Evolution of the Ly$\alpha$ forest: a consistent picture

We study the evolution with redshift of the number density of Ly$\alpha$ clouds in a CDM model using numerical simulations including photo-ionization and cooling of the baryonic component. The photo-ionizing flux is consistently taken as proportional to the rate at which material cools below 5000~K in the simulation. The number density of lines with $N$(HI)~$>$~10${}^{14}$~cm${}^{-2}$ given by the simulation can be approximated as d$n$/d$z$~$\sim$~16.8(1+$z$)${}^{0.8}$+0.687(1+z)${}^{2.5}$ which fits well the data at any redshift. The number of weaker lines is reproduced within a factor of two and does not decrease with time as fast as the number of strong lines. For $N$(HI)~$>$~10${}^{13}$~cm${}^{-2}$, we find d$n$/d$z$~$\propto$~(1+$z$)${}^{0.7}$. \par At $z$~=~0, the two-point correlation function for Lyman$\alpha$ clouds with $N$(HI)~$>$~10${}^{13}$ and $>$~10${}^{14}$~cm${}^{-2}$ has a slope $\sim$~-1.75 comparable to the slope of the two point correlation function for the galaxy distribution. The signal is higher for stronger lines for separations smaller than 1~Mpc. Part of the Lyman$\alpha$ forest is indeed associated with galaxies however the overall picture is that Ly$\alpha$ absorption lines originate in the warm gas that traces the potential wells of filamentary structures defined by the dark matter.