Large-scale structure of the universe

The large-scale structure of the universe is the spatial organization of matter into dark matter halos, galaxy clusters, sheets, filaments and cosmic voids. Galaxy clustering traces this pattern, while the underlying matter also includes material that does not shine. Gravity amplifies initial linear cosmological density perturbations, but expansion and the matter's dynamics determine which scales grow efficiently. Comparing observed statistics with models of cosmological structure formation turns this pattern into a test of the initial conditions and subsequent evolution.

A galaxy group is a gravitationally associated collection of galaxies, usually less massive and less richly populated than a galaxy cluster. Its physical system can include a shared dark matter halo and diffuse hot gas as well as its visible galaxies. A collection close together on the sky need not be a bound group: distance and velocity information are essential.

Spectroscopy measures line-of-sight velocities, imaging estimates projected separations, and diffuse X-ray emission can trace hot gas in the common gravitational potential. Under equilibrium assumptions, the scale $M\sim R{\sigma }^2/G$ relates a characteristic group size $R$ and velocity dispersion $\sigma$ to total mass. For the illustrative values $R=0.3$ Mpc and $\sigma =200$ km/s, this scale is about $2.8\times 10^{12}$ solar masses, before geometry-dependent factors. It is an inference model, not a direct mass measurement.

The ROSAT group catalogue combines optical group selection with extended X-ray measurements, illustrating why complementary observations distinguish a physical group from a projected association. Dynamically disturbed groups need not satisfy simple equilibrium mass estimates.^[ref]

A cosmic void is a large region with lower matter or galaxy density than its surroundings in the large-scale structure of the universe. It is not an exactly empty volume; voids can contain galaxies, gas and dark matter.

Voids emerge as initially underdense regions evolve relative to denser filaments and walls. An observational void depends on the tracer population, survey selection and identification algorithm, so a galaxy-defined void need not coincide exactly with a matter-defined underdensity. Their shapes, abundance and dynamics can test cosmological structure-growth models.

The nonuniform spatial distribution of galaxies, quantified for example by a two-point correlation function or a matter power spectrum related through galaxy bias. Survey geometry and selection must be separated from physical clustering.

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.

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 the Fourier transform convention ${\delta }_{\mathbf{k}}=\int d^{3}x\delta (\mathbf{x})e^{i\mathbf{k}\cdot \mathbf{x}}$ and covariance $⟨{\delta }_{\mathbf{k}}{\delta }_{{\mathbf{k}}^{\prime }}⟩=(2\pi {)}^{3}P(k){\delta }_D(\mathbf{k}+{\mathbf{k}}^{\prime })$, inversion gives $\xi (\mathbf{r})=\int d^{3}kP(k)e^{i\mathbf{k}\cdot \mathbf{r}}/(2\pi {)}^3$. Isotropy reduces the angular integral to $4\pi j_0(kr)$, where $j_0(x)=\sin x/x$ is the order-zero Spherical Bessel function. At zero separation this gives the variance if the spectrum integral converges. A nonnegative spectrum can produce negative correlations away from the origin.

For a homogeneous isotropic point process of galaxies with mean number density $\bar{n}$, the galaxy two-point correlation function gives the excess pair probability at distinct positions. Dividing the factorial pair density by the probability of a central galaxy gives conditional neighbouring-galaxy density $\bar{n}[1+\xi (r)]$. Choosing a central location uniformly in volume does not perform this conditioning.

The mean number of other galaxies within distance $R$ of a randomly chosen galaxy is the integral of its conditional number density. A random location instead has mean count $4\pi \bar{n}{R}^3/3$. For a power law $\xi (r)=(r_0/r{)}^{\gamma }$, the extra count is $4\pi \bar{n}{r}_0^{\gamma }{R}^{3-\gamma }/(3-\gamma )$ when $\gamma <3$. The central galaxy is excluded; including it adds one. This distinction matters even when the unconditional mean number density is spatially uniform.

A galaxy cluster is a gravitationally bound collection of galaxies, diffuse gas and dark matter. Much of its ordinary matter can reside in hot intracluster gas, visible through thermal bremsstrahlung and the Sunyaev-Zeldovich effect. The two signals weight the electron density differently, making the cluster useful both as a structure-formation probe and, with a geometrical model, as a distance indicator.

A galaxy cluster whose X-ray-emitting core is a well-studied example of interaction between radio bubbles and cooling gas. density fronts and ripples provide diagnostics of the mechanical energy released by the central active galaxy.^[ref]

An inward flow driven by loss of thermal pressure as hot gas undergoes radiative cooling. A classical unheated cluster model predicts substantial central cooling and deposition of cold material. active-galactic-nucleus feedback, transport and other heating can reduce the actual inflow; an X-ray cooling luminosity is not automatically a measured mass-deposition rate.

A merging galaxy cluster exhibiting the bullet Cluster dark-matter separation: peaks of weak gravitational lensing mass lie nearer the galaxies than the displaced X-ray gas. Combining the components supports a dominant less-collisional gravitating component under the usual gravitational model.^[ref]

Gravitational lensing constrains a galaxy cluster's projected mass independently of its light output or dynamical equilibrium. The lensing convergence is projected surface mass density divided by ${\Sigma }_{\mathrm{crit}}=c^2{D}_s/(4\pi G{D}_d{D}_{ds})$, using the appropriate angular diameter distances. Strong gravitational lensing constrains the central mass through multiple-image geometry, and weak gravitational lensing constrains larger-scale projected mass through coherent image distortions. Converting these observations into a three-dimensional mass needs geometrical modelling and the relevant lensing degeneracies.

The virial theorem balances a relaxed galaxy cluster's orbital kinetic energy, of order $M{\sigma }_v^2$, against its Newtonian gravitational potential energy, of order $-G{M}^2/R$. Their ratio gives the displayed mass scale. The numerical coefficient depends on the density profile, radius definition and conversion from line-of-sight velocity dispersion to three-dimensional orbital motion. Membership contamination, anisotropy and nonequilibrium dynamics can bias the inference.

The intracluster medium is diffuse hot plasma between the galaxies of a galaxy cluster. Its X-ray radiation includes thermal bremsstrahlung and atomic line emission. Measured temperature and density profiles supply a cluster hydrostatic mass estimator when approximately thermal hydrostatic equilibrium holds. The medium supplies much of a cluster's baryonic mass, but this need not equal its total gravitating mass.

For spherical thermal hydrostatic equilibrium, $dP/dr=-{\rho }_{g}GM(<r)/r^2$. Inserting the ideal gas law $P={\rho }_g{k}_{B}T/(\mu m_p)$ with constant mean molecular weight gives the displayed estimator. The density ${\rho }_g$ is gas density, while $M$ includes every gravitating component. A varying $\mu$ adds $-d\log \mu /d\log r$ inside the logarithmic bracket. Nonthermal pressure requires differentiating the total supporting pressure rather than thermal pressure alone.

A galaxy survey measures positions and other properties of a selected galaxy sample. A spectroscopic survey uses cosmological redshift to estimate radial distance, with corrections for peculiar velocity. Its volume, selection, galaxy bias, and sampling determine which matter power spectrum modes can be inferred.

Redshift-space distortions arise because observed galaxy redshifts include peculiar velocity as well as cosmological redshift. Distances inferred solely from expansion therefore distort clustering along the line of sight. Coherent large-scale motion and small-scale velocity dispersion produce different effects on the inferred matter power spectrum.

Sampling with a homogeneous Poisson point process of mean comoving number density $\bar{n}$ adds a constant $1/\bar{n}$ to the dimensional galaxy power spectrum. Subtracting its expectation leaves sampling uncertainty. Correlated tracer populations and survey weights modify this simple formula; survey depth alone does not specify $\bar{n}$.

A survey window function describes the finite spatial coverage and weights of a galaxy survey. Multiplication by this window in position space mixes Fourier modes by convolution in Fourier space. A dimension of extent $L$ cannot robustly measure wavelengths much larger than $L$; the fundamental wavenumber of a periodic box of that size is $2\pi /L$.

Cosmological structure formation is the gravitational amplification of initial linear cosmological density perturbations into the cosmic web and bound objects. Overdense regions attract extra matter while surrounding regions become relatively empty. In the cold dark matter picture, small initial particle velocities preserve fluctuations that stronger free streaming would erase. The resulting spectrum of seed fluctuations influences the populations of galaxies and clusters; gas cooling and feedback add further physics to their visible contents.^[ref]

In hierarchical galaxy formation, low-mass dark-matter structures typically collapse before larger structures because the smoothed linear density variance is larger on smaller mass scales. Accretion and mergers assemble increasingly massive haloes. Cooling, feedback and angular momentum determine how baryons form galaxies within that halo hierarchy; the ordering is statistical rather than an absolute rule for every object.

The galactic angular-momentum catastrophe is the excessive loss or redistribution of baryonic specific angular momentum in early simulated galaxy formation, producing overly compact disks or large central concentrations. Premature cooling, gravitational torques, interactions, and resolution contributed. Improved stellar feedback and gas accretion histories can mitigate it; it is a historical simulation problem rather than a theorem against cold dark matter.

Downsizing denotes the tendency for the characteristic mass of actively star-forming galaxies to decrease with time, together with related evidence that more massive quiescent systems formed their stars earlier or over shorter intervals. It does not assert that their dark-matter haloes assembled earlier: stellar formation and halo or stellar assembly are separate histories.

Monolithic galaxy formation idealizes a galaxy as forming in one early rapid dissipative collapse, followed mainly by passive evolution. Rapid formation of the stars is distinct from early assembly of all those stars into their final object; hierarchical galaxy formation can assemble old stellar populations later.

The integrated halo star-formation efficiency is the fraction of a dark matter halo's reference baryonic mass converted into stars: ${ϵ}_{\ast }=M_{\ast }/(f_b{M}_h)$, with cosmic baryon fraction $f_b$. It is distinct from star-formation efficiency per free-fall time. Cooling, gas supply and retention, stellar feedback and active-galactic-nucleus feedback make it strongly dependent on dark matter halo mass. With stellar mass-to-light ratio ${{\rm Y}}_{\ast }$, a simple central-galaxy mapping is $L={ϵ}_{\ast }{f}_b{M}_h/{{\rm Y}}_{\ast }$. Changing this efficiency changes both the mapping and the logarithmic Jacobian between the dark matter halo mass function and the luminosity function.

Selecting a baryonic mass $M_b$ from a normalized cumulative distribution $M_b(<j)\propto j$ fixes a cutoff specific angular momentum $j_d=(M_b/M_{b,\mathrm{tot}})j_{\mathrm{vir}}$. With circular support and dominant enclosed baryonic gravity, the outer radius is $r_d\simeq j_d^2/(G{M}_b)$ and speed $v_d\simeq G{M}_b/j_d$. The mean specific angular momentum is $j_d/2$, which defines a different one-zone radius. The estimate assumes energy dissipation but negligible redistribution of specific angular momentum.

The baryon conversion efficiency of a halo is the fraction of its baryonic mass incorporated into stars. If the halo retains the cosmic baryon fraction $f_b$, its stellar-to-total-mass ratio is $f_b{f}_{\ast }$. This efficiency depends on halo mass, epoch, gas supply and feedback and must not be confused with a star-formation efficiency per free-fall time.

A monotonic luminosity assignment to a counted halo population converts its differential abundance per mass into a luminosity function by a change of variable. If $n_h\propto M_h^{-a}$ and $L\propto M_h^p$, then $\Phi \propto L^{-1-(a-1)/p}$. This simple mapping assumes one counted galaxy per halo, no scatter and a fixed selection. A mass-dependent baryon conversion efficiency of a halo, satellites and stellar-population differences modify it.

The stellar-to-halo mass ratio compares a galaxy's galactic stellar mass with its host's total halo mass. In a simple baryon-conversion model it is ${ϵ}_{\ast }=f_b{f}_{\ast }$. Observational comparisons must distinguish this total-mass ratio from the fraction of baryons converted, and must specify whether central, satellite or all stellar populations are included.

If every counted halo hosts one counted galaxy and the baryon fraction and conversion efficiency are mass-independent without scatter, galactic stellar mass is a constant multiple of halo mass. The galaxy mass function is obtained by changing variable with $dM/d{M}_{\ast }=1/{ϵ}_{\ast }$. Satellites, efficiency variation and scatter prevent this mapping from being generally exact.

Gas in a collapsing halo is heated toward a virial temperature. If its cooling time is short compared with the dynamical time, thermal support is removed and rapid condensation can form a galaxy. If cooling is slower, a hot atmosphere can persist or condense only gradually. This criterion links baryonic galaxy formation to the dark-matter assembly hierarchy without making every halo into one luminous object.

Primordial atomic line cooling becomes effective near the temperature where electronic excitations are accessible, of order ten thousand kelvin. Molecular hydrogen has lower-energy transitions and can cool colder gas if it forms and survives. The corresponding virial temperature thresholds translate into epoch-dependent halo mass thresholds. Radiation, molecular chemistry, metal-line cooling and gas density qualify a single universal minimum galaxy mass.

In simple hierarchical models, increasing system mass raises the virial temperature and can carry primordial gas beyond its efficient atomic line-cooling interval. At assembly densities, the cooling-time constraint then distinguishes galaxy-sized condensations from larger hot groups or clusters. The resulting characteristic mass is approximate, depends on composition and structure, and does not prohibit subsequent mergers of already formed stellar systems.

The Zeldovich approximation maps a fluid element from its Lagrangian coordinate $\mathbf{q}$ to the comoving position $\mathbf{x}=\mathbf{q}+D(t)\mathbf{s}(\mathbf{q})$, where $D$ is the linear growth factor. It follows the initial gravitational displacement along a straight comoving trajectory until trajectories cross.

For $\mathbf{r}=a[\mathbf{q}+b(t)(p(q_1),0,0)]$, put $J=1+b{p}^{\prime }$. Before shell crossing, mass conservation gives $\rho =\bar{\rho }/J$. The dust background equation $3\ddot{a}/a=-4\pi G\bar{\rho }$ and the growing-mode equation $\ddot{b}+2H\dot{b}=4\pi G\bar{\rho }b$ imply ${\nabla }_r\cdot \ddot{\mathbf{r}}=-4\pi G\bar{\rho }+4\pi G\bar{\rho }b{p}^{\prime }/J=-4\pi G\rho$. Thus planar Zeldovich approximation trajectories satisfy the full nonlinear dust force-divergence equation until the map loses invertibility. All three physical coordinates are included in the divergence; the two unperturbed directions still contribute background acceleration.

The cosmological Lagrangian displacement sends an initially labelled matter element from its Lagrangian coordinate $\mathbf{q}$ to $\mathbf{x}(\mathbf{q},t)=\mathbf{q}+\mathbf{\psi }(\mathbf{q},t)$. Its time derivative determines the peculiar velocity through $\mathbf{v}=a\dot{\mathbf{\psi }}$.

For an irrotational displacement $\mathbf{s}=-{\nabla }_q\Phi$, the deformation tensor is the symmetric Hessian $d_{ij}=-{\partial }_i{\partial }_j\Phi$. Its real eigenvalues determine the principal collapse directions.

Shell crossing occurs when the Lagrangian-to-Eulerian map ceases to be one-to-one. Its Jacobian determinant vanishes, the pressureless single-stream density formally diverges, and several streams subsequently occupy the same position.

For the Zeldovich approximation $\mathbf{x}=\mathbf{q}-b{\nabla }_q\psi$, let ${\lambda }_i$ be the eigenvalues of the potential Hessian, with this sign convention. Before shell crossing, mass conservation gives the displayed density. The first positive eigenvalue reaches $b{\lambda }_i=1$ and collapses one direction, creating a sheetlike cosmological caustic. Extrapolating straight trajectories after crossing lets particles pass through without self-consistent gravitational trapping, so the predicted multistream sheet spreads excessively. Collisionless gravitational evolution keeps matter more tightly bound; the linear displacement extrapolation cannot determine its late thickness reliably.

A cosmological caustic is the image of a singular point of the Lagrangian map. The ideal collisionless density diverges there, although velocity dispersion and other small-scale physics regularize the singularity.

Tidal torque theory explains the early angular momentum of a protohalo through the misalignment of its mass second-moment tensor $I_{ij}$ and the Hessian matrix $T_{ij}={\partial }_i{\partial }_j\Phi$ of the surrounding peculiar gravitational potential. At leading order,

The cosmic web is the connected pattern of matter sheets, filaments and dense nodes surrounding cosmic voids. Collapse is generally anisotropic: a region can contract strongly in one direction before the others. The Zeldovich approximation makes this mechanism explicit through the principal directions of the initial deformation. Collapse along one, two or three directions gives the intuitive sequence of sheet, filament and compact concentration, although subsequent shell crossing and nonlinear dynamics complicate that picture.

For comoving position $\mathbf{x}$, canonical momentum $\mathbf{p}=am{\mathbf{x}}^{\prime }$, and Newtonian potential $\varphi$, the dark-matter phase-space density obeys

This is the gravitational, expanding-universe specialization of the Vlasov equation.

The velocity-dispersion tensor is the second central moment of a phase-space distribution,It contributes the stress-divergence term ${\rho }^{-1}{\partial }_j(\rho {\sigma }^{ij})$ to the Euler equation and vanishes in the ideal single-stream limit.

An Einstein-de Sitter universe is a spatially flat expanding model containing only pressureless matter, with zero cosmological constant. Conservation of matter gives $\rho \propto a^{-3}$; inserting this into the Friedmann equations gives $a\propto t^{2/3}$ in cosmic time. Since $dt=ad\tau$, it also gives $a\propto {\tau }^2$ and $\mathcal{H}=2/\tau$ in conformal time. These simple powers make the model useful for isolating gravitational growth, rather than representing the full matter content at every epoch of the actual universe.

For an earlier emission epoch $z_1>z_2$, the matter-only flat model gives $R\propto t^{2/3}$ and $t(z)=2(1+z{)}^{-3/2}/(3H_0)$. Subtraction gives the displayed elapsed cosmic time. It is not the comoving travel distance divided by $c$, because expansion changes the spatial scale during flight. An Einstein-de Sitter approximation fixed to the same $H_0$ differs from a matter-era approximation normalized to $H_0\sqrt{{\Omega }_{m0}}$.

Spherical secondary infall follows concentric collisionless shells around a primordial overdensity. Each shell initially expands with the Hubble flow, turns around, collapses, and subsequently crosses other shells.

The turnaround radius of a shell is its maximum physical radius before collapse. At a given cosmic time, the currently turning shell defines a natural similarity length.

If the initial fractional excess mass is a power law in enclosed mass, spherical secondary infall admits a similarity reduction in which radii scale by the current turnaround radius and masses by the mass of the currently turning shell.

Standard perturbation theory in cosmology expands the matter density contrast and peculiar-velocity divergence in powers of a small initial density field. The linear term evolves independent Fourier modes; higher terms couple them and predict departures from Gaussian statistics. The usual fluid treatment assumes pressureless, irrotational, single-stream motion. It is useful before strongly nonlinear evolution and shell crossing invalidate that approximation.^[ref]

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.

Gaussian initial conditions make every connected primordial correlation function above second order vanish. Wick theorem then reduces perturbative late-time correlation functions to sums of products of the linear power spectrum.

The continuity-equation kernel isIts symmetrization is ${\alpha }_s=1+({\mathbf{k}}_1\cdot {\mathbf{k}}_2)(k_1^{-2}+k_2^{-2})/2$.

The Euler-equation kernel is the symmetric function

The symmetrized kernel $F_n$ convolves $n$ linear density fields to produce the $n$th-order density perturbation. In an Einstein-de Sitter universe,

For plane-parallel pressureless matter in an Einstein-de Sitter universe, before shell crossing, the symmetrized growing-mode standard perturbation theory density kernel isIn particular $F_{2,s}=(q_1+q_2{)}^2/(2q_1{q}_2)$. An unsymmetrized representation of the one-dimensional beta mode-coupling kernel is $(q_1+q_2)/q_1$; its symmetric part is the expression above.

At fixed external momentum $\mathbf{k}$, mass and momentum conservation implyas $q\to \infty$. In particular, the constant term in the hard-momentum expansion must cancel.

For Gaussian initial conditions and the Einstein-de Sitter second-order standard perturbation theory density kernel, the unsmoothed density contrast obeysThere are three choices of the second-order field and two connected Wick contractions; the angular average of $F_2$ is $5/7+(2/7)/3=17/21$. Thus the reduced skewness is $6(17/21)=34/7$. Smoothing changes this result, and the unsmoothed calculation requires a finite variance.

For Gaussian initial conditions, the tree-level connected matter five-point function has perturbative-order partitionsThey correspond respectively to a four-valent star, an $F_3$--$F_2$ tree, and a chain of three $F_2$ vertices. Every topology contains four linear power spectra.

For Gaussian linear perturbations, the one-loop matter power spectrum iswhere $P_{22}$ correlates two second-order fields and $P_{13}$ correlates a linear field with a third-order field.

The $P_{22}$ contribution correlates two second-order density fields. For symmetrized kernels,

The $P_{13}$ contribution correlates a linear density field with a third-order field:Both orderings contribute $2P_{13}$ to the power spectrum.

At equal time, leading loop contributions from a very long-wavelength displacement cancel between perturbative diagrams. This is required by the equivalence principle, because a coherent long displacement merely translates short-scale structure.

A scale-free linear matter spectrum has $P(k)=A{k}^n$. Dimensional analysis then makes its one-loop $P_{22}$ contribution proportional to $A^2{k}^{2n+3}$ whenever the loop integral converges.

For a matter power spectrum $P(k)=A{k}^n$ at a fixed epoch, the Poisson equation gives potential Fourier amplitudes proportional to ${\delta }_k/k^2$, and the growing-mode peculiar velocity is proportional to ${\delta }_k\mathbf{k}/k^2$. With a spherical top-hat window function, their variances are proportional to $\int k^{n-2}W(kR{)}^{2}dk$ and $\int k^{n}W(kR{)}^{2}dk$, respectively. Rescaling $kR$ gives the displayed powers when the integrals converge: $1<n<5$ for the averaged potential and $-1<n<3$ for the bulk velocity. Absolute averaged potential has infrared sensitivity; band-limited fluctuations or potential differences can have different convergence requirements. The density variance converges only for $-3<n<1$, so these three absolute averages cannot all converge for one unbroken power law at every scale.

For an unsmoothed density spectrum $P(k)=A{k}^n$ and a scale-dependent smoothing window $\tilde{W}(kR)$, substituting $q=kR$ in the variance integral gives ${\sigma }_R^2\propto R^{-(n+3)}$, provided ${\int }_0^{\infty }{q}^{n+2}|\tilde{W}(q){|}^{2}dq$ converges. For a window tending to one at zero, infrared convergence requires $n>-3$; the ultraviolet condition depends on the window. A real-space spherical top-hat also requires $n<1$.

The symmetrized kernel $G_n$ convolves $n$ linear density fields to produce the $n$th-order peculiar-velocity divergence in cosmological standard perturbation theory.

For Gaussian initial conditions, the matter bispectrum through one loop is the sum of the tree diagram $B_{211}$ and the four one-loop topologies $B_{222}$, $B_{321}^I$, $B_{321}^{II}$, and $B_{411}$.

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 $k_3^2{F}_2(k_1,k_2)P_L(k_1)P_L(k_2)\int dq{P}_L(q)/q^2$.

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 $-k_1^2{k}_2^2{k}_3^2{P}_L(q{)}^3/q^6$.

For Gaussian initial conditions, the one-loop connected matter trispectrum contains the perturbative-order classes $T_{5111}$, $T_{4211}$, $T_{3311}$, $T_{3221}$, and $T_{2222}$. Each class includes every inequivalent external-leg assignment and Wick contraction.

The effective field theory of large-scale structure describes long-wavelength matter dynamics while representing unresolved short-scale motion through an effective stress tensor. It supplements standard perturbation theory in cosmology with symmetry-allowed terms whose coefficients can be calibrated from simulations or data. A leading correction scales as $k^2{\delta }^{(1)}$. Its role is to account for short-scale influence on long-scale predictions, not to resolve individual collapsed objects.^[ref]

For the one-dimensional B411 contribution to the one-loop matter bispectrum, contracting two hard inputs in the fourth-order field gives $-(I_{\Lambda }/2)k^2{\delta }^{(2)}$, where $I_{\Lambda }={\int }_{q_h<|q|<\Lambda }dq{P}_L(q)/(2\pi q^2)$. A counterterm $C(\Lambda ){\partial }_x^2{\delta }^{(2)}$ cancels this cutoff-dependent part when $C(\Lambda )=-I_{\Lambda }/2+C_{\mathrm{finite}}$. This sign follows from ${\partial }_x^2\leftrightarrow -k^2$; the finite coefficient describes unresolved short-scale dynamics.

The leading deterministic power-spectrum counterterm in the effective field theory of large-scale structure is proportional to $-2c_{\mathrm{eff}}^2{k}^{2}P(k)$. Its coefficient depends on the coarse-graining convention and runs so that it cancels the cutoff dependence of the one-loop $P_{13}$ contribution.

For internal momentum $q\gg k$, the P22 contribution to the one-loop matter power spectrum starts at $k^4$, while twice the P13 contribution to the one-loop matter power spectrum starts at $k^{2}P(k)$. The latter has the same momentum dependence as the effective sound-speed counterterm in large-scale structure. Short-scale stochastic contributions start at $k^4$ by mass and momentum conservation.

A bias expansion relates the density of selected tracers to the matter distribution using allowed combinations of density contrast, tidal effects, spatial derivatives and stochastic contributions. For example, ${\delta }_g=b_1\delta +\frac{1}{2}{b}_2({\delta }^2-⟨{\delta }^2⟩)+\cdots$ illustrates a local part. The coefficients encode tracer-formation physics. Including further allowed operators makes the expansion systematically improvable on sufficiently large scales.^[ref]

Galaxy bias describes the statistical relation between fluctuations in galaxy counts and the underlying matter density contrast. In the simplest large-scale limit, ${\delta }_g=b_1{\delta }_m$, so the corresponding deterministic power spectrum is $b_1^2{P}_m$. Different selections can have different $b_1$ because they sample different formation environments. More generally, the bias expansion includes nonlinear, tidal and stochastic terms, so a single multiplicative constant is an approximation.^[ref]

Number conservation under the displacement from Lagrangian coordinates to Eulerian coordinates givesThe added one is the linear matter-density response of the coordinate Jacobian.

The simplest nonlinear local bias model isFor Gaussian matter fluctuations, its leading galaxy three-point function is proportional to $b_1^2{b}_2$.

The galaxy bispectrum measures the connected three-point correlation of the Fourier modes of the galaxy density contrast. For a mean-zero field, statistical homogeneity givesThe delta function enforces a closed wavevector triangle. With the additional assumption of statistical isotropy, one can write $B_g(k_1,k_2,k_3)$ as a function of triangle size and shape. Statistical homogeneity alone leaves possible orientation dependence.

The galaxy bispectrum adds information beyond the power spectrum. For example, take a zero-mean Gaussian random field $\delta$ and a local tracer model ${\delta }_g=b_1\delta +(b_2/2)({\delta }^2-⟨{\delta }^2⟩)$. Two Wick contractions at each quadratic insertion cancel its factor of two, yielding at leading orderThus nonlinear galaxy bias can generate a three-point signal even from Gaussian matter fluctuations. Nonlinear gravitational evolution supplies a further matter-bispectrum contribution. Both effects must be modeled before interpreting a measured galaxy three-point signal as primordial non-Gaussianity.^[ref]

A dark matter halo is a gravitationally bound concentration of dark matter. Its internal motions are largely supported by a distribution of particle orbits, unlike the cooling gas that can settle into a luminous galactic disk. Halo profiles, abundances and spatial correlations provide useful ingredients for modeling nonlinear clustering. The halo model uses these ingredients as an approximate statistical description of matter.^[ref]

A population of distinct, nonoverlapping dark matter halos cannot contain more total mass per comoving volume than the cosmic mean matter density. A narrow mass population therefore obeys $n_c{M}_h\le {\bar{\rho }}_{m,0}$. Do not sum subhalo masses and their inclusive host masses in the same bound, because that double-counts material. This check can reveal incompatible inferred absorption incidence, object size and halo-mass assumptions.

The virial radius of a dark-matter halo is usually a specified spherical-overdensity boundary: $M_{\Delta }=(4\pi /3)\Delta {\rho }_{\mathrm{ref}}{r}_{\Delta }^3$. Here ${\rho }_{\mathrm{ref}}$ can be the mean matter density or the mass-equivalent cosmological critical density. The same halo therefore receives different quoted masses and radii under different conventions. This useful comparison radius is not a sharp material edge; the density and orbital population can extend beyond it. The chosen overdensity and reference density must accompany numerical values.

In a spherical-overdensity halo convention, $3M/(4\pi R^3)={\Delta }_f{\rho }_0(1+z_f{)}^3$, so $1+z_f$ follows from the displayed cube root. At fixed measured $M,R,{\rho }_0$, changing the adopted overdensity changes $1+z_f\propto {\Delta }_f^{-1/3}$. It does not change the hydrostatic temperature estimate from the same measured mass and radius. A redshift inferred this way is a model formation epoch, not a direct age measurement.

If a halo retains the mean density acquired at collapse, $3M/(4\pi R^3)={\Delta }_{\mathrm{vir}}{\bar{\rho }}_{m,0}(1+z_f{)}^3$ estimates its formation redshift. In the non-dissipative spherical-collapse model during matter domination, ${\Delta }_{\mathrm{vir}}=18{\pi }^2$. Subsequent mass growth, profile evolution and the reference-density convention limit this inference.

For halo mean density ${\Delta }_v$ times the critical density at its epoch, the circular virial velocity obeys $v_{\mathrm{vir}}^2=G{M}_h/R_h=({\Delta }_v/2)H^2{R}_h^2$. Thus $v_{\mathrm{vir}}=H{R}_h\sqrt{{\Delta }_v/2}$. Specifying whether overdensity is relative to mean matter or critical density is necessary for numerical inference.