  The warm-hot intergalactic medium (WHIM) at temperatures 1E5-1E7 K is
believed to contain 30-50% of the baryons in the local universe. However, all
current X-ray detections of the WHIM at redshifts z>0 are of low statistical
significance (<=3sigma) and/or controversial. In this work, we aim to establish
the detection limits of current X-ray observatories and explore requirements
for next-generation X-ray telescopes for studying the WHIM through X-ray
absorption lines. We analyze all available grating observations of Mrk 421 and
obtain spectra with signal-to-noise ratio (S/N) of \sim90 and 190 per 50 mA
spectral bin from Chandra and XMM observations, respectively. Although these
spectra are two of the best ever collected with Chandra and XMM, we cannot
confirm the two WHIM systems reported by Nicastro et al. in 2005. Our bootstrap
simulations indicate that spectra with such high S/N cannot constrain the WHIM
with OVII column densities N(OVII)\sim1e15 cm^{-2} (corresponding to an
equivalent widths of 2.5 mA for a Doppler velocity of 50 km s^{-1}) at >=3sigma
significance level. The simulation results also suggest that it would take >60
Ms for Chandra and 140 Ms for XMM to measure the N(OVII) at >=4sigma from a
spectrum of a background QSO with flux of \sim0.2 mCrab (1 Crab = 2E-8 erg
s^{-1} cm^{-2} at 0.5-2 keV). Future X-ray pectrographs need to be equipped
with spectral resolution R \sim 4000 and effective area A>=100 cm^2 to
accomplish the similar constraints with an exposure time of \sim2 Ms and would
require \sim11 Ms to survey the 15 QSOs with flux \sim0.2 mCrab along which
clear intergalactic OVI absorbers have been detected.
