Inphysical cosmology,cosmological perturbation theory^{[1][2][3][4][5]} is the theory by which theevolution of structure is understood in theBig Bang model. Cosmologicalperturbation theory may be broken into two categories:Newtonian orgeneral relativistic. Each case uses its governing equations to compute gravitational and pressure forces which cause small perturbations to grow and eventually seed the formation ofstars,quasars,galaxies andclusters. Both cases apply only to situations where the universe is predominantly homogeneous, such as duringcosmic inflation 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 asN-body simulations, must be used. When deciding whether to use general relativity for perturbation theory, note that Newtonian physics is only applicable in some cases such as for scales smaller than the Hubble horizon, where spacetime is sufficiently flat, and for which speeds are non-relativistic.

Due to thegauge invariance ofgeneral relativity, the correct formulation of cosmological perturbation theory is subtle. In particular, when describing an inhomogeneous spacetime, there are often no preferred coordinate choices. There are currently two distinct approaches to perturbation theory in classical general relativity:

• gauge-invariant perturbation theory based on foliating a spacetime with hyper-surfaces, and
• 1+3 covariant gauge-invariant perturbation theory based on threading a spacetime with frames.

Focusing on the effect of matter on structure formation in thehydrodynamical fluid regime is useful becausedark matter has dominated structure growth for most of the universe's history. This regime is on sub-Hubble scales (where${\displaystyle H}$ is theHubble parameter), whereupon spacetime can be taken to be flat, and the effects of general relativity can be neglected. But these scales are above a cut-off, such that perturbations in pressure and density are sufficiently linear${\displaystyle \delta P,\delta \rho \ll 1}$. Assume low pressure${\displaystyle P\ll \rho ,}$ so that radiative effects can be ignored, and low speeds${\displaystyle u\ll c}$, making relativistic effects negligible.

The first governing equation follows from matter conservation, namely, the continuity equation^[6]

${\displaystyle \frac{\mathrm{\partial }\rho }{\mathrm{\partial }t}+3H\rho +\frac{1}{a}\cdot \mathrm{\nabla }\left(\rho \mathit{v}\right)=0,}$

where${\displaystyle a}$ is thescale factor and${\displaystyle \mathit{v}}$ is thepeculiar velocity. All variables are evaluated at time${\displaystyle t}$ and the divergence${\displaystyle \mathrm{\nabla }}$ is incomoving coordinates. The second equation is the Euler equation

${\displaystyle \rho \frac{d\overrightarrow{u}}{dt}=\rho \left(\frac{\mathrm{\partial }}{\mathrm{\partial }t}+\frac{1}{a}\mathit{v}\cdot \mathrm{\nabla }\right)\mathit{u}=-\frac{1}{a}\mathrm{\nabla }P-\frac{1}{a}\rho \mathrm{\nabla }\mathrm{\Phi },}$

where${\displaystyle \mathrm{\Phi }}$ is thegravitational potential. It is a consequence of momentum conservation. In the Newtonian limit, the potential obeysPoisson's equation for gravity

${\displaystyle \frac{1}{a^2}{\mathrm{\nabla }}^2\mathrm{\Phi }=4\pi G\rho .}$

The equations so far are fully nonlinear, and can be hard to interpret intuitively. It is therefore useful to consider a perturbative expansion and examine each order separately,

${\displaystyle \rho =\overline{\rho }(1+\delta ),\mathit{u}=Ha\mathit{x}+\mathit{v},P=\overline{P}+\delta P,\mathrm{\Phi }=\overline{\mathrm{\Phi }}+\delta \mathrm{\Phi }.}$

where${\displaystyle \mathit{x}}$ is a comoving coordinate.

At linear order, the continuity equation becomes

${\displaystyle \dot{\delta }=-\frac{1}{a}\theta ,}$

where${\displaystyle \theta \equiv \mathrm{\nabla }\cdot \mathit{v}}$ is the velocity divergence. And the linear Euler equation is

${\displaystyle \overline{\rho }\left(\dot{\mathit{v}}+H\mathit{v}\right)=-\frac{1}{a}\mathrm{\nabla }\delta P-\frac{1}{a}\overline{\rho }\mathrm{\nabla }\delta \mathrm{\Phi }.}$

Combining the continuity equation, Euler, and Poisson equations yields a simple master equation governing evolution

${\displaystyle \left(\frac{{\mathrm{\partial }}^2}{{\mathrm{\partial }}^{2}t}+2H\frac{\mathrm{\partial }}{\mathrm{\partial }t}-c_s^2\frac{1}{a}{\mathrm{\nabla }}^2-4\pi G\overline{\rho }\right)\delta =0,}$

where the speed of sound is defined to be${\displaystyle c_s^2\equiv \delta P/\overline{\rho }\delta }$, giving aclosure relation. This master equation admits wave solutions in${\displaystyle \delta (\mathit{x},t)}$, describing how matter fluctuations grow over time due to a combination of competing effects: the fluctuation's self-gravity, pressure forces, the universe's expansion, and the background gravitational field.

The gauge-invariant perturbation theory is based on developments by Bardeen (1980),^[7] Kodama and Sasaki (1984)^[8] building on the work of Lifshitz (1946).^[9] This is the standard approach to perturbation theory of general relativity for cosmology.^[10] This approach is widely used for the computation of anisotropies in thecosmic microwave background radiation^[11] as part of thephysical cosmology program and focuses on predictions arising from linearisations that preserve gauge invariance with respect to Friedmann-Lemaître-Robertson-Walker (FLRW) models. This approach draws heavily on the use ofNewtonian like analogue and usually has as it starting point the FRW background around which perturbations are developed. The approach isnon-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.

Inrelativistic cosmology using the Lagrangian threading dynamics of Ehlers (1971)^[12] and Ellis (1971)^[13] it is usual to use the gauge-invariant covariant perturbation theory developed by Hawking (1966)^[14] and Ellis and Bruni (1989).^[15] Here rather than starting with a background and perturbing away from that background one starts with fullgeneral relativity and systematically reduces the theory down to one that is linear around a particular background.^[16] The approach islocal and both covariant as well as gauge invariant but can be non-linear because the approach is built around the localcomoving observer frame (seeframe bundle) 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 ofkinetic theory, because one is required to use the fulltangent bundle, it becomes convenient to use thetetrad formulation of relativistic cosmology. The application of this approach to the computation of anisotropies incosmic microwave background radiation^[17] requires the linearization of the fullrelativistic kinetic theory developed by Thorne (1980)^[18] and Ellis, Matravers and Treciokas (1983).^[19]

In relativistic cosmology there is a freedom associated with the choice of threading frame; this frame choice is distinct from the choice associated with coordinates. Picking this frame is equivalent to fixing the choice of timelike world lines mapped into each other. This reduces thegauge 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 emerge from perturbative general relativity with the choice of theNewtonian 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.

• Primordial fluctuations
• Cosmic microwave background spectral distortions

Seephysical cosmology textbooks.

• Ellis, George F. R.; van Elst, Henk (1999). "Cosmological models". In Marc Lachièze-Rey (ed.).Theoretical and Observational Cosmology: Proceedings of the NATO Advanced Study Institute on Theoretical and Observational Cosmology. Cargèse Lectures 1998. NATO Science Series: Series C. Vol. 541.Kluwer Academic. pp. 1–116.arXiv:gr-qc/9812046.Bibcode:1999ASIC..541....1E.

• Physical cosmology
• General relativity

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