inhomogeneous cosmology





physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes

theory (physics), model (physics)

experiment, measurement, computable physics



What is called inhomogeneous cosmology is the study of cosmology via cosmological solutions to Einstein's equations, without assuming or constraining these solutions to be spatially homogeneous (in the technical sense).

This is in contrast to the standard model of cosmology, based on FRW model-type solutions to Einstein's equations, where spacetime is assumed to be spatially homogeneous (an assumption also known as the cosmological principle).

Of course the observable universe is clearly not exactly homogeneous (due to initial CMB fluctuation and ensuing structure formation), but the question is whether on cosmic scales the deviation from homogeneity is small enough that it may be neglected, to first approximation, for the purpose of modelling cosmological evolution, or whether it exerts relevant “backreaction” on the global evolution of spacetime. For a clean account of the question see Kolb-Marra-Matarrese 10.

It has been shown that the effect of such backreaction is small or invisible if the inhomogeneity is modeled in a non-relativistic (i.e. Newtonian) limit, instead of taking relativity into account (Buchert 00, Buchert-Ehlers 95), which however is the standard approximation currently used in comparing the standard model of cosmology to data.

Effective dark energy from inhomogeneity?

The standard model of cosmology assumes that such inhomogeneities may be neglected to zeroth order, and studies structure formation as a perturbation about a spatially homogeneous FRW model background spacetime.

Given that the standard model of cosmology faces some issues (e.g. BCKRW 15, Riess et. al. 16) related to dark energy (a cosmological constant, and possibly related issues such as cosmic inflation), it has been suggested that these may be but an artefact of the overly idealistic approximation of cosmic homogeneity, and that a more accurate inhomogeneous cosmology would not need to assume any dark energy/cosmological constant.

References suggesting/discussing/checking this idea include the following: Célérier 00, Buchert 00, Wetterich 01, Schwarz 02, Räsänen 03, Alnes-Amarzguioui-Gron 06, Alnes-Amarzguioui 06, Buchert-Larena-Alimi 06, Enqvist-Mattsson 06 Buchert 07, Sarkar 08, Buchert 11, Buchert-Räsänen 11, Scharf 13, Smoller-Temple-Vogler 14, Moffat 16, Sarkar et al. 18, Sarkar 18.

A qualitative discussion of how inhomogeneity may cause accelerated cosmic expansion is given in Räsänen 10, section 3: “Understanding acceleration”:

In general, underdense regions [[“voids”]] are negatively curved and expand faster than the average, while overdense regions are positively curved and expand slower. (Räsänen 03, p. 15)

[...][...] as the volume occupied by [[inhomogeneous]] structures grows (along with the density contrast of typical structures), the expansion rate becomes dominated by voids, since their volume is large [...][...] overdense regions slow down more as their density contrast grows, and eventually they turn around and collapse to form stable structures. Underdense regions become ever emptier, and their deceleration decreases. Regions thus become more differentiated, and the variance of the expansion rate grows. (Räsänen 03, p. 25)

In an inhomogeneous space, different regions expand at different rates. Regions with faster expansion rate increase their volume more rapidly, by definition. Therefore the fraction of volume in faster expanding regions rises, so the average expansion rate can rise (Räsänen 10, p. 8)

The acceleration is not due to regions speeding up locally, but due to the slower region becoming less represented in the average. First the overdense region brings down the expansion rate, but its fraction of the volume falls because of the slower expansion, so eventually the underdense region takes over and the average expansion rate rises.

[...][...] After the overdense region stops being important, the expansion rate will be given by the underdense region alone, and the expansion will again decelerate. Acceleration is a transient phenomenon associated with the volume becoming dominated by the underdense region.

[...][...] Whether the expansion accelerates depends on how rapidly the faster expanding regions catch up with the slower ones, roughly speaking by how steeply the HtH t curve rises. This is why the variance contributes positively to acceleration: the larger the variance, the bigger the difference between fast and slow regions, and the more rapidly the fast regions take over.

[...][...] So there is no ambiguity: accelerated average expansion due to inhomogeneities is possible. The question is whether the distribution of structures in the universe is such that this mechanism is realised (Räsänen 10, p. 10)

An analytic proof of this qualitative picture is claimed in Smoller-Temple-Vogler 14:

Our analysis is based on the discovery of a closed ansatz for perturbations of the SM during the p=0 = 0 epoch of the Big Bang which triggers instabilities that create unexpectedly large regions of accelerated uniform expansion within Einstein’s original theory without the cosmological constant. We prove that these accelerated regions introduce precisely the same range of corrections to redshift vs luminosity as are produced by the cosmological constant in the theory of Dark Energy.

A similar conclusion is reached in Sarkar et al. 18, which in Sarkar 18, slide 44 is summarized as follows:

There is a dipole in the recession velocities of host galaxies of supernovae \Rightarrow we are in a “bulk flow” stretching out well beyond the expected scale (100Mpc\sim 100 Mpc) at which the universe is expected to become statistically homogeneous. The inference that the Hubble expansion rate is accelerating may be an artefact of the local bulk flow [...][...] The “standard” assumptions of exact isotropy and homogeneity are questionable [...][...]

Survey of the field of inhomogeneous cosmology and of attitudes in the community towards open issues is in Belejko-Korzyński 16.

If the apparent small positive cosmological constant (dark energy) were but an artefact of neglecting backreaction of inhomogeneities, some theoretical puzzlements regarding quantum gravity on de Sitter spacetimes would disappear (see Rajaraman 16 for general discussion and Danielsson-VanRiet 18, p. 27 for discussion of perturbative string theory vacua).

Numerical simulation

Numerical simulations of inhomogeneous cosmology in the required relativistic accuracy are in their infancy (see Belejko-Korzyński 16, p. 7), but include the following: Clesse-Roisin-Füzfa 17, ACDDK 17, Montanari-Räsänen 17.

The conclusion in Montanari-Räsänen 17, p. 20 is as follows:

[[the model]] shows an increase of the expansion rate of the right order of magnitude, compared to observations, at late times. [...][...]. It is nontrivial that the right order of magnitude in the amplitude and roughly right timescale of the change in the expansion rate follow simply from the known physics of structure formation. However, the model has shortcomings that would need to be overcome for the results to be more than suggestive.

Dependency of results on the choice of gauge fixing is highlighted in ACDDK 17:

We then show numerical results from the fully relativistic weak field NN-body code gevolution. (p.2)

[...][...] The conclusion of this work is therefore that there are gauges which are relatively close to what observers measure and in these gauges backreaction is small. We used the example of Poisson gauge, but there would be others, e.g. geodesic light cone gauge [53, 54]. However, comoving synchronous gauge is not well suited to describe observations in the late time clumpy universe. In this gauge backreaction becomes large and the gauge actually breaks down during structure formation. (p. 4)

The simulations in Odderskov-Koksbang-Hannestad 16, Macpherson-Lasky-Price 18 show noticeable but small effects of inhomogeneity, possibly explaining parts but not all of the measured discrepancy reported in Riess et. al. 16.

The “backreaction debate”

A seminal theoretical argument that it is consistent to neglect cosmic inhomogeneity was given by Green-Wald 10, Green-Wald 11, Green-Wald 13, Green-Wald 16. This was called into question in Buchert et al. 15, where it is concluded that the issue is more subtle and remains open. The reply to this criticism by Green-Wald 15 is summarized in Ostrowski-Roukema 15, p. 4 as follows:

Green and Wald state that their formalism does not apply to situations when

*\ast the actual metric (e.g., at recent epochs) is far from FLRW; or

*\ast one wishes to construct an effective metric (or other effective quantities) through some averaging procedure

This, in principle, ends the debate about whether backreaction has been excluded as a dark energy candidate: the Green and Wald formalism does not apply to the main body of backreaction research; backreaction remains a viable dark energy candidate.

Accordingly, the review Coley 18, section 3.5 of mathematical general relativity again regards the issue as open:

An important open question in cosmology is whether averaging of inhomogeneities can lead to significant backreaction effects on very large scales. (p. 28)

Similarly in Huang-Gao-Xu 19, p. 2:

The lack of solid proof [...][...] is a more serious concern.

Indeed, Smoller-Temple-Vogler 14 claim an analytic solution which does exhibit inhomogeneity effects mimicking dark energy (see above) and a similar conclusion is claimed in Sarkar et al. 18 (see above). Also relativistic numerical simulation, albeit in their infancy, seem to exhibit noticeable backreaction (see above).

Lemaitre-Tolman-Bondi models

A particular class of exactly soluable simple examples of inhomogeneous cosmological models are Lemaitre-Tolman-Bondi models. If taken as quasi-realistic models in themselves, these require assuming that we inhabit a position close to a singled-out “center” of the universe, usually the center of an assumed cosmic “void”, of low matter density. See e.g. Moffat 05, Enkvist 07, Moffat 16. Possible observational signatures of this scenario are discussed in Clifton-Ferreira-Land 08

It has been argued (e.g. Moffat 16, p. 2) that the apparent unlikeliness of such a “spatial coincidence” is relativized in view of the observed “temporal coincidence” that cosmic acceleration seems to start roughly with the onset of structure formation (the “coincidence problem” of cosmology), and the perceived fine-tuning of the cosmological constant required in the standard model of cosmology.

However, this may be over-interpreting the realism of these simple models. According to Räsänen 03, p. 15:

In order to evaluate the importance of backreaction in the real universe, we need statistical knowledge about complex configurations of dust, not exact information about simplified models.


Theory of backreaction

  • Thomas Buchert, Juergen Ehlers, Averaging inhomogeneous Newtonian cosmologies, Astron. Astrophys.320:1-7, 1997 (arXiv:astro-ph/9510056)

  • Thomas Buchert, On average properties of inhomogeneous cosmologies, Gen.Rel.Grav.9:306-321, 2000 (arXiv:gr-qc/0001056)

  • Thomas Buchert, Julien Larena, Jean-Michel Alimi, Correspondence between kinematical backreaction and scalar field cosmologies - the ‘morphon field’, Class. Quant. Grav.23:6379-6408, 2006 (arXiv:gr-qc/0606020)

  • Syksy Räsänen, Evaluating backreaction with the peak model of structure formation, JCAP 0804:026,2008 (arXiv:0801.2692)

  • Edward W. Kolb, Valerio Marra, Sabino Matarrese, Cosmological background solutions and cosmological backreactions, Gen.Rel.Grav.42:1399-1412, 2010 (arXiv:0901.4566)

  • Syksy Räsänen, Backreaction as an alternative to dark energy and modified gravity (arXiv:1012.0784)

  • Stephen R. Green, Robert Wald, A new framework for analyzing the effects of small scale inhomogeneities in cosmology, Phys.Rev.D83:084020, 2011 (arXiv:1011.4920)

  • Thomas Buchert, Toward physical cosmology: focus on inhomogeneous geometry and its non-perturbative effects, Class.Quant.Grav.28:164007, 2011 (arXiv:1103.2016)

  • Stephen R. Green, Robert Wald, Newtonian and Relativistic Cosmologies, Phys.Rev.D85:063512, 2012 (arXiv:1111.2997)

  • Stephen Green, Robert Wald, Examples of backreaction of small scale inhomogeneities in cosmology, Phys.Rev.D87:124037, 2013 (arxiv:1304.2318)

  • Stephen R. Green, Robert Wald, Comments on Backreaction (arXiv:1506.06452)

  • Thomas Buchert et. al, Is there proof that backreaction of inhomogeneities is irrelevant in cosmology?, Class. Quantum Grav. 32 215021, 2015 (arXiv:1505.07800)

    exposition in The Universe is inhomogeneous. Does it matter? CQG+, 2016

  • Jan J. Ostrowski, Boudewijn F. Roukema, On the Green and Wald formalism, The Fourteenth Marcel Grossmann Meeting (arXiv:1512.02947, talk slides pdf)

  • Stephen Green, Robert Wald, A Simple, Heuristic Derivation of our “No Backreaction” Results, Classical and Quantum Gravity, Volume 33, Number 12, 2016 (arXiv:1601.06789)

  • Francesco Montanari, Syksy Räsänen, Evaluating backreaction with the ellipsoidal collapse model, JCAP12(2017)008 (arXiv:1710.02451)

  • Alan Coley, section 3.5 of Mathematical General Relativity (arXiv:1807.08628)

  • Zhiqi Huang, Han Gao, Haoting Xu, Revisiting Ryskin’s Model of Cosmic Acceleration (arXiv:1905.02441)

See also

Effective dark energy from inhomogeneity

The proposal that backreaction of cosmic inhomogeneities may mimic a cosmological constant/dark energy has been discussed in the following articles:

  • Marie-Noëlle Célérier, Do we really see a cosmological constant in the supernovae data?, Astron.Astrophys.353:63-71, 2000

  • Christof Wetterich, Can Structure Formation Influence the Cosmological Evolution?, Phys.Rev. D67 (2003) 043513 (arXiv:astro-ph/0111166)

  • Dominik J. Schwarz, Accelerated expansion without dark energy (arXiv:astro-ph/0209584)

  • Syksy Räsänen, Dark energy from backreaction, JCAP 0402:003, 2004 (arXiv:astro-ph/0311257)

  • H. Alnes, M. Amarzguioui and O. Gron, An inhomogeneous alternative to dark energy?, Phys. Rev. D 73, 083519 (2006) (arXiv:astro-ph/0512006)

  • Kari Enqvist, Teppo Mattsson, The effect of inhomogeneous expansion on the supernova observations, JCAP 0702:019,2007 (arXiv:astro-ph/0609120)

  • Havard Alnes, Morad Amarzguioui, The supernova Hubble diagram for off-center observers in a spherically symmetric inhomogeneous universe, Phys. Rev. D75:023506, 2007 (arXiv:astro-ph/0610331)

  • Thomas Buchert, Dark Energy from structure: a status report, Gen.Rel.Grav.40:467-527, 2008 (arXiv:0707.2153)

  • Subir Sarkar, Is the evidence for dark energy secure?, Gen. Rel. Grav.40:269-284, 2008 (arXiv:0710.5307)

  • Alessio Notari, Can an Inhomogeneous Universe mimic Dark Energy?, 2009 (pdf)

  • Michael Blomqvist, Inhomogeneous cosmologies with clustered dark energy or a local matter void, 2010 (web)

  • Thomas Buchert, Syksy Räsänen, Backreaction in late-time cosmology, Annual Review of Nuclear and Particle Science 62 (2012) 57-79 (arXiv:1112.5335)

  • Joel Smoller, Blake Temple, Zeke Vogler, An Instability of the Standard Model Creates the Anomalous Acceleration Without Dark Energy, Proceedings of the Royal Society A, 2017 (arXiv:1412.4001, 10.1098/rspa.2016.0887, detailed talk slides: pdf, talk recording I, recording II)

  • I. Odderskov, S. M. Koksbang, S. Hannestad, The Local Value of H 0H_0 in an Inhomogeneous Universe, JCAP02(2016)001 (arXiv:1601.07356)

  • Adam G. Riess et al., A 2.4% Determination of the Local Value of the Hubble Constant, The Astrophysical Journal, Volume 826, Number 1 (arXiv:1604.01424)

  • Krzysztof Bolejko, Mikołaj Korzyński, Inhomogeneous cosmology and backreaction: Current status and future prospects, Int. J. Mod. Phys. D 26, 1730011 (2017) (arXiv:1612.08222)

  • Sebastien Clesse, Arnaud Roisin, André Füzfa, Mimicking Dark Energy with the backreactions of gigaparsec inhomogeneities (arXiv:1702.06643)

  • Julian Adamek, Chris Clarkson, David Daverio, Ruth Durrer, Martin Kunz, Safely smoothing spacetime: backreaction in relativistic cosmological simulations (arXiv:1706.09309)

  • Ulf Danielsson, Thomas Van Riet, What if string theory has no de Sitter vacua? (arXiv:1804.01120)

  • Hayley Macpherson, Paul D. Lasky, Daniel J. Price, The trouble with Hubble: Local versus global expansion rates in inhomogeneous cosmological simulations with numerical relativity, ApJ Letters (arXiv:1807.01714)

  • J. Colin, R. Mohayaee, M. Rameez, Subir Sarkar, Apparent cosmic acceleration due to local bulk flow (arXiv:1808.04597)

  • Subir Sarkar, Is the universe isotropic?, talk at Current Themes in High Energy Physics and Cosmology 2018 (pdf)

  • Lucas Lombriser, On the cosmological constant problem, Phys. Lett. B 797, 134804 (2019) (arXiv:1901.08588, doi:10.1016/j.physletb.2019.134804)

No effective dark energy from inhomogeneity

In constrast, arguments that cosmic inhomogeneity can not be the cause of any sizeable amount of effective dark energy are advanced in the following articles:

  • Ghazal Geshnizjani, Daniel J.H. Chung, Niayesh Afshordi, Do Large-Scale Inhomogeneities Explain Away Dark Energy?, Phys.Rev. D72 (2005) 023517 (arXiv:astro-ph/0503553)

  • E. R. Siegel, J. N. Fry, Effects of Inhomogeneities on Cosmic Expansion, Astrophys.J. 628 (2005) L1-L4 (arXiv:astro-ph/0504421)

  • Eanna E. Flanagan, Can superhorizon perturbations drive the acceleration of the Universe?, Phys.Rev. D71 (2005) 103521 (arXiv:hep-th/0503202)

  • Giovanni Marozzi, Jean-Philippe Uzan, Late time anisotropy as an imprint of cosmological backreaction (arXiv:1206.4887)

  • Ido Ben-Dayan, Maurizio Gasperini, Giovanni Marozzi, Fabien Nugier, Gabriele Veneziano, Do stochastic inhomogeneities affect dark-energy precision measurements?, Phys. Rev. Lett. 110, 021301 (2013) (arXiv:1207.1286)

Lemaitre-Tolman-Bondi models

See also

Last revised on September 7, 2019 at 04:07:05. See the history of this page for a list of all contributions to it.