Testing the Copernican and Cosmological Principles in the local universe with galaxy surveys
aa r X i v : . [ a s t r o - ph . C O ] J un JHEP00(2007)000
Published by Institute of Physics Publishing for SISSA/ISAS
Received:
October 22, 2018
Accepted:
October 22, 2018
Testing the Copernican and Cosmological Principles inthe local universe with galaxy surveys
Francesco Sylos Labini
Centro Enrico Fermi, Piazza del Viminale 1, 0084 Rome ItalyandIstituto dei Sistemi Complessi CNR, - Via dei Taurini 19, 00185 Rome, ItalyE-mail:
Yuri V. Baryshev
Institute of Astronomy, St.Petersburg State University, Staryj Peterhoff, 198504,St.Petersburg, RussiaE-mail: [email protected]
Abstract:
Cosmological density fields are assumed to be translational and rotationalinvariant, avoiding any special point or direction, thus satisfying the Copernican Principle.A spatially inhomogeneous matter distribution can be compatible with the CopernicanPrinciple but not with the stronger version of it, the Cosmological Principle which requiresthe additional hypothesis of spatial homogeneity. We establish criteria for testing thata given density field, in a finite sample at low redshifts, is statistically and/or spatiallyhomogeneous. The basic question to be considered is whether a distribution is, at differentspatial scales, self-averaging. This can be achieved by studying the probability densityfunction of conditional fluctuations. We find that galaxy structures in the SDSS samples,the largest currently available, are spatially inhomogeneous but statistically homogeneousand isotropic up to ∼
100 Mpc/h. Evidences for the breaking of self-averaging are foundup to the largest scales probed by the SDSS data. The comparison between the resultsobtained in volumes of different size allows us to unambiguously conclude that the lackof self-averaging is induced by finite-size effects due to long-range correlated fluctuations.We finally discuss the relevance of these results from the point of view of cosmologicalmodeling.
Keywords: redshift surveys,cosmic web,cosmology of theories beyond the SM. c (cid:13) SISSA/ISAS 2018 http://jhep.sissa.it/JOURNAL/JHEP3.tar.gz
HEP00(2007)000
Contents
1. Introduction 12. Ergodicity and self-averaging 33. Breaking of self-averaging properties 44. Galaxy Catalogs 65. Discussion 86. Conclusions 10
1. Introduction
The attempts to construct cosmological models including spatial inhomogeneities have ex-perienced a renewed interest in connection with the evidences for a speeding up expansionof the universe as shown by the supernovae observations [1, 2]. Indeed, the deduction ofthe existence of dark energy is based on the assumption that the universe has a Friedmann-Robertson-Walker (FRW) geometry. There have been various claims that these observa-tions can at least in principle be accounted for without the presence of any dark energy, if weconsider the possibility of inhomogeneities. This can happen in two different ways: locallyvia back-reaction [3, 4, 5] or by placing the observer in a special point of the local universe[6, 7]. Direct observational tests of the basic assumptions used in the derivation of the FRWmodels are thus of considerable importance. A widespread idea in cosmology is that theso-called concordance model of the universe combines two fundamental assumptions. Thefirst is that the dynamics of space-time is determined by Einstein’s field equations. Thesecond is that the universe is homogeneous and isotropic. This hypothesis, usually calledthe Cosmological Principle, is though to be a generalization of the Copernican Principlethat “the Earth is not in a central, specially favored position” [8, 9]. The FRW model isderived under these two assumptions and it describes the geometry of the universe in termsof a single function, the scale factor, which obeys to the Friedmann equation [10]. Thereis a subtlety in the relation between the Copernican Principle (all observes are equivalentand there are no special points and directions) and the Cosmological Principle (the uni-verse is homogeneous and isotropic). Indeed, the fact that the universe looks the same, atleast in a statistical sense, in all directions and that all observers are alike does not implyspatial homogeneity of matter distribution. It is however this latter condition that allowsus to treat, above a certain scale, the density field as a smooth function, a fundamental– 1 –
HEP00(2007)000 hypothesis used in the derivation of the FRW metric. Thus there are distributions whichsatisfy the Copernican Principle and which do not satisfy the Cosmological Principle [11].These are statistically homogeneous and isotropic distributions which are also spatially in-homogeneous. Therefore the Cosmological Principle represents a specific case, holding forspatially homogeneous distributions, of the Copernican Principle which is, instead, muchmore general. Statistical and spatial homogeneity refer to two different properties of agiven density field. The problem of whether a fluctuations field is compatible with the con-ditions of the absence of special points and direction can be reformulated in terms of theproperties of the probability density functional (PDF) which generates the stochastic field.In what follows we precisely discuss this point, both from a theoretical and observationalpoint of view.Different strategies have been proposed to test the large scale isotropy of matter distri-bution and the basic predictions of homogeneous models [12]. (i) If the cosmic microwavebackground radiation (CMBR) is anisotropic around distant observers, Sunyaev-Zeldovichscattered photons have a distorted spectrum that reflects the spatial inhomogeneity [13, 14].(ii) Tests, based on future supernovae surveys, to determine whether there is a geometriccusp at the origin [15, 6]. (iii) Geometric effects on distance measurements [16, 17]. (iv)There are then some indirect tests [18]. All these approaches thus consider mainly datafrom the CMBR and from supernovae surveys and they do not directly test for spatialhomogeneity.However Ellis [19] pointed out that ”Spatial homogeneity is one of the foundations ofstandard cosmology, so any chance to check those foundations observationally should bewelcomed with open arms” . As recently it became possible to measure directly the natureof the spatial galaxy distribution by using galaxy redshift surveys, in this paper we presenta new test focused to determine whether matter distribution is statistically homogeneousand isotropic and whether it is spatially homogeneous. Testing these two hypotheses canbe achieved by characterizing galaxy distribution from the latest data of the Sloan DigitalSky Survey (SDSS) [20].The paper is organized as follows. In Sect.2 we recall some basic statistical propertiesof spatially homogeneous and inhomogeneous distributions. Particularly we discuss thatan inhomogeneous distribution can be fully compatible with the Copernican Principle thatthere are not special points or directions. The compatibility of a fluctuating density field,regardless of whether it is spatially homogeneous, is encoded in the properties of its PDF.This is a very fundamental issue which, in our opinion, has been overlooked in the literature.For example, in ref. [10] it is stated that ”The visible universe seems the same in alldirections around us, at least if we look out to distances larger than about 300 million lightyears” , to mean that it is spatially homogeneous as then the standard FRW modelingis used to derive a number of properties. We point out instead that the fact that theobservable galaxy distribution looks the same in all directions around us implies statisticalhomogeneity and not necessarily spatial homogeneity. This is the key point which requiresa more detailed investigation from the point of view of theoretical modeling, as the lackof spatial homogeneities has a deep impact on it. For instance the works on back-reaction[3, 4, 5] consider precisely the effect statistically homogeneous large-amplitude fluctuations– 2 –
HEP00(2007)000 (i.e. spatial inhomogeneities) up to few hundreds Mpc on the geometrical properties of thelarge scale universe.In Sect.3 we present two simple examples which may clarify this point further. Notethat the discussion in Sects.2-3 refers to an ideal case of a distribution in an Euclideanspace and does not consider the additional complication introduced by a curved and timedependent geometry. However this treatment is fully valid in the galaxy samples we con-sider, as they are limited to low redshifts, i.e. z < .
2. It is clear that any conclusion wecan draw about the statistical properties of galaxy fluctuations is limited to the range ofscales we considered.We then pass, in Sect.4, to the discussion of the observed galaxy redshift samplesprovided by the data release 7 (DR7) of the Sloan Digital Sky Survey (SDSS). Our mainresult is that galaxy distribution as observed by current surveys is inhomogeneous butnot characterized by any special point of direction, i.e. it is statistically homogeneousbut spatially inhomogeneous. This fact, was overlooked in the past [21] when only theprojection on the sky of the galaxy density field was available. Having three-dimensionalmaps allows us to test statistically homogeneity and isotropy from many points (observers),which was not possible for projection on the sky.In Sect.5 we discuss the relevance of the results obtained in the low-redshift galaxysurveys which respect to the theoretical modeling and the extension to the test we intro-duced to higher redshift. Particularly we consider the fact that we make observations onour past light-cone which is not a space-like surface. Finally we draw our conclusions inSect.6.
2. Ergodicity and self-averaging
Mass density fields can be represented as stationary stochastic processes. The stochasticprocess consists in extracting the value of the microscopic density function ρ ( ~r ) at anypoint of the space. This is completely characterized by its probability density functional P [ ρ ( ~r )]. This functional can be interpreted as the joint probability density function ofthe random variables ρ ( ~r ) at every point ~r . If the functional P [ ρ ( ~r )] is invariant underspatial translations then the stochastic process is statistically homogeneous or translationalinvariant (stationary) [11]. When P [ ρ ( ~r )] is also invariant under spatial rotation then thedensity field is statistically isotropic [11].Matter distribution in cosmology then is considered to be a realization of a stationary stochastic point process. This is enough to satisfy the Copernican Principle i.e., thatthere are no special points or directions; however this does not imply spatial homogeneity.Spatially homogeneous stationary stochastic processes satisfy the special and stronger caseof the Copernican Principle described by Cosmological Principle. Indeed, isotropy aroundeach point together with the hypothesis that the matter distribution is a smooth functionof position i.e., that this is analytical, implies spatial homogeneity. (A formal proof can befound in [22].) This is no longer the case for a non-analytic structure (i.e., not smooth),for which the obstacle to applying the FRW solutions has in fact solely to do with the lackof spatial homogeneity [23]. – 3 – HEP00(2007)000
The condition of spatial homogeneity ( uniformity ) is satisfied if the ensemble averagedensity of the field h ρ i is strictly positive. Otherwise, when h ρ i = 0 the distribution isinhomogeneous. We are interested in the finite sample properties of a given density fieldand for this reason we should introduce the concept of spatial average. First, we remindthat a crucial assumption usually used is that stochastic fields are required to satisfyspatial ergodicity . Let us take a generic observable F = F ( ρ ( ~r ) , ρ ( ~r ) , ... ) function ofthe mass distribution ρ ( ~r ) at different points in space ~r , ~r , ... . Ergodicity implies that hF i = F = lim V →∞ F V , where F V is the spatial average in a finite volume V [11].When considering a finite sample realization of a stochastic process, and thus statisticalestimators of asymptotic quantities, the first question to be sorted out concerns whether acertain observable is self-averaging in a given finite volume [24, 25]. In general a stochasticvariable F is self-averaging if F = hF i (see [25] for a more detailed discussion). Thus ifthis is ergodic, F = hF i , then it is also self-averaging as F = (cid:10) F (cid:11) : finite sample spatialaverages must be self-averaging in order to satisfy spatial ergodicity.A simple test to determine whether a distribution is stationary and self-averaging in agiven sample of linear size L consists in studying the probability density function (PDF)of conditional fluctuations G (which contains, in principle, all information about momentsof any order) in sub-samples of linear size L ′ < L placed in different and non-overlappingspatial regions of the sample (i.e., S , S , ...S N ). That the self-averaging property holdsis shown by the fact that P ( G , L ′ ; S i ) is the same, modulo statistical fluctuations, in thedifferent sub-samples, i.e., P ( G , L ′ ; S i ) ≈ P ( G , L ′ ; S j ) ∀ i = j. On the other hand, if deter-minations of P ( G , L ′ ; S i ) in different sample regions S i show systematic differences, thenthere are two different possibilities: (i) the lack of the property of stationarity or (ii) thebreaking of the property of self-averaging due to a finite-size effect related to the presence oflong-range correlated fluctuations. Therefore while the breaking of statistical homogeneityand/or isotropy imply the lack of self-averaging property the reverse is not true. However,if the determinations of the spatial averages give sample-dependent results, this impliesthat those statistical quantities do not represent the asymptotic properties of the givendistribution [25].To test statistical and spatial homogeneity it is necessary to employ statistical quan-tities that do not require the assumption of spatial homogeneity inside the sample andthus avoid the normalization of fluctuations to the estimation of the sample average [25].These are conditional quantities, which describe local properties of the distribution. Forinstance, we consider the number of points N i ( r ) contained in a sphere of radius r centeredon the i th point. This depends on the scale r and on the spatial position of the i th sphere’scenter, namely its radial distance R i from a given origin and its angular coordinates ~α i .Integrating over ~α i for fixed radial distance R i , we obtain that N i ( r ) = N ( r ; R i ) [25].
3. Breaking of self-averaging properties
In order to illustrate an example let us consider a case where translation invariance isbroken. We generate a Poisson-Radial distribution (PRD) which is a inhomogeneous dis-tribution that can mimic the effect of a “local hole” around the origin. In a sphere of– 4 –
HEP00(2007)000 radius R = 1 we place, for instance, N = 2 · points. In each bin at radial distance fromthe sphere center [ R i , R i +1 ], and with thickness ∆ R , the distribution is Poissonian with adensity varying as n ( R ) = n · R , where n is a constant. We determine the PDF P ( N ; r )of conditional fluctuations obtained by making an histogram of the values of N ( r ; R ) atfixed r (see the upper panels of Fig.1). The whole-sample PDF is clearly left-skewed: thisoccurs because the peak of the PDF corresponds to the most frequent counts which areat large radial distance simply because shells far-way from the origin contain more points.The spread of the PDF can easily be related to the difference in the density between smalland large radial distances in the sample. By computing the PDF into two non-overlappingsub-samples, nearby to and faraway from the origin, one may clearly identify the systematicdependence of this quantity on the specific region where this is measured. This breakingof the self-averaging properties is caused by the radial-distance dependence of the densityand thus by the breaking of translational invariance.Let us now consider a stationary stochastic distribution, where the breaking self-averaging properties is due to the effect of large scale fluctuations. An example is rep-resented by the inhomogeneous toy model (ITM) constructed as follows. We generate astochastic point distribution by randomly placing, in a two-dimensional box of side L ,structures represented by rectangular sticks. We first distribute randomly N s points whichare the sticks centers: they are characterized by a mean distance Λ ≈ ( L /N s ) / . Thenthe orientation of each stick is chosen randomly. The points belonging to each stick arealso placed randomly within the stick area, that for simplicity we take to be ℓ × ℓ/
10. Thelength-scale ℓ can vary, for example being extracted from a given PDF. The number ofsticks placed in the box fixes Λ. This distribution is by construction stationary i.e., thereare no special points or directions. When ℓ ≥ L and Λ ≤ L but with ℓ varying in such away that there can been large differences in its size, the resulting distribution is long-rangecorrelated, spatially inhomogeneous and it can be not self-averaging. This latter case oc-curs when, by measuring the PDF of conditional fluctuations in different regions of a givensample, one finds, for large enough r , systematic differences in the PDF shape and peaklocation (see the bottom panels Fig.1). These are due to the strong correlations extendingwell over the size of the sample.How can we distinguish between the case in which a distribution is not self-averagingbecause it is not statistically translational invariant and when instead this is stationary butfluctuations are too extended in space and have too large amplitude ? The clearest testis to change the scale r where P ( N, r ) is measured, and determining whether the PDF isself-averaging. Indeed, in the case of the PRD the strongest differences between the PDFmeasured in regions placed at small and large radial distance from the structure center,occur for small r . This is because the local density has the largest variations at small andlarge radial distances by construction. When r grows, different radial scales are mixed asthe generic sphere of radius r pick up contributions both from points nearby the originand from those far away from it, resulting in a smoothing of local differences. Instead,in the ITM for small r the difference is negligible while for large enough r the differentdeterminations of the local density start to feel the presence of a few large structures which– 5 – HEP00(2007)000 N P ( N ; r ) PRD r=0.1 P ( N ; r ) PRD; r=0.3 N P ( N ; r ) ITM; r=0.02 N P ( N ; r ) ITM; r=0.1
Figure 1:
Upper panels: The PDF for r = 0 . r = 0 . r = 0 .
02 (left)and r = 0 . dominate the large scale distribution in the sample.
4. Galaxy Catalogs
Let us now consider two (volume limited [25]) samples constructed from the data release6 (DR6) and DR7 [20] of the SDSS (see [25, 26] for details). We cut each sample volumeinto two regions, one nearby us (small R ) and the other faraway from us (large R ).We determine the PDF P ( N ; r ) separately in both regions, and at two different r scales.In a first case (left panels of Fig.2), at small scales ( r = 10 Mpc/h), the distribution isself-averaging both in the DR6 sample (that covers a solid angle Ω DR = 0 .
94 sr.) thanin the sample extracted from DR7 (Ω DR = 1 .
85 sr. ≈ × Ω DR sr). Indeed, the PDF isstatistically the same in the two sub-samples considered. Instead, for larger sphere radiii.e., r = 80 Mpc/h, (right panels of Fig.2) in the DR6 sample, the two PDF show clearly asystematic difference. Not only the peaks do not coincide, but the overall shape of the PDF R = R ( z ) is the metric distance for which we used the standard cosmological parameters Ω M = 0 . Λ = 0 .
7. Given that the redshift is limited to z ≤ .
2, different values of Ω M , Ω Λ have little effects onour results – 6 – HEP00(2007)000 is not smooth and different. On the other hand, for the sample extracted from DR7, thetwo determinations of the PDF are in very good agreement. We conclude therefore that,in DR6 for r = 80 Mpc/h there are large density fluctuations which are not self-averagingbecause of the limited sample volume [25]. They are instead self-averaging in DR7 becausethe volume is increased by a factor two. N P ( N ; r ) DR6; r=10 P ( N ; r ) DR6; r=80 N P ( N ; r ) DR7; r=10 N P ( N ; r ) DR7; r=80
Figure 2:
PDF of conditional fluctuations in the sample defined by R ∈ [125 , M ∈ [ − . , − .
2] in the DR6 (upper panels) and DR7 (lower panels) data, for two differentvalues of the sphere radii r = 10 Mpc/h and r = 80 Mpc/h. In each panel, the black line representsthe full-sample PDF, the red line (green) the PDF measured in the half of the sample closer to(farther from) the origin. The lack of self-averaging properties at large scales in the DR6 sample is due to thepresence of large scale galaxy structures which correspond to density fluctuations of largeamplitude and large spatial extension, whose size is limited only by the sample boundaries.The appearance of self-averaging properties in the larger DR7 sample volumes is the un-ambiguous proof that the lack of them is induced by finite-size effects due to long-rangecorrelated fluctuations.For the deepest sample we consider, which include mainly bright galaxies, the breakingof self-averaging properties does not occur as well for small r but it is found for large r .This can be due to the same effects i.e., that the sample volumes are still too small aseven in DR7 for r = 120 Mpc/h we do not detect self-averaging properties (right panels– 7 – HEP00(2007)000 N P ( N ; r ) DR6; r=20 P ( N ; r ) DR6; r=120 N P ( N ; r ) DR7; r=20 N P ( N ; r ) DR7; r=120
Figure 3:
The same of Fig.2 but for the sample defined by R ∈ [200 , M ∈ [ − . , − .
8] and for r = 20 ,
120 Mpc/h. of Fig.3). Other radial distance-dependent selections, like galaxy evolution [27], could inprinciple give an effect in the same direction. However this would not affect the conclusionthat, on large enough scales, self-averaging is broken. Note that, contrary to the PRDcase, in the SDSS samples for small values of r the PDF is found to be statistically stablein different sub-regions of a given sample. For this reason we do not interpret the lackof self-averaging properties as due to a “local hole” around us. As discussed above, thiswould affect all samples and all scales, which is indeed not the case. Because of these largefluctuations in the galaxy density field, self-averaging properties are well-defined only in alimited range of scales. Only in that range it will be statistically meaningful to measurewhole-sample average quantities [25, 26].
5. Discussion
The discussion in the previous sections was meant to treat the statistical properties of thegalaxy density field in a spatial hyper-surface. As mentioned above, this is an approxima-tion valid when considering the galaxy distribution limited to relatively low redshifts, i.e. z < .
2. In particular, we have developed a test to focus on the properties of statistical andhomogeneity homogeneity in nearby redshift surveys. The assumptions of the cosmologicalmodel enter in the data analysis when calculating the metric distance from the redshift– 8 –
HEP00(2007)000 and the absolute magnitude from the apparent one and the redshift. However, given thatsecond order corrections are small for z < .
2, our results are basically independent on thechosen underlying model to reconstruct metric distances and absolute magnitudes fromdirect observables. In practice we can use just a linear dependence of the metric distanceon the redshift (which is, to a very good approximation, compatible with observations atlow redshift). For this same reason we can approximate the observed galaxies as lying in aspatial hyper-surface.In the ideal case of having a very deep survey, up to z ≈
1, we should consider thatwe make observations on our past light-cone which is not a space-like surface. In order toevolve our observations onto a spatial surface we would need a cosmological model, whichat such high redshift can play an important role in the whole determination of statisticalquantities. A sensible question is whether we can to reformulate the statistical test givenso that it can be applied to data on our past light-cone, and not on an assumed spatialhyper-surface. Going to higher redshift poses a number of question, first of the all the oneof checking the effect of the assumptions used to construct metric distances and absolutemagnitudes from direct observables. Testing these effects can be simply achieved by usingdifferent distance-redshift relations.However, we note that a smooth change of the distance-redshift relation as implied bya given cosmological model, may change the average behavior of the conditional densityas a function of redshift but it cannot smooth out fluctuations, i.e. it cannot substantiallychange the PDF of conditional fluctuations when they are measured locally. Indeed, our testis based on the characterization of the PDF of conditional fluctuations and not only of thebehavior of the conditional average density as a function of distance. The PDF provides,in principle, with a complete characterization of the fluctuations statistical properties.We have shown that the PDF of fluctuations has a clear imprint when the distribution isspherically symmetric or when it is spatially inhomogeneous but statistically homogeneous.The fact that we analyze conditional fluctuations means that we consider only localproperties of the fluctuations: local with respect to an observer placed at different radial(metric) distances from the us, i.e. at different redshifts. For the determination of the PDFwe have to consider two different length scale: the first is the (average) metric distance R of the galaxies on which we center the sphere and the second is the sphere radius r .Irrespective of the value of R when r is smaller than a few hundreds Mpc (i.e., when itssize is much smaller than any cosmological length scale), we can always locally neglect thespecific R ( z ) relation induced by a specific cosmology. In other words, when the sphereradius is limited to a few hundreds Mpc we can approximate the measurements of theconditional density to be performed on a spatial hyper-surface.The whole description of the matter density field in terms of FRW or even Lemaitre-Tolman-Bondi (LTB) cosmologies, refer to the behavior of, for instance, the average matterdensity as a function of time (in the LTB case also as a function of scale) but it saysanything on the fluctuation properties of the density field. Thus, when looking at differentepochs in the evolution of the universe, we should detect that the average density varies(being higher in early epochs). This means only that the peak of the PDF will be locatedat different N values, but the shape of the PDF is unchanged by this overall (smooth)– 9 – HEP00(2007)000 evolution. Fluctuations are simply not present in the FRW or LTB models, and the wholeissue of back-reaction studies is to understand what is their effect.Note that models which explain dark energy through inhomogeneity do so using aspatial under-density in the matter density which varies on Gpc scales — out to z ≈ z > .
6. Conclusions
We have presented tests on both the Copernican and Cosmological Principles at lowredshift, where we can neglect the important complications of evolving observations ontoa spatial surface for which we need a specific cosmological model. We have discussedhowever that the statistical properties of the matter density field up to a few hundredsMpc is crucially important for the theoretical modeling.We have discussed that these are achieved by considering the properties of the prob-ability density function of conditional fluctuations in the available galaxy samples. Wehave shown that galaxy distribution in different samples of the SDSS is compatible withthe assumptions that this is transitionally invariant, i.e. it satisfies the requirement of theCopernican Principle that there are no spacial points or directions. On the other hand, wefound that there are no clear evidences of spatial homogeneity up to scales of the order ofthe samples sizes, i.e. ∼
100 Mpc/h . This implies that galaxy distribution is not com-patible with the stronger assumption of spatial homogeneity, encoded in the CosmologicalPrinciple. In addition, at the largest scales probed by these samples (i.e., r ≈
120 Mpc/h) These results are compatible with those found by [30, 31, 32] in the Two Degree Field Galaxy RedshiftSurvey. – 10 –
HEP00(2007)000 we found evidences for the breaking of self-averaging properties, i.e. that the distribution isnot statistically homogeneous. Forthcoming redshift surveys will allow us to clarify whetheron such large scales galaxy distribution is still inhomogeneous but statistically stationary,or whether the evidences for the breaking of spatial translational invariance found in theSDSS samples were due to selection effects in the data.
Acknowledgments
We thank T. Antal and N. L. Vasilyev for fruitful collaboration, A. Gabrielli and M.Joyce for interesting discussions and comments. We also thank T. Clifton, R. Durrerand D. Wiltshire for useful remarks. An anonymous referee made a list of interestingcomments and criticisms which have allowed us to improve the presentation of our results.We acknowledge the use of the Sloan Digital Sky Survey data ( ). References [1] Riess, A. G., et al. , Astron. J. (1998) 1009[2] Perlmutter, S. et al. , Astrophys. J. (1999) 565[3] Buchert, T.
Gen. Rel. Grav. (2000) 105[4] Wiltshire, D.L., Phys. Rev. Lett. (2007) 251101[5] R¨as¨anen, S., J. Cosm. Astr. Phys. (2006) 003[6] Clifton T., Ferreira, P. G., Land, K. Phys. Rev. Lett. (2008) 131302[7] C´el´erier M.N.,
New Advan. Phys. (2007) 29[8] Bondi, H. Cosmology , (Cambridge University Press, Cambridge, 1952).[9] Clifton T., Ferreira, P. G.,
Phys. Rev.
D 80 (2009) 103503[10] Weinberg, S.,
Cosmology (Oxford University Press, Oxford, 2008)[11] Gabrielli A., Sylos Labini F., Joyce M., Pietronero L.,
Statistical Physics for CosmicStructures (Springer Verlag, Berlin, 2005)[12] Ellis, G.F.R, 2008, in the proc. of the Conference “Dark Energy and Dark Matter”[13] Goodman, J.,
Phys. Rev.
D 52 (1995) 1821[14] Caldwell R., Stebbins, A.,
Phys. Rev. Lett. (2008) 191302[15] Vanderveld, R.A., et al. , Phys. Rev.
D 74 (023506) 2006[16] Clarkson, C., Bassett, B.A., Lu T.C.,
Phys. Rev. Lett. (2008) 011301[17] Wiltshire, D.L.,
Phys. Rev.
D 80 (2009) 123512[18] Lima, L. M., Vitenti S., Reboucas, M.J.,
Phys. Rev.
D 77 (2008) 083518[19] Ellis, G.,
Nature (2008) 158[20] Adelman-McCarthy, J.K., et al. , Astrophys. J. Suppl. (2008) 297 – 11 –
HEP00(2007)000 [21] Peebles, P.J.E.,
Principles Of Physical Cosmology , (Princeton University Press, Princeton,New Jersey, 1993)[22] Straumann, N.,
Helv. Phys. Acta (197) 379[23] Joyce, M., et al. , Europhys. Lett. (2000) 416[24] Aharony, A., Harris, B., Phys. Rev. Lett. (1996) 3700[25] Sylos Labini, F. Vasilyev, N.L., Baryshev, Y.V., Astron. Astrophys. (2009) 17[26] Antal, T., Sylos Labini, F., Vasilyev, N.L., Baryshev, Yu. V.,
Europhys. Lett. (2009)59001[27] Loveday, J., Mon. Not. R. Acad. Soc (2004) 601[28] Sylos Labini F., Montuori, M. Pietronero, L.
Phys. Rept. (1998) 61[29] Wiltshire, D.L.,
Int. J. Mod. Phys.
D 17 (2008) 641[30] Vasilyev, N.L., Baryshev, Yu. V., Sylos Labini, F.,
Astron. Astrophys. (2006) 431[31] Sylos Labini, F., Vasilyev, N.L., Baryshev, Yu. V.,
Europhys. Lett. (2009) 29002[32] Sylos Labini, F., Vasilyev, N.L., Baryshev, Yu. V., Astron. Astrophys. (2009) 7(2009) 7