We use numerical simulations to predict the soft X-ray ([0.4-0.6] keV) and Sunyaev-Zel'dovich (SZ) signal (at 150 GHz) from a large-scale structure in the universe and then compute two-point statistics to study the spatial distribution and time evolution of the signals. The average X-ray signal predicted for the warm-hot intergalactic medium (WHIM) is in good agreement with observational constraints that set it at about 10% of the total diffuse X-ray background. The characteristic angle computed with the autocorrelation function is of the order of some arcminutes and becomes smaller at higher redshift. The power spectrum peak of the SZ due to the WHIM is at l ~ 10,000 and has an amplitude of ~0.2 μK2, about one order of magnitude below the signal measured with telescopes like Planck, Atacama Cosmology Telescope, and South Pole Telescope. Even if the high-redshift WHIM signal is too weak to be detected using X-rays only, the small-scale correlation between X-ray and SZ maps is dominated by the high-redshift WHIM. This makes the analysis of the SZ signal in support of X-rays a promising tool to study the early time WHIM.
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