We make use of a 500 ks Chandra HRC-S/LETG spectrum of the blazar H 2356-309 , combined with a lower S/N ( 100 ks ) pilot LETG 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 , and to constrain its physical state and geometry .
Strong ( logN _ { OVII } \geq 10 ^ { 16 } cm ^ { -2 } ) OVII K \alpha absorption associated with a third LSS crossed by this line of sight ( the Sculptor Wall , SW ) , at z = 0.03 , has already been detected in a previous work .
Here we focus on two additional prominent filamentary LSSs along the same line of sight , at z=0.062 ( the Pisces-Cetus Supercluster , PCS ) and at z=0.128 ( the “ Farther Sculptor Wall ” , FSW ) .

The combined LETG spectrum has a S/N of \sim 11.6 - 12.6 per resolution element in the 20 - 25 Å , and an average 3 \sigma sensitivity to intervening OVII K \alpha absorption line equivalent widths of EW _ { OVII } { { } _ { > } \atop { } ^ { \sim } } 14 mÅ in the available redshift range ( z < 0.165 ) .
No statistically significant ( i.e . \geq 3 \sigma ) individual absorption is detected from any of the strong He- or H-like transitions of C , O and Ne ( the most abundant metals in gas with solar-like composition ) at the redshifts of the PCS and FSW structures , and down to the above EW thresholds .
However we are still able to constrain the physical and geometrical parameters of the putative absorbing gas associated with these structures , by performing joint spectral fit of various 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 \log { T } = 5.35 _ { -0.13 } ^ { +0.07 } ~ { } K and \log { N _ { H } } = ( 19.1 \pm 0.2 ) ~ { } cm ^ { -2 } possibly coexisting with a much hotter and statistically less significant phase with \log { T } = 6.9 ^ { +0.1 } _ { -0.8 } ~ { } K and \log { N _ { H } } = 20.1 ^ { +0.3 } _ { -1.7 } ~ { } cm ^ { -2 } ( 1 \sigma errors ) .
These two separate physical phases are identified through , and mainly constrained by , CV K \alpha ( warm phase ) and OVIII K \alpha ( hot phase ) absorption , with single line significances of 1.5 \sigma each .

For the second LSS , at z \simeq 0.128 , only one hot component is hinted in the data , through OVIII K \alpha ( 1.6 \sigma ) and NeIX K \alpha ( 1.2 \sigma ) .
For this system , we estimate \log { T } = 6.6 _ { -0.2 } ^ { +0.1 } ~ { } K and \log { N _ { H } } = 19.8 _ { -0.8 } ^ { +0.4 } ~ { } cm ^ { -2 } .

Our column density and temperature constraints on the warm-hot gaseous content of these two LSSs , combined with the measurements obtained for the hot gas permeating the SW , allow us to estimate the cumulative number density per unit redshifts of OVII WHIM absorbers at 3 different equivalent width thresholds of 0.4 m \AA , 7 m \AA and 25.8 m \AA .
This is consistent with expectations only at the very low end of EW thresholds , but exceed predictions at 7mÅ and 25.8 mÅ ( by more than 2 \sigma ) .
We also estimate the cosmological mass density of the WHIM based on the 4 absorbers we tentatively detect along this line of sight , obtaining \Omega _ { b } ^ { WHIM } = ( 0.021 ^ { +0.031 } _ { -0.018 } ) ( Z / Z _ { \odot } ) ^ { -1 } , consistent with the cosmological mass density of the intergalactic ’ missing baryons ’ only if we assume high metallicities ( Z \sim Z _ { \odot } ) .