There is a growing consensus that in the present universe most baryons reside in galaxy clusters and groups in the form of highly ionized gas at temperatures of 10 ^ { 6 } \sim 10 ^ { 8 } K. The H-like and He-like ions of the heavy elements can produce absorption features - the so-called “ X-ray Forest ” - in the X-ray spectrum of a background quasar . We investigate the distribution of the X-ray absorption lines produced by this gas under three different cosmological models : the standard CDM with \Omega _ { 0 } = 1 , a flat model with \Omega _ { 0 } = 0.3 and an open model with with \Omega _ { 0 } = 0.3 . We give a semi-analytic calculation of the X-ray forest distribution based on the Press-Schechter formalism , following Perna & Loeb ( 36 ) . We choose three ions ( O viii , Si xiv and Fe xxv ) and calculate the distribution functions , the number of absorbers along the line-of-sight ( LOS ) to a distant quasar vs. redshift and column density in a given ion . We find that significant differences in the evolution of the distribution functions among the three cosmological models . Using Monte Carlo simulations , we simulate the distribution of X-ray absorption lines for 10,000 random LOS . We find there are at least several O viii lines with column density higher than 10 ^ { 16 } { cm ^ { -2 } } . Finally we explore the possibility of detecting the X-ray forest with current and upcoming X-ray missions and we present an XMM RGS simulation of a representative quasar X-ray spectrum .