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The cosmic ionizing background is the diffuse radiation field capable of photoionization of intergalactic gas. Its specific intensity depends on source luminosity functions, ionizing photon escape fractions, cosmological redshifting, and radiative transfer through absorbers. Quasars and star formation supply major contributions with different spectral shapes and evolution.

The Lyman-alpha forest is the multitude of Lyman-alpha absorption lines blueward of the broad emission line in a distant quasar spectrum. Neutral hydrogen along the line of sight absorbs at its rest wavelength ${\lambda }_{\alpha }\simeq 1216\mathring{\mathrm{A}}$, so each absorber appears at ${\lambda }_{\mathrm{obs}}=(1+z_c){\lambda }_{\alpha }$. Much of the ordinary forest traces modest overdensities in the mostly ionized intergalactic medium, rather than compact objects made primarily of neutral gas. Strong self-shielded systems form a distinct absorption regime. Line strengths constrain neutral-hydrogen column density, while line widths contain thermal and velocity information.

The redshift evolution of the Lyman-alpha forest is normally described, for a fixed line selection, by an incidence $d\mathcal{N}/dz=A(1+z{)}^{\gamma }$. At redshifts of a few the line incidence and mean absorption increase rapidly with redshift; typical selected populations have $\gamma$ of order $2$–$3$, not a universal exponent. Low-redshift evolution is appreciably flatter. Geometry alone cannot explain the full change: for a fixed comoving abundance and fixed proper cross section one obtains $d\mathcal{N}/dz\propto (1+z{)}^2/H(z)$, only $(1+z{)}^{1/2}$ in matter domination. Density evolution, thermal evolution and the cosmic ionizing background change how many structures exceed the chosen absorption threshold. In optically thin photoionization equilibrium,so for roughly fixed overdensity the rising cosmic density alone makes $n_{\mathrm{HI}}$ scale approximately as $\alpha (T)(1+z{)}^6/{\Gamma }_{\mathrm{HI}}$. At low redshift, the declining ionizing radiation partly offsets the decrease in gas density. Near the end of reionization the forest becomes very opaque and the simple individual-line description ceases to suffice.

The quasar proximity effect is the statistically reduced absorption near the quasar's own redshift, contrary to the general increase of absorption towards higher redshift. It results from additional ionizing radiation from the nearby quasar. At fixed density and temperature the local photoionization rate becomes ${\Gamma }_{\mathrm{bg}}+{\Gamma }_Q$. Accordinglywhere $N_{\mathrm{HI}}^{(0)}$ is the value with the background alone. Lines weaken and some fall below the detection threshold; the absorbers need not have been physically removed. This gives the proximity-effect estimator of the ionizing background.

For an approximately isotropic quasar and a nearby cloud at proper distance $d$, calculate its flux from $F_{\nu ,Q}\simeq L_{\nu ,Q}/(4\pi d^2)$, allowing for absorption and relative redshift if necessary. The quasar and isotropic background ionization rates arewhere ${\nu }_0$ is the hydrogen ionization threshold, $h_P$ is the Planck constant and $J_{\nu }$ is the background specific intensity. Establish the ordinary absorption statistics far from quasars, then fit their reduction as a function of distance and known quasar luminosity. The scale at which $\omega \sim 1$ gives the background rate; an assumed spectral shape converts this to an intensity near the ionization threshold. For matching spectral shapes, $\omega \simeq L_{{\nu }_0,Q}/(16{\pi }^2{d}^2{J}_{{\nu }_0})$. A column-density distribution proportional to $N^{-p}$ gives a line-incidence suppression $(1+\omega {)}^{1-p}$ above a fixed threshold, by shifting that threshold to $(1+\omega )N_{\min }$ in the unperturbed distribution. This is the physical measurement principle established in adsabs.harvard.edu/pdf/1988ApJ...327..570B .

The inference is statistical. Quasars live in overdense environments, which enhance absorption and can mask extra ionization. Emission-redshift errors smear the effect, as part (iii) demonstrates. Finite source lifetimes, anisotropic emission, variability and departures from ionization equilibrium can also alter the radial profile. These effects must be modeled rather than attributed entirely to the cosmic ionizing background.

The background is produced mainly by ionizing radiation from accretion-powered quasars and hot massive stars in star-forming galaxies. Their relative importance changes with redshift. Quasars have hard spectra and can ionize helium as well as hydrogen; stellar photons depend strongly on their ionizing photon escape fraction from galaxies. Integrating source luminosities over the luminosity function and propagating radiation through absorbing gas gives the background. At high redshift the observed quasar population alone need not supply the required hydrogen ionization, so star-forming galaxies can be essential, as discussed in arxiv.org/abs/0807.4177 . Diffuse recombination radiation can contribute, but a residual primordial radiation field is not an adequate ionizing source.

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.

A star-forming galaxy is a galaxy with ongoing star formation. Its hot young stars can supply hydrogen-ionizing radiation to the intergalactic medium, with the contribution controlled by the ionizing photon escape fraction.