nLab inhomogeneous cosmology

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 – see for instance the KBC void, KBC 13, large observed an-isotropies SHRMS22, and see Aluri et al. 2022, Peebles 2022 for extensive survey of observed inhomogeneity, anisotropy and other deviations) – but the question is whether on cosmic scales the deviation from homogeneity is small enough that it may be neglected, to good 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, for review see Buchert-Räsänen 11, Ellis 18.

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 cosmic 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, MRS 21) related to dark energy (a cosmological constant, and possibly related issues such as cosmic inflation and the observational preference for phantom dark energy), 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, Notari 09, Blomqvist 10, Buchert 11, Buchert-Räsänen 11, Scharf 13, Smoller-Temple-Vogler 14, Ludwick 15 Moffat 16, Ludwick 18 Colin-Mohayaee-Rameez-Sarkar 19, Colin-Mohayaee-Rameez-Sarkar 19b, Sarkar 18, Lombriser 19, Deledicque 19, Heinesen-Buchert 20.

Discussion specifically of cosmic inhomogenity as the possible cause of the H0-tension is in Bolejko 17, Sikora-Glod 20, SnowmassWhitePaper22, Section VII.H.

From Koksbang 19, p. 3:

Cosmic backreaction is particularly interesting because it in principle has the potential to explain the apparent accelerated expansion of the Universe without introducing any exotic dark energy component as well as possibly being able to mimic dark matter.

Less ambitiously, cosmic backreaction might solve the $H_0$ -problem through the emergence of curvature (Bolejko 17), or a small backreaction may bias the values obtained from analyses of data based on FLRW models and must therefore be identified and taken into account in an era of precision cosmology. Yet another option is that cosmic backreaction is entirely negligible in the real universe.

Whichever is the case, a theoretical quantification of cosmic backreaction is necessary for getting the foundations of cosmology onto solid ground; the mathematics clearly shows that in principle backreaction terms affect the overall dynamics of the Universe. It is therefore an important goal of cosmologists to obtain a theoretical understanding of the size of cosmic backreaction in the real universe similarly to e.g. the desire to theoretically understand the value of the vacuum energy density.

From CosmoBack 18:

Since the early 2000’s a large debate emerged in the cosmology community around the so-called «averaging problem» (Buchert 2000; Ellis & Buchert 2005), a question introduced on a more general ground in general relativity by G.F.R. Ellis in 1962. Because on the non-linearity of the Einstein equations, a non-trivial backreaction effect of the small-scale matter inhomogeneities is expected on the average large-scale dynamics of the spacetime. The occurrence of the late-time accelerated expansion of the universe at the same epoch when structure formation becomes non-linear is a tempting coincidence encouraging the backreaction conjecture against the real need of a dark energy component, which is needed instead in a Friedmann-Lemaître-Robertson-Walker setting. Besides, the same backreaction mechanism also accounts for an energy source dynamically equivalent to the dark matter. These arguments are often supported by toy models but not demonstrated in general, both from a theoretical and observational point-of-view. Simplified inhomogeneous models have been proposed to interpret the non-trivial dynamics of structures and the light propagation on cosmological scales. Only recently cosmological fully general-relativistic N-body simulations have been realised, offering a valuable ground to investigate the geometry of a realistic, “lumpy” universe.

Qualitative mechanism

A qualitative discussion of how cosmic 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 $H t$ 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)

Data and models

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 $= 0$ 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 survey of the field of inhomogeneous cosmology and of attitudes in the community towards open issues, as of 2016, is in Belejko-Korzyński 16. More recent survey of the field and the prevailing attitude is in Sikora-Glod 20, Section 1.

A similar conclusion is reached in Colin-Mohayaee-Rameez-Sarkar 19, 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 ( $\sim 100 Mpc$ ) 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 $[...]$

and a potential fallacy in the widely accepted interpretation of the data is claimed, which in Rameez 19 is summarized as follows:

The Fitting Problem in cosmology was written in 1987. In the context of this work and the significant theoretical difficulties involved in inferring fundamental physics from the real Universe, any claims of having measured a cosmological constant from directionally skewed, sparse samples of intrinsically scattered observations should have been taken with a grain of salt. By honouring this claim with a Nobel Prize, the Swedish Academy may have induced runaway prestige bias in favour of some of the least principled analyses in science, strengthening the confirmation bias that seems prevalent in cosmology.

A conclusion along these lines, albeit weaker, is also reached independently in Lin-Mack-Hou 19, p. 5:

some proposals suggest some local/environmental factors ( $z \leq 0.03$ ) can bias the local determinations. This would mean the locally measured $H_0$ cannot be interpreted as the global Hubble constant of the homogeneous universe. An example is a local underdense region (Lombriser 2019; Shanks et al. 2019). Recent studies have shown observational evidence supporting a small-scale local underdense region (Boehringer et al. 2019; Pustilnik et al. 2019), though it has been argued that the likelihood for a local void to substantially affect the local measurement may be low (Kenworthyet al. 2019). These local factors do not pose a problem tothe standard ΛCDM model at large scales, but instead point to the need for a more detailed description of our local environment to account for such a systematic effect that can shift all local measurements in the same way. If all local measurements produce high values of $H_0$ , it would favor such a local/environmental-factor explanation over systematic effects that may be unique to each observation.

A strong claim is due to Heinesen-Buchert 20, who present a model of inhomogeneous cosmology that is claimed to be (p. 14):

a natural and consistent explanation of

(i) dark energy,

(ii) the coincidence problem (here conceptually,not quantitatively),

(iii) positive initial curvature,

(iv) the small matter density cosmological parameterfound in local probes of the matter density,

(v) the large angular diameter distance tothe CMB consistent with JLA supernova sample parameter constraints,

and (vi) the local expansion rate measurements (removal of the ‘Hubble tension’).

We believe that this model architecture needs convincing arguments to be rejected as a physically viable show-case, on the basis of which the model ingredients can be improved in order to build a physical cosmology in the future.

Impact

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 in terms of perturbative string theory vacua) and some theoretically attractive models could become phenomenologically viable again, such as “Witten's Dark Fantasy”.

In this context, notice that even a negative cosmological constant has been argued to still be compatible with observation if suitable scalar fields are present (Sen, Adil & Sen 2021)

Numerical simulation

Numerical simulations of inhomogeneous cosmology in the required relativistic accuracy is in its infancy (see Belejko-Korzyński 16, p. 7), but includes the following results:

ADDK 13, Clesse-Roisin-Füzfa 17, ACDDK 17, Montanari-Räsänen 17, Adamek 18 (“gevolution”).

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 $N$ -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.

Similarly Adamek 18, slide 14 on simulations obtained with gevolution:

Backreaction is a real phenomenon that…

can be quantified accurately with numerical experiments

quantitatively cannot explain observed data without dark energy

may nevertheless be relevant for precision cosmology with future surveys

For more see also the pointers in Räsänen 18, slide 7.

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.

Later, Ostrowski 19 summarizes this as follows:

Green and Wald formalism, being a special case of two-scale asymptotic homogenization is not applicable to gravitational systems with hierarchical structures

any features of backreaction (including backreaction being trace-free) based on Green and Wald formalism are unjustified

Open problem

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.

and CosmoBack 18:

Indeed, the actual amplitude of the backreaction effect, whether it requires a fully nonlinear general relativistic treatment or whether a perturbative approach is sufficient, the impact of the gauge choice, of coarse-graining, and of averaging procedures in defining the observables are still open problems.

and Koksbang 19, p. 3:

Whichever is the case, a theoretical quantification of cosmic backreaction is necessary for getting the foundations of cosmology onto solid ground; the mathematics clearly shows that in principle backreaction terms affect the overall dynamics of the Universe.

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 Colin-Mohayaee-Rameez-Sarkar 19 (see above). Also relativistic numerical simulation, albeit in their infancy, seem to exhibit noticeable backreaction (see above).

A new analytical approach is suggested by Ginat 20, where non-negligible but small effects due to ingomogeneity are found.

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

From KBC 13:

In particular, “minimal void” scenarios (e.g., Alexander et al. 2009; Bolejko & Sussman 2011) have shown that very simple models placing the observer near the center of an under-density that is ∼ 300 Mpc in radius and roughly half the density of its surroundings are sufficient to explain the acceleration observed via type Ia supernovae. While these models are simplistic, they make clear that an observer’s location with respect to large-scale structures could have profound implications for that observer’s measurement of cosmological observables.

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.

Related concepts

References

General

George Ellis, W. Stoeger, The ‘fitting problem’ in cosmology, Classical and Quantum Gravity, Volume 4, Number 6, 1987 (doi:10.1088/0264-9381/4/6/025)

General review:

George Ellis, summary talk at CosmoBack 2018 (pdf)
Pavan Kumar Aluri et al.: Is the Observable Universe Consistent with the Cosmological Principle?, Classical and Quantum Gravity 40 9 (2023) [arXiv:2207.05765, doi:10.1088/1361-6382/acbefc]
Subir Sarkar: A challenge to the cosmological standard model, lectures (2023) [webpage, pdf]

On computer experiment:

Julian Adamek, The Numerical Challenge – Backreaction in Relativistic N-body Simulations of Cosmic Structure Formation, talk at CosmoBack 2018 (pdf)

(on gevolution)

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)
Julian Adamek, David Daverio, Ruth Durrer, Martin Kunz, General Relativistic N-body simulations in the weak field limit, Phys. Rev. D 88, 103527 (2013) (arxiv:1308.6524)
Stephen R. Green, Robert Wald, Comments on Backreaction (arXiv:1506.06452)
Thomas Buchert, Syksy Räsänen, Boudewijn F. Roukema et al.: Is there proof that backreaction of inhomogeneities is irrelevant in cosmology?, Class. Quantum Grav. 32 (2015) 215021 [arXiv:1505.07800, doi:10.1088/0264-9381/32/21/215021]

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)
S. M. Koksbang, Towards statistically homogeneous and isotropic perfect fluid universes with cosmic backreaction, Class. Quantum Grav. 36 185004, 2019 (arxiv:1907.08681)
CosmoBack 2018 , From inhomogeneous gravity to cosmological backreaction. Theoretical opportunity? Observational evidence?
Jan Ostrowski, Green and Wald formalism: The aftermath of the “backreaction debate”, talk at CosmoBack 2018 (pdf)
Syksy Räsänen, Ways to settle the backreaction conjecture, talk at CosmoBack 2018 (pdf)
Hayley J. Macpherson, Inhomogeneous cosmology in an anisotropic Universe (arxiv:1910.13380)
Yonadav Barry Ginat, Multiple-Scales Approach for Addressing The Averaging Problem in Cosmology (arXiv:2005.03026)
Martin Bojowald, Ding Ding, Canonical description of cosmological backreaction (arXiv:2011.03018)

In relation to structure formation during cosmic inflation:

Martin Bojowald, Ding Ding, Canonical description of cosmological backreaction, JCAP 03 (2021) 083
(arXiv:2011.03018)

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 (arxiv:astro-ph/9907206)
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)
Kevin J. Ludwick, Examining the Viability of Phantom Dark Energy, Phys. Rev. D 92, 063019 (2015) (arXiv:1507.06492)

(on phantom dark energy)
I. Odderskov, S. M. Koksbang, S. Hannestad, The Local Value of $H_0$ 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 826 1 (2016) [arXiv:1604.01424, doi:10.3847/0004-637X/826/1/56]
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)
Krzysztof Bolejko, Emerging spatial curvature can resolve the tension between high-redshift CMB and low-redshift distance ladder measurements of the Hubble constant, Phys. Rev. D 97, 103529 (2018) (arxiv:1712.02967)
Ulf Danielsson, Thomas Van Riet, What if string theory has no de Sitter vacua?, International Journal of Modern Physics D 27 12 (2018) 1830007 [arXiv:1804.01120, doi:10.1142/S0218271818300070]
Kevin J. Ludwick, The Viability of Phantom Dark Energy as a Quantum Field in 1st-Order FLRW Space, Phys. Rev. D 98, 043519 (2018) (arXiv:1804.02987)

(on phantom dark energy)
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)
Jacques Colin, Roya Mohayaee, Mohamed Rameez, Subir Sarkar, Evidence for anisotropy of cosmic acceleration, Astronomy & Astrophysics 631 L13 (2019) doi:10.1051/0004-6361/201936373, (arXiv:1808.04597)
Subir Sarkar, Is the universe isotropic?, talk at Current Themes in High Energy Physics and Cosmology 2018 (pdf)
Mohamed Rameez, On the Real Inhomogeneous Universe and the Weirdness of ‘Dark Energy’ (Nov. 14, 2019)
Jacques Colin, Roya Mohayaee, Mohamed Rameez, Subir Sarkar, A response to Rubin & Heitlauf: “Is the expansion of the universe accelerating? All signs still point to yes” (arXiv:1912.04257)
Lucas Lombriser, On the cosmological constant problem, Phys. Lett. B 797, 134804 (2019) (arXiv:1901.08588, doi:10.1016/j.physletb.2019.134804)
Vincent Deledicque, Theoretical developments on the adequacy of the fitting of the FLRW metric on the universe’s real metric (arxiv:1907.01580)
Weikang Lin, Katherine J. Mack, Liqiang Hou, Investigating the Hubble Constant Tension – Two Numbers in the Standard Cosmological Model (arXiv:1910.02978)
Léo Brunswic, Thomas Buchert, Gauss-Bonnet-Chern approach to the averaged Universe, Class. Quantum Grav. 37 (2020) 215022 (arXiv:2002.08336)
Asta Heinesen, Thomas Buchert, Solving the curvature and Hubble parameter inconsistencies through structure formation-induced curvature, Class. Quantum Grav. 37 (2020) 164001 (arXiv:2002.10831)
Szymon Sikora, Krzysztof Głód, Construction of the cosmological model with periodically distributed inhomogeneities with growing amplitude (arXiv:2005.06385)

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

John Moffat, Late-time Inhomogeneity and Acceleration Without Dark Energy, JCAP 0605 (2006) 001 (arXiv:astro-ph/0505326)
Kari Enqvist, Lemaitre-Tolman-Bondi model and accelerating expansion, Gen. Rel. Grav.40:451-466, 2008 (arXiv:0709.2044)
Timothy Clifton, Pedro G. Ferreira, Kate Land, Living in a Void: Testing the Copernican Principle with Distant Supernovae, Phys. Rev. Lett. 101 (2008) 131302 (arXiv:0807.1443)
Günter Scharf, Inhomogeneous cosmology in the cosmic rest frame without dark stuff, chapter 6 in the latest edition of Quantum Gauge Theories – A True Ghost Story, Wiley 2001 (arXiv:1312.2695)
John Moffat, Inhomogeneous Cosmology Redux (arXiv:1608.00534)

Observed inhomogeneity

Introducing observation of the KBC void:

Ryan C. Keenan, Amy J. Barger, Lennox L. Cowie, Evidence for a ~300 Mpc Scale Under-density in the Local Galaxy Distribution, 2013, ApJ, 775, 62 (arXiv:1304.2884)

Its potential relevance for the Hubble tension (apparently too small):

Hao-Yi Wu, Dragan Huterer, Sample variance in the local measurements of the Hubble constant, Monthly Notices of the Royal Astronomical Society, Volume 471, Issue 4, November 2017, Pages 4946–4955 (arXiv:1706.09723)

Anisotropy:

Nathan Secrest, Sebastian von Hausegger, Mohamed Rameez, Roya Mohayaee, Subir Sarkar, A Challenge to the Standard Cosmological Model, ApJ Letters (2022) [arXiv2206.05624]

The joint significance of this rejection of the cosmological principle is 5.2σ.

Applying topological data analysis/persistent homology to the CMB:

Pratyush Pranav, Thomas Buchert, Homology reveals significant anisotropy in the cosmic microwave background (arXiv:2308.10738)

Survey of deviations:

Phillip James E. Peebles, Anomalies in Physical Cosmology [arXiv:2208.05018]