Asymptotic connection between full- and flat-sky angular correlators
Zucheng Gao, Alvise Raccanelli, Zvonimir Vlah
Abstract
We investigate the connection between the full- and flat-sky angular power spectra. First, we revisit this connection established on the geometric and physical grounds, namely, that the angular correlations on the sphere and in the plane (flat-sky approximation) correspond to each other in the limiting case of small angles and a distant observer. To establish the formal conditions for this limit, we first resort to a simplified shape of the 3D power spectrum, which allows us to obtain analytic results for both the full- and flat-sky angular power spectra. Using a saddle point approximation, we find that the flat-sky results are obtained in the limit when the comoving distance and wave modes $\ensuremath{\ell}$ approach infinity at the same rate. This allows us to obtain an analogous asymptotic expansion of the full-sky angular power spectrum for general 3D power spectrum shapes, including the $\mathrm{\ensuremath{\Lambda}}$ cold dark matter universe. In this way, we find a robust limit of correspondence between the full- and flat-sky results. These results also establish a mathematical relation, i.e., an asymptotic expansion of the ordinary hypergeometric function of a particular choice of arguments that physically corresponds to the flat-sky approximation of a distant observer. This asymptotic form of the ordinary hypergeometric function is obtained in two ways: relying on our saddle point approximation and using some of the known properties of the hypergeometric function.