On the Cross‐Correlation between Dark Matter Distribution and Secondary Cosmic Microwave Background Anisotropies of the Ionized Intergalactic Medium

Abstract

We investigate a cross-correlation between the weak gravitational lensing field of the large-scale structure κ and the secondary temperature fluctuation field Δ of the cosmic microwave background (CMB) induced by Thomson scattering of CMB photons off the ionized medium in the mildly nonlinear structure. The cross-correlation is expected to observationally unveil the biasing relation between the dark matter and ionized medium distributions in the large-scale structure. We develop a formalism for calculating the cross-correlation function and its angular power spectrum based on the small angle approximation. As a result, we find that the leading contribution to the cross-correlation comes from the secondary CMB fluctuation field induced by the quadratic Doppler effects of the bulk velocity field v of the ionized medium since the cross-correlations with the O(v) Doppler effect and the Ostriker-Vishniac effect of O(vδ) are suppressed on relevant angular scales because the linear dependence on the bulk velocity field leads to cancellations between positive and negative contributions among the scatterers for the ensemble average. The magnitude of the cross-correlation can be estimated as κΔ ~ 10-10 on small angular scales (l 1000) under the currently favored cold dark matter model of the structure formation and the simplest scenario of homogeneous reionization after a given redshift of zion 5. Although the magnitude turns out to be small, we find several interesting aspects of the effect. One of them is that the density-modulated quadratic Doppler effect of O(δev2) produces the cross-correlation with the weak lensing field that has linear dependence on the electron density fluctuation field δe or, equivalently, on the biasing relation between the dark matter and electron distributions. In other words, the angular power spectrum could be either positive or negative, depending on the positive biasing or antibiasing, respectively. Detection of the cross-correlation thus offers a new opportunity to observationally understand the reionization history of intergalactic medium connected to the dark matter clustering.