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At an unattenuated inner boundary, define photon-weighted cross section moments $⟨{\sigma }^k⟩=Q^{-1}\int {\sigma }_{\nu }^k{L}_{\nu }/(h\nu )d\nu$. Across a layer $dr$, the inverse-square geometrical factor contributes $-2dr/r$, and expanding $e^{-n_H(1-\zeta ){\sigma }_{\nu }dr}$ contributes $-n_H(1-\zeta )dr⟨{\sigma }^2⟩/⟨\sigma ⟩$. Both are first order; a plane-parallel treatment can suppress the former only as a separate geometrical approximation. The second moment occurs because the photoionization rate weights an already attenuated spectrum by another cross section.

For pure hydrogen illuminated by an isotropic central source with hydrogen-ionizing photon production rate $Q$, an empty cavity of physical radius $r_{\mathrm{in}}$ gives photoionization rate $q=Q⟨\sigma ⟩/(4\pi r_{\mathrm{in}}^2)$ per neutral atom. Local photoionization equilibrium is $(1-\zeta )q={\alpha }_B{n}_H{\zeta }^2$, so $\Theta ={\alpha }_B{n}_H/q$ and the physical quadratic equation root is the displayed rationalized expression. If an unextinguished total recombination-line observed radiative flux $F_{\ell }$ has photon yield $y_{\ell }$, then $Q=4\pi D^2{F}_{\ell }{\lambda }_{\ell }/(y_{\ell }hc)$ and $\Theta =y_{\ell }{\alpha }_B{n}_{H}hc(r_{\mathrm{in}}/D{)}^2/(⟨\sigma ⟩F_{\ell }{\lambda }_{\ell })$. The ratio $r_{\mathrm{in}}/D$, rather than a product of lengths, makes $\Theta$ dimensionless.

A neutral-hydrogen column density $N$ traces total hydrogen column $N/x_{\mathrm{HI}}$ only after specifying its neutral fraction $x_{\mathrm{HI}}$. Integrating the latter against the distribution per absorption distance gives the comoving gas mass density in the displayed expression. The factor ${\mu }_H$ includes helium if desired. In photoionization equilibrium for optically thin, almost fully ionized hydrogen, $x_{\mathrm{HI}}\simeq \alpha (T)n_H/\Gamma$, so gas density, temperature, photoionization rate and absorber size are required. An observed distribution alone does not fix the total baryon inventory.

The Lyman-alpha forest is the multitude of foreground Lyman-alpha absorption features in a distant quasar's spectrum, at wavelengths shorter than its own redshifted Lyman-alpha emission. The quasar supplies a bright background; intervening neutral hydrogen atoms remove photons near their local Lyman-alpha line resonance. Many dips can therefore come from one atomic transition encountered at different redshifts, rather than from many different chemical elements. Some features are instead transitions of other ions, so a dip alone does not identify Lyman-alpha absorption. This makes faint foreground gas detectable through the light it removes, even when its own emission would be difficult to see. ^[ref].

Ignoring peculiar velocity for the moment, an absorber at cosmological redshift $z_a$ puts its Lyman-alpha line atThe rest wavelength is in the ultraviolet; at $z_a=2$ it appears at $3647.01\text{A˚}$ in the optical spectrum. In a toy sightline to a quasar at $z_Q=3$, absorbers at $z_a=2,2.35,2.7$ appear at $3647.01,4072.4945,4497.979\text{A˚}$, all shortward of the quasar's Lyman-alpha emission position $4862.68\text{A˚}$. The absorbing atoms see the selected photons at their own resonance; the observed dip's redshift belongs to the foreground gas. Peculiar velocity adds a Doppler shift, so the inferred $z_a$ is not a pure distance coordinate. The rest wavelength and resonance interpretation are given in Rauch (1998), section 1.

The diagram uses invented optical depths and 26 Doppler cores; three selected foreground regions are marked. Their drawn positions and sizes are schematic. The dashed comparison imposes a uniform optical depth $\tau =2$, giving transmission $e^{-2}\simeq 0.135$ throughout the chosen foreground band. It is an idealized smooth-absorption model, not a measured spectrum or a prediction of the actual intergalactic medium.

The smooth comparison explains the Gunn-Peterson effect: neutral hydrogen distributed throughout an expanding intergalactic medium can absorb continuously over a wavelength interval, because different parts of the path resonate with different observed wavelengths. The resolved Lyman-alpha forest instead has strong transmission fluctuations and distinguishable absorption lines. These need not represent isolated compact clouds: diffuse sheets and filaments in the cosmic web can generate peaks in optical depth along one line of sight. A continuously varying gas distribution can look like a collection of lines. ^[ref]. ^[ref].

An early observation makes the identification concrete. Lynds's 1971 paper analyzed six spectrograms of the quasar 4C 05.34, with emission redshift $2.877$, taken in spring 1970 using an image-tube spectrograph on Kitt Peak's 84-inch telescope. Five covered $3200$–$7500\text{A˚}$ with approximately $10\text{A˚}$ resolution; the most informative covered about $3500$–$6000\text{A˚}$ at $5\text{A˚}$ resolution. Its reproduced spectrogram shows the strong Lyman-alpha emission flanked mostly on the short-wavelength side by numerous absorption lines. Lynds tabulated their wavelengths and estimated strengths, identified five absorption redshift systems, and proposed that many remaining unidentified lines were the same Lyman-alpha line at additional redshifts. He explicitly warned about blending, an incomplete census of weak lines, and wavelength errors around $0.5$–$1\text{A˚}$. The result established a striking spectral pattern; the paper's proposed interpretation did not measure a separate physical cloud for every dip. ^[ref].

A contrasting observation demonstrates smooth, very strong absorption. Becker and collaborators (2001) used the Echelle Spectrograph and Imager on Keck II to observe the quasar SDSSp J103027.10+052455.0 at $z_Q=6.28$. Sky lines gave an instrumental full width at half maximum of about $1.8\text{A˚}$ near $8500\text{A˚}$, or spectral resolving power $R\simeq 4700$. After sky subtraction and flux calibration, the mean continuum-normalized transmission over $8450$–$8710\text{A˚}$ was $0.0038\pm 0.0026$, consistent with zero; this corresponds to absorption redshifts approximately $5.95$–$6.16$. They identified a Gunn-Peterson trough. Crucially, resonant Lyman-alpha absorption is so sensitive that even a small neutral fraction can make transmission undetectable: the trough alone does not establish a mostly neutral intergalactic medium or uniquely date reionization. ^[ref].

What a spectrum constrains is a radiative transfer problem. For an ideal foreground screen with negligible emission into the observed beam, the intrinsic continuum-normalized transmission isHere $F$ is spectral radiative flux, $F_{\mathrm{cont}}$ is its unabsorbed continuum, $\tau$ is optical depth, $n_{\mathrm{HI}}$ is neutral hydrogen number density, and $N_{\mathrm{HI}}$ is neutral hydrogen column density along proper path length $\ell$. A Voigt profile model constrains redshift, column density and a velocity width together. A nearly black line has lost much of its sensitivity to further increases in optical depth; blending and an uncertain continuum can also admit different fitted components. Finite spectral resolving power and noise limit which widths and weak features can be measured. ^[ref]. ^[ref].

For a purely thermal Doppler core, use the velocity convention $\tau (v)\propto \exp [-(v-v_0{)}^2/b^2]$. The Maxwell-Boltzmann velocity distribution then giveswhere $k_B$ is the Boltzmann constant, $T$ is temperature and $m_{\mathrm{H}}$ is a hydrogen atom's mass. This $b$ is $\sqrt{2}$ times the Gaussian distribution's standard deviation. Higher temperature spreads the same column density over more velocities. An instrument also broadens a line: locally, for a nearly flat continuum, the recorded transmission is approximately the convolution of $\mathcal{T}$ with the normalized instrumental response, not the exponential of a convolved optical depth. A Gaussian response with full width at half maximum $c/R$ has standard deviation $c/[R\sqrt{8\log 2}]$, where $c$ is the speed of light and $R$ the spectral resolving power.

The upper model fixes the integral $\int \tau (v)dv$, hence the neutral hydrogen column density for this transition, while changing temperature. A cooler line blurred by an instrument resembles a warmer intrinsic line. The lower model has two equal Doppler cores separated by $24\mathrm{km}{\mathrm{s}}^{-1}$: reducing spectral resolving power merges them into one apparent feature. Both panels use invented profiles, Gaussian instrumental responses, and no noise or damping wings. They illustrate an inverse problem, rather than measurements of temperature or a universal relation between a dip and a cloud.

Better observations provide the inputs to that inverse problem. Kim, Cristiani and D'Odorico (2001) analyzed three quasars with the VLT's UVES spectrograph. For HE2217–2818, Table 1 records $z_Q=2.413$ and $16200\mathrm{s}$ of exposure in the $3050$–$3860\text{A˚}$ setup in October 1999. Isolated thorium-argon lamp lines measured $R\simeq 45000$; the data were resampled into $0.05\text{A˚}$ bins, with signal-to-noise ratios about $45$–$50$ per pixel over $3550$–$4050\text{A˚}$. They fitted a local polynomial continuum, fitted Voigt profiles, and compared statistics of the transmitted flux. Their analysis excluded the region near the quasar to avoid the quasar proximity effect and used the interval between its Lyman-beta and Lyman-alpha emission to reduce confusion with other Lyman-series absorption. Features below $N_{\mathrm{HI}}=10^{12.5}\mathrm{c}{\mathrm{m}}^{-2}$ became confused with noise, setting their adopted lower column density limit. Figure 1 shows the measured spectrum, fitted profiles and residuals. This was a calibrated absorption measurement, with explicit selection limits. ^[ref].

Even after instrumental correction, a width need not be purely thermal Doppler broadening. Hubble flow across extended gas, peculiar velocity structure and unresolved blends can contribute. If an isolated line's intrinsic width obeys $b^2=b_{\mathrm{th}}^2+b_{\mathrm{nt}}^2$ with an independent Gaussian nonthermal contribution $b_{\mathrm{nt}}^2\ge 0$, then $T\le m_{\mathrm{H}}{b}^2/(2k_B)$; $b=25\mathrm{km}{\mathrm{s}}^{-1}$ gives a computed upper bound about $3.79\times 10^4\mathrm{K}$. A complex blend need not satisfy this single-component model. Noise can also bias the fitted widths of weak lines. This is why the thermal cutoff of Lyman-alpha line widths, calibrated against gas models, is more informative than assigning a temperature to every dip. ^[ref].

Finally, neutral hydrogen column density is not total gas mass density. In a pure hydrogen approximation, with gas optically thin to ionizing radiation and nearly fully ionized in photoionization equilibrium, the Lyman-alpha forest photoionization scaling giveswhere $n_{\mathrm{H}}$ counts all hydrogen nuclei, $\Gamma$ is the photoionization rate, $\alpha (T)$ is the radiative recombination coefficient and $x_{\mathrm{HI}}=n_{\mathrm{HI}}/n_{\mathrm{H}}$ is the neutral fraction. At fixed temperature and $\Gamma$, doubling $n_{\mathrm{H}}$ quadruples $n_{\mathrm{HI}}$; doubling $\Gamma$ halves it. These computed examples explain both the sensitivity to density contrast and the ambiguity: stronger absorption can mean denser gas, weaker ionizing radiation, a different temperature, or more path length. The Jeans-length ionization correction for the Lyman-alpha forest adds a size model rather than an independently measured depth. Optically thick Lyman limit systems and damped Lyman-alpha systems require different treatment. ^[ref]. ^[ref].

The beauty of the Lyman-alpha forest is that a spectrum converts otherwise faint foreground gas into a sequence of markers through cosmic time. With a model of radiative transfer, velocities, ionizing radiation and the temperature-density relation of photoionized intergalactic gas, its transmission statistics probe the cosmic web and gas thermal history. A single line of sight remains one narrow sample, not a unique three-dimensional density map. ^[ref].

Let $q_{\mathrm{in}}$ be the photoionization rate per neutral hydrogen atom just outside the empty cavity. With an isotropic central source and no attenuation inside that cavity,The cross section is zero below threshold; this permits the lower integration limit zero. Pure hydrogen with hydrogen ionization fraction $\zeta$ has $n_{\mathrm{HI}}=n_H(1-\zeta )$ and $n_e=n_p=n_H\zeta$. Local photoionization equilibrium is thereforeDefine the dimensionless quantity ${\Theta }_{\mathrm{phys}}={\alpha }_r{n}_H/q_{\mathrm{in}}$. Solving the resulting quadratic equation, ${\Theta }_{\mathrm{phys}}{\zeta }_{\mathrm{in}}^2+{\zeta }_{\mathrm{in}}-1=0$, gives the physical rootThe rationalized form avoids numerical cancellation for small ${\Theta }_{\mathrm{phys}}$. Strong illumination gives ${\zeta }_{\mathrm{in}}\simeq 1-{\Theta }_{\mathrm{phys}}$; weak illumination gives ${\zeta }_{\mathrm{in}}\simeq {\Theta }_{\mathrm{phys}}^{-1/2}$.

Eliminating $Q$ with the H-alpha luminosity, $Q=2.2(4\pi D^2{F}_{\mathrm{H}\alpha }){\lambda }_{\mathrm{H}\alpha }/(hc)$, yieldsThe printed definition of $\Theta$ is inconsistent with the defined radii. The source calls $r_{\ast }$ a physical stellar radius, but its factor $(2\pi r_{\ast }D{)}^2$ gives $\Theta$ units of length to the fourth power, rather than a dimensionless value. It also omits the physically necessary cavity radius $r_{\mathrm{in}}$. For a concrete counterexample, double $r_{\mathrm{in}}$ while holding the source, $D$, and $n_H$ fixed: the incident photoionization rate falls by four, so ${\Theta }_{\mathrm{phys}}$ rises by four; the printed expression does not change. The root has the requested form with the corrected ${\Theta }_{\mathrm{phys}}$ above. There is no supplied relation allowing $r_{\mathrm{in}}$ to be replaced by $r_{\ast }$.

Retain the consistent ${\Theta }_{\mathrm{phys}}$ of d. Across a sufficiently thin layer, the neutral hydrogen column density is $N_{\mathrm{HI}}=n_H(1-{\zeta }_{\mathrm{in}})\Delta r+O(\Delta r^2)$. The direct stellar continuum acquires optical depth ${\tau }_{\nu }=N_{\mathrm{HI}}{\sigma }_{\nu }$. Define the photon-weighted second momentExpanding the attenuating exponential in the photoionization rate givesThe second cross section moment appears because the radiation both undergoes absorption and is weighted by the photoionization cross section in computing the rate.

In a local plane-parallel approximation that suppresses geometrical dilution, setThen local photoionization equilibrium becomes ${\Theta }_{\mathrm{phys}}{\zeta }^2=\Gamma (1-\zeta )$, soThis gives the requested algebraic dependence, with the corrected definition of $\Theta$. Besides $\Delta r\ll r_{\mathrm{in}}$, linear attenuation requires $N_{\mathrm{HI}}{\sigma }_{\nu }\ll 1$ over the frequencies contributing appreciably to the rate. In particular $\Gamma$ must remain positive.

The source's $\Gamma$ neglects $-2\Delta r/r_{\mathrm{in}}$, which is also first order in layer thickness. A consistent spherical first-order answer uses ${\Gamma }_{\mathrm{sph}}=\Gamma -2\Delta r/r_{\mathrm{in}}$ in the same root. Neglecting dilution relative to absorption additionally requires $n_H(1-{\zeta }_{\mathrm{in}})⟨{\sigma }^2⟩/⟨\sigma ⟩\gg 2/r_{\mathrm{in}}$. As a concrete limiting test, with negligible absorption the source's $\Gamma =1$ predicts no change, while inverse-square dilution still decreases $q$ and thus $\zeta$.

Use absorption distance $X$ and its distribution $f_X$. Summing hydrogen columns per unit $X$ gives $\int N{f}_{X}dN$. The physical light-path length per $X$ isThe mean neutral-hydrogen number density is therefore $(H_0/c)(1+z{)}^3\int N{f}_{X}dN$. Divide by $(1+z{)}^3$ to obtain its comoving density, multiply by hydrogen atomic mass, and divide by today's mass critical density ${\rho }_{\mathrm{crit},0}=3H_0^2/(8\pi G)$. This derives the neutral gas density from an absorption distribution:If “neutral gas” includes helium, replace $m_H$ by ${\mu }_H{m}_H$, with a stated abundance correction approximately ${\mu }_H\simeq 1.3$. Equivalently, for the distribution per redshift the prefactor inside the integral is $m_{H}H(z)/[c{\rho }_{\mathrm{crit},0}(1+z{)}^2]$ multiplying $N{f}_z$. In the PDF's energy-density convention, both the mass density and critical density are multiplied by $c^2$, leaving their ratio unchanged. The usual quoted abundance uses comoving mass density relative to today's critical density; a ratio of physical density to the epoch's critical density instead multiplies this answer by $(1+z{)}^3{H}_0^2/H(z{)}^2$.

To infer total gas, one needs the neutral fraction $x_{\mathrm{HI}}=N_{\mathrm{HI}}/N_H$ for each absorber. The ionized gas density from an absorption distribution isThis is an ionization-corrected integral, not an inference available from the observed columns alone. For optically thin, almost fully ionized hydrogen, photoionization equilibrium gives $\Gamma n_{\mathrm{HI}}=\alpha (T)n_e{n}_{\mathrm{HII}}\simeq \alpha (T)n_H^2$. If the absorbing depth $L$ is specified, $N=\alpha (T)n_H^{2}L/\Gamma$ andThus photoionization rate, temperature and depth are all needed.

One useful physical closure sets $L$ to a local Jeans length, $L\propto (f_{g}T/n_H{)}^{1/2}$, with local gas fraction $f_g$. Solving $N\propto [\alpha (T)/\Gamma ](f_{g}T{)}^{1/2}{n}_H^{3/2}$ then gives the Jeans-length ionization correction for the Lyman-alpha forest:This illustrates why weak, highly ionized forest lines can dominate the total gas inventory even though damped Lyman-alpha systems dominate the neutral-column-weighted inventory. It is a model for locally confined optically thin gas; optically thick Lyman limit systems and damped Lyman-alpha systems require a separate radiative transfer treatment. The physical basis and observational application of this closure are developed in Schaye's absorption-based matter-density calculation.

Near the background quasar, the quasar proximity effect typically reduces the number and strength of Lyman-alpha forest features relative to regions farther along the sightline. It is an ionization effect, not evidence that the quasar lies in an empty region. At fixed gas density and temperature, photoionization equilibrium givesIf the unperturbed neutral-hydrogen column-density distribution is $A{N}^{-\beta }$, $\beta >1$, detecting a column above $N_{\min }$ now requires its unperturbed column to exceed $(1+\omega )N_{\min }$. Integrating that distribution shows that the number above threshold is reduced by $(1+\omega {)}^{1-\beta }$. This supplies an explicit model for the decline of line counts.

The known quasar luminosity estimates its photoionization rate at proper separation $R$:These expressions assume unattenuated, steady, isotropic quasar emission and a specified ionizing spectrum. Fitting the measured suppression as a function of separation determines the ratio ${\Gamma }_Q/{\Gamma }_{\mathrm{bg}}$ and hence constrains the cosmic ionizing background. A weaker background gives a larger region in which the quasar dominates. For fixed background spectral shape, this constrains its specific intensity near the Lyman limit system ionization edge, not an arbitrary spectral function. The proximity-effect estimator of the ionizing background is biased if the host gas is overdense, the quasar's past luminosity differs from its observed luminosity, its radiation is anisotropic, or systemic-redshift errors distort the inferred separation; absorption between source and gas also changes ${\Gamma }_Q$. These effects explain why the method needs environmental and source modeling, as discussed in the proximity-effect measurements.

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.

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.

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.

In photoionization equilibrium at fixed gas density and temperature, the extra quasar photoionization rate reduces neutral-hydrogen column density by the displayed factor. Its local rate is calculated from the quasar flux, while the isotropic background rate is ${\Gamma }_{\mathrm{bg}}=4\pi \int J_{\nu }{\sigma }_{\mathrm{HI}}(\nu )/(h\nu )d\nu$. Fitting the absorption suppression versus proper distance constrains the cosmic ionizing background. For an unperturbed column-density distribution proportional to $N^{-p}$ with $p>1$, the line abundance above a fixed threshold is reduced by $(1+\omega {)}^{1-p}$. Density enhancement, quasar age, anisotropy and redshift errors must be distinguished from this ionization effect.