We use numerical simulations to predict the soft X-ray ([0.4-0.6] keV) and Sunyaev-Zeldovich signal (at 150 GHz) from the large scale structure in the Universe and then compute 2-point statistics to study the spatial distribution and time evolution o
f the signals. The average X-ray signal predicted for the 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~10000 and has amplitude of ~0.2 muK^2, about one order of magnitude below the signal measured with telescopes like Planck, ACT, and SPT. 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.
Recent analyses of the fluctuations of the soft Diffuse X-ray Background (DXB) have provided indirect detection of a component consistent with the elusive Warm Hot Intergalactic Medium (WHIM). In this work we use theoretical predictions obtained from
hydrodynamical simulations to investigate the angular correlation properties of the WHIM in emission and assess the possibility of indirect detection with next-generation X-ray missions. Our results indicate that the angular correlation signal of the WHIM is generally weak but dominates the angular correlation function of the DXB outside virialized regions. Its indirect detection is possible but requires rather long exposure times [0.1-1] Ms, large (~1{deg} x1{deg}) fields of view and accurate subtraction of isotropic fore/background contributions, mostly contributed by Galactic emission. The angular correlation function of the WHIM is positive for {theta} < 5 and provides limited information on its spatial distribution. A satisfactory characterization of the WHIM in 3D can be obtained through spatially resolved spectroscopy. 1 Ms long exposures with next generation detectors will allow to detect ~400 O VII+O VIII X-ray emission systems that we use to trace the spatial distribution of the WHIM. We predict that these observations will allow to estimate the WHIM correlation function with high statistical significance out to ~10 Mpc h^-1 and characterize its dynamical state through the analysis of redshift-space distortions. The detectable WHIM, which is typically associated with the outskirts of virialized regions rather than the filaments has a non-zero correlation function with slope {gamma} = -1.7 pm 0.1 and correlation length r0 = 4.0 pm 0.1 Mpc h^-1 in the range r = [4.5, 12] Mpc h^-1. Redshift space distances can be measured to assess the dynamical properties of the gas, typically infalling onto large virialized structures.
