Prospects of detecting soft X-ray emission from typical WHIM filaments around massive clusters and the Coma cluster soft excess

While hot ICM in galaxy clusters makes these objects powerful X-ray sources, the cluster's outskirts and overdense gaseous filaments might give rise to much fainter sub-keV emission. Cosmological simulations show a prominent "focusing" effect of rich clusters on the space density of the Warm-Hot Intergalactic Medium (WHIM) filaments up to a distance of $\sim 10\,{\rm Mpc}$ ($\sim$ turnaround radius, $r_{ta}$) and beyond. Here, we use Magneticum simulations to characterize their properties in terms of integrated emission measure for a given temperature and overdensity cut and the level of contamination by the more dense gas. We suggest that the annuli $(\sim 0.5-1)\times \,r_{ta}$ around massive clusters might be the most promising sites for the search of the gas with overdensity $\lesssim 50$. We model spectral signatures of the WHIM in the X-ray band and identify two distinct regimes for the gas at temperatures below and above $\sim 10^6\,{\rm K}$. Using this model, we estimate the sensitivity of X-ray telescopes to the WHIM spectral signatures. We found that the WHIM structures are within reach of future high spectral resolution missions, provided that the low-density gas is not extremely metal-poor. We then consider the Coma cluster observed by SRG/eROSITA during the CalPV phase as an example of a nearby massive object. We found that beyond the central $r\sim 40'$ ($\sim 1100\,{\rm kpc}$) circle, where calibration uncertainties preclude clean separation of the extremely bright cluster emission from a possible softer component, the conservative upper limits are about an order of magnitude larger than the levels expected from simulations.