The signature of the Warm Hot Intergalactic Medium in WMAP and the
  forthcoming PLANCK data
  We compute the cross-correlation between the Warm-Hot Intergalactic Medium
and maps of Cosmic Microwave Background temperature anisotropies using a
log-normal probability density function to describe the weakly non-linear
matter density field. We search for this contribution in the data measured by
the Wilkinson Microwave Anisotropy Probe. We use a template of projected matter
density reconstructed from the Two-Micron All-Sky Redshift Survey as a tracer
of the electron distribution. The spatial distribution of filaments is modeled
using the recently developed Augmented Lagrangian Perturbation Theory. On the
scales considered here, the reconstructed density field is very well described
by the assumed log-normal distribution function. We predict that the
cross-correlation will have an amplitude of $0.03-0.3\mu$K. The measured value
is close to $1.5\mu$K, compatible with random alignments between structure in
the template and in the temperature anisotropy data. Using the W1 Differencing
Assembly to remove this systematic gives a residual correlation dominated by
Galactic foregrounds. Planck could detect the Warm-Hot Medium if it is well
traced by the density field reconstructed from galaxy surveys. The 217GHz
channel will allow to eliminate spurious contributions and its large frequency
coverage can show the sign change from the Rayleigh-Jeans to the Wien part of
the spectrum, characteristic of the thermal Sunyaev-Zeldovich effect.
