CMB B polarization to map the large-scale structures of the universe

We explore the possibility of using B-type polarization of the cosmic microwave background to map the large-scale structures of the Universe taking advantage of the lens effects on the CMB polarization. The functional relation between the $B$ component with the primordial CMB polarization and the line-of-sight mass distribution is explicated. Noting that a sizable fraction (at least 40%) of the dark halo population which is responsible for this effect can also be detected in a galaxy weak lensing survey, we present statistical quantities that should exhibit a strong sensitivity to this overlapping. We stress that it would be a sound test of the gravitational instability picture, independent of many systematic effects that may hamper lensing detection in CMB or a galaxy survey alone. Moreover, we estimate the intrinsic cosmic variance of the amplitude of this effect to be less than 8% for a $100\hspace{0.5em}{\mathrm{deg}}^2$ survey with a ${10}^{\prime }$ CMB beam. Its measurement would then provide us with an original means for constraining the cosmological parameters, more particularly, as it turns out, the cosmological constant $\Lambda .$