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\begin{document}

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\section*{inhomogeneous cosmology}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{gravity}{}\paragraph*{{Gravity}}\label{gravity}

[[!include gravity contents]]

\hypertarget{physics}{}\paragraph*{{Physics}}\label{physics}

[[!include physicscontents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{effective_dark_energy_from_inhomogeneity}{Effective dark energy from inhomogeneity?}\dotfill \pageref*{effective_dark_energy_from_inhomogeneity} \linebreak
\noindent\hyperlink{NumericalSimulation}{Numerical simulation}\dotfill \pageref*{NumericalSimulation} \linebreak
\noindent\hyperlink{BackreactionDebate}{The ``backreaction debate''}\dotfill \pageref*{BackreactionDebate} \linebreak
\noindent\hyperlink{LemaitreTolmanBondi}{Lemaitre-Tolman-Bondi models}\dotfill \pageref*{LemaitreTolmanBondi} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{theory_of_backreaction}{Theory of backreaction}\dotfill \pageref*{theory_of_backreaction} \linebreak
\noindent\hyperlink{EffectiveDarkEnergyFromInhomogeneity}{Effective dark energy from inhomogeneity}\dotfill \pageref*{EffectiveDarkEnergyFromInhomogeneity} \linebreak
\noindent\hyperlink{ReferencesNoEffect}{No effective dark energy from inhomogeneity}\dotfill \pageref*{ReferencesNoEffect} \linebreak
\noindent\hyperlink{lemaitretolmanbondi_models_2}{Lemaitre-Tolman-Bondi models}\dotfill \pageref*{lemaitretolmanbondi_models_2} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

What is called \emph{inhomogeneous cosmology} is the study of [[cosmology]] via cosmological solutions to [[Einstein's equations]], \emph{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]] \emph{is} assumed to be spatially homogeneous (an assumption also known as the \emph{cosmological principle}).

Of course the [[observable universe]] is clearly not \emph{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 \hyperlink{KolbMarraMatarrese10}{Kolb-Marra-Matarrese 10}, for review see \hyperlink{BuchertRasanen11}{Buchert-Räsänen 11}, \hyperlink{Ellis18}{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 field theory|relativistic]] (i.e. Newtonian) limit, instead of taking [[general relativity|relativity]] into account (\hyperlink{Buchert00}{Buchert 00}, \hyperlink{BuchertEhlers95}{Buchert-Ehlers 95}), which however is the standard approximation currently used in comparing the [[standard model of cosmology]] to data.

\hypertarget{effective_dark_energy_from_inhomogeneity}{}\subsection*{{Effective dark energy from inhomogeneity?}}\label{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 field|background]] [[spacetime]].

Given that the [[standard model of cosmology]] faces some issues (e.g. \href{standard+model+of+cosmology#BCKRW15}{BCKRW 15}, \hyperlink{RiessEtAl16}{Riess et. al. 16}) related to \emph{[[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: \hyperlink{Celerier00}{Célérier 00}, \hyperlink{Buchert00}{Buchert 00}, \hyperlink{Wetterich01}{Wetterich 01}, \hyperlink{Schwarz02}{Schwarz 02}, \hyperlink{Rasanen03}{Räsänen 03}, \hyperlink{AlnesAmarzguiouiGron06}{Alnes-Amarzguioui-Gron 06}, \hyperlink{AlnesAmarzguioui06}{Alnes-Amarzguioui 06}, \hyperlink{BuchertLarenaAlimi06}{Buchert-Larena-Alimi 06}, \hyperlink{EnqvistMattsson06}{Enqvist-Mattsson 06} \hyperlink{Buchert07}{Buchert 07}, \hyperlink{Sarkar08}{Sarkar 08}, \hyperlink{Buchert11}{Buchert 11}, \hyperlink{BuchertRasanen11}{Buchert-Räsänen 11}, \hyperlink{Scharf13}{Scharf 13}, \hyperlink{SmollerTempleVogler14}{Smoller-Temple-Vogler 14}, \hyperlink{Moffat16}{Moffat 16}, \hyperlink{SarkarEtAl18}{Sarkar et al. 18}, \hyperlink{Sarkar18}{Sarkar 18}, \hyperlink{Lombriser19}{Lombriser 19}, \hyperlink{Deledicque19}{Deledicque 19}.

Discussion specifically of cosmic inhomogenity as the possible cause of the $H_0$-tension: \hyperlink{Bolejko17}{Bolejko 17}.

From \hyperlink{Koksbang19}{Koksbang 19, p. 3}

\begin{quote}%
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 (\hyperlink{Bolejko17}{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.


\end{quote}
From \hyperlink{CosmoBack18}{CosmoBack 18}:

\begin{quote}%
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.


\end{quote}
 A qualitative discussion of how inhomogeneity may cause accelerated cosmic expansion is given in \hyperlink{Rasanen10}{Räsänen 10, section 3: ``Understanding acceleration''}:

\begin{quote}%
In general, underdense regions $[$``voids''$]$ are negatively curved and expand faster than the average, while overdense regions are positively curved and expand slower. (\hyperlink{Rasanen03}{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. (\hyperlink{Rasanen03}{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 (\hyperlink{Rasanen10}{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 (\hyperlink{Rasanen10}{Räsänen 10, p. 10})


\end{quote}
 An analytic proof of this qualitative picture is claimed in \hyperlink{SmollerTempleVogler14}{Smoller-Temple-Vogler 14}:

\begin{quote}%
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.


\end{quote}
A similar conclusion is reached in \hyperlink{SarkarEtAl18}{Sarkar et al. 18}, which in \hyperlink{Sarkar18}{Sarkar 18, slide 44} is summarized as follows:

\begin{quote}%
 There is a dipole in the recession velocities of host galaxies of supernovae $\Rightarrow$ we are in a ``bulk flow'' stretching out \emph{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 \emph{questionable $[...]$}


\end{quote}
Survey of the field of inhomogeneous cosmology and of attitudes in the community towards open issues is in \hyperlink{BelejkoKorzynski16}{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 \href{de+Sitter+spacetime#Rajaraman16}{Rajaraman 16} for general discussion and \hyperlink{DanielssonVanRiet18}{Danielsson-VanRiet 18, p. 27} for discussion of [[perturbative string theory vacua]]).

\hypertarget{NumericalSimulation}{}\subsection*{{Numerical simulation}}\label{NumericalSimulation}

[[computer experiment|Numerical simulations]] of inhomogeneous cosmology in the required [[relativistic field theory|relativistic]] accuracy is in its infancy (see \hyperlink{BelejkoKorzynski16}{Belejko-Korzyński 16, p. 7}), but includes the following results: \hyperlink{ADDK13}{ADDK 13}, \hyperlink{ClesseRoisinFufza17}{Clesse-Roisin-Füzfa 17}, \hyperlink{ACDDK17}{ACDDK 17}, \hyperlink{MontanariRasanen17}{Montanari-Räsänen 17}, \hyperlink{Adamek18}{Adamek 18} (``[[gevolution]]'').

The conclusion in \hyperlink{MontanariRasanen17}{Montanari-Räsänen 17, p. 20} is as follows:

\begin{quote}%
$[$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.


\end{quote}
Dependency of results on the choice of [[gauge fixing]] is highlighted in \hyperlink{ACDDK17}{ACDDK 17}:

\begin{quote}%
We then show numerical results from the fully relativistic weak field $N$-body code \emph{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)


\end{quote}
The simulations in \hyperlink{OdderskovKoksbangHannestad16}{Odderskov-Koksbang-Hannestad 16}, \hyperlink{MacphersonLaskyPrice18}{Macpherson-Lasky-Price 18} show noticeable but small effects of inhomogeneity, possibly explaining parts but not all of the measured discrepancy reported in \hyperlink{RiessEtAl16}{Riess et. al. 16}.

Similarly \hyperlink{Adamek18}{Adamek 18, slide 14} on simulations obtained with [[gevolution]]:

\begin{quote}%
Backreaction is a real phenomenon that\ldots{}

\begin{itemize}%
\item can be quantified accurately with [[computer experiment|numerical experiments]]


\item quantitatively cannot explain observed data without [[dark energy]]


\item may nevertheless be relevant for precision cosmology with future surveys



\end{itemize}

\end{quote}
For more see also the pointers in \hyperlink{Rasanen18}{Räsänen 18, slide 7}.

\hypertarget{BackreactionDebate}{}\subsection*{{The ``backreaction debate''}}\label{BackreactionDebate}

A seminal theoretical argument that it \emph{is} consistent to neglect cosmic inhomogeneity was given by \hyperlink{GreenWald10}{Green-Wald 10}, \hyperlink{GreenWald11}{Green-Wald 11}, \hyperlink{GreenWald13}{Green-Wald 13}, \hyperlink{GreenWald16}{Green-Wald 16}. This was called into question in \hyperlink{BuchertEtAl15}{Buchert et al. 15}, where it is concluded that the issue is more subtle and remains open. The reply to this criticism by \hyperlink{GreenWald15}{Green-Wald 15} is summarized in \hyperlink{OstrowskiRoukema15}{Ostrowski-Roukema 15, p. 4} as follows:

\begin{quote}%
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.


\end{quote}
Later, \hyperlink{Ostrowski19}{Ostrowski 19} summarizes this as follows:

\begin{quote}%
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


\end{quote}
 \textbf{Open problem}

Accordingly, the review \hyperlink{Coley18}{Coley 18, section 3.5} of [[mathematical physics|mathematical]] [[general relativity]] again regards the issue as open:

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


\end{quote}
Similarly in \hyperlink{HuangGaoXu19}{Huang-Gao-Xu 19, p. 2}:

\begin{quote}%
The lack of solid proof $[...]$ is a more serious concern.


\end{quote}
and \hyperlink{CosmoBack18}{CosmoBack 18}:

\begin{quote}%
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.


\end{quote}
and \hyperlink{Koksbang19}{Koksbang 19, p. 3}:

\begin{quote}%
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.


\end{quote}
Indeed, \hyperlink{SmollerTempleVogler14}{Smoller-Temple-Vogler 14} claim an analytic solution which does exhibit inhomogeneity effects mimicking dark energy (see \hyperlink{AbstractSmollerTempleVogler14}{above}) and a similar conclusion is claimed in \hyperlink{SarkarEtAl18}{Sarkar et al. 18} (see \hyperlink{SarkarFlowQuote}{above}). Also relativistic numerical simulation, albeit in their infancy, seem to exhibit noticeable backreaction (see \hyperlink{NumericalSimulation}{above}).

\hypertarget{LemaitreTolmanBondi}{}\subsection*{{Lemaitre-Tolman-Bondi models}}\label{LemaitreTolmanBondi}

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. \hyperlink{Moffat05}{Moffat 05}, \hyperlink{Enkvist07}{Enkvist 07}, \hyperlink{Moffat16}{Moffat 16}. Possible observational signatures of this scenario are discussed in \hyperlink{CliftonFerreiraLand08}{Clifton-Ferreira-Land 08}

It has been argued (e.g. \hyperlink{Moffat16}{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 \hyperlink{Rasanen03}{Räsänen 03, p. 15}:

\begin{quote}%
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.


\end{quote}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[structure formation]]

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

General review:

\begin{itemize}%
\item [[George Ellis]], summary talk at \href{https://cosmoback.sciencesconf.org/}{CosmoBack 2018} ([[EllisCosmoBack18.pdf:file]])

\end{itemize}
On [[computer experiment]]:

\begin{itemize}%
\item Julian Adamek, \emph{The Numerical Challenge -- Backreaction in Relativistic N-body Simulations of Cosmic Structure Formation}, talk at \href{https://cosmoback.sciencesconf.org/}{CosmoBack 2018} ([[AdamekCosmoBack18.pdf:file]])

(on [[gevolution]])



\end{itemize}
See also

\begin{itemize}%
\item Wikipedia, \emph{\href{https://en.wikipedia.org/wiki/Inhomogeneous_cosmology}{Inhomogeneous cosmology}}


\item Wikipedia, \emph{\href{https://en.wikipedia.org/wiki/Accelerating_expansion_of_the_universe#Alternative_theories}{Accelerating expansion of the universe -- Alternative theories}}



\end{itemize}
\hypertarget{theory_of_backreaction}{}\subsubsection*{{Theory of backreaction}}\label{theory_of_backreaction}

\begin{itemize}%
\item [[Thomas Buchert]], Juergen Ehlers, \emph{Averaging inhomogeneous Newtonian cosmologies}, Astron. Astrophys.320:1-7, 1997 (\href{https://arxiv.org/abs/astro-ph/9510056}{arXiv:astro-ph/9510056})


\item [[Thomas Buchert]], \emph{On average properties of inhomogeneous cosmologies}, Gen.Rel.Grav.9:306-321, 2000 (\href{https://arxiv.org/abs/gr-qc/0001056}{arXiv:gr-qc/0001056})


\item [[Thomas Buchert]], Julien Larena, Jean-Michel Alimi, \emph{Correspondence between kinematical backreaction and scalar field cosmologies - the `morphon field'}, Class. Quant. Grav.23:6379-6408, 2006 (\href{https://arxiv.org/abs/gr-qc/0606020}{arXiv:gr-qc/0606020})


\item [[Syksy Räsänen]], \emph{Evaluating backreaction with the peak model of structure formation}, JCAP 0804:026,2008 (\href{https://arxiv.org/abs/0801.2692}{arXiv:0801.2692})


\item Edward W. Kolb, Valerio Marra, Sabino Matarrese, \emph{Cosmological background solutions and cosmological backreactions}, Gen.Rel.Grav.42:1399-1412, 2010 (\href{https://arxiv.org/abs/0901.4566}{arXiv:0901.4566})


\item [[Syksy Räsänen]], \emph{Backreaction as an alternative to dark energy and modified gravity} (\href{https://arxiv.org/abs/1012.0784}{arXiv:1012.0784})


\item Stephen R. Green, [[Robert Wald]], \emph{A new framework for analyzing the effects of small scale inhomogeneities in cosmology}, Phys.Rev.D83:084020, 2011 (\href{https://arxiv.org/abs/1011.4920}{arXiv:1011.4920})


\item [[Thomas Buchert]], \emph{Toward physical cosmology: focus on inhomogeneous geometry and its non-perturbative effects}, Class.Quant.Grav.28:164007, 2011 (\href{https://arxiv.org/abs/1103.2016}{arXiv:1103.2016})


\item Stephen R. Green, [[Robert Wald]], \emph{Newtonian and Relativistic Cosmologies}, Phys.Rev.D85:063512, 2012 (\href{https://arxiv.org/abs/1111.2997}{arXiv:1111.2997})


\item Stephen Green, [[Robert Wald]], \emph{Examples of backreaction of small scale inhomogeneities in cosmology}, Phys.Rev.D87:124037, 2013 (\href{https://arxiv.org/abs/1304.2318}{arxiv:1304.2318})


\item Julian Adamek, David Daverio, Ruth Durrer, Martin Kunz, \emph{General Relativistic N-body simulations in the weak field limit}, Phys. Rev. D 88, 103527 (2013) (\href{https://arxiv.org/abs/1308.6524}{arxiv:1308.6524})


\item Stephen R. Green, [[Robert Wald]], \emph{Comments on Backreaction} (\href{https://arxiv.org/abs/1506.06452}{arXiv:1506.06452})


\item [[Thomas Buchert]] et. al, \emph{Is there proof that backreaction of inhomogeneities is irrelevant in cosmology?}, Class. Quantum Grav. 32 215021, 2015 (\href{https://arxiv.org/abs/1505.07800}{arXiv:1505.07800})

exposition in \emph{\href{https://cqgplus.com/2016/01/20/the-universe-is-inhomogeneous-does-it-matter/}{The Universe is inhomogeneous. Does it matter?}} CQG+, 2016


\item Jan J. Ostrowski, Boudewijn F. Roukema, \emph{On the Green and Wald formalism}, The Fourteenth Marcel Grossmann Meeting (\href{https://arxiv.org/abs/1512.02947}{arXiv:1512.02947}, \href{https://cosmoback.sciencesconf.org/data/program/Ostrowski.pdf}{talk slides pdf})


\item Stephen Green, [[Robert Wald]], \emph{A Simple, Heuristic Derivation of our ``No Backreaction'' Results}, Classical and Quantum Gravity, Volume 33, Number 12, 2016 (\href{https://arxiv.org/abs/1601.06789}{arXiv:1601.06789})


\item Francesco Montanari, [[Syksy Räsänen]], \emph{Evaluating backreaction with the ellipsoidal collapse model}, JCAP12(2017)008 (\href{https://arxiv.org/abs/1710.02451}{arXiv:1710.02451})


\item [[Alan Coley]], section 3.5 of \emph{Mathematical General Relativity} (\href{https://arxiv.org/abs/1807.08628}{arXiv:1807.08628})


\item Zhiqi Huang, Han Gao, Haoting Xu, \emph{Revisiting Ryskin's Model of Cosmic Acceleration} (\href{https://arxiv.org/abs/1905.02441}{arXiv:1905.02441})


\item S. M. Koksbang, \emph{Towards statistically homogeneous and isotropic perfect fluid universes with cosmic backreaction}, Class. Quantum Grav. 36 185004, 2019 (\href{https://arxiv.org/abs/1907.08681}{arxiv:1907.08681})


\item \href{https://cosmoback.sciencesconf.org/}{CosmoBack 2018} , \emph{From inhomogeneous gravity to cosmological backreaction. Theoretical opportunity? Observational evidence?}


\item Jan Ostrowski, \emph{Green and Wald formalism: The aftermath of the ``backreaction debate''}, talk at \href{https://cosmoback.sciencesconf.org}{CosmoBack 2018} ([[OstrowskiBackreactionDebate19.pdf:file]])


\item [[Syksy Räsänen]], \emph{Ways to settle the backreaction conjecture}, talk at \href{https://cosmoback.sciencesconf.org}{CosmoBack 2018} ([[RasanenWaysToSettle18.pdf:file]])


\item Hayley J. Macpherson, \emph{Inhomogeneous cosmology in an anisotropic Universe} (\href{https://arxiv.org/abs/1910.13380}{arxiv:1910.13380})



\end{itemize}
\hypertarget{EffectiveDarkEnergyFromInhomogeneity}{}\subsubsection*{{Effective dark energy from inhomogeneity}}\label{EffectiveDarkEnergyFromInhomogeneity}

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

\begin{itemize}%
\item Marie-Noëlle Célérier, \emph{Do we really see a cosmological constant in the supernovae data?}, Astron. Astrophys. 353:63-71, 2000 (\href{https://arxiv.org/abs/astro-ph/9907206}{arxiv:astro-ph/9907206})


\item [[Christof Wetterich]], \emph{Can Structure Formation Influence the Cosmological Evolution?}, Phys.Rev. D67 (2003) 043513 (\href{https://arxiv.org/abs/astro-ph/0111166}{arXiv:astro-ph/0111166})


\item Dominik J. Schwarz, \emph{Accelerated expansion without dark energy} (\href{https://arxiv.org/abs/astro-ph/0209584}{arXiv:astro-ph/0209584})


\item [[Syksy Räsänen]], \emph{Dark energy from backreaction}, JCAP 0402:003, 2004 (\href{https://arxiv.org/abs/astro-ph/0311257}{arXiv:astro-ph/0311257})


\item H. Alnes, M. Amarzguioui and O. Gron, \emph{An inhomogeneous alternative to dark energy?}, Phys. Rev. D 73, 083519 (2006) (\href{https://arxiv.org/abs/astro-ph/0512006}{arXiv:astro-ph/0512006})


\item Kari Enqvist, Teppo Mattsson, \emph{The effect of inhomogeneous expansion on the supernova observations}, JCAP 0702:019,2007 (\href{https://arxiv.org/abs/astro-ph/0609120}{arXiv:astro-ph/0609120})


\item Havard Alnes, Morad Amarzguioui, \emph{The supernova Hubble diagram for off-center observers in a spherically symmetric inhomogeneous universe}, Phys. Rev. D75:023506, 2007 (\href{https://arxiv.org/abs/astro-ph/0610331}{arXiv:astro-ph/0610331})


\item [[Thomas Buchert]], \emph{Dark Energy from structure: a status report}, Gen.Rel.Grav.40:467-527, 2008 (\href{http://xxx.lanl.gov/abs/0707.2153}{arXiv:0707.2153})


\item [[Subir Sarkar]], \emph{Is the evidence for dark energy secure?}, Gen. Rel. Grav.40:269-284, 2008 (\href{https://arxiv.org/abs/0710.5307}{arXiv:0710.5307})


\item Alessio Notari, \emph{Can an Inhomogeneous Universe mimic Dark Energy?}, 2009 ([[NotariInhomogeneousCosmology.pdf:file]])


\item Michael Blomqvist, \emph{Inhomogeneous cosmologies with clustered dark energy or a local matter void}, 2010 (\href{http://www.diva-portal.org/smash/record.jsf?pid=diva2%3A353689&dswid=2010}{web})


\item [[Thomas Buchert]], [[Syksy Räsänen]], \emph{Backreaction in late-time cosmology}, Annual Review of Nuclear and Particle Science 62 (2012) 57-79 (\href{https://arxiv.org/abs/1112.5335}{arXiv:1112.5335})


\item Joel Smoller, Blake Temple, Zeke Vogler, \emph{An Instability of the Standard Model Creates the Anomalous Acceleration Without Dark Energy}, Proceedings of the Royal Society A, 2017 (\href{https://arxiv.org/abs/1412.4001}{arXiv:1412.4001}, \href{http://rspa.royalsocietypublishing.org/content/473/2207/20160887}{10.1098/rspa.2016.0887}, detailed talk slides: [[Temple16.pdf:file]], talk \href{https://www.youtube.com/watch?v=fV8KPj8vmGw}{recording I}, \href{http://cdsweb.cern.ch/record/1371553}{recording II})


\item I. Odderskov, S. M. Koksbang, S. Hannestad, \emph{The Local Value of $H_0$ in an Inhomogeneous Universe}, JCAP02(2016)001 (\href{https://arxiv.org/abs/1601.07356}{arXiv:1601.07356})


\item Adam G. Riess et al., \emph{A 2.4\% Determination of the Local Value of the Hubble Constant}, The Astrophysical Journal, Volume 826, Number 1 (\href{https://arxiv.org/abs/1604.01424}{arXiv:1604.01424})


\item Krzysztof Bolejko, Mikołaj Korzyński, \emph{Inhomogeneous cosmology and backreaction: Current status and future prospects}, Int. J. Mod. Phys. D 26, 1730011 (2017) (\href{https://arxiv.org/abs/1612.08222}{arXiv:1612.08222})


\item Sebastien Clesse, Arnaud Roisin, André Füzfa, \emph{Mimicking Dark Energy with the backreactions of gigaparsec inhomogeneities} (\href{https://arxiv.org/abs/1702.06643}{arXiv:1702.06643})


\item Julian Adamek, Chris Clarkson, David Daverio, Ruth Durrer, Martin Kunz, \emph{Safely smoothing spacetime: backreaction in relativistic cosmological simulations} (\href{https://arxiv.org/abs/1706.09309}{arXiv:1706.09309})


\item Krzysztof Bolejko, \emph{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) (\href{https://arxiv.org/abs/1712.02967}{arxiv:1712.02967})


\item [[Ulf Danielsson]], [[Thomas Van Riet]], \emph{What if string theory has no de Sitter vacua?} (\href{https://arxiv.org/abs/1804.01120}{arXiv:1804.01120})


\item Hayley Macpherson, Paul D. Lasky, Daniel J. Price, \emph{The trouble with Hubble: Local versus global expansion rates in inhomogeneous cosmological simulations with numerical relativity}, ApJ Letters (\href{https://arxiv.org/abs/1807.01714}{arXiv:1807.01714})


\item J. Colin, R. Mohayaee, M. Rameez, [[Subir Sarkar]], \emph{Evidence for anisotropy of cosmic acceleration}, Astronomy \& Astrophysics Letters (\href{https://arxiv.org/abs/1808.04597}{arXiv:1808.04597})


\item [[Subir Sarkar]], \emph{Is the universe isotropic?}, talk at \emph{\href{https://indico.nbi.ku.dk/event/973/}{Current Themes in High Energy Physics and Cosmology 2018}} (\href{https://indico.nbi.ku.dk/event/973/contributions/8344/attachments/2648/3858/Sarkar_Copenhagen18_cosmo.pdf}{pdf})


\item Lucas Lombriser, \emph{On the cosmological constant problem}, Phys. Lett. B 797, 134804 (2019) (\href{https://arxiv.org/abs/1901.08588}{arXiv:1901.08588}, \href{https://doi.org/10.1016/j.physletb.2019.134804}{doi:10.1016/j.physletb.2019.134804})


\item Vincent Deledicque, \emph{Theoretical developments on the adequacy of the fitting of the FLRW metric on the universe's real metric} (\href{https://arxiv.org/abs/1907.01580}{arxiv:1907.01580})



\end{itemize}
\hypertarget{ReferencesNoEffect}{}\subsubsection*{{No effective dark energy from inhomogeneity}}\label{ReferencesNoEffect}

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:

\begin{itemize}%
\item Ghazal Geshnizjani, Daniel J.H. Chung, Niayesh Afshordi, \emph{Do Large-Scale Inhomogeneities Explain Away Dark Energy?}, Phys.Rev. D72 (2005) 023517 (\href{https://arxiv.org/abs/astro-ph/0503553}{arXiv:astro-ph/0503553})


\item E. R. Siegel, J. N. Fry, \emph{Effects of Inhomogeneities on Cosmic Expansion}, Astrophys.J. 628 (2005) L1-L4 (\href{https://arxiv.org/abs/astro-ph/0504421}{arXiv:astro-ph/0504421})


\item Eanna E. Flanagan, \emph{Can superhorizon perturbations drive the acceleration of the Universe?}, Phys.Rev. D71 (2005) 103521 (\href{https://arxiv.org/abs/hep-th/0503202}{arXiv:hep-th/0503202})


\item Giovanni Marozzi, Jean-Philippe Uzan, \emph{Late time anisotropy as an imprint of cosmological backreaction} (\href{https://arxiv.org/abs/1206.4887}{arXiv:1206.4887})


\item Ido Ben-Dayan, Maurizio Gasperini, Giovanni Marozzi, Fabien Nugier, [[Gabriele Veneziano]], \emph{Do stochastic inhomogeneities affect dark-energy precision measurements?}, Phys. Rev. Lett. 110, 021301 (2013) (\href{https://arxiv.org/abs/1207.1286}{arXiv:1207.1286})



\end{itemize}
\hypertarget{lemaitretolmanbondi_models_2}{}\subsubsection*{{Lemaitre-Tolman-Bondi models}}\label{lemaitretolmanbondi_models_2}

\begin{itemize}%
\item [[John Moffat]], \emph{Late-time Inhomogeneity and Acceleration Without Dark Energy}, JCAP 0605 (2006) 001 (\href{https://arxiv.org/abs/astro-ph/0505326}{arXiv:astro-ph/0505326})


\item Kari Enqvist, \emph{Lemaitre-Tolman-Bondi model and accelerating expansion}, Gen. Rel. Grav.40:451-466, 2008 (\href{https://arxiv.org/abs/0709.2044}{arXiv:0709.2044})


\item Timothy Clifton, Pedro G. Ferreira, Kate Land, \emph{Living in a Void: Testing the Copernican Principle with Distant Supernovae}, Phys. Rev. Lett. 101 (2008) 131302 (\href{https://arxiv.org/abs/0807.1443}{arXiv:0807.1443})


\item [[Günter Scharf]], \emph{Inhomogeneous cosmology in the cosmic rest frame without dark stuff}, chapter 6 in the latest edition of \emph{[[Quantum Gauge Theories -- A True Ghost Story]]}, Wiley 2001 (\href{https://arxiv.org/abs/1312.2695}{arXiv:1312.2695})


\item [[John Moffat]], \emph{Inhomogeneous Cosmology Redux} (\href{https://arxiv.org/abs/1608.00534}{arXiv:1608.00534})



\end{itemize}
See also

\begin{itemize}%
\item Wikipedia, \emph{\href{https://en.wikipedia.org/wiki/Lema%C3%AEtre%E2%80%93Tolman_metric}{Lemaître–Tolman metric}}

\end{itemize}
[[!redirects inhomogeneous cosmologies]]

[[!redirects Lemaitre-Tolman-Bondi cosmology]] [[!redirects Lemaitre-Tolman-Bondi model]] [[!redirects Lemaitre-Tolman-Bondi models]]

[[!redirects LTB model]] [[!redirects LTB models]]



\end{document}