Inhomogeneous cosmology

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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.[1]

This usually includes the study of structure in the Universe by means of exact solutions of Einstein's field equations (i.e. metrics)[1] or by spatial or spacetime averaging methods.[2]


Such models are not homogeneous,[3] but may allow effects which can be interpreted as dark energy, or can lead to cosmological structures such as voids or galaxy clusters.[1][2]

Perturbative approach[edit]

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[edit]

Work towards a non-perturbative approach includes the Relativistic Zel'dovich Approximation.[4] As of 2016, Thomas Buchert, George Ellis, Edward Kolb and their colleagues,[5] 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.[6]

Exact solutions[edit]

The first historically examples of inhomogeneous (though spherically symmetric) solutions are the Lemaître–Tolman metric (or LTB model - Lemaître–Tolman-Bondi [7][8][9]). Stephani metric [10][11][12] can be spherically symmetric or totally inhomogeneous. Some other examples are the Szekeres metric, Szafron metric, Barnes metric, Kustaanheimo-Qvist metric, and Senovilla metric.[1] The Bianchi metrics as given in Bianchi classification and Kantowski-Sachs metrics are homogeneous.

Averaging methods[edit]

The best-known[according to whom?] averaging approach is the scalar averaging approach[further explanation needed], leading to the kinematical backreaction and mean 3-Ricci curvature functionals;[2] the main equations are often referred to as the set of Buchert equations.


  1. ^ a b c d Krasinski, A., Inhomogeneous Cosmological Models, (1997) Cambridge UP, ISBN 0-521-48180-5
  2. ^ a b c Buchert, Thomas (2008). "Dark Energy from structure: a status report". General Relativity and Gravitation. 40 (2–3): 467–527. arXiv:0707.2153. Bibcode:2008GReGr..40..467B. doi:10.1007/s10714-007-0554-8.
  3. ^ Ryan, M.P., Shepley, L.C., Homogeneous Relativistic Cosmologies, (1975) Princeton UP, ISBN 0-691-08146-8
  4. ^ Buchert, Thomas; Nayet, Charly; Wiegand, Alexander (2013). "Lagrangian theory of structure formation in relativistic cosmology II: average properties of a generic evolution model". Physical Review D. 87 (12): 123503. arXiv:1303.6193. Bibcode:2013PhRvD..87l3503B. doi:10.1103/PhysRevD.87.123503.
  5. ^ Buchert, Thomas; Carfora, Mauro; Ellis, George F.R.; Kolb, Edward W.; MacCallum, Malcolm A.H.; Ostrowski, Jan J.; Räsänen, Syksy; Roukema, Boudewijn F.; Andersson, Lars; Coley, Alan A.; Wiltshire, David L. (2015-10-13). "Is there proof that backreaction of inhomogeneities is irrelevant in cosmology?". Classical and Quantum Gravity. 32 (21): 215021. arXiv:1505.07800. Bibcode:2015CQGra..32u5021B. doi:10.1088/0264-9381/32/21/215021.
  6. ^ Buchert, Thomas; Carfora, Mauro; Ellis, George F.R.; Kolb, Edward W.; MacCallum, Malcolm A.H.; Ostrowski, Jan J.; Räsänen, Syksy; Roukema, Boudewijn F.; Andersson, Lars; Coley, Alan A.; Wiltshire, David L. (2016-01-20). "The Universe is inhomogeneous. Does it matter?". CQG+. Institute of Physics. Archived from the original on 2016-01-21. Retrieved 2016-01-21.
  7. ^ Lemaître, George (1933). "L'univers en expansion". Ann. Soc. Sci. Bruxelles. A53: 51. Bibcode:1933ASSB...53...51L.
  8. ^ Tolman, Richard C. (1934). "Effect of Inhomogeneity on Cosmological Models" (PDF). Proc. Natl. Acad. Sci. U.S.A. 20 (3): 169–176. Bibcode:1934PNAS...20..169T. doi:10.1073/pnas.20.3.169.
  9. ^ Bondi, Hermann (1947). "Spherically Symmetrical Models in General Relativity". Mon. Not. R. Astron. Soc. 107 (5–6): 410–425. Bibcode:1947MNRAS.107..410B. doi:10.1093/mnras/107.5-6.410.
  10. ^ Stephani, Hans (1947). "Über Lösungen der Einsteinschen Feldgleichungen, die sich in einen fünfdimensionalen flachen Raum einbetten lassen". Commun. Math. Phys. 4 (2): 137–142. doi:10.1007/BF01645757.
  11. ^ Dabrowski, Mariusz P. (1993). "Isometric Embedding of the Spherically Symmetric Stephani Universe. Some Explicit Examples". J. Math. Phys. 34 (4): 1447–1479. Bibcode:1993JMP....34.1447D. doi:10.1063/1.530166.
  12. ^ Balcerzak, Adam; Dabrowski, Mariusz P.; Denkiewicz, Tomasz; Polarski, David; Puy, Denis (2015). "Critical assessment of some inhomogeneous pressure Stephani models". Physical Review. D91 (8): 0803506. arXiv:1409.1523. Bibcode:2015PhRvD..91h3506B. doi:10.1103/PhysRevD.91.083506.