Probing cosmic microwave background non-Gaussianity using local curvature

It is possible to classify pixels of a smoothed cosmic microwave background (CMB) fluctuation map according to their local curvature in ‘hill’, ‘lake’ and ‘saddle’ regions. In the Gaussian case, fractional areas occupied by pixels of each kind can be computed analytically for families of excursion sets as functions of threshold and moments of the fluctuation power spectrum. We show how the shape of these functions can be used to constrain accurately the level of non-Gaussianity in the data by applying these new statistics to an hypothetical mixed model suggested by Bouchet et al. (2001). According to our simple test, with only one 12.5 × 12.5 deg2 map, Planck should be able to detect with a high significance a non-Gaussian level as weak as 10 per cent in temperature standard deviation (rms; 5 per cent in Cℓ), whereas a marginal detection would be possible for MAP with a non-Gaussian level around 30 per cent in temperature (15 per cent in Cℓ).