  We expect a detectable correlation between two seemingly unrelated
quantities: the four point function of the cosmic microwave background (CMB)
and the amplitude of flux decrements in quasar (QSO) spectra. The amplitude of
CMB convergence in a given direction measures the projected surface density of
matter. Measurements of QSO flux decrements trace the small-scale distribution
of gas along a given line-of-sight. While the cross-correlation between these
two measurements is small for a single line-of-sight, upcoming large surveys
should enable its detection. This paper presents analytical estimates for the
signal to noise (S/N) for measurements of the cross-correlation between the
flux decrement and the convergence and for measurements of the
cross-correlation between the variance in flux decrement and the convergence.
For the ongoing BOSS (SDSS III) and Planck surveys, we estimate an S/N of 30
and 9.6 for these two correlations. For the proposed BigBOSS and ACTPOL
surveys, we estimate an S/N of 130 and 50 respectively. Since the
cross-correlation between the variance in flux decrement and the convergence is
proportional to the fourth power of $\sigma_8$, the amplitude of these
cross-correlations can potentially be used to measure the amplitude of
$\sigma_8$ at z~2 to 2.5% with BOSS and Planck and even better with future data
sets. These measurements have the potential to test alternative theories for
dark energy and to constrain the mass of the neutrino. The large potential
signal estimated in our analytical calculations motivate tests with non-linear
hydrodynamical simulations and analyses of upcoming data sets.
