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Determining the column density of a cloud of chemical atoms from the attenuation of a probe beam of light; the measured image integrates through the cloud along the viewing direction.

A curve of growth describes the equivalent width $E(m)=\int (1-e^{-m{\kappa }_{\nu }})d\nu$ as a function of absorber mass path or column density. Its weak-line limit is $E\simeq mS$. Saturation makes growth sublinear: a pure Gaussian core gives $E\sim 2{\Delta }_D\sqrt{\log {\tau }_0}$, while saturated Lorentzian line shape wings give $E\sim 2\sqrt{mS{\Delta }_L}$. The spectral coordinate and coefficient convention must be consistent.

A damped Lyman-alpha system has a sufficiently high neutral-hydrogen column density that the Lorentzian wings of its Voigt profile are conspicuous. These wings come from the finite lifetime of the excited atomic state, not from unusually high gas temperature. Damped systems usually trace substantial neutral reservoirs associated with galaxies.

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 $f_X={\partial }^2\mathcal{N}/(\partial N\partial X)$ count neutral hydrogen absorbers per column density and absorption distance. The summed column per unit proper light-path length is the mean neutral number density. Since $d\ell =cdz/[(1+z)H(z)]$ and $dX/dz=H_0(1+z{)}^2/H(z)$, that density is $(H_0/c)(1+z{)}^3\int N{f}_{X}dN$. Dividing the proper mass density by $(1+z{)}^3$ gives the comoving density in the displayed formula. The denominator is today's critical density; including neutral helium requires a stated mass correction. For a distribution per redshift, replace $H_0{f}_X$ by $H(z)f_z/(1+z{)}^2$.

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.

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 $\zeta$.

With the spontaneous emission term suppressed, $j_{\nu }=0$. A uniform absorption coefficient over length $\ell$ gives directlyThe normalized specific intensity deficit is unchanged by conversion between wavelength and frequency, since both numerator and denominator acquire the same Jacobian determinant. The Boltzmann distribution and the Einstein coefficients give $n_2{B}_{21}/(n_1{B}_{12})=e^{-h{\nu }_{12}/(k_B{T}_e)}\ll 1$. Consequently, in the optically thin limit,Here $N_1=n_1\ell$ is the column density, and $\nu \simeq {\nu }_{12}$ has been used in slowly varying factors. This linear relation between equivalent width and column density fails once the central optical depth becomes appreciable.

For the unresolved Lyman-alpha line, the lower $1s$ and combined upper $2p$ state degeneracies are $g_1=2$ and $g_2=6$, so the Einstein coefficients giveThe measured equivalent width $W=7.0\times 10^{-12}\mathrm{m}$ then impliesAssume a central transverse line of sight through a uniform filament, so $\ell =d=0.5\mathrm{Mpc}=1.54\times 10^{22}\mathrm{m}$. Nearly all neutral hydrogen is in its ground state at this temperature, henceThe cylinder's length $L$ determines its total volume but does not enter a transverse column density. Geometry must be specified: an axial line of sight has $\ell =L$ and gives $n_{\mathrm{HI}}=4.2\times 10^{-7}{\mathrm{m}}^{-3}$ for the same measured equivalent width. Intermediate chords give intermediate path-dependent estimates. The numbers above treat $W$ as a rest-frame equivalent width; an observed value at redshift $z$ must first be divided by $1+z$.

Adopt the stipulated neutral fraction $f_{\mathrm{HI}}=10^{-6}$, independently of the equilibrium estimate in e. For pure hydrogen, $n_e=(1-f_{\mathrm{HI}})n_{\mathrm{HI}}/f_{\mathrm{HI}}\simeq 8.3{\mathrm{m}}^{-3}$. The thermal Sunyaev-Zeldovich effect measures integrated electron pressure:Using the same absorbing line of sight as in d,Thus one such filament is below the stated measurement sensitivity. If the number density inferred transversely is instead used to view the same cylinder end-on, its path is twenty times longer and $y=2.9\times 10^{-10}$, still below the stated sensitivity. Changing the assumed path of the original absorption measurement while keeping its column density fixed does not change its predicted $y$. The quoted neutral-fraction floor is a premise of this estimate, not a universal bound derivable from thermal equilibrium.

Define neutral-hydrogen column density along a ray by $N=\int n_{\mathrm{HI}}d\ell$, in atoms per unit area. The neutral-hydrogen column-density distribution per redshift isso $f_{z}dNdz$ is the expected number of absorbers in those intervals along one sightline. A logarithmic column bin instead has distribution $N\ln (10)f_z$; the bin convention must be specified.

Let $n_c$ be a constant comoving number density and ${\sigma }_p$ a constant physical interception area. The corresponding physical number density is $n_p=n_c(1+z{)}^3$. Proper light-path length satisfies $d\ell =c|dt|=cdz/[(1+z)H(z)]$. The expected intersection count, or first-order probability in infinitesimal $dz$, is thereforeThe Friedmann equation, the constant-equation-of-state density scaling for pressureless matter, and the present cosmological density parameters giveCombining them yields the absorber incidence and comoving number density relationHere the constant cross-section is physical, as required for this result; a fixed comoving cross-section would instead decrease in physical area as $(1+z{)}^{-2}$. It is convenient to define absorption distance by $dX/dz=(H_0/H(z))(1+z{)}^2$ and $f_X=f_z/(dX/dz)$, so a population with these non-evolution assumptions has constant incidence per $X$.

Typical weak Lyman-alpha forest absorbers have neutral hydrogen column densities roughly $10^{12}$–$10^{16}{\mathrm{cm}}^{-2}$, with stronger forest absorption extending towards $10^{17}{\mathrm{cm}}^{-2}$. Common Doppler broadening parameters are a few tens of ${\mathrm{km}\mathrm{s}}^{-1}$, consistent with gas at approximately $10^4$ K together with bulk velocity gradients. The thermal contribution isA line width is consequently a temperature upper bound if all other broadening is neglected, not a unique thermometer. The gas commonly lies in diffuse sheets and filaments at modest density contrast; only the strongest systems need substantial self-shielding or a galactic environment.

For optically thin gas illuminated by the cosmic ionizing background, photoionization equilibrium requireswhere the last approximation uses almost complete hydrogen ionization. For example, $n_H=10^{-5}{\mathrm{cm}}^{-3}$, $\alpha \simeq 4\times 10^{-13}{\mathrm{cm}}^3{\mathrm{s}}^{-1}$ and ${\Gamma }_{\mathrm{HI}}=10^{-12}{\mathrm{s}}^{-1}$ give $x_{\mathrm{HI}}\simeq 4\times 10^{-6}$. A $100$ kpc path then has $N_{\mathrm{HI}}\sim 10^{13}{\mathrm{cm}}^{-2}$ despite its tiny neutral fraction. This illustrates the Lyman-alpha forest photoionization scaling: low density makes the two-body radiative recombination rate weak relative to the one-body photoionization rate, but a long path and a strong resonance still produce measurable absorption.

Independent information is important because a column density alone is not a volume density. Similar absorption in close pairs of quasars indicates substantial transverse coherence, while an ionizing optical depth $N_{\mathrm{HI}}{\sigma }_{912}\ll 1$ shows that these weak systems do not shield themselves from the background. The continuum-shielding threshold is $1/{\sigma }_{912}\simeq 1.6\times 10^{17}{\mathrm{cm}}^{-2}$. Together with the observed line widths and ionization balance, these facts favor extended, low-density, highly ionized gas over small cold neutral clouds. The physical interpretation and characteristic scales are consistent with Rauch's forest analysis and the measured column and line-width distributions.

For a fixed column density or rest-equivalent-width threshold, the number of ordinary Lyman-alpha forest features per unit redshift generally increases towards higher redshift. Over much of $z\sim 1$–$4$, a rough description is $d\mathcal{N}/dz\propto (1+z{)}^{\gamma }$ with an index of order $2$–$3$, dependent on selection; the evolution is much slower at low redshift. Near reionization, blending and widespread opacity make a decomposition into individually detected lines increasingly inappropriate.

First separate physical evolution from geometrical path length. For comoving absorber density $n_c$ and proper interception area ${\sigma }_p$, absorber incidence and comoving number density givesEven a non-evolving population is not constant per unit redshift. Using absorption distance removes this particular geometrical effect.

The remaining redshift evolution of the Lyman-alpha forest reflects several competing processes. Cosmic expansion dilutes the gas; at fixed overdensity, photoionization equilibrium gives $n_{\mathrm{HI}}\propto \alpha (T)(1+z{)}^6/\Gamma$, so dilution can move many features below a fixed threshold. The cosmic ionizing background evolves as quasars, stellar sources and ionizing-photon absorption evolve; a decline of its intensity at late times partly offsets dilution. Photoheating changes temperature and radiative recombination, while gravitational instability redistributes gas into denser structures and hot shocked phases and changes its velocity field. Thus evolving line counts are not a direct count of conserved clouds; the density, thermal state, radiation field, geometry and observational selection must all be accounted for. The high-redshift increase and its physical interpretation are documented in the forest observations and models.

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.

In an Einstein-de Sitter universe, $E(z)=(1+z{)}^{3/2}$. Under the non-evolution assumptions of the previous part, the incidence above a fixed neutral-hydrogen column density threshold consequently scales as $(1+z{)}^{1/2}$. Between the two specified epochs,The higher-redshift rate would be about $41\%$ larger.

Observed Lyman-alpha forest incidence above substantial column thresholds rises much more rapidly. For example, the survey at Kim and collaborators' measurements finds an exponent $2.19\pm 0.27$ for $10^{13.64}\lesssim N_{\mathrm{HI}}\lesssim 10^{16}{\mathrm{cm}}^{-2}$ over this redshift range, giving a representative increase $2^{2.19}\simeq 4.56$, rather than $\sqrt{2}$. The empirical exponent is threshold dependent. The forest therefore cannot be treated as a conserved population of identical fixed-size neutral clouds: gas expansion, structure evolution and the cosmic ionizing background change neutral fractions, absorption strengths and the effective area above the threshold. In photoionization equilibrium, $n_{\mathrm{HI}}\Gamma \simeq \alpha (T)n_e{n}_p$, so changing density or ionization rate changes a counted neutral column even without creating or destroying its baryons.

The neutral-hydrogen column density is $N_{\mathrm{HI}}=\int n_{\mathrm{HI}}d\ell$, with units ${\mathrm{cm}}^{-2}$. Define the neutral-hydrogen column-density distribution byHere $\mathcal{N}$ counts absorption systems along a quasar sightline. A histogram per logarithmic column interval instead measures $N\ln (10)f(N,z)$; forgetting this factor changes the inferred slope. Another common convention uses the absorption distance $X$, with $dX/dz=H_0(1+z{)}^2/H(z)$, and $f_X=f/(dX/dz)$.

At a fixed epoch, weak Lyman-alpha forest lines have an approximately falling power-law distribution, $f(N,z)\simeq A(z)N^{-\beta }$ with $\beta \simeq 1.4$–$1.5$ over representative low-column ranges. Stronger absorbers deviate from one universal slope, and the normalization and fitted slopes depend on epoch and sample. The low-column fit is measured, for example, in Kim and collaborators' forest survey. The sketch distinguishes that empirical low-column trend from an illustrative higher-column continuation.

Ordinary Lyman-alpha forest lines cover roughly $10^{12}\lesssim N_{\mathrm{HI}}\lesssim 10^{17.2}{\mathrm{cm}}^{-2}$. They are numerous resonant Lyman-alpha absorption features from intervening gas at different cosmological redshifts, collectively resembling a forest in a quasar spectrum. Many are optically thin at the ionization edge even when their line centers are saturated.

A Lyman limit system becomes optically thick at the $91.2\mathrm{nm}$ hydrogen ionization edge. With ${\sigma }_{912}\simeq 6.3\times 10^{-18}{\mathrm{cm}}^2$, ${\tau }_{912}=N_{\mathrm{HI}}{\sigma }_{912}\gtrsim 1$ requiresIts spectrum consequently has a strong continuum decrement below the redshifted Lyman edge. In a classification separating the highest columns, Lyman limit systems extend up to the damped-system boundary.

A damped Lyman-alpha system conventionally has $N_{\mathrm{HI}}\gtrsim 2\times 10^{20}{\mathrm{cm}}^{-2}$. Its Lyman-alpha absorption is sufficiently strong that the Lorentzian wings of the Voigt profile are prominent. “Damped” refers to the finite radiative lifetime of the atomic excited state, not an inference that the gas has a high temperature. These systems trace substantial neutral reservoirs, usually associated with galaxies.

Use high-resolution quasar spectra to fit Voigt profiles to Lyman-alpha forest lines near $z=3$, where the transition appears near $121.6(1+3)\simeq 486.4\mathrm{nm}$. Each fit estimates the neutral-hydrogen column density and the velocity-width parameter $b$. The one-dimensional Maxwell-Boltzmann velocity distribution has variance $k_{B}T/m_{\mathrm{H}}$; consequently the thermal optical-depth profile is proportional to $\exp [-(v-v_0{)}^2/b^2]$, with the thermal cutoff of Lyman-alpha line widthsFor example, a purely thermal $b=20\mathrm{km}{\mathrm{s}}^{-1}$ corresponds to $T\simeq 2.4\times 10^4\mathrm{K}$.

An observed broad line is not by itself a thermometer: peculiar velocities, bulk gradients, blending and instrumental resolution also broaden it. For independent Gaussian contributions, $b^2=b_{\mathrm{th}}^2+b_{\mathrm{nonthermal}}^2+b_{\mathrm{inst}}^2$. After modeling the instrumental response, an uncorrected total width gives an upper bound on the thermal temperature, not a lower bound. The narrowest lines at each column density are least affected by additional broadening. Their lower envelope in the $b$–$N_{\mathrm{HI}}$ plane constrains the temperature-density relation of photoionized intergalactic gas, often written $T=T_0(1+\delta {)}^{\gamma -1}$.

Calibrate the conversion between column density, density and width with hydrodynamic models incorporating the cosmic ionizing background, pressure smoothing, resolution and selection. It estimates $T_0$ and the slope rather than assigning one temperature to every absorber. Where aligned metal lines trace the same gas, comparing species of different atomic mass can separately estimate a common nonthermal contribution because thermal $b^2$ scales as $m^{-1}$. The forest lower-envelope method and its numerical calibration are developed in Schaye and collaborators' temperature analysis.

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.