Neutral-hydrogen column-density distribution ($f(N,z)={\partial }^2\mathcal{N}/(\partial N\partial z)$)

The neutral-hydrogen column-density distribution counts Lyman-alpha absorption systems per unit column density and redshift. A count in logarithmic column bins is $N\ln (10)f(N,z)$. At low forest columns an approximate power law $f\propto N^{-\beta }$ with $\beta$ near $1.5$ is useful, but the slope and normalization depend on column range and epoch. Lyman limit systems are optically thick at the hydrogen photoionization edge; damped Lyman-alpha systems have strong radiative damping wings. A distribution per absorption distance instead removes the geometrical path factor.