Absorption studies of the most diffuse gas in the Large Scale Structure
  As the Universe evolves, it develops a web of filamentary structure of
matter. This cosmic web is filled with gas, with the most diffuse gas lying in
the intergalactic regions. At low redshift, the gas is predominantly warm-hot,
and one of its best tracers is X-ray absorption in sightlines to background
quasars. In this Chapter, we present the theoretical background for the
formation of the warm-hot intergalactic medium (WHIM) and present the physical
properties of the WHIM from cosmological hydro-dynamical simulations. We
discuss the feasibility of detecting the WHIM with X-ray absorption lines, with
high-resolution and high signal-to-noise spectra. We present detailed
discussion of observing techniques, including the WHIM ionization balance,
observable lines, the curve of growth, and the diagnostics using the X-ray
lines. We present the current efforts of detecting the WHIM with gratings
on-board Chandra and XMM-Newton observatories. We discuss the criticality of
WHIM detections reported in literature, where robust detections are likely from
the circumgalactic medium of intervening galaxies, or intra-group medium,
rather than truly diffuse gas in the intergalactic medium. Secure detections of
the most diffuse gas in the low redshift large scale structure may have to
await next generation of X-ray telescopes. We end our Chapter with the
discussion of future missions carrying dispersive and non-dispersive
spectrometers. We present figure-of-merit parameters for line detectibility as
well as for the number of WHIM systems that can be detected with future
missions. These will define our ability to account for the missing low-redshift
baryons and to understand the evolution of the Universe over half of its life.
