Cosmological perturbation theory
Encyclopedia
In physical cosmology
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...

, cosmological perturbation theory is the theory by which the evolution of structure is understood in the big bang
Big Bang
The Big Bang theory is the prevailing cosmological model that explains the early development of the Universe. According to the Big Bang theory, the Universe was once in an extremely hot and dense state which expanded rapidly. This rapid expansion caused the young Universe to cool and resulted in...

 model. It uses 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...

 to compute the gravitational forces causing small perturbations to grow and eventually seed the formation of star
Star formation
Star formation is the process by which dense parts of molecular clouds collapse into a ball of plasma to form a star. As a branch of astronomy star formation includes the study of the interstellar medium and giant molecular clouds as precursors to the star formation process and the study of young...

s, quasar
Quasar
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, galaxies and clusters. It only applies to situations in which the universe is predominantly homogeneous, such as during cosmic inflation
Cosmic 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...

 and large parts of the big bang. The universe is believed to still be homogeneous enough that the theory is a good approximation on the largest scales, but on smaller scales more involved techniques, such as N-body simulation
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...

s, must be used.

Because of the gauge invariance of 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...

, the correct formulation of cosmological perturbation theory is subtle. There are currently two distinct approaches to perturbation theory in classical general relativity:
  • gauge-invariant perturbation theory based on foliating a space-time with hyper-surfaces, and
  • 1+3 covariant gauge-invariant perturbation theory based on threading a space-time with frames.

Gauge-invariant Perturbation Theory

The gauge-invariant perturbation theory is based on developments by Bardeen (1980), Kodama and Sasaki (1984) building on the work of Lifshitz (1946). This is the standard approach to perturbation theory of general relativity for cosmology . This approach is widely used for the computation of anisotropies in the cosmic microwave background radiation
Cosmic microwave background radiation
In cosmology, cosmic microwave background radiation is thermal radiation filling the observable universe almost uniformly....

 as part of the physical cosmology
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...

 program and focuses on predictions arising from linearisations that preserve gauge invariance with respect to Friedmann-Lemaitre-Robertson-Walker (FLRW) models. This approach draws heavily on the use of Newtonian
Newtonian
Newtonian refers to the work of Isaac Newton, in particular:* Newtonian mechanics, also known as classical mechanics* Newtonian telescope, a type of reflecting telescope* Newtonian cosmology* Newtonian dynamics...

 like analogue and usually has as it starting point the FRW background around which perturbations are developed. The approach is non-local and coordinate dependent but gauge invariant as the resulting linear framework is built from a specified family of background hyper-surfaces which are linked by gauge preserving mappings to foliate the space-time. Although intuitive this approach does not deal well with the nonlinearities natural to general relativity.

1+3 Covariant Gauge-invariant Perturbation Theory

In relativistic cosmology using the Lagrangian threading dynamics of Ehlers (1971) and Ellis (1971) it is usual to use the gauge-invariant covariant perturbation theory developed by Hawking (1966) and Ellis and Bruni (1989). Here rather than starting with a background and perturbing away from that background one starts with full 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...

 and systematically reduces the theory down to one that is linear around a particular background . The approach is local and both covariant as well as gauge invariant but can be non-linear because the approach is built around the local comoving observer frame (see frame bundle
Frame bundle
In mathematics, a frame bundle is a principal fiber bundle F associated to any vector bundle E. The fiber of F over a point x is the set of all ordered bases, or frames, for Ex...

) which is used to thread the entire space-time. This approach to perturbation theory produces differential equations that are of just the right order needed to describe the true physical degrees of freedom and as such no non-physical gauge modes exist. It is usual to express the theory in a coordinate free manner. For applications of kinetic theory
Kinetic theory
The kinetic theory of gases describes a gas as a large number of small particles , all of which are in constant, random motion. The rapidly moving particles constantly collide with each other and with the walls of the container...

, because one is required to use the full tangent bundle
Tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is the disjoint unionThe disjoint union assures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector...

, it becomes convenient to use the tetrad
Tetrad
Tetrad may refer to:* Tetrad , Bivalents or Tetrad of homologous chromosomes consisting of four synapsed chromatids that become visible during the Pachytene stage of meiotic prophase...

 formulation of relativistic cosmology. The application of this approach to the computation of anisotropies in cosmic microwave background radiation
Cosmic microwave background radiation
In cosmology, cosmic microwave background radiation is thermal radiation filling the observable universe almost uniformly....

 requires the linearization of the full relativistic kinetic theory developed by Thorne (1980) and Ellis, Matravers and Treciokas (1983).

Gauge freedom and frame fixing

In relativistic cosmology there is a freedom associated with the choice of threading frame, this frame choice is distinct from choice associated with coordinates. Picking this frame is equivalent to fixing the choice of timelike world lines mapped into each other, this reduces the gauge freedom it does not fix the gauge but the theory remains gauge invariant under the remaining gauge freedoms. In order to fix the gauge a specification of correspondences between the time surfaces in the real universe (perturbed) and the background universe are required along with the correspondences between points on the initial spacelike surfaces in the background and in the real universe. This is the link between the gauge-invariant perturbation theory and the gauge-invariant covariant perturbation theory. Gauge invariance is only guaranteed if the choice of frame coincides exactly with that of the background; usually this is trivial to ensure because physical frames have this property.

Newtonian-like equations

Newtonian-like equations emerge from perturbative general relativity with the choice of the Newtonian gauge; the Newtonian gauge provides the direct link between the variables typically used in the gauge-invariant perturbation theory and those arising from the more general gauge-invariant covariant perturbation theory.
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