Extended Dyer-Roeder Approach Improves the Cosmic Concordance Model

0.22 MB·English

by  J. A. S. Lima

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.
JASL is partially supported by CNPq and FAPESP
whileVCBandRCSaresupportedbyCNPqandINCT-
[1] Ya.B.Zel’dovich,Astrofizika6,164(1973).Seealso,Ya. [16] V.C. Busti, R.C. Santosand J. A.S.Lima, Phys.Rev.
B. Zel’dovich and S. F. Shandarin, Sov. Astron. Lett. 8, D 85, 103503 (2012), arXiv:1202.0449[astro-ph].
67 (1982). [17] J. S. Alcaniz, J. A. S. Lima, R. Silva, Int. J. Mod.
[2] V. Springel et al., Nature 435, 629 (2005), Phys. D 13, 1309 (2004); R. C. Santos, J. V. Cunha
arXiv:0504097[astro-ph]. and J. A. S. Lima, Phys. Rev. D 77, 023519 (2008),
[3] S. Rasanen, J. Cosmol. Astropart. Phys. 2, 11 (2009), arXiv:0709.3679[astro-ph].
arXiv:0812.2872[astro-ph]. [18] R.C.SantosandJ.A.S.Lima,Phys.Rev.D77,083505
[4] L. Amendola, K. Kainulainen, V. Marra and M. (2008), arXiv:0803.1865[astro-ph].
Quartin, Phys. Rev. Lett. 105, 121302 (2010), [19] V.C.BustiandR.C.Santos,ResearchinAstronomyand
arXiv:1002.1232[astro-ph]. Astrophysics 11, 637 (2011), arXiv:1103.0244[astro-ph].
[5] K.Bolejko, Mon.Not.R.Astron.Soc.412,1937(2011), [20] P.R.K.Sachs,Proc.R.Soc.LondonA264,309(1961);
arXiv:1011.3876[astro-ph]; J. Cosmol. Astropart. Phys. P.Jordan,J.Ehlers,andR.K.Sachs,Akad.Wiss.Mainz
1102, 025 (2011), arXiv:1101.3338[astro-ph]. 1, 1 (1961).
[6] E. W. Kolb, Class. Quant. Grav. 28, 1 (2011). [21] E. V. Linder, Astron. Astrophys. 206, 190 (1988); As-
[7] C. Clarkson et al., Mon. Not. R.Astron. Soc. 426, 1121 trophys. J. 497, 28 (1998), arXiv:9707349[astro-ph].
(2012), arXiv:1109.2484[astro-ph]. [22] R. Amanullah et al., Astrophys. J. 716, 712 (2010),
[8] V.C.BustiandJ.A.S.Lima,Mon.Not.R.Astron.Soc. arXiv:1004.1711[astro-ph].
426,L41(2012),arXiv:1204.1083[astro-ph];C.Clarkson, [23] A.Shafieloo,V.Sahni,andA.A.Starobinsky,Phys.Rev.
B. Bassett and T. Hui-Ching Lu, Phys. Rev. Lett. 101, D 80, 101301(R) (2009), arXiv:0903.5141[astro-ph].
011301 (2008), arXiv:0712.3457[astro-ph]. [24] I. M. H. Etherington, Phil. Mag. 15, 761 (1933); B. A.
[9] Ya.B. Zel’dovich, Sov.Astron. 8, 13 (1964). Bassett and M. Kunz, Phys. Rev. D 69, 101305 (2004),
[10] B. Bertotti, Proc. R.Soc. London A 294, 195 (1966). arXiv:0312443[astro-ph].
[11] J. E. Gunn,Astrophys. J. 150, 737 (1967). [25] R. F. L. Holanda, J. A. S. Lima and M. B. Ribeiro,
[12] R.Kantowski, Astrophys. J. 155, 89 (1969). Astrophys. J. 722, L233 (2010), arXiv:1005.4458[astro-
[13] C. C. Dyer and R. C. Roeder, Astrophys. J. 174, L115 ph]; Astron. and Astrophys. 528, L14 (2011),
(1972); Astrophys.J. 180, L31 (1973). arXiv:1003.5906[astro-ph].
[14] C. Bonvin, R. Durrer and M. A. Gasparini, Phys. Rev. [26] G. Ellis, R. Poltis, J.-P. Uzan, and A. Weltman,
D.73,023523(2006),arXiv:0511183[astro-ph];K.Kain- arXiv:1301.1312[astro-ph].
ulainen and V. Marra, Phys. Rev. D 80, 123020 (2009), [27] P. M. Sutter, G. Lavaux, B. D. Wandelt, and
arXiv:0909.0822[astro-ph]; T. Mattsson, Gen. Relativ. D. H. Weinberg, Astrophys. J. 761, 44 (2012),
Gravit. 42, 567 (2010), arXiv:0711.4264[astro-ph]. arXiv:1207.2524[astro-ph].
[15] S. Rasanen, J. Cosmol. Astropart. Phys. 1003, 018
(2010), arXiv:0912.3370[astro-ph].