoothness of the $C_l$ and of the ${\mathrm{\Gamma }}_m$, we provide analytical expressions which allow to recover one of the spectrum at the 1% level, the other one being known.