= Solution
= Solution
{id=a/solution}
{synonym}
A useful description of the low-column <Lyman-alpha forest> at $z\simeq3$ is a <power law> $f_X(N,z)=A(z)N^{-\beta}$, with $\beta$ around $1.4$–$1.5$ over the ordinary unsaturated forest range. The same column dependence holds for $f_z$ at a fixed redshift. It is not a universal unbroken <power law>: saturation, increased neutral fraction and changing absorber populations produce breaks. <Lyman limit systems> become optically thick at $N\simeq10^{17.2}\,\mathrm{cm}^{-2}$ because $N\sigma_{912}\simeq1$ with $\sigma_{912}\simeq6.3\times10^{-18}\,\mathrm{cm}^2$. <Damped Lyman-alpha systems> have $N\gtrsim2\times10^{20}\,\mathrm{cm}^{-2}$ and strong radiative damping wings. They are rare in number but contain substantial neutral mass. The highest-column distribution steepens or cuts off rather than extending indefinitely.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2004/iii/paper-67-absorber-columns.png]
{title=Schematic neutral-hydrogen column distribution showing the forest, optically thick Lyman-limit systems and damped systems}
This sketch shows representative shapes and transition scales, not an observational fit; its ordinate is distribution per unit column, not counts per logarithmic bin. In an <Einstein-de Sitter universe>, $H=H_0(1+z)^{3/2}$, so a fixed comoving population with fixed physical cross-section above a fixed column threshold has $d\mathcal N/dz\propto(1+z)^{1/2}$. Thus
$$
\boxed{\frac{(d\mathcal N/dz)_{z=4}}{(d\mathcal N/dz)_{z=1.5}}
=\left(\frac5{2.5}\right)^{1/2}=\sqrt2\simeq1.414.}
$$
The observed <redshift evolution of the Lyman-alpha forest> is much stronger for commonly selected forest lines. For example, https://arxiv.org/abs/astro-ph/0101005[Kim, Cristiani and D'Odorico's absorption survey] finds an index about $2.19$ in $d\mathcal N/dz\propto(1+z)^\gamma$ for $10^{13.64}<N/\mathrm{cm}^{-2}<10^{16}$ over $1.5<z<4$, corresponding to an increase by approximately $2^{2.19}\simeq4.56$. This number is selection-dependent. Expansion changes the gas <density>, and the evolving <cosmic ionizing background> changes its neutral fraction through <photoionization equilibrium>, moving systems across a fixed neutral-column threshold. Structure also evolves. A non-evolving population with fixed physical cross-section is therefore not an adequate physical model of the observed forest.
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