  The intergalactic neutral hydrogen which is responsible for the Lyman alpha
forest of quasar absorption is a tracer of much larger amounts of ionised
hydrogen. The ionised component has yet to be detected directly, but is
expected to scatter CMB photons via the Sunyaev-Zel'dovich (SZ) effect. We use
hydrodynamic simulations of a LambdaCDM universe to create mock quasar spectra
and CMB sky maps. We find that the high-z Lya forest gas causes temperature
fluctuations of the order of 1 muK rms in the CMB on arcmin scales. The kinetic
and thermal SZ effects have a similar magnitude at z=3, with the thermal effect
becoming relatively weaker as expected at higher z. The CMB signal associated
with lines of sight having HI column densities > 10^18 cm^-2 is only marginally
stronger than that for lower column densities. The strong dependence of rms
temperature fluctuation on mean Lya absorbed flux, however, suggests that the
CMB signal effectively arises in lower density material. We investigate the use
of the cross-correlation of the Lya forest and the microwave background to
detect the SZ effect at redshifts 2-4. In so doing we are able to set direct
limits on the density of diffuse ionised intergalactic baryons. We carry out a
preliminary comparison at a mean redshift z=3 of 3488 quasar spectra from SDSS
Data Release 3 and the WMAP first year data. Assuming that the baryons are
clustered as in a LambdaCDM cosmology, and have the same mean temperature, the
cross-correlation yields a weak limit on the cosmic density of ionised baryons
Omega_(b,I), which is Omega_(b,I) < 0.8 at 95% confidence. With data from
upcoming CMB telescopes, we anticipate that a direct detection of the high
redshift ionised IGM will soon be possible, providing an important consistency
check on cosmological models.
