«HIP-2007-64/TH Preprint typeset in JHEP style - HYPER VERSION Dark energy as a mirage arXiv:0711.4264v2 [astro-ph] 23 Dec 2007 Teppo Mattsson1,2,∗ ...»
Preprint typeset in JHEP style - HYPER VERSION
Dark energy as a mirage
arXiv:0711.4264v2 [astro-ph] 23 Dec 2007
Helsinki Institute of Physics, P.O. Box 64, FIN-00014 University of Helsinki, Finland
Department of Physical Sciences, P.O. Box 64, FIN-00014 University of Helsinki,
Abstract: We show that the observed inhomogeneities in the universe have a quintessential eﬀect on the observable distance-redshift relations. The eﬀect is modeled quantitatively by an extended Dyer-Roeder method that allows for two crucial physical properties of the universe: inhomogeneities in the expansion rate and the growth of nonlinear structures. On large scales, the universe is homogeneous, but due to the initially smooth matter forming opaque clumps with time, the regions the detectable light traverses get emptier and emptier compared to the average. As space expands the faster the lower the local matter density, the expansion can then accelerate along our line of sight. This phenomenon provides both a natural physical interpretation and a quantitative match for the observations from the cosmic microwave background anisotropy, the position of the baryon oscillation peak, the magnitude-redshift relations of type Ia supernovae, the local Hubble ﬂow and the nucleosynthesis, resulting in a concordant model with 90% dark matter, 10% baryons, no dark energy and 14.8 Gyr as the age of the universe. The model is based only on the observed inhomogeneities so, unlike a large local void, it respects the cosmological principle, further explaining the late onset of the perceived acceleration as a consequence of the forming nonlinear structures. Altogether, the results seem to imply that dark energy is a mirage.
Keywords: Inhomogeneous Cosmological Models, Dark Energy, Cosmology, Gravitation.
1. Introduction The widely adopted framework of modern cosmology is the spatially homogeneous and isotropic Friedman-Robertson-Walker spacetime, with the growth of structure described as linear perturbations evolving on that smooth FRW background. Accordingly, the standard model of cosmology is built on the assumption that the eﬀects of the evident nonlinear inhomogeneities on the detectable light are averaged out1 at cosmological distances. This assumption — although perhaps in concordance with Newtonian intuition — lacks a convincing demonstration within general relativity, not to mention a mathematically rigorous proof. In fact, the assumption was criticized already in the 60’s by Zel’dovich , Feynman2, Bertotti , Gunn , Kantowski  and in the 70’s by Dyer & Roeder [5, 6], who Apart from gravitational lenses that are occasionally taken into account as additional corrections.
Apparently, R. P. Feynman was also one of the pioneers of this subject, as [3, 4] cite a colloquium by Feynman in 1964.
–1– suggested that the clumping of matter in the real universe has consequences on the observable distance-redshift relations that the simpliﬁed FRW description fails to capture.
At the time of this pioneering research, the cosmological observations were too inaccurate to distinguish the predictions of the proposed inhomogeneous models from the simple and elegant FRW solutions. Consequently, the eﬀorts towards a more thorough description of the eﬀects of the nonlinear structures on the distance-redshift relations never became part of the mainstream research in cosmology. Although the structure formation itself has developed into a major part of modern cosmology, the emphasis is usually given either to the general relativistic but perturbative description of structures on large scales, or to the Newtonian description of the small scale structure e.g. in the context of galaxy formation.
However, the unparalleled precision of the recent cosmological observations has made the debate about the role of the nonlinear structures in the distance-redshift relations topical. Indeed, the observations indicate the existence of inhomogeneities that are on large enough scales to have cosmological signiﬁcance and still too strong to be described within the linear theory — in particular the observed voids and the clustering of galaxies as ﬁlaments between the voids [7–11]. The increase in the precision of the data naturally demands a corresponding enhancement in the theoretical model; otherwise there is an increased risk of drawing incorrect conclusions from the observations.
In fact, the recent discovery that the standard FRW cosmology needs a mysterious, severely ﬁne-tuned dark energy component [12–14] in order to account for the observations [15–18], suggests that the precision of the data already has surpassed the precision of the employed theoretical model. This is also backed up by a strange coincidence in the standard
cosmology, pointed out by Schwarz  and Wetterich  and elaborated by R¨s¨nen :
aa the eﬀects of the elusive dark energy happen to appear just around the same time when nonlinear structures start to form at cosmologically signiﬁcant scales. As in general, a dynamical explanation would be preferred.
Whereas the ﬁne-tuning of dark energy only suggests that something fundamental is lacking from the standard picture, the coincidence problem provides the smoking gun evidence for a causal connection between dark energy and structure formation. Indeed, the most natural explanation seems to be, that the standard FRW description breaks down when the nonlinearities become dominant in the late-time universe and the need for a tiny cosmological constant is only a manifestation of this breakdown, not evidence for dark energy. The missing piece from the ﬁnal solution has been a quantitative model that would both ﬁt the observations and give a natural physical interpretation for how the structure formation actually mimics dark energy — we aim to provide such a model in this work.
The paper is organized as follows. In Sect. 2, we clarify various ways how nonlinear inhomogeneities can be seen to aﬀect the cosmological observations and potentially mimic dark energy; Sects. 2.1 and 2.2 are mostly review and can be skipped. Most importantly, in Sect. 2.3 we introduce light propagation in voids as the physical mechanism on which the principal model of this work, the optically altered nonlinear universe presented in Sect.
3, is based. The physical foundations of this model are provided in Sect. 3.1, whereas the employed extended Dyer-Roeder method is presented in Sect. 3.2. In Sect. 3.3, we construct a coarse grained description of the structure formation in terms of a function –2– that measures the deviation of the optically perceived expansion from the FRW expansion, forming a key part of the quantitative properties of the model summarized in Sect. 3.4. A comparison with the cosmological observations is performed in Sect. 3.5 and a discussion of the results is given in Sect. 3.6. Finally, Sect. 4 contains our main conclusions.
2. A classiﬁcation of inhomogeneous extensions to FRW cosmology
For clariﬁcation, we provide here the main reasons given in the literature as to why the cosmological inhomogeneities would not be properly taken into account in the linearly perturbed FRW models. Formally, the problem is the following: we have a physical system, the universe, with 1080 degrees of freedom3 but we want a model with 10 degrees of freedom, represented by the cosmological parameters. For a mathematical description of this system, we thus need a coarse graining map, a physiomorphism: 1080 dof → 10 dof, that preserves the relevant physical structure of the spacetime but removes its mathematical complexity. As our cosmological knowledge is ﬁrmly based on observing light, the relevant structure refers to the optical properties of the universe. That is, we want the observable distance-redshift relations to be unaltered by the coarse graining — the common goal for all the diﬀerent approaches presented next.
2.1 Backreaction of nonlinear structures A natural choice to construct the coarse grained description of the universe is via averaging. For this, one needs to average dynamical quantities of Einstein’s gravitational theory.
As ﬁrst pointed out by Shirokov and Fisher  and later made more popular in the observational cosmology programme by Ellis and collaborators [23–26], it seems physically more correct to ﬁrst calculate the Einstein ﬁeld G(g) for the exact metric g and only then average G(g), than to calculate the Einstein ﬁeld for the averaged metric G( g ) as in the standard FRW approach. The reason is that the Einstein ﬁeld is more closely related to physical quantities whereas the metric corresponds to gravitational potentials, whose derivatives determine the physics. In general relativity, the ﬁeld G depends nonlinearly on the metric g, so its evaluation does not commute with averaging and the non-vanishing difference, G(g) − G( g ) = 0, gives rise to what is known as the cosmological backreaction;
see . Hence the issue is not only a conceptual one, but in general, the two approaches yield identical results only in the absence of nonlinear inhomogeneities. Indeed, the backreaction is a way to quantitatively demonstrate that the observed nonlinear inhomogeneities have cosmological signiﬁcance not taken into account in the standard FRW description.
Even if performed in the above explained order, the averaging is not free of problems.
Firstly, both a covariant and practical deﬁnition for the averages of tensors seems to be possible only for the inherently invariant rank 0 tensors or scalars. Secondly, the desire in cosmology to divide the spacetime into temporal and spatial parts leads to the issue of gauge dependence: in order to take spatial averages, we must artiﬁcially break the symmetry between time and space inherent in general relativity.
(H0 Gmp )−1 c3 ∼ 1080 gives a rough estimate for the number of protons in the universe.
see e.g. [29, 30]. Although the form of the metric (2.7) is the same as in the perfectly homogeneous FRW universe, the time evolution of the scale factor aD (τ ) and the spatial curvature kD (τ ) are in general diﬀerent from the FRW case.
The average expansion accelerates, if the right hand side of Eq. (2.1) is positive; this is realized with large enough variance of the expansion rate if the counterbalancing average –4– shear is not in turn too large. The variance becomes large when contracting (θ 0) and expanding (θ 0) regions coexist. This is exactly what one would expect in the late universe with structures forming via gravitational collapse [31, 32], so it has been conjectured that the average acceleration could explain the cosmological observations .
However, leaving the issue of gauge-dependence aside, the approach has other, perhaps more critical shortcomings. First a practical problem: due to the fact that the Einstein equation can be only partially decomposed into scalar equations, the Buchert equations (2.1), (2.2) and (2.3) contain three equations for four unknowns; the required fourth equation is the deﬁnition of the backreaction (2.4), which can be calculated only once the exact solution is known. Therefore, additional information is in general needed to solve the equations, which might make it diﬃcult to calculate observables within this formalism4.
Another, perhaps a more crucial issue is that the sign of the backreaction term in the Friedman equation (2.2) is negative, in contrast with dark energy. As Eq. (2.2) determines the Hubble distance — having the most prominent role in the observable distance-redshift relations; see Eq. (2.21) and the paragraph around — one could speculate that negative backreaction is needed to account for the observations. This, in turn, would be against the conjecture that backreaction is dominated by the large variance of the expansion rate induced by collapsing structures. Perhaps a possibility is that decreasing backreaction due to virialization of the structures and/or the average spatial curvature would mimic acceleration. Overall, it is still an open question whether it is practicable to describe the observational eﬀects of the nonlinear inhomogeneities via a backreaction term .
2.2 A large local void
The major part of the universe is taken up by voids of size 10...100 Mpc [7–10], nearly empty regions expanding faster than the whole universe on average. Recently, even larger voids (up to size ∼ 300 Mpc) have been observed, the largest of which also manifests itself as a cold spot in the cosmic microwave background . Naturally, there is also the possibility that we would happen to live inside such a void [33, 34], also known as Hubble Bubble in this context.
Due to the observed inhomogeneities at relatively large scales, a description which includes averaging over a single scale may fail5 [21, 35]. In that case, a possibility is to give up the notion of a global scale factor and replace the FRW form of the metric (2.7) by an inhomogeneous metric; a particularly useful metric for describing voids is the spherically symmetric Lemaitre-Tolman-Bondi metric (3.3), an exact solution of the Einstein equations that Lemaitre discovered in 1933 .
In 1997, Mustapha, Hellaby and Ellis argued that any isotropic set of observations can be explained by appropriate inhomogeneities in the LTB model . Two years later — after the ﬁrst indications of accelerated expansion in the supernova observations — C´l´rier ee made use of this to explicitly demonstrate that the supernova data can as well be accounted for by suitable inhomogeneities in the LTB solution without accelerated expansion and dark The reason this issue is not explicit in the FRW cosmology is that it disregards the backreaction.
However, see  for an extended averaging method to describe inhomogeneities also at large scales.
–5– energy . In 2005, Alnes, Amarzguioui and Grøn showed that the suitable inhomogeneity proﬁle was in fact physically describing a local void , whose existence had a few years earlier been speculated by Zehavi et. al.  and Tomita . During the recent years, also several other authors have pointed out interesting novel aspects of this subject [35, 40–47].