  We present a theoretical estimate for a new observable: the cross-correlation
between the Lyman-alpha flux fluctuations in quasar (QSO) spectra and the
convergence of the cosmic microwave background (CMB) as measured along the same
line-of-sight. As a first step toward the assessment of its detectability, we
estimate the signal-to-noise ratio using linear theory. Although the
signal-to-noise is small for a single line-of-sight and peaks at somewhat
smaller redshifts than those probed by the Lyman-alpha forest, we estimate a
total signal-to-noise of 9 for cross-correlating QSO spectra of SDSS-III with
Planck and of 20 for cross-correlating with a future polarization based CMB
experiment. The detection of this effect would be a direct measure of the
neutral hydrogen-matter cross-correlation and could provide important
information on the growth of structures at large scales in a redshift range
which is still poorly probed by observations.
