We make use of a 500ks Chandra HRC-S/LETG spectrum of the blazar H2356-309,
combined with a lower S/N spectrum of the same target, to search for the
presence of warm-hot absorbing gas associated with two Large-Scale Structures
(LSSs) crossed by this sightline at z=0.062 (the Pisces-Cetus Supercluster,
PCS) and at z=0.128 ("Farther Sculptor Wall", FSW). No statistically
significant (>=3sigma) individual absorption is detected from any of the strong
He- or H-like transitions of C, O and Ne at the redshifts of the structures.
However we are still able to constrain the physical and geometrical parameters
of the associated putative absorbing gas, by performing joint spectral fit of
marginal detections and upper limits of the strongest expected lines with our
self-consistent hybrid ionization WHIM spectral model. At the redshift of the
PCS we identify a warm phase with logT=5.35_-0.13^+0.07 K and log N_H
=19.1+/-0.2 cm^-2 possibly coexisting with a hotter and less significant phase
with logT=6.9^+0.1_-0.8 K and log N_H=20.1^+0.3_-1.7 cm^-2 (1sigma errors). For
the FSW we estimate logT=6.6_-0.2^+0.1 K and log N_H=19.8_-0.8^+0.4 cm^-2. Our
constraints allow us to estimate the cumulative number density per unit
redshifts of OVII WHIM absorbers. We also estimate the cosmological mass
density obtaining Omega_b(WHIM)=(0.021^+0.031_-0.018) (Z/Z_sun)^-1, consistent
with the mass density of the intergalactic 'missing baryons' for high
metallicities.Comment: 29 pages, 8 figures, 4 tables. Accepted for publication in Ap