On the Cross-Correlation between Dark Matter Distribution and Secondary CMB Anisotropies by Ionized Intergalactic Medium

We investigate a cross-correlation between the weak gravitational lensing field of the large-scale structure, $\kappa$, and the secondary temperature fluctuation field, $\Delta$, of 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 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 ionized medium. The magnitude of the cross-correlation can be estimated as $\kaco{\kappa\Delta}\sim 10^{-10}$ on small angular scales ($l\simgt 1000$) under the currently favored cold CDM model of the structure formation and the simplest scenario of homogeneous reionization after a given redshift of $z_{\rm ion}\simgt 5$. Interestingly, we find that the density-modulated quadratic Doppler effect of $O(\delta_e v^2)$ produces the cross-correlation with the weak lensing field that has {\em linear} dependence on the electron density fluctuation field $\delta_e$ or equivalently on the biasing relation between the dark matter and electron distributions. In other words, the angular power spectrum could be both 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.