  We discuss the four-point correlation function, or the trispectrum in Fourier
space, of CMB temperature and polarization anisotropies due to the weak
gravitational lensing effect by intervening large scale structure. We discuss
the squared temperature power spectrum as a probe of this trispectrum and, more
importantly, as an observational approach to extracting the power spectrum of
the deflection angle associated with the weak gravitational lensing effect on
the CMB. We extend previous discussions on the trispectrum and associated weak
lensing reconstruction from CMB data by calculating non-Gaussian noise
contributions, beyond the previously discussed dominant Gaussian noise.
Non-Gaussian noise contributions are generated by lensing itself and by the
correlation between the lensing effect and other foreground secondary
anisotropies in the CMB such as the Sunyaev-Zel'dovich (SZ) effect. When the SZ
effect is removed from temperature maps using its spectral dependence, we find
these additional non-Gaussian noise contributions to be an order of magnitude
lower than the dominant Gaussian noise. If the noise-bias due to the dominant
Gaussian part of the temperature squared power spectrum is removed, then these
additional non-Gaussian contributions provide the limiting noise level for the
lensing reconstruction. The temperature squared power spectrum allows a high
signal-to-noise extraction of the lensing deflections and a confusion-free
separation of the curl (or B-mode) polarization due to inflationary
gravitational waves from that due to lensed gradient (or E-mode) polarization.
The small angular scale temperature and polarization anisotropy measurements
provide a novel approach to weak lensing studies, complementing the approach
based on galaxy ellipticities.
