Large-scale structure of the universe

The large-scale structure of the universe is the cosmic distribution of matter on scales where galaxies and halos form a web of clusters, filaments, sheets, and voids.

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.

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 growth of primordial density perturbations into sheets, filaments, halos, galaxies, and clusters under gravity.

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 network of matter sheets, filaments, nodes, and intervening voids produced by anisotropic gravitational collapse.

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 spatially flat and contains only pressureless matter. In conformal time, $a\propto {\tau }^2$ and $\mathcal{H}=a^{\prime }/a=2/\tau$.

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 expands the density contrast and velocity divergence of a pressureless irrotational fluid in powers of the linear density field.

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 represents unresolved short-scale dynamics by an effective stress tensor. Its leading deterministic correction is proportional to $k^2{\delta }^{(1)}$ and renormalizes ultraviolet-sensitive loop contributions.

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 expresses a tracer density, such as a galaxy or halo density, as a renormalized sum of local and nonlocal operators built from matter density, tidal fields, derivatives, and stochastic fields. Its coefficients encode unresolved tracer formation physics.

Galaxy bias describes how fluctuations in galaxy number density differ from fluctuations in the underlying matter density. On sufficiently large scales, the leading deterministic relation is ${\delta }_g=b_1{\delta }_m$.

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 is the connected momentum-space three-point function of the galaxy density contrast after removing its momentum-conserving delta function.

A dark-matter halo is a gravitationally bound concentration of dark matter that hosts galaxies and larger structures.

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 conventionally the radius within which the mean density is a chosen overdensity $v$ times a reference critical or matter density.

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.

For a fixed mass under a spherical-overdensity definition, $R_h\propto ({\Delta }_c{H}^2{)}^{-1/3}$ and $v_h^2=G{M}_h/R_h$. Thus equivalent virial mass scales at two epochs have the displayed velocity ratio. With fixed overdensity during matter domination, $v_h\propto (1+z{)}^{1/2}$. This converts an epoch-specific mass–velocity calibration; it does not require a previously collapsed bound halo to physically expand with the later background density.

If the virial radius is defined relative to critical density ${\rho }_c$, the virial mass is

The concentration of a spherical dark-matter halo compares its outer radius with an inner scale radius. For a Navarro--Frenk--White profile, $c=r_v/r_s$.

The mass-concentration relation describes the statistical dependence of halo concentration on virial mass and epoch. In hierarchical cold-dark-matter cosmologies, lower-mass haloes generally formed earlier against a denser background and are therefore more concentrated at fixed epoch.

The Navarro--Frenk--White profile isIt has an inner $r^{-1}$ density cusp and an outer $r^{-3}$ tail.

For an NFW profile, $x=r/r_s$ and $v_c^2=GM(r)/r$. The circular speed rises as $r^{1/2}$ at small radius and eventually decreases as $[\log (r/r_s)/r{]}^{1/2}$. Its maximum occurs at $x\simeq 2.16258$, with $v_{c,\max }^2\simeq 2.71706G{\rho }_s{r}_s^2$.

The NFW scale radius is where the logarithmic density slope equals $-2$.

The NFW characteristic density fixes the normalization of the halo profile at a given scale radius.

The cusp--core problem is the tension between the central cusps of collisionless cold-dark-matter halo profiles and the approximately constant-density cores inferred for many dwarf and low-surface-brightness galaxies.

A dark-matter halo merger joins previously distinct dark matter halos. Orbital angular momentum can become internal halo spin, while escaping material carries away some mass and angular momentum. Halo mergers and accretion are important reasons that early tidal torque theory is only an approximate predictor of the spin of a later virialized remnant.

A top-hat filter averages a field with uniform weight inside a ball and zero weight outside it.

The spherical-collapse model applies a spherical top-hat filter to an overdense region and predicts collapse when its linearly extrapolated smoothed overdensity exceeds the threshold ${\delta }_c\simeq 1.686$ in an Einstein-de Sitter universe.

For ideal isolated spherical collapse, energy conservation and the virial theorem give $R_{\mathrm{vir}}=R_{\mathrm{ta}}/2$ and ${\rho }_{\mathrm{vir}}=8{\rho }_{\mathrm{ta}}$ if the density-profile shape is unchanged. The cycloid gives $t_{\mathrm{coll}}=2t_{\mathrm{ta}}$ and ${\rho }_{\mathrm{ta}}/\bar{\rho }(t_{\mathrm{ta}})=9{\pi }^2/16$. Since the Einstein-de Sitter universe has $\bar{\rho }\propto t^{-2}$, the overdensity at the nominal collapse time is $32(9{\pi }^2/16)=18{\pi }^2$. Hence ${\rho }_{\mathrm{vir}}=18{\pi }^2{\rho }_0(1+z_{\mathrm{coll}}{)}^3$. The collapse time is that of the formal cold solution; actual virialization replaces its point collapse.

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 $4/3-{\tau }_{\mathrm{expand}}(y)$.

For a homologous, isolated collapsing sphere, write its Newtonian gravitational potential energy as $U=-aG{M}^2/R$. At spherical-collapse turnaround the coherent kinetic energy vanishes. Conserving total energy and applying the virial theorem to the final equilibrium gives $U_{\mathrm{ta}}=U_{\mathrm{vir}}/2$ and hence $R_{\mathrm{vir}}=R_{\mathrm{ta}}/2$. If the gravitational structure coefficient changes, the ratio is $a_{\mathrm{vir}}/(2a_{\mathrm{ta}})$ instead. Mass loss, surface pressure, external tides and energy exchange also change this estimate.

A pressureless bound spherical shell first expands, reaches $\dot{R}=0$ at its maximum radius, and recollapses. In the cycloidal spherical-collapse model, turnaround occurs at $\theta =\pi$, with $R_{\max }=2A$ and $t_{\max }=\pi B$. Its mean density is $9{\pi }^2/16$ times the background density in an Einstein-de Sitter universe.

For fixed mass $M$, a uniform sphere of physical radius $R$ has $W_G=-3G{M}^2/(5R)$ and $W_{\Lambda }=-\Lambda M{R}^2/10$, using the convention in which its outward acceleration is $\Lambda r/3$. The virial theorem reads $2T+W_G=2W_{\Lambda }$. Defining $x=R_f/R_{\mathrm{ta}}$ and $\eta =\Lambda /(4\pi G{\rho }_{\mathrm{ta}})$, conservation of energy gives $2\eta x^3-(2+\eta )x+1=0$. The collapsing branch has $x=1/2-\eta /8+O({\eta }^2)$. For small positive cosmological constant, it is more compact at fixed turnaround data, even though the outward force inhibits turnaround and collapse.