Extended Dyer-Roeder Approach Improves the Cosmic Concordance Model J. A. S. Lima1,∗ V. C. Busti1,† and R. C. Santos2‡ 1Departamento de Astronomia, Universidade de S˜ao Paulo, 05508-900 S˜ao Paulo, SP, Brazil 2DepartamentodeCiˆenciasExatasedaTerra,UniversidadeFederaldeS˜aoPaulo(UNIFESP),Diadema,09972-270SP,Brazil AnewinterpretationoftheconventionalDyer-Roeder(DR)approachbyallowinglightreceivedfrom distant sources to travelin regions denser than average is proposed. It is argued that theexistence of a distribution of small and moderate cosmic voids (or “black regions”) implies that its matter content was redistributed to the homogeneous and clustered matter components with the former 3 becoming denser than the cosmic average in the absence of voids. Phenomenologically, this means 1 thattheDRsmoothnessparameter(denotedherebyαE)canbegreaterthanunity,and,therefore, 0 all previous analyses constraining it should be rediscussed with a free upperlimit. Accordingly, by 2 performingastatisticalanalysisinvolving557typeIasupernovae(SNeIa)fromUnion2compilation n datainaflatΛCDMmodelweobtainfortheextendedparameter,αE =1.26+−00..6584 (1σ). Theeffects a of αE are also analyzed for generic ΛCDM models and flat XCDM cosmologies. For both models, J we find that a value of αE greater than unity is able to harmonize SNe Ia and cosmic microwave 2 background (CMB) observations thereby alleviating the well known tension between low and high 2 redshift data. Finally, a simple toy model based on the existence of cosmic voids is proposed in order tojustify why αE can be greater than unity as required by Supernovaedata. ] O PACSnumbers: Darkenergy,cosmicdistance,supernovae, inhomogeneities C . h I. INTRODUCTION of small-scale inhomogeneities in the light propagation p from distant sources. Later on, Dyer and Roeder (DR) - [13]assumedexplicitlythatonlyafractionoftheaverage o The accelerating cosmic concordance model (flat r matter density must affect the light propagation in the ΛCDM) is in agreement with all the existing observa- t intergalacticmedium. Phenomenologically,theunknown s tions both at the background and perturbative levels. a physical conditions along the path, associated with the [ However,whilemoredataarebeinggathered,thereisan clumpinesseffects,weredescribedbythesmoothnesspa- 1 accumulating evidence that a more realistic description rameter: beyond the “precisionera” requires a better comprehen- v sion of systematic effects in order to have the desirable 0 accuracy. ρ 6 h α= , (1) 3 Local inhomogeneities are not only possible sources ρh+ρcl 5 of different systematics, but may also be signalizing for 1. an intrinsic incompleteness of the cosmic description. where ρh and ρcl are the fractions of homogeneous and 0 This occurs because the Universe is homogeneous and clumped densities, respectively. This parameter quan- 3 isotropic only on large scales (&100Mpc). However, on tifies the fraction of homogeneously distributed matter 1 smaller scales, a variety of structures involving galaxies, within a given light cone. For α = 0 (empty beam), : v clusters, andsuperclusters of galaxiesareobserved. Per- all matter is clumped while for α = 1 the fully homo- i meating these structures there are also voids or “black geneous case is recovered, and for a partial clumpiness X regions” (as dubbed long ago by Zel’dovich [1]) where the smoothness parameter is restricted over the interval ar galaxiesarealmostortotallyabsentasrecentlysuggested [0,1]. Thereadershouldkeepinmindthatsucharestric- by the N-body Millenium simulations [2]. This means tionclearlyexcludesthepossibilityoflightraystraveling that statistically uniform cosmologies are only coarse- inregionsdenser thanaverage. Inprinciple, it shouldbe grainedrepresentationsofwhatisactuallypresentinthe veryinterestingtoseehowthepresenceofcosmicvoids- real Universe. As a consequence, the description of light akeyentity nowadays-couldbe consideredinthe above propagationbytakingintoaccountsuchrichnessofstruc- prescription. tures is a challengingtask to improvethe cosmic concor- Morerecently,manystudiesconcerningthelightprop- dance model, but the correct method still remains far agationanditseffectsonthederiveddistanceshavebeen from a consensus [3, 4, 5, 6, 7, 8]. performed [5, 7, 14, 15]. Current constraints on the smoothnessparameterarestillweak[16,17,18,19],how- Zel’dovich[9],followedbyBertotti[10],Gunn[11]and ever, it is intriguing that the quoted analyses had their Kantowski[12] were the first to investigate the influence bestfits for αequaltounity whichcorrespondsto a per- fectly ΛCDM homogeneous model at all scales [16, 17]. Morerecently,someauthorshavealsoarguedforacrucial deficiencyoftheDRapproach,and,assuch,itshouldbe ∗Electronicaddress: [email protected] †Electronicaddress: [email protected] replacedby a more detailed description, probably, based ‡Electronicaddress: [email protected] on the weak lensing approach [5, 15]. 2 In this letter we advocate a slightly different but com- the validity of the duality relationbetween the an- • plementary point of view. It will be assumed that the gulardiameterandluminositydistances[24,25,26] DR approach is an useful tool in the sense that it pro- vides the simplest one-parametric description of the ef- dL(z)=(1+z)2dA(z). (5) fects causedby localinhomogeneities,but its initial con- ception needs to be somewhat extended. This is done For a general XCDM model, where the dark energy in two steps: (i) by allowing α (here denoted by αE) to componentisdescribedbyaperfectfluidwithequationof begreaterthanunityinthestatisticaldataanalyses,(ii) state p = wρ (w constant), the Dyer-Roeder distance x X by interpreting the obtained results in terms of the exis- (d =H−1D ) can be written as: L 0 L tence of an uncompensated distribution of cosmic voids or “black regions” in the Universe (see section V). As 3 we shall see, by performing a statistical analysis involv- α (z)Ω (1+z)3+Ω (1+w)(1+z)3(1+w) D (z) E m X L ing557SNeIafromtheUnion2compilationdata[22],we 2h i obtainαE =1.26+−00..6584(1σ)foraflatΛCDMmodel. This +(1+z)2E(z) d (1+z)2E(z) d DL(z) =0,(6) 1σ confidence regionshowsthat α>1 has a verysignifi- dz (cid:20) dz(1+z)2(cid:21) cantprobability. We alsoshowthatαgreaterthanunity isalsoabletoharmonizethelowredshift(SupernovaeIa) where αE(z) denotes the extended Dyer-Roeder param- andbaryonacousticoscillations(BAO)datawiththeob- eter, ΩX, w, are the density and equation of state pa- servations from cosmic microwavebackground (CMB). rameters of dark energy while the dimensionless Hubble parameter, E(z)=H/H , reads: 0 II. THE DYER-ROEDER DISTANCE E(z)= Ω (1+z)3+Ω (1+z)3(1+w)+Ω (1+z)2, m X k q (7) The differentialequationdrivingthe lightpropagation where Ω = (1 Ω Ω ) and the limiting case k m X in curved spacetimes is the Sachs optical equation − − (ω = 1, Ω =Ω )ofalltheaboveexpressionsdescribe X Λ − an arbitrary ΛCDM model. The above Eq.(6) must be √A′′+ 12Rµνkµkν√A=0, (2) s0olavneddwddDiztLh|zt=w0oi=nit1ia.lAcosndinititohnes,onraigminealyl:DDRLa(zpp=ro0a)ch=, from now on it will be assumed that α is a constant E whereaprimedenotesdifferentiationwithrespecttothe parameter (see, however, [18, 21]). affine parameter λ, A is the cross-sectional area of the light beam, R the Ricci tensor, kµ the photon four- µν momentum(kµk =0),andtheshearwasneglected[20]. µ III. DETERMINING αE FROM SUPERNOVA Five steps are needed to achieve the luminosity dis- DATA tance in the Dyer-Roeder approach: In order to show the physical interest of the approach the assumptionthat the angular diameter distance • d √A, proposed here we have performed a statistical analysis A ∝ involving 557 SNe Ia from the Union2 compilation data the relation between the Ricci tensor and the [22]. Following standardlines, we have applied the max- • energy-momentum tensor T through the Ein- imum likelihood estimator [we refer the reader to Ref. µν tein’s field equations [16, 22] for details on statistical analysis involving Su- pernovae data]. In Fig. 1(a) we display the results obtained by as- 1 suming a flat ΛCDM model. The contours correspond R Rg =8πGT , (3) µν µν µν − 2 to 68.3% (1σ) and 95.4% (2σ) confidence levels. The best fits are Ω = 0.25 and α = 1.26. As we can see where inour units c=1, R is the scalarcurvature, m E from Figs. 1(b) and 1(c) the matter density parame- g is the metric describing a FRW geometry, G is µν Newton’s constant and R kµkν =8πGT kµkν. ter is well constrained, being restricted over the interval µν µν 0.21 Ω 0.29 (1σ), while the smoothness parameter m ≤ ≤ therelationbetweentheaffineparameterλandthe isintheinterval0.72 α 1.94(1σ). Althoughα be- E E • ≤ ≤ redshift z ingpoorlyconstrained,weseethattheprobabilitypeaks in α > 1, and, therefore, denser than average regions dz H(z) E =(1+z)2 , (4) in the line of sightare fully compatible with the data. It dλ H 0 is interesting to compare the bounds over Ωm with our previous analysis with the restriction α 1.0 [16]. The whereH(z)istheHubbleparameterwhosepresent ≤ interval 0.24 Ω 0.35 (2σ) was obtained. As should day value, H , is the Hubble’s constant, m 0 ≤ ≤ be expected, by dropping the restriction α 1.0 lesser the ansatz ρ goes to αρ , and, finally, values of Ω are allowed by data. ≤ m m m • 3 4.0 1.0 1.0 a) b) c) 3.5 0.8 0.8 3.0 2.5 ood0.6 1 ood0.6 1 h h 2.0 eli eli k k Li0.4 Li0.4 1.5 1.0 0.2 0.2 2 2 0.5 0.0 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 0.0 0.2 0.4 0.6 0.8 1.0 m E m FIG. 1: (color online). a) The (Ωm,αE) plane for a flat ΛCDM model. The contours represent the 68.3% and 95.4% confidence levels. The best fit is Ωm = 0.25 and αE = 1.26. Note that a flat model with only matter and inhomogeneities (Ωm = 1) is ruled out with great statistical confidence. b) Likelihood of αE. The smoothness parameter is restricted on the interval 0.72 ≤ αE ≤ 1.94 (1σ). c) Likelihood of Ωm. We see that the density parameter Ωm is restricted to the interval 0.21≤Ωm ≤0.29 (1σ). IV. SUPERNOVAE-CMB TENSION AND αE previous ΛCDM analysis. Again, we see that for higher values of α , the contours are displaced towards regions E The tension between low and high redshift data has withhighervaluesforwandsmallervaluesforΩm,again been reported by many authors (see, for instance, [23]). contributing to cancel the tension between the low and Anumericalweaklensingapproachtosolvethisproblem high redshift data. wasrecentlydiscussedby Amendolaetal. [4]basedona In Table II, we summarize the best fits for Ωm and w meatballmodel. Cansuchatensionbe alleviatedby our along with their respective minimum reduced χ2red. extended DR approach? In order to answer that, let us consider an arbitrary TABLE II: Best fits for Ωm and w. ΛCDM model and plot the bounds on the (Ω ,Ω ) m Λ plane by fixing three different values of αE. By select- αE Ωm w χ2red ing α = 0.7, 1.0 and 1.3 we may study what happens E 0.7 0.35 -1.18 0.978 with the (Ωm,ΩΛ) contours when higher values are con- 1.0 0.29 -1.06 0.978 sidered. In Fig. 2(a) we show the contours obtained for 1.3 0.23 -0.96 0.977 the chosenvalues of α . Note that when α growsfrom E E 0.7 to 1.3 the best fit moves of around 1σ towards lower valuesofthe pair(Ω ,Ω )therebybecomingmorecom- m Λ patible with the cosmic concordance flat ΛCDM model. This is a remarkable result since it improves the agree- V. WHY IS αE BIGGER THAN UNITY? ment with independent constraints coming from baryon acoustic oscillations (BAO) and the angular power spec- Hereweproposeasimpletoymodelbasedontheexis- trum of the cosmic microwave background (CMB), and, tence of cosmic voids in order to explain why αE can be more important, maintaining the same reduced χ2 . bigger than unity. Recent studies have pointed out that red cosmic voids not only represent a key constituent of the cosmic mass distribution, but, potentially, may become TABLEI: Best fits for Ωm and ΩΛ. one of the cleanest probes to constrain cosmological pa- rameters [27]. The idea is to consider that very large αE Ωm ΩΛ χ2red voids are relatively rare entities, i.e. their formationsuf- 0.7 0.39 0.83 0.978 ferfromthesamekindofsize(mass)segregation‘felt’by 1.0 0.30 0.78 0.977 thelargestgalaxiesandclusters. Byassumingthatthe 3 1.3 0.24 0.74 0.977 basicentitiesfillingtheobservedUniverseare: (i)matter homogeneously distributed (ρ ), (ii) the clustered com- h In Table I, the basic results are summarized. Note ponent (ρcl) and (iii) voids (ρvd) of small and moderate thatthegreatestvalueofα yieldstheminimumreduced sizes,wedefinethe extendedDRparameter(seeEq.(1)): E χ2 =χ2/ν (ν is number of d.o.f). red InFig. 2(b), wedisplaythestatisticalresultsforaflat ρ h α = . (8) XCDMmodelandthesamevaluesforαE adoptedinthe E ρh+ρcl+ρvd 4 2 0 α =1.3 α =1.3 a) αE=1.0 b) αE=1.0 αE=0.7 αE=0.7 E E 1.5 -0.5 ΩΛ 1 w -1 0.5 -1.5 0 -2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Ω Ω m m FIG. 2: (color online) a) The influence of the smoothness parameter on the (Ωm,ΩΛ) plane. The contours for three values of the smoothness parameter αE using 557 SNe Ia from the Union2 compilation data [22] correspond to 1, 2 and 3σ. Greater values of αE provide results more compatible with a flat model. b) Contours for the (Ωm,w) plane in a flat XCDM model. The same trend is observed,greater valuesof αE imply greater values of w thereby alleviating thetension among thelow and high redshift data. The important task now is to quantify the contribution VI. CONCLUSIONS ofvoids representingthe localunderdensities inthe Uni- verse. The presence of a void means that its matter was In this Letter we have discussed the role played by lo- somehow redistributed to the clustered and the homo- cal inhomogeneities on the light propagation based on geneous components. The gravitational effect of a void an extended Dyer-Roeder approach. In the new inter- in an initially homogeneous distribution is equivalent to pretationlightcantravelinregionsdenser thanaverage, superimpose a negative density (for small densities the a possibility phenomenologically described by a smooth- nonrelativistic superposition principle is approximately ness parameter α >1. valid). For simplicity, it will be assumed here that the E overall contribution of the void component can be ap- In order to test such a hypothesis we have performed proximatedby the linear expression, ρvd = δ(ρh+ρcl), a statisticalanalysis in a flat ΛCDM model and the best − where δ is a positive number smaller than unity. There- fit achieved was α = 1.26 and Ω = 0.25, the param- E m fore, αE given Eq. (8) can be rewritten as: eters being restricted to the intervals 0.72 αE 1.94 ≤ ≤ and0.21 Ω 0.29withinthe 68.3%confidence level. m ≤ ≤ Although α being poorly constrained, the results are E fully compatible with the hypothesis of light traveling ρ α α = h , (9) in denser than average regions. We have also analyzed E (ρh+ρcl)(1 δ) ≡ 1 δ how different values for the smoothness parameter affect − − the bounds over(Ω ,Ω )inanarbitraryΛCDMmodel. m Λ Interestingly, α > 1 improves the cosmic concordance which clearly satisfies the inequality α α, where E E ≥ model since it provides a better agreement between low α is the standard DR parameter. In particular, when and high redshift data (Supernovae, CMB and BAO). the clustered component does not contribute we find The same happened when a flat XCDM model was con- α = 1 1. The previous analyses using supernovae daEta im1−pδlie≥s that we have effectively constrained the ex- sidered with the assumption that αE >1. tendedparameter,α . Howtoroughlyestimatethevoid Suchresultssuggestthatthehypothesisoflighttravel- E contribution from this crude model? By applying the inginregionsdenserthanthecosmicaverageseemstobe standard DR approach to the Union2 sample, the best quiterealistic. Atoymodeljustifyingwhythismayoccur fit is α = 1, and combining with the result for a flat with values of α greater than unity was also discussed E ΛCDM model (section III), one may check that the void by taking into account the possible influence of cosmic contribution has a best fit of δ 0.2. It should be im- voids on the Dyer-Roeder approach. The simplicity of ∼ portant to search for a possible connection between the the model and the obtained results reinforce the interest present approach and more sophisticated methods from on the influence of local inhomogeneities and may pave weak lensing. the way for a more fundamental description. 5 Acknowledgments Astrof´ısica,respectively. 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