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Structure formation

Structure formation refers to a fundamental problem in physical cosmology. The universe, as is now known from observations of the cosmic microwave background radiation, began in a hot, dense, nearly uniform state approximately 13.7 Gyr ago. However, looking in the sky today, we see structures on all scales, from stars and planets to galaxies and, on much larger scales still, galaxy clusters, and enormous voids between galaxies. How did all of this come about from the nearly uniform early universe?


Under present models, the structure of the visible universe was formed in the following stages:

  • The very early universe In this stage, some mechanism, such as cosmic inflation is responsible for establishing the initial conditions of the universe: homogeneity, isotropy and flatness.
  • The primordial plasma The universe is dominated by radiation for most of this stage, and due to free-streaming structures cannot be amplified gravitationally. Nonetheless, important evolution takes place, such as big bang nucleosynthesis creates the primordial elements and the cosmic microwave background is emitted. The detailed anisotropy structure of the cosmic microwave background is also created in this epoch.
  • Linear growth of structure Once matter, in particular cold dark matter, dominates the universe gravitational collapse can start to amplify the tiny inhomogeneities left by cosmic inflation, causing matter to fall towards dense regions and making rarefied regions more rarefied. In this epoch, the density inhomogeneities are described by a simple linear differential equation.
  • Non-linear growth of structure As the dense regions become denser, the linear approximation describing density inhomogeneities begins to break down – adjacent particles may even begin to cross in caustics – and a more detailed treatment, using the full Newtonian theory of gravity, becomes necessary. (Aside from the background expansion of the universe, which is due to general relativity, evolution on these comparatively small scales is usually well approximated by the Newtonian theory.) This is where structures, such as galaxy clusters and galaxy haloes begin to form. Still, in this regime only gravitational forces are significant because dark matter, which is thought to have very weak interactions, is the dominant player.
  • "Gastrophysical" evolution The final step of the evolution is when electromagnetic forces become important in the evolution of structure, where baryonic matter clusters densely, as in galaxies and stars. In some cases, such as active galactic nuclei and quasars, Newtonian theory works poorly and general relativity becomes significant. It is called "gastrophysical" because of its complexity: many different, complicated effects, including gravity, magnetohydrodynamics and nuclear reactions must be taken into account.

The last three stages of evolution occur at different times depending on the scale. The largest scales in the universe are still well-approximated by linear theory, whereas galaxy clusters and superclusters are non-linear, and many phenomena in the local galaxy must be modelled by a more nuanced approach, accounting for all the forces. This is what is called hierarchical structure formation: the smallest gravitationally bound structures – quasars and galaxies – form first, followed by groups, clusters and superclusters of galaxies. It is thought that, because of the presence of dark energy in our universe, no larger structures will be able to form.

Very early universe

The very early universe is still a poorly-understood epoch, from the viewpoint of fundamental physics. The prevailing theory, cosmic inflation, does a good job explaining the observed flatness, homogeneity and isotropy of the universe, as well as the absence of exotic relic particles (such as magnetic monopoles). In addition, it has made a crucial prediction that has been borne out by observation: that the primordial universe would have tiny perturbations which seed the formation of structure in the later universe. These fluctuations, while they form the foundation for all structure in the universe, appear most clearly as tiny temperature fluctuations at one part in 100,000. (To put this in perspective, the same level of fluctuations on a topographic map of the United States would show no feature higher than a few meters high.) These fluctuations are critical, because they provide the seeds from which the largest structures within the universe can grow and eventually collapse to form galaxies and stars. COBE (Cosmic Background Explorer) provided the first detection of the intrinsic fluctuations in the cosmic microwave background radiation in the 1990s.

These perturbations are thought to have a very specific character: they form a Gaussian random field whose covariance function is diagonal and nearly scale-invariant. The observed fluctuations appear to have exactly this form, and in addition the spectral index measured by WMAP – the spectral index measures the deviation from a scale-invariant (or Harrison-Zel'dovich) spectrum – is very nearly the value predicted by the simplest and most robust models of inflation. Another important property of the primordial perturbations, that they are adiabatic (or isentropic between the various kinds of matter that compose the universe), is predicted by cosmic inflation and has been confirmed by observations.

Other theories of the very early universe, which are claimed to make very similar predictions, have been proposed, such as the brane gas cosmology, cyclic model, pre-big bang model and holographic universe, but they remain in their nascency and are not as widely accepted. Some theories, such as cosmic strings have largely been falsified by increasingly precise data.

The horizon problem

An extremely important concept in the theory of structure formation is the notion of the Hubble radius, often called simply the horizon as it is closely related to the particle horizon. The Hubble radius, which is related to the Hubble parameter H as R=c/H, where c is the speed of light, defines, roughly speaking, the volume of the nearby universe that has recently (in the last expansion time) been in causal contact with an observer. Since the universe is continually expanding, its energy density is continually decreasing (in the absence of truly exotic matter such as phantom energy). The Friedmann equation relates the energy density of the universe to the Hubble parameter, and shows that the Hubble radius is continually increasing.

The horizon problem of the big bang cosmology says that, without inflation, perturbations were never in causal contact before they entered the horizon and thus the homogeneity and isotropy of, for example, the large scale galaxy distributions cannot be explained. This is because, in an ordinary Friedmann-Robertson-Walker cosmology, the Hubble radius increases more rapidly than space expands, so perturbations are only ever entering the Hubble radius, and they are not being pushed out by the expansion of space. This paradox is resolved by cosmic inflation, which suggests that there was a phase of very rapid expansion in the early universe in which the Hubble radius was very nearly constant. Thus, the large scale isotropy that we see today is due to quantum fluctuations produced during cosmic inflation being pushed outside the horizon.

Primordial plasma

The end of inflation is called reheating, when the inflation particles decay into a hot, thermal plasma of other particles. In this epoch, the energy content of the universe is entirely radiation, with standard model particles having relativistic velocities. As the plasma cools, baryogenesis and leptogenesis are thought to occur, as the quark-gluon plasma cools, electroweak symmetry breaking occurs and the universe becomes principally composed of ordinary protons, neutrons and electrons. As the universe cools further, big bang nucleosynthesis occurs and small quantities of deuterium, helium and lithium nuclei are created. As the universe cools and expands, the energy in photons begins to redshift away, particles become non-relativistic and ordinary matter begins to dominate the universe. Eventually, atoms begin to form as free electrons bind to nuclei. This suppresses Thompson scattering of photons. Combined with the rarefaction of the universe (and consequent increase in the mean free path of photons), this makes the universe transparent and the cosmic microwave background is emitted at recombination (the surface of last scattering).

Acoustic oscillations

The amplitude of structures does not grow substantially during this epoch. For dark matter the expansion of space (which is caused by the large radiation component) is so rapid that growth is highly suppressed for the non-relativistic dark matter particles. Moreover, because dark matter is pressureless, free-streaming prevents the growth of small structures. In the relativistic fluid, on the other hand, the very large pressure prevents the growth of structures larger than the Jeans length, which is very nearly equal to the Hubble radius for radiation. This causes perturbations to be damped.

These perturbations are still very important, however, as they are responsible for the subtle physics that result in the cosmic microwave background anisotropy. In this epoch, the amplitude of perturbations which 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 which determines the detailed peak structure of the microwave background is complicated, but these oscillations provide the essence.

Linear structure

One of the key realizations made by cosmologists in the 1970s and 1980s was that the majority of the matter content of the universe was composed not of atoms, but rather a mysterious form of matter known as dark matter. Dark matter interacts through the force of gravity, but it is not composed of baryons and it is known with very high accuracy that it does not emit or absorb radiation. It may be composed of particles that interact through the weak interaction, such as neutrinos, but it cannot be composed entirely of the three known kinds of neutrinos (although some have suggested it is a sterile neutrino). Recent evidence suggests that there is about five times as much dark matter as baryonic matter, and thus the dynamics of the universe in this epoch are dominated by dark matter.

Dark matter plays a key role in structure formation because it feels only the force of gravity: the gravitational Jeans instability which allows compact structures to form is not opposed by any force, such as radiation pressure. As a result, dark matter begins to collapse into a complex network of dark matter halos well before ordinary matter, which is impeded by pressure forces. Without dark matter, the epoch of galaxy 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 different wavelengths evolve independently. As the Hubble radius grows in the expanding universe, it encompasses larger and larger perturbations. During matter domination, all causal dark matter perturbations grow through gravitational clustering. However, the shorter-wavelength perturbations that are encompassed during radiation domination have their growth retarded until matter domination. At this stage, luminous, baryonic matter is expected to simply mirror the evolution of the dark matter, and their distributions should closely trace one another.

It is a simple matter to calculate this "linear power spectrum" and, as a tool for cosmology, it is of comparable importance to the cosmic microwave background. The power spectrum has been measured by galaxy surveys, such as the Sloan Digital Sky Survey, and by surveys of the Lyman-α forest. Since these surveys 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 closely mirror the distribution of dark matter. 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.

Non-linear structure

When the perturbations have grown sufficiently, a small region might become substantially more dense 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 more dense 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 black hole, and the speed 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 are that dark matter starts to form caustics in which the trajectories of adjacent particles cross, or particles start to form orbits. These dynamics are generally best understood using N-body simulations (although a variety of semi-analytic schemes, such as the Press-Schechter formalism, can be used in some cases). While in principle these simulations are quite simple, in practice they are very difficult to implement, as they require simulating millions or even billions of particles. Moreover, despite the large number of particles, each particle typically weighs 109 solar masses and discretization effects may become significant. The largest such simulation is the recent Millennium simulation.

The result of N-body simulations suggest that the universe is composed largely of voids, whose densities might be as low as one tenth the cosmological mean. The matter condenses in large filaments and haloes which have an intricate web-like structure. These form galaxy groups, clusters and superclusters. 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. In particular, many more small haloes form than we see in astronomical observations as dwarf galaxies and globular clusters. This is known as the galaxy bias problem, and a variety of explanations have been proposed. Most account for it as an effect in the complicated physics of galaxy formation, but some have suggested that it is a problem with our model of dark matter and that some effect, such as warm dark matter, prevents the formation of the smallest haloes.

Gastrophysical evolution

The final stage in evolution comes when baryons condense in the centers of galaxy haloes to form galaxies, stars and quasars. A paradoxical aspect of structure formation is that while dark matter greatly accelerates the formation of dense haloes, because dark matter does not have radiation pressure, the formation of smaller structures from dark matter is impossible because dark matter cannot dissipate angular momentum, whereas ordinary baryonic matter can collapse to form dense objects by dissipating angular momentum through radiative 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 even general 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 star formation.

Modelling structure formation

Cosmological perturbations

Much of the difficulty, and many of the disputes, in understanding the large-scale structure of the universe can be resolved by understanding the choice of gauge in general relativity better. By the scalar-vector-tensor decomposition, the metric includes four scalar perturbations, two vector perturbations, and one tensor 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 primordial gravitational 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 possible gauge fixings. The most popular gauge is Newtonian 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, including synchronous gauge, which can be an efficient gauge for numerical computation (it is used by CMBFAST). 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 rho(mathbf{x},t) in space is then given by an isotropic, homogeneous Gaussian random field of mean zero. This means that the spatial Fourier transform of rhohat{rho}(mathbf{k},t) has the following correlation functions

where delta^{(3)} is the three dimensional Dirac delta function and k=|mathbf{k}| is the length of mathbf{k}. Moreover, the spectrum predicted by inflation is nearly scale invariant, which means
where 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.


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