In cosmology and general relativity, inhomogeneous
cosmology in the most general sense (totally
inhomogeneous universe) is modelling the universe
as a whole with the spacetime which does not
possess any spacetime symmetries.
Typically considered cosmological spacetimes
have either the maximal symmetry which composes
of three translational symmetries and three
rotational symmetries (homogeneity and isotropy
with respect to every point of spacetime),
the translational symmetry only (homogeneous
models), or the rotational symmetry only (spherically
symmetric models).
Models with less symmetries (e.g. axisymmetric)
are also considered as symmetric.
However, it is common to call spherically
symmetric models or non-homogeneous models
as inhomogeneous.
In inhomogeneous cosmology the large-scale
structure of the universe is modelled by exact
solutions of the Einstein field equations
(i.e. non-perturbatively) unlike in the theory
of cosmological perturbations which is the
study of the Universe that takes structure
formation (galaxies, galaxy clusters, the
cosmic web) into account, but in a perturbative
way.This usually includes the study of structure
in the Universe by means of exact solutions
of Einstein's field equations (i.e. metrics)
or by spatial or spacetime averaging methods.
== Homogeneity ==
Such models are not homogeneous, but may allow
effects which can be interpreted as dark energy,
or can lead to cosmological structures such
as voids or galaxy clusters.
== Perturbative approach ==
In contrast, perturbation theory, which deals
with small perturbations from e.g. a homogeneous
metric, only holds as long as the perturbations
are not too large, and N-body simulations
use Newtonian gravity which is only a good
approximation when speeds are low and gravitational
fields are weak.
== Non-perturbative approach ==
Work towards a non-perturbative approach includes
the Relativistic Zel'dovich Approximation.
As of 2016, Thomas Buchert, George Ellis,
Edward Kolb and their colleagues, judged that
if the Universe is described by cosmic variables
in a backreaction scheme that includes coarse-graining
and averaging, then the question of whether
dark energy is an artefact of the way of using
the Einstein equation is an unanswered question.
== Exact solutions ==
The first historically examples of inhomogeneous
(though spherically symmetric) solutions are
the Lemaître–Tolman metric (or LTB model
- Lemaître–Tolman-Bondi ). Stephani metric
can be spherically symmetric or totally inhomogeneous.
Some other examples are the Szekeres metric,
Szafron metric, Barnes metric, Kustaanheimo-Qvist
metric, and Senovilla metric.
The Bianchi metrics as given in Bianchi classification
and Kantowski-Sachs metrics are homogeneous.
== Averaging methods ==
The best-known averaging approach is the scalar
averaging approach, leading to the kinematical
backreaction and mean 3-Ricci curvature functionals;
the main equations are often referred to as
the set of Buchert equations
