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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.

Gravitational lensing is the deflection and focusing of light by spacetime curvature along the line of sight.

A useful first model is a point-mass gravitational lens: one displaced source gives two images of a point-mass lens, while exact alignment gives an Einstein ring. The Einstein angular radius sets the angular scale, and image parity in gravitational lensing explains why the two images have different handedness. The spectroscopic test of the Twin Quasar lens shows how optical spectra can test whether two apparent quasars represent one source.

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].

The Lyman-alpha forest is the large collection of narrow Lyman-alpha absorption features in the spectrum of a distant quasar. Neutral hydrogen in foreground gas absorbs at its own redshifted $1s\to 2p$ resonance, so an absorber at $z_{\mathrm{abs}}$ places its feature atOrdinary intervening absorbers have $z_{\mathrm{abs}}<z_Q$, putting their Lyman-alpha lines blueward of the quasar's own broad Lyman-alpha emission. Many different absorbing patches along the same line of sight make the forest; they are not different transitions of one cloud. At sufficiently short wavelengths higher Lyman-series transitions can overlap the forest, so an actual analysis identifies these blends rather than treating every dip as isolated Ly-alpha.

The upper panel is an original schematic spectrum for $z_Q=3$, with absorption features at smaller cosmological redshifts and a broad emission feature near $4863$ Å. The lower panel anticipates the column-density discussion: both curves are teaching sketches, not measured spectra or fitted survey distributions. The ordinary Lyman-alpha forest mainly traces the fluctuating, photoionized intergalactic medium, rather than a collection of identical isolated, neutral clouds.

Hydrogen is the most abundant element, and its Lyman-alpha line is a strong resonance from the ground state. Very small neutral fractions in an otherwise ionized intergalactic medium therefore produce detectable Lyman-alpha absorption. A long quasar sightline samples a great many modest density enhancements in the cosmic gas, with a much larger covering area than the compact high-column regions that produce damped Lyman-alpha systems.

Heavy-element absorption additionally requires metal enrichment and enough ions in the relevant ionization state. Many weak Lyman-alpha forest absorbers contain too few such ions to produce detectable metal lines at the same sensitivity, while high-column Lyman limit systems and damped Lyman-alpha systems are intrinsically rarer. Thus abundant hydrogen, a sensitive resonance and the large interception area of diffuse gas together explain the forest's numerical dominance. The comparison is between absorber classes at specified observational sensitivity; it does not mean that a metal-rich system has only one spectral transition.

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.

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 primordial light-element abundance determination uses minimally processed material and estimates its residual processing rather than directly observing the universe at a few minutes old.

For deuterium, high-redshift, low-metallicity quasar absorption systems resolve isotope-shifted D I lines from the H I Lyman series. The deuterium feature is displaced by about $82{\mathrm{km}\mathrm{s}}^{-1}$ from hydrogen; fitting several transitions and the velocity components distinguishes genuine deuterium from an unrelated hydrogen blend. The ratio of the inferred columns estimates D/H after the ionization and component structure are checked. Low metallicity minimizes stellar destruction, giving a representative primordial ratio of roughly $(2.5\text{–}3)\times 10^{-5}$.

For helium-4, helium and hydrogen recombination-emission lines in metal-poor H II regions of nearby dwarf galaxies give a mass fraction. Several lines constrain temperature, electron number density and corrections for collisional excitation, ionization structure, radiative transfer and underlying stellar absorption. Extrapolation of helium enrichment against metallicity towards zero metallicity estimates $Y_p$, approximately $0.24$–$0.25$, corresponding to helium/hydrogen by number near $0.08$.

For helium-3, the approximately $8.665$ GHz hyperfine line of ${}^3{\mathrm{He}}^{+}$ in Galactic H II regions probes the abundance, with ionization and nebular structure corrections. Stellar production and destruction make its primordial value hard to extract; a characteristic abundance near $10^{-5}$ relative to hydrogen is compatible with standard nucleosynthesis but is not a comparably clean primordial measurement.

For lithium-7, the approximately $6708$ Å Li I absorption doublet in warm, old, metal-poor halo stars gives a photospheric abundance after stellar-atmosphere and effective-temperature corrections. The Spite plateau suggests an initial number ratio of order $(1\text{–}2)\times 10^{-10}$ if depletion is small; transport below the photosphere, nuclear destruction and atmosphere systematics can lower the measured value relative to the birth abundance. Standard Big Bang nucleosynthesis at the deuterium- and microwave-background-favored baryon density instead predicts several $10^{-10}$, roughly $(4\text{–}5)\times 10^{-10}$ in exam-era calculations. This discrepancy is the cosmological lithium problem, not a measurement of an independently adjustable primordial lithium parameter.

The approximate values deliberately represent exam-era orders of magnitude rather than a claim of current precision; the contemporary abundance assessment discusses the inference and tensions.

The Lyman-alpha forest supplies many relatively narrow absorption features along each bright distant quasar sightline. These arise in intervening intergalactic hydrogen rather than in the quasar's rapidly varying emission region. A repeat high-resolution spectrum can be fitted against the same absorption template, and the information from many lines and many sightlines can be combined to estimate a common spectroscopic velocity drift. Diffuse intergalactic absorbers also generally have smaller peculiar accelerations than gas close to galactic nuclei, although their peculiar accelerations and changing line profiles still require control.

The preceding calculation requires a relative wavelength displacement $|\Delta \lambda |/\lambda \simeq 6\times 10^{-10}$ over a decade. This is much smaller than one instrumental resolution element. Accurate line centroids can nevertheless be measured to a fraction of a resolution element when the signal-to-noise ratio and template information are sufficient. The real obstacles are long-term wavelength calibration, instrumental drift, sufficient photon counts and corrections for observer motion, along with astrophysical variability. Statistical averaging reduces independent errors; a common calibration error remains common to all lines. The Sandage-Loeb test therefore needs large collecting area, stable high-resolution spectroscopy and a long time baseline. The original absorption-forest strategy is described in Loeb's 1998 proposal.

The measurement is feasible in principle but demands exceptional long-term stability. It directly probes the change of redshift with observer time, giving $H_0(1+z)-H(z)$ rather than a distance inferred from a standard candle. Measuring its sign and redshift dependence would distinguish the expansion histories in part (ii).

For a radial null geodesic in a spatially flat universe, the comoving radial distance increment is $d\chi =cdz/H(z)$. If the cloud and quasar redshifts differ only slightly, take $H$ at their common approximate epoch $z_{\ast }$, givingMultiplying by the scale factor at that epoch gives the local proper distance in cosmology relevant to the quasar's radiation flux:If distance is instead expressed in comoving units, it is $(1+z_{\ast })d_{cq}$. Equivalently the small rest-frame velocity difference is $\delta v=c\delta z/(1+z_{\ast })$, and $d_{cq}\simeq \delta v/H(z_{\ast })$. This quasar-cloud redshift separation assumes the redshift difference is due to homogeneous expansion, with negligible peculiar velocities. Since the two observations lie on a light cone, the proper separation is understood as the short separation on a nearly common cosmic-time slice; specifying the epoch matters at higher order.

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 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.

A primordial light-element abundance determination seeks gas or stars with little subsequent chemical processing. The three abundances require different observations and corrections.

For helium, measure helium and hydrogen recombination emission in metal-poor extragalactic ionized regions. Correct line ratios for electron temperature, density, extinction, collisional excitation, underlying stellar absorption and unobserved ionization stages. Convert the number ratio $y=n_{\mathrm{He}}/n_{\mathrm{H}}$ to helium mass fraction $Y\simeq 4y/(1+4y)$, allowing for heavy elements when necessary. Since stars produce helium and metals, extrapolate a sample toward zero gas-phase metallicity to infer $Y_p$. The sensitivity to atomic data and these corrections is illustrated by Peimbert and collaborators' helium determination.

For deuterium, fit isotope-shifted hydrogen and deuterium absorption lines in relatively simple, metal-poor quasar absorption systems. The deuterium feature is displaced by about $-82\mathrm{km}{\mathrm{s}}^{-1}$ from the corresponding hydrogen line. Multiple transitions and velocity components help distinguish deuterium from unrelated hydrogen contamination. The fitted neutral-column ratio estimates D/H in predominantly neutral gas. Since stellar processing destroys deuterium, low-metallicity, minimally processed absorbers are favored; the technique is exemplified by Pettini and collaborators' deuterium measurement.

For lithium, measure the ${}^7\mathrm{Li}$ absorption feature near $670.8\mathrm{nm}$ in warm, old, metal-poor halo stars. Stellar-atmosphere models convert line strength to Li/H. The Spite plateau supplies a nearly constant low-metallicity level, but stellar metallicity trends, atmospheric temperature scales, departures from local thermal equilibrium and stellar lithium depletion must be considered. Thus the photospheric level is not automatically the initial abundance. Ryan and collaborators' halo lithium measurements illustrate the measured abundance and chemical-evolution corrections.

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.

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.

A quasar is a luminous active galactic nucleus powered by gas accretion onto a supermassive black hole. The hot disc produces much of the optical/ultraviolet continuum; surrounding gas gives broad and narrow emission lines, while some systems have relativistic jets. A small emitting region is inferred from rapid variability, $R\lesssim c\Delta t_{\mathrm{obs}}/(1+z)$ when beaming is absent. Sustaining a high luminosity from such a compact region favours accretion over ordinary stellar power. For approximately spherical, ionized hydrogen with electron-scattering opacity,Comparing luminosity with this Eddington scale constrains mass only after assumptions about anisotropy, opacity and accretion state. The emitted power obeys $L=\eta \dot{M}{c}^2$; high binding-energy efficiency makes growth of a massive compact accretor energetically plausible.

Broad-line reverberation mapping estimates a gas radius from a continuum-to-line time lag, $R\sim c\tau$, and combines it with line velocity width to infer $M\sim f_{\mathrm{vir}}R{v}^2/G$. Geometry, inclination and nongravitational gas motions enter the factor $f_{\mathrm{vir}}$. Independent evidence for compact massive nuclei comes from resolved stellar or gas orbits in nearby galaxies. Andrea M. Ghez and collaborators' Galactic-centre stellar-orbit study fitted a common central potential and distance, demonstrating a compact dark mass of order four million solar masses; the Galactic centre is a low-luminosity example rather than a quasar.^[ref]

quasars are an accretion phase of massive nuclear black holes; luminosity, variability and dynamical evidence constrain different aspects of that interpretation. A finite-resolution dark-mass measurement by itself is not a direct image of an event horizon.

The Penrose singularity theorem states that a globally hyperbolic spacetime with a noncompact Cauchy hypersurface, a closed trapped surface, and the null convergence condition is future null-geodesically incomplete. Time reversal replaces the trapped surface by an anti-trapped surface and gives past incompleteness.

In his 1965 paper Gravitational Collapse and Space-Time Singularities, Roger Penrose used trapped surfaces and global causal structure to address whether nonspherical perturbations could avert a breakdown during gravitational collapse. Quasars supplied the opening astrophysical motivation. The paper used metric signature $(+,-,-,-)$ and a stronger timelike local-energy assumption; the statement above uses the modern null convergence condition.^[ref]

For this version, take a smooth connected four-dimensional spacetime with a time orientation. A Cauchy hypersurface meets every inextendible causal curve exactly once; its existence expresses global hyperbolicity. The hypersurface must be noncompact, meaning not compact; this is a global spatial condition, not an assumption about the size of the collapsing body. The null convergence condition is $R_{ab}{k}^a{k}^b\ge 0$ for every null vector $k$. Through the Einstein field equations it follows from the null energy condition; a cosmological constant drops out of the null contraction.^[ref]

The conclusion is that at least one future-inextendible null geodesic has a finite upper endpoint of its affine parameter. It neither asserts infinite mass density or divergent curvature at every endpoint, nor says that such an endpoint is visible. Locating an event horizon requires additional global information; the theorem is not a horizon-finding rule for arbitrary spacetimes. Weak cosmic censorship conjecture is an additional issue.^[ref]

The supermassive black hole account describes the motivating observational evidence from stellar orbits at the Galactic centre. Such measurements infer a compact gravitating mass. They do not observe a trapped surface, measure a geodesic's complete future, or experimentally prove the Penrose singularity theorem.