Scalar statistics on the sphere: application to the cosmic microwave background

A method to compute several scalar quantities of cosmic microwave background (CMB) maps on the sphere is presented. We consider here four type of scalars: the Hessian matrix scalars, the distortion scalars, the gradient-related scalars and the curvature scalars. Such quantities are obtained directly from the spherical harmonic coefficients aℓm of the map. We also study the probability density function of these quantities for the case of a homogeneous and isotropic Gaussian field, which are functions of the power spectrum of the initial field. From these scalars it is possible to construct a new set of scalars which are independent of the power spectrum of the field. We test our results using simulations and find good agreement between the theoretical probability density functions and those obtained from simulations. Therefore, these quantities are proposed to investigate the presence of non-Gaussian features in CMB maps. Finally, we show how to compute the scalars in the presence of anisotropic noise and realistic masks.