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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.

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 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 spectral line has finite width when the emitted or absorbed field loses coherence, or when different emitters have different transition frequencies. It is useful to distinguish homogeneous spectral broadening of each emitter from inhomogeneous spectral broadening across the ensemble.

Finite lifetime and dephasing. For an upper-state population lifetime $\tau$ and a stable lower state, the radiating amplitude decays as $e^{-t/(2\tau )}$. Its Fourier spectrum is Lorentzian, with full width at half maximumMore generally, an optical coherence time $T_2$ gives $\Delta {\omega }_{\mathrm{hom}}=2/T_2$. Both level lifetimes and pure dephasing can contribute to $1/T_2$. Distinguishing population lifetime from amplitude coherence time prevents the usual factor-of-two error.

Thermal motion. To first order in velocity, $\delta \nu ={\nu }_0{v}_{\parallel }/c$. A Maxwell-Boltzmann velocity distribution therefore produces a Gaussian profile, withThis is inhomogeneous doppler broadening: different velocity classes have different resonance frequencies. A distribution of bulk velocities can add further broadening, while a common bulk velocity shifts the line.

Interactions with the surroundings. Collisions can interrupt the dipole wave phase and perturb the energy levels, producing pressure broadening and shifts. In the impact approximation, approximately exponential wave phase relaxation produces a Lorentzian contribution; its rate increases with perturber density times an appropriate velocity-averaged collision cross section. Slowly varying electric microfields can instead produce non-Lorentzian Stark profiles. Neutral perturbers, charged perturbers and resonance interactions need not produce the same line shape. These mechanisms and the Gaussian thermal limit are distinguished in NIST's atomic-spectroscopy compendium.^[ref]

Finite observation time limits coherence; transit-time broadening also occurs for particles crossing a narrow optical beam. Strong resonant excitation causes power broadening: for an ideal lifetime-limited two-level transition, $\Delta \omega =\Gamma \sqrt{1+s}$ with population decay rate $\Gamma$ and saturation parameter $s$. Unresolved fine structure, hyperfine structure, isotope components, or spatially varying Zeeman and Stark shifts may look like a single broadened feature even though several frequencies are present.

The observed spectrum is additionally convolved with the instrument response. A Gaussian Doppler contribution convolved with Lorentzian homogeneous broadening gives a Voigt profile; their full widths do not generally add in quadrature. An informative experiment varies temperature, perturber pressure and excitation intensity separately, calibrates the spectrometer response with a narrower reference, and checks optical thickness or self-absorption. Extrapolation to weak excitation and low pressure helps isolate the lifetime contribution, while a Doppler-free technique is needed if velocity spread still dominates.

Use $u$ as a spectroscopic wavenumber, in the same units throughout. Write the mass absorption coefficient of the absorbing species asHere $S$ is the spectral line strength, the integrated mass absorption coefficient; it is not the line-centre coefficient. If ${\kappa }_{\nu }$ has units ${\mathrm{m}}^2\mathrm{k}{\mathrm{g}}^{-1}$ and $\nu$ is in $\mathrm{c}{\mathrm{m}}^{-1}$, then $S$ has units ${\mathrm{m}}^2\mathrm{k}{\mathrm{g}}^{-1}\mathrm{c}{\mathrm{m}}^{-1}$. For absorber mass density ${\rho }_a$, the volume coefficient in Beer-Lambert law is ${\rho }_a{\kappa }_{\nu }$.

Lifetime broadening. Finite lifetimes of the upper and lower states destroy optical coherence, even for an isolated molecule. Exponential coherence decay gives a Lorentzian line shape. For population decay rates ${\Gamma }_u,{\Gamma }_l$ and no extra dephasing, its wavenumber half-width is $({\Gamma }_u+{\Gamma }_l)/(4\pi c)$, with $c$ expressed in the length unit reciprocal to $\nu$. This sets a natural linewidth and can matter at extremely low collision rates; it is generally much smaller than atmospheric collisional or thermal widths.

Thermal Doppler broadening. A Maxwell-Boltzmann distribution of molecular line-of-sight velocities givesThis Gaussian distribution has a $1/e$ half-width ${\Delta }_D$, determined by absorber molecular mass $m_a$ and temperature. It is especially relevant in the rarefied upper atmosphere, where collisions are infrequent.

Pressure broadening. Collisions perturb transition frequencies and phases. In the impact regime, exponential phase decorrelation givesHere ${\Delta }_L$ is the half-width at half maximum, proportional to the coherence-destroying collision rate. It depends on perturber density, molecular collision properties, and temperature; it is often parameterized by ${\Delta }_L\propto p(T_{\ast }/T{)}^{n_T}$ for a specified broadener. It dominates much of the dense lower atmosphere. The Lorentzian line shape has algebraic $\delta {\nu }^{-2}$ wings, whereas the Doppler profile has rapidly decreasing Gaussian wings. Where both mechanisms matter, their convolution is a Voigt profile. A pure pressure-broadened profile should not be extrapolated to arbitrarily low pressure while discarding the finite Doppler core.

Take a weak line in the radiative sense: the integrated slant optical depth is small wherever absorption materially contributes, not merely a small greenhouse-gas mass mixing ratio. Normalize every local spectral line profile so that $\int {\kappa }_{\nu }(p)d\nu =S$, with the line strength independent of pressure and, as stipulated, temperature. Keep a finite Doppler or natural core, or a Voigt profile, so that a pressure-only zero-width core at the top does not invalidate a uniform optical-depth expansion. The line must also be narrow enough to use $B_{{\nu }_0}$ across it.

To first order,Thus the detailed normalized line shape drops out, even if its width varies with pressure. Substituting in the deficit in c givesThe neglected term is second order when the relevant squared optical depths are integrable and uniformly small. A very narrow saturated line need not be weak simply because $q\ll 1$. More generally the displayed first-order integrated limit follows from the finite integrated opacity even if an idealized line centre makes a pointwise expansion nonuniform; a finite physical core is the straightforward sufficient assumption.

For an explicitly perturbative temperature estimate, evaluate the bracket at $T_{\mathrm{bare}}$:At fixed $p_s,g,q,S$ and radiative conditions, increasing $\kappa =R_b/c_{p,b}$ makes every level $0<r<1$ cooler relative to the surface, enlarging the bracket and the warming. Increasing $c_{p,b}$ at fixed $R_b$ has the opposite effect. The limit $\kappa \to 0$ is isothermal and gives zero line-induced deficit. A change of background molecular mass alone need not change this leading result: for ideal gases with the same active degrees of freedom, $R_b$ and $c_{p,b}$ scale together and their ratio is unchanged. Comparing only the height-coordinate lapse rate $g/c_{p,b}$ would miss that the absorber column is naturally a pressure-coordinate quantity.

Pressure broadening is proportional to collision frequency and therefore becomes small in the upper atmosphere. Doppler broadening remains finite at finite temperature, with a characteristic width of order ${\nu }_0\sqrt{k_{B}T/(m{c}^2)}$ up to the stated width convention. Where that exceeds the collisional width, the line core is Doppler dominated. A Voigt profile combines the two mechanisms; natural broadening can contribute at still lower pressures. Thus a Lorentz pressure-broadened profile cannot hold all the way to $p=0$.

At line centre, replacing the fictitious $1/p$ opacity by a finite Doppler-core coefficient removes the divergent column optical depth. The emitting pressure is then finite rather than arbitrarily close to zero. If that level still lies in the isothermal stratosphere and the core is opaque, the central brightness temperature stays approximately $T_{\mathrm{strat}}$, so the flux floor changes little. If it lies below the stratosphere, or the Doppler-core column becomes thin, warmer layers or the surface contribute and the core brightens. The core-to-wing transition changes shape and width, but Lorentz wings formed mainly at higher pressures remain approximately as in part (c). The size of the correction needs the missing spectroscopic widths, but no unique numerical correction is requested.

A Voigt profile is the convolution of a Gaussian distribution and a Cauchy distribution. For an absorption line, thermal Doppler broadening contributes the Gaussian core while finite excited-state lifetime contributes the Lorentzian wings. Instrumental broadening and unresolved motions must also be modeled when inferring column density and temperature.