Structure formation

In physical cosmology, structure formation is the formation of galaxies, galaxy clusters and larger structures from small early density fluctuations. The universe, as is now known from observations of the cosmic microwave background radiation, began in a hot, dense, nearly uniform state approximately 13.8 billion years ago.^[1] However, looking at the night sky today, structures on all scales can be seen, from stars and planets to galaxies. On even larger scales, galaxy clusters and sheet-like structures of galaxies are separated by enormous voids containing few galaxies.^[2] Structure formation attempts to model how these structures were formed by gravitational instability of small early ripples in spacetime density^[3][4][5][6] or another emergence.^[7]

The modern

An important concept in structure formation is the notion of the

. The Hubble radius, which is related to the Hubble parameter ${\displaystyle H}$ as ${\displaystyle R=c/H}$, where ${\displaystyle c}$ is the

Friedmann equation

relates the energy density of the universe to the Hubble parameter and shows that the Hubble radius is continually increasing.

The

Friedmann–Lemaître–Robertson–Walker cosmology

, the Hubble radius increases more rapidly than space expands, so perturbations only enter the Hubble radius, and are not pushed out by the expansion. This paradox is resolved by cosmic inflation, which suggests that during a phase of rapid expansion in the early universe the Hubble radius was nearly constant. Thus, large scale isotropy is due to quantum fluctuations produced during cosmic inflation that are pushed outside the horizon.

Primordial plasma

The end of inflation is called

The primordial plasma would have had very slight overdensities of matter, thought to have derived from the enlargement of quantum fluctuations during inflation. Whatever the source, these overdensities gravitationally attract matter. But the intense heat of the near constant photon-matter interactions of this epoch rather forcefully seeks thermal equilibrium, which creates a large amount of outward pressure. These counteracting forces of gravity and pressure create oscillations, analogous to sound waves created in air by pressure differences.

These perturbations are important, as they are responsible for the subtle physics that result in the cosmic microwave background anisotropy. In this epoch, the amplitude of perturbations that enter the horizon oscillate sinusoidally, with dense regions becoming more rarefied and then becoming dense again, with a frequency which is related to the size of the perturbation. If the perturbation oscillates an integral or half-integral number of times between coming into the horizon and recombination, it appears as an acoustic peak of the cosmic microwave background anisotropy. (A half-oscillation, in which a dense region becomes a rarefied region or vice versa, appears as a peak because the anisotropy is displayed as a power spectrum, so underdensities contribute to the power just as much as overdensities.) The physics that determines the detailed peak structure of the microwave background is complicated, but these oscillations provide the essence.^{[9][10][11][12][13]}

Linear structure

One of the key realizations made by cosmologists in the 1970s and 1980s was that the majority of the

The physics of structure formation in this epoch is particularly simple, as dark matter perturbations with different wavelengths 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 retarded 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 the Sloan Digital Sky Survey, and by surveys of the Lyman-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.

Nonlinear structure

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 a

The final stage in evolution comes when baryons condense in the centres of galaxy haloes to form galaxies, stars and

Cosmological perturbations

Much of the difficulty, and many of the disputes, in understanding the large-scale structure of the universe can be resolved by better understanding the choice of

). 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.

Inflation and initial conditions

The initial conditions for the universe are thought to arise from the scale invariant quantum mechanical fluctuations of

cosmic inflation

. The perturbation of the background energy density at a given point ${\displaystyle \rho (\mathbf {x} ,t)}$ in space is then given by an

${\displaystyle \rho }$

${\displaystyle {\hat {\rho }}(\mathbf {k} ,t)}$

${\displaystyle \langle {\hat {\rho }}(\mathbf {k} ,t){\hat {\rho }}(\mathbf {k} ',t)\rangle =f(k)\delta ^{(3)}(\mathbf {k} -\mathbf {k'} )}$,

where ${\displaystyle \delta ^{(3)}}$ is the three-dimensional Dirac delta function and ${\displaystyle k=|\mathbf {k} |}$ is the length of ${\displaystyle \mathbf {k} }$. Moreover, the spectrum predicted by inflation is nearly

${\displaystyle \langle {\hat {\rho }}(\mathbf {k} ,t){\hat {\rho }}(\mathbf {k} ',t)\rangle =k^{n_{s}-1}\delta ^{(3)}(\mathbf {k} -\mathbf {k'} )}$,

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.

See also

References