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In an attractor with one adiabatic clock, the leading squeezed primordial bispectrum is the response to a coordinate dilation. Holding a halo's physical mass and smoothing scale fixed cancels this dilation in its abundance response. Thus the consistency-relation term does not generate the physical $k^{-2}$ scale-dependent halo bias from local non-Gaussianity. For example, a top-hat filter of fixed physical radius has coordinate radius $R_c=e^{-{\zeta }_L}{R}_0$, while the short density contrast is ${\delta }_s(\mathbf{x})={\delta }_s^{(0)}(e^{{\zeta }_L}\mathbf{x})$. The substitution $\mathbf{y}=e^{{\zeta }_L}\mathbf{x}$ shows that its normalized average over $|\mathbf{x}|<R_c$ equals the original average over $|\mathbf{y}|<R_0$. Genuine local effects start at two spatial derivatives of the long mode. Non-attractor backgrounds or nonstandard initial states need not satisfy this argument.

The connected correlation of three second-order density contrasts forms a triangle of three linear power spectra. It has eight Wick contractions. In one dimension, with $k_1+k_2+k_3=0$ and $P_L$ the power at the evaluation time,For the one-dimensional cosmological density kernel, its hard-loop integrand starts as $-k_1^2{k}_2^2{k}_3^2{P}_L(q{)}^3/q^6$.

The connected correlation of one fourth-order and two first-order density contrasts contracts a pair of inputs at the fourth-order vertex into a loop. For each choice of the fourth-order external field there are twelve Wick contractions. In one dimension,For the one-dimensional cosmological density kernel, $F_4=-k_3^4/(24k_1{k}_2{q}^2)$ exactly. Its leading cutoff dependence is therefore proportional to $k_3^2{F}_2(k_1,k_2)P_L(k_1)P_L(k_2)\int dq{P}_L(q)/q^2$.

In a dark-matter-dominated growing mode with a constant relative offset and initially negligible baryon density contrast, ${\Delta }_B/{\Delta }_D=1-R_r/R=1-(1+z)/(1+z_r)$. Replacing this by $1-z/z_r$ requires high redshift. The initial relative velocity and finite baryon self-gravity alter transients, although the relative difference becomes small as the common growing mode increases.

For a flat Newtonian expanding background $R\propto t^{2/3}$ with polytropic equation of state $p=K{\rho }^{\gamma }$, a comoving density contrast satisfies $\ddot{\delta }+4\dot{\delta }/(3t)+[{\Lambda }^2{t}^{-2\nu -2}-2/(3t^2)]\delta =0$. Here $\Lambda =kc(t_0)t_0^{\gamma -1/3}$ and $\nu =\gamma -4/3$. For $\nu >0$, substitution $z=\Lambda /(\nu t^{\nu })$ and $\delta =t^{-1/6}y(z)$ gives the Bessel equation of order $\lambda$. A universally valid basis uses the Bessel function of the first kind and Bessel function of the second kind, $t^{-1/6}{J}_{\lambda }(z)$ and $t^{-1/6}{Y}_{\lambda }(z)$. When $\lambda$ is not an integer, $J_{-\lambda }$ can replace $Y_{\lambda }$. For $4/3<\gamma <5/3$, the early oscillatory envelope is $t^{(3\gamma -5)/6}$; late modes have leading powers $t^{2/3}$ and $t^{-1}$. Late growth requires a nonzero coefficient of the growing branch.

The virial theorem and hydrostatic equilibrium relate a cluster's gas temperature to $GM/R$, while $M=(4\pi /3){\Delta }_v{\rho }_c{R}^3$. A calibrated mass-temperature relation converts the Press-Schechter halo mass function into $dn/dT=(dn/dM)|dM/dT|$. Its exponential rare-cluster tail is sensitive to the density contrast variance on cluster scales, hence to the power-spectrum amplitude, often parameterized by ${\sigma }_8$. Mass calibration, temperature scatter, nonthermal pressure, sample selection and the matter density must be included rather than treating temperature counts as an independent measurement of amplitude alone.

A comoving gauge in cosmology chooses hypersurfaces with zero scalar momentum density for the specified fluid or total system. For cold pressureless matter it can be implemented by taking the matter velocity potential to vanish. The choice of component must be stated in a multi-fluid system; density contrasts on this slicing can differ from those in a Newtonian or spatially flat gauge.

The cosmological two-point correlation function measures excess pairing or correlated density fluctuations at a separation. For a statistically homogeneous and isotropic mean-zero density contrast, $\xi (r)=⟨\delta (\mathbf{x})\delta (\mathbf{x}+\mathbf{r})⟩$. For distinct points in a discrete galaxy population it instead specifies the factorial pair density, ${\bar{n}}^2[1+\xi (r)]$. The latter excludes self-pairs; an unsmoothed discrete-density covariance also contains a Dirac delta at zero separation. These definitions connect large-scale galaxy clustering with the matter power spectrum while keeping galaxy power spectrum shot noise separate.

For pressureless homologous spherical collapse, let $\varphi$ be its cycloid parameter. The density contrast equation becomes ${\delta }_{\varphi \varphi }-\tan (\varphi /2){\delta }_{\varphi }-3\delta /(1+\cos \varphi )=0$. One solution is ${\delta }_g=\sin \varphi /(1+\cos \varphi {)}^2$. Reduction of order gives an independent solution that decays at collapse,For $ϵ=\pi -\varphi$, the integral is ${ϵ}^5/40+O({ϵ}^7)$, while ${\delta }_g\sim 4{ϵ}^{-3}$. Therefore ${\delta }_d\sim {ϵ}^2/10$. Since $R\sim R_0{ϵ}^2/4$, the displayed powers follow. The growing branch is unique up to scale and addition of the decaying branch; many mixed solutions grow, so it is not literally the only function with growth. Finite perturbations eventually leave the linear regime.

For spherical collapse in an Einstein-de Sitter universe, take $y=R/R_{\mathrm{ta}}$ and $\tau =H_{\mathrm{ta}}t$, with ${\Delta }_{\mathrm{ta}}=(3\pi /4{)}^2$. The expanding branch has $\tau ={\Delta }_{\mathrm{ta}}^{-1/2}{\int }_0^y[u/(1-u){]}^{1/2}du$. Its small-radius expansion is $\tau =(8/9\pi )y^{3/2}(1+3y/10+O(y^2))$, and its turnaround time is $2/3$. These constants are linked: using another leading coefficient without changing the time normalization no longer gives a density contrast approaching zero at early times. The collapsing branch is $4/3-{\tau }_{\mathrm{expand}}(y)$.

The parameter ${\sigma }_8$ is the present linear rms density contrast averaged in a spherical top-hat of comoving radius $8h^{-1}\mathrm{Mpc}$, with $h=H_0/(100\mathrm{km}{\mathrm{s}}^{-1}\mathrm{Mp}{\mathrm{c}}^{-1})$. Thus ${\sigma }_8^2=(2{\pi }^2{)}^{-1}{\int }_0^{\infty }{k}^2{P}_m(k,0)|W(k{R}_8){|}^{2}dk$. It specifies an amplitude on a chosen scale rather than the complete matter power spectrum; converting cluster abundance into this parameter also requires a spectral shape and background cosmology.

A halo is associated with the first upcrossing of the linear spherical-collapse threshold by the smoothed density contrast as its variance increases. With a sharp-k smoothing filter, the Brownian reflection principle makes the crossing probability twice the one-scale Gaussian upper tail. This handles trajectories that crossed earlier but finish below the barrier and leads to the Press-Schechter halo mass function.

The galaxy density contrast measures a fractional departure of galaxy number density from its ensemble or survey mean. Linear galaxy bias relates its large-scale three-dimensional component to the matter density contrast by ${\delta }_g=b_g{\delta }_m$. An angular sample averages this field with its normalized redshift selection; a probability density $p(z)$ per unit cosmological redshift corresponds to a distance kernel $H(z)p(z)/c$ in a spatially flat background.

For a pressureless fluid withthe first-order density contrast $\delta =\delta \rho /\bar{\rho }$ obeys

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 ${\delta }_c=C_g{\tau }^2+C_d{\tau }^{-3}$ be the matter-era growing and decaying density modes, with radiation neglected in the background gravity. The radiation density contrast isThe formula follows by the Green's function of the forced harmonic oscillator. For $k\tau \gg 1$, the slowly varying particular response to the decaying mode begins at $16C_d{\tau }^{-5}/{\omega }^2$; lower-endpoint oscillations are absorbed into $C_1,C_2$. The growing mode drives a constant radiation offset, while cold dark matter continues growing as ${\tau }^2$.

During matter domination in a flat FLRW metric, the cosmological continuity equation gives $\rho \propto a^{-3}$, and the Friedmann equation gives $a\propto {\tau }^2$ and $8\pi G\rho a^2=12/{\tau }^2$. The CDM density equation in a matter-radiation universe, with radiation neglected, is therefore ${\delta }^{\prime \prime }+2{\delta }^{\prime }/\tau -6\delta /{\tau }^2=0$. Its Euler-Cauchy equation powers satisfy $(p-2)(p+3)=0$, proving the displayed general solution. The growing density contrast is proportional to the scale factor, while the decaying mode is proportional to $a^{-3/2}$. This all-scale form uses the CDM-comoving synchronous gauge; other density slices require their corresponding gauge transformation.

In matter domination with the constant growing potential, the full Newtonian-gauge energy constraint gives the stated density contrast. The second term dominates inside the Hubble radius and grows as $a$, because ${\mathcal{H}}^2\propto a^{-1}$. Outside the Hubble radius, the constant $-2\Phi$ term dominates and is gauge dependent. The comoving density combination ${\Delta }_c={\delta }_c+3\mathcal{H}{\theta }_c/k^2$ cancels it for ${\theta }_c=2k^2\Phi /(3\mathcal{H})$, leaving the cosmological Poisson equation relation. Matter-spectrum power laws must specify whether they refer to this comoving density or to subhorizon Newtonian-gauge density.

The nonlinear wavenumber is the inverse scale at which linear density contrast fluctuations become of order unity. For the one-dimensional convention $⟨\delta (k)\delta (k^{\prime })⟩=2\pi {\delta }^{(D)}(k+k^{\prime })P_L(k)$, a usual estimate is $k_{\mathrm{NL}}{P}_L(k_{\mathrm{NL}})/\pi \sim 1$. Loop momenta above this scale require the effective field theory of large-scale structure even when external modes are perturbative.

Cooling oceanic lithosphere contracts and increases its column mass, causing isostatic deepening. Integrating a half-space cooling profile gives an excess mass proportional to $\sqrt{t}$. Converting this to seafloor depth includes the density contrast between mantle and seawater. Finite-plate cooling and basal heat supply limit indefinite square-root subsidence at old ages.