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 nonlinear 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 μK. The measured value is close to 1.5 μK, 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 217?GHz 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-Zel'dovich effect.
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