Lensing-induced non-Gaussian signatures in the cosmic microwave background

We propose a new method for extracting non-Gaussian signatures on isotemperature statistics in the cosmic microwave background (CMB) sky, which are induced by the gravitational lensing due to the intervening large-scale structure of the universe. To develop the method, we focus on a specific statistical property of the intrinsic Gaussian CMB field: a field point in the map that has a larger absolute value of the temperature threshold tends to have a larger absolute value of the curvature parameter defined by a trace of the second-derivative matrix of the temperature field, while the ellipticity parameter similarly defined is uniformly distributed independently of the threshold because of the isotropic nature of the Gaussian field. Weak lensing then causes a stronger distortion effect on the isotemperature contours with higher thresholds and especially induces a coherent distribution of the ellipticity parameter correlated with the threshold as a result of the coupling between the CMB curvature parameter and the gravitational tidal shear in the observed map. These characteristic patterns can be statistically picked up by considering three independent characteristic functions, which are obtained from the averages of quadratic combinations of the second-derivative fields of the CMB over isotemperature contours with each threshold. Consequently, we find that the lensing effect generates non-Gaussian signatures on those functions that have a distinct functional dependence on the threshold. We test the method using numerical simulations of CMB maps and show that the lensing signals can be measured definitely, provided that we use CMB data with sufficiently low noise and high angular resolution.