Structure formation
Encyclopedia
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 star
s and planet
s to galaxies
and, on much larger scales still, galaxy cluster
s, and enormous voids between galaxies. How did all of this come about from the nearly uniform early universe?
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 nonlinear, 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.
, 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 monopole
s). 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 string
s have largely been refuted by increasingly precise data.
as 
, where 
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-Lemaitre-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.
and leptogenesis
are thought to occur, as the quark-gluon plasma
cools, electroweak symmetry breaking occurs and the universe becomes principally composed of ordinary proton
s, neutron
s and electron
s. 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).
, 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.
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 baryon
s 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 neutrino
s, 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 halo
s 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 wavelength
s 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.
, 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 mass
es and discretization
effects may become significant. The largest such simulation as of 2005 is the Millennium simulation
.
The result of N-body simulations suggest that the universe is composed largely of void
s, whose densities might be as low as one tenth the cosmological mean. The matter condenses in large filament
s and haloes which have an intricate web-like structure. These form galaxy
groups, clusters
and supercluster
s. 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 cluster
s. 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.
s. 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.
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 fixing
s. 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.
. The perturbation of the background energy density at a given point
in space is then given by an isotropic
, homogeneous
Gaussian random field
of mean
zero. This means that the spatial Fourier transform of
– 
has the following correlation function
s
,
where
is the three dimensional Dirac delta function
and
is the length of 
. Moreover, the spectrum predicted by inflation is nearly scale invariant, which means
,
where
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.
Physical cosmology
Physical cosmology, as a branch of astronomy, is the study of the largest-scale structures and dynamics of the universe and is concerned with fundamental questions about its formation and evolution. For most of human history, it was a branch of metaphysics and religion...
. The universe
Universe
The Universe is commonly defined as the totality of everything that exists, including all matter and energy, the planets, stars, galaxies, and the contents of intergalactic space. Definitions and usage vary and similar terms include the cosmos, the world and nature...
, as is now known from observations of the cosmic microwave background radiation
Cosmic microwave background radiation
In cosmology, cosmic microwave background radiation is thermal radiation filling the observable universe almost uniformly....
, began in a hot, dense, nearly uniform state approximately 13.7 Gyr ago
Age of the universe
The age of the universe is the time elapsed since the Big Bang posited by the most widely accepted scientific model of cosmology. The best current estimate of the age of the universe is 13.75 ± 0.13 billion years within the Lambda-CDM concordance model...
. However, looking in the sky today, we see structures on all scales, from star
Star
A star is a massive, luminous sphere of plasma held together by gravity. At the end of its lifetime, a star can also contain a proportion of degenerate matter. The nearest star to Earth is the Sun, which is the source of most of the energy on Earth...
s and planet
Planet
A planet is a celestial body orbiting a star or stellar remnant that is massive enough to be rounded by its own gravity, is not massive enough to cause thermonuclear fusion, and has cleared its neighbouring region of planetesimals.The term planet is ancient, with ties to history, science,...
s to galaxies
Galaxy
A galaxy is a massive, gravitationally bound system that consists of stars and stellar remnants, an interstellar medium of gas and dust, and an important but poorly understood component tentatively dubbed dark matter. The word galaxy is derived from the Greek galaxias , literally "milky", a...
and, on much larger scales still, galaxy cluster
Galaxy cluster
A galaxy cluster is a compact cluster of galaxies. Basic difference between a galaxy group and a galaxy cluster is that there are many more galaxies in a cluster than in a group. Also, galaxies in a cluster are more compact and have higher velocity dispersion. One of the key features of cluster is...
s, and enormous voids between galaxies. How did all of this come about from the nearly uniform early universe?
Overview
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 inflationCosmic inflationIn physical cosmology, cosmic inflation, cosmological inflation or just inflation is the theorized extremely rapid exponential expansion of the early universe by a factor of at least 1078 in volume, driven by a negative-pressure vacuum energy density. The inflationary epoch comprises the first part...
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 streamingFree streamingIn astronomy, free streaming is a term used to describe a particle, most often a photon, which propagates through a medium without scattering.-Use in defining surfaces:...
structures cannot be amplified gravitationally. Nonetheless, important evolution takes place; big bang nucleosynthesisBig Bang nucleosynthesisIn physical cosmology, Big Bang nucleosynthesis refers to the production of nuclei other than those of H-1 during the early phases of the universe...
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 matterDark matterIn astronomy and cosmology, dark matter is matter that neither emits nor scatters light or other electromagnetic radiation, and so cannot be directly detected via optical or radio astronomy...
, 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 equationLinear differential equationLinear differential equations are of the formwhere the differential operator L is a linear operator, y is the unknown function , and the right hand side ƒ is a given function of the same nature as y...
. - Nonlinear 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 causticsCaustic (optics)In optics, a caustic or caustic network is the envelope of light rays reflected or refracted by a curved surface or object, or the projection of that envelope of rays on another surface. The caustic is a curve or surface to which each of the light rays is tangent, defining a boundary of an...
– 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 clusterGalaxy clusterA galaxy cluster is a compact cluster of galaxies. Basic difference between a galaxy group and a galaxy cluster is that there are many more galaxies in a cluster than in a group. Also, galaxies in a cluster are more compact and have higher velocity dispersion. One of the key features of cluster is...
s 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 baryonBaryonA baryon is a composite particle made up of three quarks . Baryons and mesons belong to the hadron family, which are the quark-based particles...
ic matter clusters densely, as in galaxiesGalaxyA galaxy is a massive, gravitationally bound system that consists of stars and stellar remnants, an interstellar medium of gas and dust, and an important but poorly understood component tentatively dubbed dark matter. The word galaxy is derived from the Greek galaxias , literally "milky", a...
and starStarA star is a massive, luminous sphere of plasma held together by gravity. At the end of its lifetime, a star can also contain a proportion of degenerate matter. The nearest star to Earth is the Sun, which is the source of most of the energy on Earth...
s. In some cases, such as active galactic nucleiActive galactic nucleusAn active galactic nucleus is a compact region at the centre of a galaxy that has a much higher than normal luminosity over at least some portion, and possibly all, of the electromagnetic spectrum. Such excess emission has been observed in the radio, infrared, optical, ultra-violet, X-ray and...
and quasarQuasarA quasi-stellar radio source is a very energetic and distant active galactic nucleus. Quasars are extremely luminous and were first identified as being high redshift sources of electromagnetic energy, including radio waves and visible light, that were point-like, similar to stars, rather than...
s, Newtonian theory works poorly and general relativity becomes significant. It is called "gastrophysical" because of its complexity: many different, complicated effects, including gravity, magnetohydrodynamicsMagnetohydrodynamicsMagnetohydrodynamics is an academic discipline which studies the dynamics of electrically conducting fluids. Examples of such fluids include plasmas, liquid metals, and salt water or electrolytes...
and nuclear reactionNuclear reactionIn nuclear physics and nuclear chemistry, a nuclear reaction is semantically considered to be the process in which two nuclei, or else a nucleus of an atom and a subatomic particle from outside the atom, collide to produce products different from the initial particles...
s 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 nonlinear, 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
Galaxy groups and clusters
Galaxy groups and clusters are the largest known gravitationally bound objects to have arisen thus far in the process of cosmic structure formation. They form the densest part of the large scale structure of the universe...
and superclusters of galaxies. It is thought that, because of the presence of dark energy
Dark energy
In physical cosmology, astronomy and celestial mechanics, dark energy is a hypothetical form of energy that permeates all of space and tends to accelerate the expansion of the universe. Dark energy is the most accepted theory to explain recent observations that the universe appears to be expanding...
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 inflationCosmic inflation
In physical cosmology, cosmic inflation, cosmological inflation or just inflation is the theorized extremely rapid exponential expansion of the early universe by a factor of at least 1078 in volume, driven by a negative-pressure vacuum energy density. The inflationary epoch comprises the first part...
, does a good job explaining the observed flatness, homogeneity and isotropy of the universe, as well as the absence of exotic relic particles
Relic particles
Relic particles are superheavy particles hypothesized to be remnants of early cosmological expansion.-See also:*Big Bang*Cosmological Constant*Physics*Quantum Physics*Subatomic Particles...
(such as magnetic monopole
Magnetic monopole
A magnetic monopole is a hypothetical particle in particle physics that is a magnet with only one magnetic pole . In more technical terms, a magnetic monopole would have a net "magnetic charge". Modern interest in the concept stems from particle theories, notably the grand unified and superstring...
s). 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
Temperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...
fluctuations at one part in 100,000. (To put this in perspective, the same level of fluctuations on a topographic map
Topographic map
A topographic map is a type of map characterized by large-scale detail and quantitative representation of relief, usually using contour lines in modern mapping, but historically using a variety of methods. Traditional definitions require a topographic map to show both natural and man-made features...
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
COBE
The COsmic Background Explorer , also referred to as Explorer 66, was a satellite dedicated to cosmology. Its goals were to investigate the cosmic microwave background radiation of the universe and provide measurements that would help shape our understanding of the cosmos.This work provided...
(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
Gaussian random field
A Gaussian random field is a random field involving Gaussian probability density functions of the variables. A one-dimensional GRF is also called a Gaussian process....
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
Cyclic model
A cyclic model is any of several cosmological models in which the universe follows infinite, self-sustaining cycles. For example, the oscillating universe theory briefly considered by Albert Einstein in 1930 theorized a universe following an eternal series of oscillations, each beginning with a...
, pre-big bang model and holographic universe, but they remain in their nascency and are not as widely accepted. Some theories, such as cosmic string
Cosmic string
Cosmic strings are hypothetical 1-dimensional topological defects which may have formed during a symmetry breaking phase transition in the early universe when the topology of the vacuum manifold associated to this symmetry breaking is not simply connected. It is expected that at least one string...
s have largely been refuted 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 as , where is the speed of lightSpeed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...
, defines, roughly speaking, the volume of the nearby universe that has recently (in the last expansion time) been in causal
Causality (physics)
Causality is the relationship between causes and effects. It is considered to be fundamental to all natural science, especially physics. Causality is also a topic studied from the perspectives of philosophy and statistics....
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
Phantom energy
Phantom energy is a hypothetical form of dark energy with equation of state \! w Phantom energy is a hypothetical form of dark energy with equation of state...
). 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
Horizon problem
The horizon problem is a problem with the standard cosmological model of the Big Bang which was identified in the 1970s. It points out that different regions of the universe have not "contacted" each other because of the great distances between them, but nevertheless they have the same temperature...
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-Lemaitre-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, baryogenesisBaryogenesis
In physical cosmology, baryogenesis is the generic term for hypothetical physical processes that produced an asymmetry between baryons and antibaryons in the very early universe, resulting in the substantial amounts of residual matter that make up the universe today.Baryogenesis theories employ...
and leptogenesis
Leptogenesis
*In physics, leptogenesis is the process of creating a lepton-antilepton asymmetry in the early stages of the universe.*In biblical studies, leptogenesis is an alternative name for book of Jubilees....
are thought to occur, as the quark-gluon plasma
Quark-gluon plasma
A quark–gluon plasma or quark soup is a phase of quantum chromodynamics which exists at extremely high temperature and/or density. This phase consists of asymptotically free quarks and gluons, which are several of the basic building blocks of matter...
cools, electroweak symmetry breaking occurs and the universe becomes principally composed of ordinary proton
Proton
The proton is a subatomic particle with the symbol or and a positive electric charge of 1 elementary charge. One or more protons are present in the nucleus of each atom, along with neutrons. The number of protons in each atom is its atomic number....
s, neutron
Neutron
The neutron is a subatomic hadron particle which has the symbol or , no net electric charge and a mass slightly larger than that of a proton. With the exception of hydrogen, nuclei of atoms consist of protons and neutrons, which are therefore collectively referred to as nucleons. The number of...
s and electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
s. As the universe cools further, big bang nucleosynthesis
Big Bang nucleosynthesis
In physical cosmology, Big Bang nucleosynthesis refers to the production of nuclei other than those of H-1 during the early phases of the universe...
occurs and small quantities of deuterium
Deuterium
Deuterium, also called heavy hydrogen, is one of two stable isotopes of hydrogen. It has a natural abundance in Earth's oceans of about one atom in of hydrogen . Deuterium accounts for approximately 0.0156% of all naturally occurring hydrogen in Earth's oceans, while the most common isotope ...
, helium
Helium
Helium is the chemical element with atomic number 2 and an atomic weight of 4.002602, which is represented by the symbol He. It is a colorless, odorless, tasteless, non-toxic, inert, monatomic gas that heads the noble gas group in the periodic table...
and lithium
Lithium
Lithium is a soft, silver-white metal that belongs to the alkali metal group of chemical elements. It is represented by the symbol Li, and it has the atomic number 3. Under standard conditions it is the lightest metal and the least dense solid element. Like all alkali metals, lithium is highly...
nuclei
Atomic nucleus
The nucleus is the very dense region consisting of protons and neutrons at the center of an atom. It was discovered in 1911, as a result of Ernest Rutherford's interpretation of the famous 1909 Rutherford experiment performed by Hans Geiger and Ernest Marsden, under the direction of Rutherford. The...
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
Mean free path
In physics, the mean free path is the average distance covered by a moving particle between successive impacts which modify its direction or energy or other particle properties.-Derivation:...
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 lengthJeans length
Jeans' length is the critical radius of a cloud where thermal energy, which causes the cloud to expand, is counteracted by gravity, which causes the cloud to collapse...
, 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
Matter
Matter is a general term for the substance of which all physical objects consist. Typically, matter includes atoms and other particles which have mass. A common way of defining matter is as anything that has mass and occupies volume...
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 baryon
Baryon
A baryon is a composite particle made up of three quarks . Baryons and mesons belong to the hadron family, which are the quark-based particles...
s and it is known with very high accuracy that it does not emit or absorb radiation
Radiation
In physics, radiation is a process in which energetic particles or energetic waves travel through a medium or space. There are two distinct types of radiation; ionizing and non-ionizing...
. It may be composed of particles that interact through the weak interaction
Weak interaction
Weak interaction , is one of the four fundamental forces of nature, alongside the strong nuclear force, electromagnetism, and gravity. It is responsible for the radioactive decay of subatomic particles and initiates the process known as hydrogen fusion in stars...
, such as neutrino
Neutrino
A neutrino is an electrically neutral, weakly interacting elementary subatomic particle with a half-integer spin, chirality and a disputed but small non-zero mass. It is able to pass through ordinary matter almost unaffected...
s, but it cannot be composed entirely of the three known kinds of neutrinos (although some have suggested it is a sterile neutrino
Sterile neutrino
Sterile neutrinosIn scientific literature, these particles are also variously referred to as right-handed neutrinos, inert neutrinos, heavy neutrinos, or neutral heavy leptons . are a hypothetical type of neutrino that do not interact via any of the fundamental interactions of the Standard Model...
). 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
Jeans instability
In physics, the Jeans instability causes the collapse of interstellar gas clouds and subsequent star formation. It occurs when the internal gas pressure is not strong enough to prevent gravitational collapse of a region filled with matter...
which allows compact structures to form is not opposed by any force, such as radiation pressure
Radiation pressure
Radiation pressure is the pressure exerted upon any surface exposed to electromagnetic radiation. If absorbed, the pressure is the power flux density divided by the speed of light...
. As a result, dark matter begins to collapse into a complex network of dark matter halo
Dark matter halo
A dark matter halo is a hypothetical component of a galaxy, which extends beyond the edge of the visible galaxy and dominates the total mass. Since they consist of dark matter, halos cannot be observed directly, but their existence is inferred through their effects on the motions of stars and gas...
s 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 wavelength
Wavelength
In physics, the wavelength of a sinusoidal wave is the spatial period of the wave—the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...
s 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
Sloan Digital Sky Survey
The Sloan Digital Sky Survey or SDSS is a major multi-filter imaging and spectroscopic redshift survey using a dedicated 2.5-m wide-angle optical telescope at Apache Point Observatory in New Mexico, United States. The project was named after the Alfred P...
, and by surveys of the Lyman-α forest
Lyman-alpha forest
In astronomical spectroscopy, the Lyman-alpha forest is the sum of absorption lines arising from the Lyman-alpha transition of the neutral hydrogen in the spectra of distant galaxies and quasars....
. 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.
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 black holeBlack hole
A black hole is a region of spacetime from which nothing, not even light, can escape. The theory of general relativity predicts that a sufficiently compact mass will deform spacetime to form a black hole. Around a black hole there is a mathematically defined surface called an event horizon that...
, and the speed of gravity
Speed of gravity
In the context of classical theories of gravitation, the speed of gravity is the speed at which changes in a gravitational field propagate. This is the speed at which a change in the distribution of energy and momentum of matter results in subsequent alteration, at a distance, of the gravitational...
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
Caustic (optics)
In optics, a caustic or caustic network is the envelope of light rays reflected or refracted by a curved surface or object, or the projection of that envelope of rays on another surface. The caustic is a curve or surface to which each of the light rays is tangent, defining a boundary of an...
in which the trajectories of adjacent particles cross, or particles start to form orbits. These dynamics are generally best understood using N-body simulations
N-body simulation
An N-body simulation is a simulation of a dynamical system of particles, usually under the influence of physical forces, such as gravity . In cosmology, they are used to study processes of non-linear structure formation such as the process of forming galaxy filaments and galaxy halos from dark...
(although a variety of semi-analytic schemes, such as the Press-Schechter formalism
Press-Schechter formalism
The Press–Schechter formalism is a mathematical model for predicting the number of objects of a certain mass within a given volume of the Universe. It was described in a famous paper by William H...
, 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 mass
Solar mass
The solar mass , , is a standard unit of mass in astronomy, used to indicate the masses of other stars and galaxies...
es and discretization
Discretization
In mathematics, discretization concerns the process of transferring continuous models and equations into discrete counterparts. This process is usually carried out as a first step toward making them suitable for numerical evaluation and implementation on digital computers...
effects may become significant. The largest such simulation as of 2005 is the Millennium simulation
Millennium simulation
The Millennium Run, or Millennium Simulation referring to its size, was a computer N-body simulation used to investigate how matter in the Universe evolved over time...
.
The result of N-body simulations suggest that the universe is composed largely of void
Void (astronomy)
In astronomy, voids are the empty spaces between filaments, the largest-scale structures in the Universe, that contain very few, or no, galaxies. They were first discovered in 1978 during a pioneering study by Stephen Gregory and Laird A. Thompson at the Kitt Peak National Observatory...
s, whose densities might be as low as one tenth the cosmological mean. The matter condenses in large filament
Galaxy filament
In physical cosmology, galaxy filaments, also called supercluster complexes or great walls, are, so far, the largest known cosmic structures in the universe. They are massive, thread-like structures with a typical length of 50 to 80 megaparsecs h-1 that form the boundaries between large voids in...
s and haloes which have an intricate web-like structure. These form galaxy
Galaxy
A galaxy is a massive, gravitationally bound system that consists of stars and stellar remnants, an interstellar medium of gas and dust, and an important but poorly understood component tentatively dubbed dark matter. The word galaxy is derived from the Greek galaxias , literally "milky", a...
groups, clusters
Galaxy groups and clusters
Galaxy groups and clusters are the largest known gravitationally bound objects to have arisen thus far in the process of cosmic structure formation. They form the densest part of the large scale structure of the universe...
and supercluster
Supercluster
Superclusters are large groups of smaller galaxy groups and clusters and are among the largest known structures of the cosmos. They are so large that they are not gravitationally bound and, consequently, partake in the Hubble expansion.-Existence:...
s. 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
Dwarf galaxy
A dwarf galaxy is a small galaxy composed of up to several billion stars, a small number compared to our own Milky Way's 200-400 billion stars...
and globular cluster
Globular cluster
A globular cluster is a spherical collection of stars that orbits a galactic core as a satellite. Globular clusters are very tightly bound by gravity, which gives them their spherical shapes and relatively high stellar densities toward their centers. The name of this category of star cluster is...
s. 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
Dark matter
In astronomy and cosmology, dark matter is matter that neither emits nor scatters light or other electromagnetic radiation, and so cannot be directly detected via optical or radio astronomy...
and that some effect, such as warm dark matter
Warm dark matter
Warm dark matter is a hypothesized form of dark matter that has properties intermediate between those of hot dark matter and cold dark matter, causing structure formation to occur bottom-up from above their free-streaming scale, and top-down below their free streaming scale. The most common WDM...
, 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 quasarQuasar
A quasi-stellar radio source is a very energetic and distant active galactic nucleus. Quasars are extremely luminous and were first identified as being high redshift sources of electromagnetic energy, including radio waves and visible light, that were point-like, similar to stars, rather than...
s. 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
Radiative cooling
Radiative cooling is the process by which a body loses heat by thermal radiation.- Earth's energy budget :In the case of the earth-atmosphere system it refers to the process by which long-wave radiation is emitted to balance the absorption of short-wave energy from the sun.The exact process by...
. Understanding these processes is an enormously difficult computational problem, because they can involve the physics of gravity, magnetohydrodynamics
Magnetohydrodynamics
Magnetohydrodynamics is an academic discipline which studies the dynamics of electrically conducting fluids. Examples of such fluids include plasmas, liquid metals, and salt water or electrolytes...
, atomic physics
Atomic physics
Atomic physics is the field of physics that studies atoms as an isolated system of electrons and an atomic nucleus. It is primarily concerned with the arrangement of electrons around the nucleus and...
, nuclear reactions, turbulence
Turbulence
In fluid dynamics, turbulence or turbulent flow is a flow regime characterized by chaotic and stochastic property changes. This includes low momentum diffusion, high momentum convection, and rapid variation of pressure and velocity in space and time...
and even general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
. 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.
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 gaugeGauge theory
In physics, gauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a gauge theory...
in general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
better. By the scalar-vector-tensor decomposition
Scalar-vector-tensor decomposition
In cosmological perturbation theory, the scalar-vector-tensor decomposition is a decomposition of the most general linearized perturbations of the Friedmann-Lemaitre-Robertson-Walker metric into components according to their transformations under spatial rotations. It was first discovered by E. M....
, the metric includes four scalar
Scalar field
In mathematics and physics, a scalar field associates a scalar value to every point in a space. The scalar may either be a mathematical number, or a physical quantity. Scalar fields are required to be coordinate-independent, meaning that any two observers using the same units will agree on the...
perturbations, two vector
Vector field
In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...
perturbations, and one tensor
Tensor field
In mathematics, physics and engineering, a tensor field assigns a tensor to each point of a mathematical space . Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis of stress and strain in materials, and in numerous applications in the physical...
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 fixing
Gauge fixing
In the physics of gauge theories, gauge fixing denotes a mathematical procedure for coping with redundant degrees of freedom in field variables. By definition, a gauge theory represents each physically distinct configuration of the system as an equivalence class of detailed local field...
s. 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
CMBFAST
In physical cosmology, CMBFAST is a computer code, written by Uros Seljak and Matias Zaldarriaga, for computing the anisotropy of the cosmic microwave background...
). 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 inflationCosmic inflation
In physical cosmology, cosmic inflation, cosmological inflation or just inflation is the theorized extremely rapid exponential expansion of the early universe by a factor of at least 1078 in volume, driven by a negative-pressure vacuum energy density. The inflationary epoch comprises the first part...
. The perturbation of the background energy density at a given point
in space is then given by an isotropicIsotropy
Isotropy is uniformity in all orientations; it is derived from the Greek iso and tropos . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix an, hence anisotropy. Anisotropy is also used to describe situations where properties vary...
, homogeneous
Homogeneous space
In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a non-empty manifold or topological space X on which G acts continuously by symmetry in a transitive way. A special case of this is when the topological group,...
Gaussian random field
Gaussian random field
A Gaussian random field is a random field involving Gaussian probability density functions of the variables. A one-dimensional GRF is also called a Gaussian process....
of mean
Mean
In statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....
zero. This means that the spatial Fourier transform of
– has the following correlation functionCorrelation function
A correlation function is the correlation between random variables at two different points in space or time, usually as a function of the spatial or temporal distance between the points...
s
,where
is the three dimensional Dirac delta functionDirac delta function
The Dirac delta function, or δ function, is a generalized function depending on a real parameter such that it is zero for all values of the parameter except when the parameter is zero, and its integral over the parameter from −∞ to ∞ is equal to one. It was introduced by theoretical...
and
is the length of . Moreover, the spectrum predicted by inflation is nearly scale invariant, which means,where
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.