Inphysical cosmology,structure formation describes the creation of galaxies, galaxy clusters, and larger structures via gravitational and hydrodynamic processes operating on cosmological inhomogeneities.^[1]: 458 Theuniverse, as is now known from observations of thecosmic microwave background radiation, began in a hot, dense, nearly uniform state approximately13.8 billion years ago.^[2] However, looking at the night sky today, structures on all scales can be seen, fromstars andplanets togalaxies. On even larger scales,galaxy clusters and sheet-like structures of galaxies are separated by enormous voids containing few galaxies.^[3] Structure formation applies models of gravitational instability to small ripples in mass density to predict these shapes.^[4]: 6

The modernLambda-CDM model is successful at predicting the observed large-scale distribution of galaxies, clusters and voids; but on the scale of individual galaxies there are many complications due to highly nonlinear processes involving baryonic physics, gas heating and cooling, star formation and feedback. Understanding the processes of galaxy formation is a major topic of modern cosmology research, both via observations such as theHubble Ultra-Deep Field and via large computer simulations.

Structure formation began some time after recombination, when the early universe cooled enough from expansion to allow the formation of stable hydrogen and helium atoms.^[5]: 6At this point thecosmic microwave background(CMB) is emitted; many careful measurements of the CMB provide key information about the initial state of the universe before structure formation. The measurements support a model of small fluctuations in density, critical seeds for structures to come.

In this stage, some mechanism, such ascosmic inflation, was responsible for establishing the initial conditions of the universe: homogeneity, isotropy, and flatness.^[4][6]Cosmic inflation also would have amplified minute quantum fluctuations (pre-inflation) into slight density ripples of overdensity and underdensity (post-inflation).

The early universe was dominated by radiation; in this case density fluctuations larger than the cosmic horizon grow proportional to the scale factor, as the gravitational potential fluctuations remain constant. Structures smaller than the horizon remained essentially frozen due to radiation domination impeding growth. As the universe expanded, the density of radiation drops faster than matter (due to redshifting of photon energy); this led to a crossover called matter-radiation equality at ~ 50,000 years after the Big Bang. After this all dark matter ripples could grow freely, forming seeds into which the baryons could later fall. Theparticle horizon at this epoch induces a turnover in the matterpower spectrum which can be measured in largeredshift surveys.

The universe was dominated by radiation for most of this stage, and due to the intense heat and radiation, the primordial hydrogen and helium were fully ionized into nuclei and free electrons. In this hot and dense situation, the radiation (photons) could not travel far beforeThomson scattering off an electron. The universe was very hot and dense, but expanding rapidly and therefore cooling. Finally, at a little less than 400,000 years after the 'bang', it became cool enough (around 3000 K) for the protons to capture negatively charged electrons, forming neutral hydrogen atoms. (Helium atoms formed somewhat earlier due to their larger binding energy). Once nearly all the charged particles were bound in neutral atoms, the photons no longer interacted with them and were free to propagate for the next 13.8 billion years; we currently detect those photons redshifted by a factor 1090 down to 2.725 K as the Cosmic Microwave Background Radiation (CMB) filling today's universe. Several remarkable space-based missions (COBE,WMAP,Planck), have detected very slight variations in the density and temperature of the CMB. These variations were subtle, and the CMB appears very nearly uniformly the same in every direction. However, the slight temperature variations of order a few parts in 100,000 are of enormous importance, for they trace variations in the density that were the early "seeds" from which all subsequent complex structures in the universe ultimately developed.

After the first matter condensed, the radiation traveled away, leaving a slightly inhomogeneous dark matter subject to gravitational interaction. The interaction eventually collapses the dark matter into "halos" that then attracts the normal orbaryonic matter, primarily hydrogen. As the density of hydrogen increases due gravitational attraction, stars ignite, emittingultraviolet light that re-ionizes any surrounding atoms.^[5]: 6 The gravitational interaction continues in hierarchical structure formation: the smaller gravitationally bound structures such as the first stars and stellar clusters form, then galaxies, followed bygroups, clusters andsuperclusters of galaxies.

Dark matter plays a crucial role in structure formation because it feels only the force of gravity: the gravitationalJeans instability which allows compact structures to form is not opposed by any force, such asradiation pressure. As a result, dark matter begins to collapse into a complex network ofdark matter halos well before ordinary matter, which is impeded by pressure forces. Without dark matter, the epoch ofgalaxy formation would occur substantially later in the universe than is observed.

The physics of structure formation in this epoch is particularly simple, as dark matter perturbations with differentwavelengths evolve independently. As the Hubble radius grows in the expanding universe, it encompasses larger and larger disturbances. During matter domination, all causal dark matter perturbations grow through gravitational clustering. However, the shorter-wavelength perturbations that are included during radiation domination have their growth suppressed until matter domination. At this stage, luminous, baryonic matter is expected to mirror the evolution of the dark matter simply, and their distributions should closely trace one another.

It is straightforward to calculate this "linear power spectrum" and, as a tool for cosmology, it is of comparable importance to the cosmic microwave background. Galaxy surveys have measured the power spectrum, such as theSloan Digital Sky Survey, and by surveys of theLyman-α forest. Since these studies observe radiation emitted from galaxies and quasars, they do not directly measure the dark matter, but the large-scale distribution of galaxies (and of absorption lines in the Lyman-α forest) is expected to mirror the distribution of dark matter closely. This depends on the fact that galaxies will be larger and more numerous in denser parts of the universe, whereas they will be comparatively scarce in rarefied regions.

When the perturbations have grown sufficiently, a small region might become substantially denser than the mean density of the universe. At this point, the physics involved becomes substantially more complicated. When the deviations from homogeneity are small, the dark matter may be treated as a pressureless fluid and evolves by very simple equations. In regions which are significantly denser than the background, the full Newtonian theory of gravity must be included. (The Newtonian theory is appropriate because the masses involved are much less than those required to form ablack hole, and thespeed of gravity may be ignored as the light-crossing time for the structure is still smaller than the characteristic dynamical time.) One sign that the linear and fluid approximations become invalid is that dark matter starts to formcaustics in which the trajectories of adjacent particles cross, or particles start to form orbits. These dynamics are best understood usingN-body simulations (although a variety of semi-analytic schemes, such as thePress–Schechter formalism, can be used in some cases). While in principle these simulations are quite simple, in practice they are tough to implement, as they require simulating millions or even billions of particles. Moreover, despite the large number of particles, each particle typically weighs 10⁹solar masses anddiscretization effects may become significant. The largest such simulation as of 2005 was theMillennium simulation.^[7]

The result ofN-body simulations suggests that the universe is composed largely ofvoids, whose densities might be as low as one-tenth the cosmological mean. The matter condenses in largefilaments andhaloes which have an intricate web-like structure. These formgalaxygroups, clusters andsuperclusters. While the simulations appear to agree broadly with observations, their interpretation is complicated by the understanding of how dense accumulations of dark matter spur galaxy formation.

The final stage in evolution comes when baryons condense in the centres of galaxy haloes to form galaxies, stars andquasars. As dark matter does not have electromagnetic interactions, the formation of smaller structures from dark matter is impossible. This is because dark matter cannot dissipate angular momentum, whereas ordinary baryonic matter can collapse to form dense objects by dissipating angular momentum throughradiative cooling. Understanding these processes is an enormously difficult computational problem, because they can involve the physics of gravity,magnetohydrodynamics,atomic physics,nuclear reactions,turbulence and evengeneral relativity. In most cases, it is not yet possible to perform simulations that can be compared quantitatively with observations, and the best that can be achieved are approximate simulations that illustrate the main qualitative features of a process such as a star formation.

Much of the difficulty in understanding the large-scale structure of the universe is associated with the choice ofgauge ingeneral relativity. By thescalar-vector-tensor decomposition, the metric includes fourscalar perturbations, twovector perturbations, and onetensor perturbation. Only the scalar perturbations are significant: the vectors are exponentially suppressed in the early universe, and the tensor mode makes only a small (but important) contribution in the form of primordialgravitational radiation and the B-modes of the cosmic microwave background polarization. Two of the four scalar modes may be removed by a physically meaningless coordinate transformation. Which modes are eliminated determine the infinite number of possiblegauge fixings. The most popular gauge isNewtonian gauge (and the closely related conformal Newtonian gauge), in which the retained scalars are the Newtonian potentials Φ and Ψ, which correspond exactly to the Newtonian potential energy from Newtonian gravity. Many other gauges are used, includingsynchronous gauge, which can be an efficient gauge for numerical computation (it is used byCMBFAST). Each gauge still includes some unphysical degrees of freedom. There is a so-called gauge-invariant formalism, in which only gauge invariant combinations of variables are considered.

The initial conditions for the universe are thought to arise from the scale invariant quantum mechanical fluctuations ofcosmic inflation. The perturbation of the background energy density at a given point${\displaystyle \rho (\mathbf{x},t)}$ in space is then given by anisotropic,homogeneousGaussian random field ofmean zero. This means that the spatial Fourier transform of${\displaystyle \rho }$ –${\displaystyle \hat{\rho }(\mathbf{k},t)}$ has the followingcorrelation functions

${\displaystyle ⟨\hat{\rho }(\mathbf{k},t)\hat{\rho }({\mathbf{k}}^{\prime },t)⟩=f(k){\delta }^{(3)}(\mathbf{k}-{\mathbf{k}}^{\prime })}$,

where${\displaystyle {\delta }^{(3)}}$ is the three-dimensionalDirac delta function and${\displaystyle k=|\mathbf{k}|}$ is the length of${\displaystyle \mathbf{k}}$. Moreover, the spectrum predicted by inflation is nearlyscale invariant, which means

${\displaystyle ⟨\hat{\rho }(\mathbf{k},t)\hat{\rho }({\mathbf{k}}^{\prime },t)⟩=k^{n_s-1}{\delta }^{(3)}(\mathbf{k}-{\mathbf{k}}^{\prime })}$,

where${\displaystyle n_s-1}$ is a small number. Finally, the initial conditions are adiabatic or isentropic, which means that the fractional perturbation in the entropy of each species of particle is equal.The resulting predictions fit very well with observations.

• Padmanabhan, T. (1993).Structure formation in the universe.Cambridge University Press.ISBN 978-0-521-42486-8.
• Peebles, P. J. E. (1980).The Large-Scale Structure of the Universe.Princeton University Press.ISBN 978-0-691-08240-0.

• Physical cosmological concepts
• Structure

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