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Cosmological hydrodynamical simulations predict that a large fraction of allbaryons reside within mildly non-linear structures with temperatures in therange $10^{5}-10^{7}$K. As the gas is highly ionized, it could be detected bythe temperature anisotropies generated on the Cosmic Microwave Backgroundradiation. We refine our previous estimates of the thermal Sunyaev-Zeldovicheffect by introducing a non-polytropic equation of state to model thetemperature distribution of the shock heated gas derived fromtemperature-density phase diagrams of different hydrodynamical simulations.Depending on the specific model, the Comptonization parameter varies in therange $10^{-7}\le y_c \le 2\times 10^{-6}$, compatible with the FIRAS upperlimit. This amplitude is in agreement with a simple toy model constructed toestimate the average effect induced by filaments of ionized gas. Using thelog-normal probability density function we calculate the correlation functionand the power spectrum of the temperature anisotropies generated by the WHIMfilaments. For a wide range of the parameter space, the maximum amplitude ofthe radiation power spectrum is $(\ell+1)\ell C_{\ell}/2\pi =0.7-70(\mu K)^2$at $\ell \approx 200-500$. This amplitude scales with baryon density, Hubbleconstant and the amplitude of the matter power spectrum $\sigma_8$ as$[(\ell+1)\ell C_\ell]_{\mathrm{max}}/2\pi \propto \sigma_8^{2.6}(\Omega_bh)^2$. Since the thermal Sunyaev-Zeldovich effect has a specific frequencydependence, we analyze the possibility of detecting this component with theforthcoming Planck data.

The contribution of the Warm-Hot Intergalactic Medium to the CMB anisotropies via the Sunyaev-Zeldovich effect