Cosmological perturbation theory may be broken into two categories: Newtonian or general relativistic.
Each case uses its governing equations to compute gravitational and pressure forces which cause small perturbations to grow and eventually seed the formation of stars, quasars, galaxies and clusters.
Both cases apply only to situations where the universe is predominantly homogeneous, such as during cosmic inflation and large parts of the Big Bang.
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.
Because of the gauge invariance of general relativity, the correct formulation of cosmological perturbation theory is subtle.
There are currently two distinct approaches to perturbation theory in classical general relativity: In this section, we will focus on the effect of matter on structure formation in the hydrodynamical fluid regime.
This regime is useful because dark matter has dominated structure growth for most of the universe's history.
is the Hubble parameter) so we can take spacetime to be flat, and ignore general relativistic corrections.
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 the cosmic microwave background radiation[11] as part of the physical 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 of Newtonian 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.
[16] 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) 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.
For applications of kinetic theory, because one is required to use the full tangent bundle, it becomes convenient to use the tetrad formulation of relativistic cosmology.
The application of this approach to the computation of anisotropies in cosmic microwave background radiation[17] requires the linearization of the full relativistic kinetic theory developed by Thorne (1980)[18] and Ellis, Matravers and Treciokas (1983).
Picking this frame is equivalent to fixing the choice of timelike world lines mapped into each other.