We assess the possibility of detecting the warm-hot intergalactic medium in emission and characterizing its physical conditions and spatial distribution through spatially resolved X-ray spectroscopy, in the framework of the recently proposed DIOS, EDGE, Xenia, and ORIGIN missions, all of which are equipped with microcalorimeter-based detectors. For this purpose, we analyze a large set of mock emission spectra, extracted from a cosmological hydrodynamical simulation. These mock X-ray spectra are searched for emission features showing both the O VII Kα triplet and O VIII Lyα line, which constitute a typical signature of the warm-hot gas. Our analysis shows that 1 Ms long exposures and energy resolution of 2.5 eV will allow us to detect about 400 such features per deg2 with a significance >=5σ and reveals that these emission systems are typically associated with density ~100 above the mean. The temperature can be estimated from the line ratio with a precision of ~20%. The combined effect of contamination from other lines, variation in the level of the continuum, and degradation of the energy resolution reduces these estimates. Yet, with an energy resolution of 7 eV and all these effects taken into account, one still expects about 160 detections per deg2. These line systems are sufficient for tracing the spatial distribution of the line-emitting gas, which constitute an additional information, independent from line statistics, to constrain the poorly known cosmic chemical enrichment history and the stellar feedback processes.
