Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 66 / 3

The Lyman-alpha forest is the collection of absorption lines blueward of a distant quasar's own Lyman-alpha line. Foreground neutral hydrogen removes photons at rest wavelength ${\lambda }_{\alpha }\simeq 1215.67$ angstroms, observed at $\lambda =(1+z){\lambda }_{\alpha }$ because of cosmological redshift. Gas at many different foreground redshifts therefore produces many lines. The weak forest traces diffuse, mostly ionized intergalactic medium, rather than a population of fully neutral clouds; much stronger, self-shielded absorbers are a separate regime. Madau's account of quasar absorption systems distinguishes these column-density regimes.

For highly ionized hydrogen in photoionization equilibrium, balance the photoionization rate against radiative recombination:Thus the neutral fraction is $n_{\mathrm{HI}}/n_H\simeq \alpha (T)n_H/{\Gamma }_{\mathrm{HI}}$ and the neutral column density across a proper path $L$ isFor an illustrative diffuse gas with $T\sim 10^4\mathrm{K}$, $\alpha \sim 4\times 10^{-13}\mathrm{c}{\mathrm{m}}^3{\mathrm{s}}^{-1}$, ${\Gamma }_{\mathrm{HI}}\sim 10^{-12}{\mathrm{s}}^{-1}$, and $n_H\sim 10^{-5}\mathrm{c}{\mathrm{m}}^{-3}$, the neutral fraction is only $4\times 10^{-6}$. A proper path of $100$ kiloparsecs then gives $N_{\mathrm{HI}}\sim 10^{13}\mathrm{c}{\mathrm{m}}^{-2}$, enough to produce a weak Lyman-alpha absorption feature even though the gas is almost entirely ionized.

This hydrogen number density is of the order of the cosmic mean at redshift a few. Since the weak forest is common along most high-redshift lines of sight, substantial path length passes through such diffuse gas. Much denser material would give much larger column density for comparable path lengths and ionizing flux. These arguments support near-mean or modestly overdense intergalactic medium as the origin of weak forest lines, not the claim that every absorber has exactly the cosmic mean density: path length, temperature, and ionizing background must also be constrained.

A radial null geodesic in the FLRW metric obeys $d\chi =c|dt|/a$. The redshift-time relation $dt=-dz/[(1+z)H(z)]$, with $a_0=1$, therefore givesThis is comoving radial distance. The proper distance in cosmology at the clouds' approximately common epoch is insteadThe distinction is essential when quoting a cloud separation. The approximation neglects the change of $H$ across the interval and assumes the observed spacing is dominated by cosmological redshift; peculiar velocity can perturb the inferred separation. It is the redshift version of proper absorber separation from a wavelength interval.

The quasar proximity effect is weaker Lyman-alpha forest absorption near a quasar's emission redshift. Extra photons from the quasar increase the photoionization rate and reduce the neutral fraction without necessarily reducing the total gas density.

For isotropic specific intensity $J_{\nu }$ and a steady, isotropically radiating quasar with spectral luminosity $L_{\nu }$, the background and local rates areHere $r$ is a proper separation small enough that attenuation and expansion across it can initially be neglected, and ${\sigma }_{\nu }$ is the hydrogen photoionization cross-section. At fixed density and temperature, photoionization equilibrium makes the neutral column density, and hence a given line's optical depth, scale asThe observed quasar luminosity supplies ${\Gamma }_Q(r)$; the distance at which the absorption changes significantly estimates where ${\Gamma }_Q\sim {\Gamma }_{\mathrm{bg}}$. Adopting a background spectral shape then converts the inferred rate to its specific intensity, often quoted near the hydrogen ionization energy threshold.

Important uncertainties include an enhanced gas density contrast around the host, the unobserved ionizing spectrum, anisotropic emission, source lifetime and variability, the systemic redshift and continuum placement, attenuation between source and absorber, and fluctuations between lines of sight. Higher gas density enhances recombination and can partially hide the effect, biasing a simple background estimate upward. Dall'Aglio, Wisotzki and Worseck's 2008 measurements illustrate the observed sightline scatter and the roles of variability and clustering.

The ionizing background is produced mainly by hot stars in star-forming galaxies and by accretion-powered quasars. The stellar contribution depends on star formation and the ionizing photon escape fraction; the quasar contribution depends on its evolving luminosity function. Radiative transfer through the intergalactic medium, including absorption, recombination emission, and cosmological redshifting, determines the radiation actually available for photoionization.

In the picture relevant to this 2009 paper, increasingly rare quasars cannot by themselves explain the high-redshift hydrogen ionizing field, so galaxies become especially important above $z\sim 3$. Harder quasar radiation is important for doubly ionizing helium around $z\sim 3$. Hydrogen reionization must be largely accomplished by $z\sim 6$. At lower redshift the evolution reflects declining source emissivity together with the increasing propagation distance of ionizing photons; near reionization, short absorption lengths and spatially uneven ionized regions prevent a simple uniform-background description. The background rate need not follow the quasar population alone: a roughly flat hydrogen photoionization rate over $2\lesssim z\lesssim 4$ is compatible with a changing mixture of galaxies and quasars. Faucher-Giguère and collaborators' 2009 background calculation and their preceding opacity analysis give this interpretation. These redshifts describe the broad historical picture, rather than precise universal transition times.