Detectability of the Gravitational Lensing Effect on the Two‐Point Correlation Function of Hot Spots in Cosmic Microwave Background Maps

Abstract

We present quantitative investigations of the weak-lensing effect on the two-point correlation functions of local maxima (hot spots), ξpk-pk(θ), in cosmic microwave background (CMB) maps. The lensing effect depends on the projected mass fluctuations between today and redshift zrec ≈ 1100. If we adopt the Gaussian assumption for the primordial temperature fluctuations field, the peak statistics can provide additional information about the intrinsic distribution of hot spots: that those pairs have some characteristic separation angles. The weak lensing then redistributes hot spots in the observed CMB maps from the intrinsic distribution and consequently imprints non-Gaussian signatures onto ξpk-pk(θ). In particular, since the intrinsic ξpk-pk(θ) has a pronounced depression feature around the angular scale of θ ≈ 70' for a flat universe, the weak lensing induces a large smoothing at that scale. We show that the lensing signature therefore has an advantage for effectively probing mass fluctuations with large wavelength modes around λ ≈ 50 h-1 Mpc. To reveal the detectability, we performed numerical experiments with specifications of the Microwave Anisotropy Probe (MAP) and Planck Surveyor, including the instrumental effects of beam smoothing and detector noise. We find that our method can successfully provide constraints on the amplitude of the mass fluctuations and cosmological parameters in a flat universe with and without the cosmological constant, provided that we use maps with the 65% sky coverage expected from Planck.