We study the evolution with redshift , from z \sim 5 to z = 0 , of the Lyman \alpha forest in a CDM model using numerical simulations including collisionless particles only .
The baryonic component is assumed to follow the dark matter distribution .

We distinguish between two populations of particles : Population P _ { s } traces the filamentary structures of the dark matter , evolves slowly with redshift and , for N ( \mbox { H { \sc i } } ) \ga 10 ^ { 14 } \mbox { cm$ { } ^ { -2 } $ } , dominates the number density of lines at z < 3 ; most of population P _ { u } is located in underdense regions and for the same column densities , disappears rapidly at high redshift .

We generate synthetic spectra from the simulation and show that the redshift evolution of the Lyman \alpha forest ( decrement , N ( \mbox { H { \sc i } } ) distribution ) is well reproduced over the whole redshift range for \Omega _ { b } h ^ { 2 } \sim 0.0125 and J _ { -21 } \approx 0.1 at z \sim 3 where J _ { -21 } is the UV background flux intensity in units of 10 ^ { -21 } erg cm ^ { -2 } s ^ { -1 } Hz ^ { -1 } sr ^ { -1 } .

The total number of lines with N ( \mbox { H { \sc i } } ) \ga 10 ^ { 12 } \mbox { cm$ { } ^ { -2 } $ } remains approximately constant from z \sim 4 to z = 1 .
At z \sim 0 , the number density of lines per unit redshift with \log N ( \mbox { H { \sc i } } ) > 12 , 13 , 14 is of the order of 400 , 100 , and 20 respectively .
Therefore , at low redshift , if most of the strong ( w _ { r } > 0.3 \mbox { \AA } ) lines are expected to be associated with galaxies , the bulk of the Lyman \alpha forest however should have lower equivalent width and should not be tightly correlated with galaxies .