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.

Cosmological structure formation is the growth of primordial density perturbations into sheets, filaments, halos, galaxies, and clusters under gravity.

The Zel'dovich 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 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.

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.

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

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 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,

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, 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.

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.

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 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.

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

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

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.

The spherical-collapse model predicts collapse when a linearly extrapolated smoothed overdensity exceeds the threshold ${\delta }_c\simeq 1.686$ in an Einstein-de Sitter universe.

The peak height of a halo of mass $M$ is $\nu ={\delta }_c/\sigma (M)$, where ${\sigma }^2(M)$ is the variance of the linear density field smoothed on the corresponding mass scale.

The Press-Schechter formalism estimates the halo abundance from the Gaussian probability that a smoothed linear overdensity crosses the spherical-collapse threshold, with a factor of two enforcing total mass conservation.

The Press-Schechter differential abundance is

The peak-background split decomposes density fluctuations into long- and short-wavelength parts. A long overdensity lowers the effective local collapse threshold and thereby changes the abundance of halos.

Linear Lagrangian halo bias is the fractional response of halo abundance to a long-wavelength overdensity in the initial coordinates. For Press-Schechter halos, $b_L=({\nu }^2-1)/{\delta }_c$.

Mapping matter and halos from their initial positions adds the matter displacement contribution, giving $b=1+b_L$. Mass conservation requires the mass-weighted consistency relation $\int (dn/dM)(M/\bar{\rho })b(M)dM=1$.

The halo model represents the matter density as a sum of normalized halo profiles and decomposes its power spectrum into same-halo and distinct-halo contributions.

The one-halo term comes from pairs of mass elements in the same halo:

At linear order in halo correlations,