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This content was downloaded from IP address 131.169.4.70 on 28/11/2017 at 11:25 JCAP12(2014)049 SNLS c,b hysics ave a higher P ic proposed in le e Ia from the dovich effect, cosmo- ic t and the Dark Energy pa- ar 0 , D m 10.1088/1475-7516/2014/12/049 gular diameter distances from ese results with constraints ob- ip of the CMB power spectrum, verse with lower matter density, e´orica, doi: . The same behavior is confirmed strop . We conclude exploring qualitatively A and Elisabetta Majerotto WMAP-9 a [email protected] , data. We find that an inconsistency arises in predicting the osmology and and osmology Davide Maino Planck C a,b catalogs, confirming that the backreacted Universe should h and 1405.7911 ] constraining the matter density parameter Ω 1 gravity, supernova type Ia - standard candles, Sunyaev-Zel
We explore the backreaction model based on the template metr with recent data. We provide constraints based on Supernova [email protected] rnal of rnal Union2.1 w WMAP-9 ou An IOP and SISSA journal An IOP and 2014 IOP Publishing Ltd and Sissa Medialab srl University of Milan, Physicsvia Department, Giovanni Celoria 16, Milan,INAF-Osservatorio Italy Astronomico di Brera, via Emilio Bianchi 46, Merate,Departamento and Italy de Te´orica Instituto F´ısica de T F´ısica Universidad de Aut´onoma Madrid IFT-UAM/CSIC, 28049 Cantoblanco, Madrid, Spain E-mail: [email protected] b c a c Received June 3, 2014 Revised November 7, 2014 Accepted November 28, 2014 Published December 22, 2014 Abstract.
Matteo Chiesa, J Observational tests of backreaction with recent data
Larena et al. [ and the rameter matter density than theclusters corresponding data Friedmaniann confirm one. the same An tained feature. from the Finally position we of combinefitting the th first three peaks and the first d logical parameters from CMBR ArXiv ePrint: the motivations of this inconsistency. Keywords: in contradiction with results producedby by analyzing SnIa the and CMB-shift clusters parameters from scale factor at recombination, leading to a backreacted Uni JCAP12(2014)049 – 6 7 7 7 ]. n 1 2 6 6 10 10 12 10 19 21 15 , ] con- 9 3 , 2 onian gravity [ pe Ia (SnIa) [ models is still open. ation as an effect of the l features of DE (see e.g [ atter and radiation lead to otropic. General Relativity nstant which enters Einstein content such as a scalar field fluid is needed to explain the equations, allowing in principle stulating any new cosmological in terms of a homogeneous and time metric plays a crucial role: count for about 70% of the whole ies and their gravitational effects e domain, and its global dynamics be described by a modification of rtson-Walker (FLRW) metric. ] and references therein) ] and references therein). Indeed the 5 13 , – 4 9 – 1 – ], focusing on the interpretation of the averaging problem i 18 – 16 ] for an overview). 14 3.3.2 CMB shift parameters 3.3.1 Acoustic peaks The accelerated expansion we can observe from supernovae ty Recently it has been proposed to describe the cosmic acceler Models of backreaction were first built in the context of Newt Many different models have been built to describe the physica 4.1 FLRW cosmology 4.2 Backreaction cosmology 3.1 SZ3.2 clusters Supernovae3.3 Ia CMB ] for a review). The simplest one is based on a cosmological co 8 5 Conclusions A Fitting procedure 1 Introduction The Universe on large(GR) scales provides a appears simple to andisotropic be clear spacetime homogeneous description metric: of and the the is Friedmann-Lemaitre-Robe Universe 4 Data analysis and results 3 Data sets Contents 1 Introduction 2 Backreaction template metric equations, while other modelscalled assume quintessence. some The exotic apparent newGR acceleration energy on could large also scales. The debate between DEbackreaction and of Modified the Gravity local inhomogeneities (see [ Generalizations followed [ procedure of averaging the local inhomogeneties over a finit trasts such a simplea view deceleration of of the ouracceleration. expansion, Universe: This and fluid, ordinary called a Darkenergy types new Energy content of (DE), negative of should pressure m the ac Universe. (see for example [ are non-commutative operations which modify thefor Friedmann an apparent accelerated expansionfluid without (see the [ need of po the context of Generalthe Relativity. full In metric these which models describes the the space non linear inhomogeneit JCAP12(2014)049 ] 1 (2.1) . metric ] and it w ] 21 19 ]. Unfortu- , , we consider 1 22 nt catalogs: the . we describe the data . s is handled by intro- 3 5 uation of state 2 sible to the backreaction Ω nstraints. Also the likeli- ut of this problem. d ffers from the homogeneous ackreaction template metric is devoted to the analysis of + osition of the peaks and dips introduces an approximated e parameter is assumed to be . The likelihood function is ckreaction: the set of angular esponds to a constant spatial . For instance a perturbative volve in time. It is introduced 2 4 te tighter than those produced cy in the results produced by n ect proposed by [ mation due to the large errors metric looks like [ r mogeneous FLRW background, h special emphasis on the CMB ) n n section . Under these assumptions, in [ t ]. ( 2 ion of the backreaction problem on tours of the present matter density 19 D dr k − 1 we summarize the features of the template 2 0 2 D 2 D a a ] data. The CMB data are used in two different 0 – 2 – 2 25 D 1 [ ], the nine-year Wilkinson Microwave Anisotropy H 23 + 2 dt Planck ) and the scaling exponent − 0 D m = ν dx ] and cosmic microwave background (CMB) data from [ µ ] and the 20 24 dx ] in the context of backreaction, while in section = 1 are assumed everywhere. ] with present background cosmological data. In particular ] performed a test on a backreaction model based on a template 1 µν 1 1 c g ], that we will use in our data analysis. ) data [ 1 SnIa catalog presented in [ ] the problem of testing averaged inhomogeneous cosmologie 1 WMAP In [ Natural units Larena et al. [ In this paper we aim to put the most stringent constraints pos The paper is organized as follows. In section Moreover, we update the SnIa and CMB data sets to the most rece The new SnIa dataset brings noticeable improvement to the co is unknown, requiring toapproach make could strong be assumptions used up onbut to its non-perturbative linear form schemes scales, have starting been from suggested, a ho too [ built from the SnIa of [ introduced by [ 2 Backreaction template metric Here we summarize the features and the main equations of the b metric proposed by [ used to constrain it, i.e. clusters, SnIa and the CMB. Sectio ducing a smoothed templatecurvature metric. model at The any template time,to metric but describe corr the the curvature path iscounterpart. of enabled to light Since e in the ansmoothed averaged true lightcone model for metric and the is how computation it unknown, of di the observables. template The is compared to a standard FLRW model with a DE with constanta eq dataset which has neverdiameter been distances used measured before from innately, Sunyaev the it Zeldovich context (SZ) turns of eff out ba associated that with these them. data do not add relevant infor ways: by exploiting the CMBof shift the parameters temperature or power by spectrum. using the p the data and todata the in presentation the of context the of results backreaction. obtained, wit Conclusions are given i Probe ( whose constant time slices aretime-dependent FLRW-like. in The order FLRW to curvatur encodethe the authors effect perform of a backreaction likelihoodtheir analysis template of metric, a providing scaling likelihoodparameter solut confidence of con the Universe (Ω model proposed in [ hood contours produced with recentby CMB older observations are data. qui CMB Interestingly data. however, We we hence propose find an an explanation inconsisten and possible ways o Union2.1 JCAP12(2014)049 1, − (2.4) (2.3) (2.5) (2.2) (2.8) (2.6) (2.9) (2.7) = a u a as follows: k D k . The parameter ] in order to make at different cosmic D 1 ) written as ) is simply the speed 2.5 . 2.4 2 wing [ . x is related to t-hand side) and an extra ) ) , re, cannot be arbitrary. It t 2 the curvature parameter. If D D ive redshift ( on the same slice is given by r i pect to cosmic time, in order a dx D ) f eq. ( D ( 2 R k y with the FLRW model. The , D that at the observer x h D da equation ( a 2 ging to different slices, separated . − ( inate distance travelled by a pho- , x dk = 0, which can be written in the 0 2 ) D 1 t r , 2 D k ( µ )) ) r a t . S 0 k O | D D ( − ) ) S ν Z 0 k a b d ) dx D 1 ( ) 0 D u ∇ u k 2 t − i a ν c k ( r 2 D p ) k k k ) 1 R − a D ) D dt D a D ab cd 1 D |h p a g g a ) dk a ( ( ( ( t 0 r ( ) p 2 ( =( 0 t D D ) r – 3 – D ( Z t D = k ( D H ) D k H 1 z t D a D − ( 2 D z a = a D is the comoving observer 4-velocity, S labels the source a 1+ )+ D − 2(1 = t i b 1+ ) is the dimensionless coordinate distance, which can be : ( − u l = R D dt ) dr h )= dt t ) t a = ( ( ( D l r 0 D D a D ˆ k ( H a da d dr 0 dimensionless. = 1 ˆ k r dl dt indicates quantities averaged over a spatial domain D ] let us compute the distance between two points on two slices , representing the Hubble function today, is inserted follo 1 2 0 is the light wave-vector, − D a k H Then, we need to assume that the averaged Ricci scalar ), which describes the time variation of the 3-space curvatu t ( D We can identify theterm, standard which Hubble arises flow aswe (first a are term consequence interested on on of the the the righ path time of dependence a of photon, the left-hand side o and so we canto compute evaluate the the derivative infinitesimal ofby distance this an between distance infinitesimal points with time belon res interval k which is a differential equationton. for In the analogy dimensionless with coord the standard model we introduce an effect Here the subscript Following [ of light, so the equation reduces to the coordinate distance time. First we remember that the distance between two points derived with respect to the effective scale factor by solving In this equation the function where and O the observer. Since the wave-vector is normalized such The wave-vector satisfies the geodesic equation must be related tofactor the full spacetime Ricci scalar, in analog we simply have context of the template metric as JCAP12(2014)049 term. (2.19) (2.15) (2.12) (2.16) (2.10) (2.14) (2.13) (2.17) (2.11) (2.18) : L and the d n and the A ackreaction d . at the given and D 0 i A D r d R . ] it is shown that h ) D 17 z ( πG 1 A lized Larena’s equations 48 , d ) 2 ponent Ω ion to the last constraint ) + annian form if an effective n D rite . D , a e pressure field, but here we D iameter distance z 0 er to obtain dynamical backreaction n D n D D X Q a a endent density and pressure are 0 pletely negligible. We assumed i . D X D πG 1 i , + Ω , 4 n D 2 R 16 h + Ω a r − D 2 )=(1+ D ) ) = 0 − i n D a 4 D = 0 a H , D D − D , D eff z = i i a D r a ( ) we find: p D D eff ( 0 D R i L | +2 ρ i h D eff D D r + 0 n D + 6) 2.2 R D k R + Ω a D X h h n 0 w D eff 3 − , d 2 + 6 + 2 ρ D X + Ω − D , p 3( ) – 4 – ( 1 = a 3 n n shows that it is kinematically undistinguishable D D − 0 D − D z i , + 6)Ω − D eff D m ( a H n D = p D R p r 0 n a h + 6)Ω ) a = ( (Ω D m i ] showed that the effective inhomogeneous cosmology | D X 0 ) into eq. ( D + 3 n Ω D D z 2 ( Ω D 26 ( = 0) = 0. The previous model allows the computation πG 1 Q Q D eff H s − ˙ D ρ 2.11 z 16 a = ( = 0 r − D D ) = ) = 1 D Q D D D H dr a a Q da ( ( 2 D D )= , that corresponds to a direct coupling between backreactio : k -component as πG 1 H p D L z X d 16 -component, accounting for the combined contributions of b ( = A − X n d D i ρ ], we consider solutions with exact scaling h 1 = based on the analogy with the usual homogeneous case. In [ D eff 4 ρ − D As in [ a 0 D r and curvature, as In analogy with the standard homogeneous cosmology, we can w with the boundary condition and defining the The scaling of the energy-momentum tensor is defined. Its effective domain-dep based on the averaging procedure could be written in a Friedm the effective Friedmann equationsThis are has modified the effect byneglect of the the smoothing extra latter. out the Larena inhomogeneities et in th al. [ of important observables in cosmology, such as the angular d one can derive the equations that will have to be solved in ord scalar curvature. Inserting eqs. ( In terms of the effective sources, where the exponentscharacterized are by reals. We are interested in the solut from DE in aby FLRW adding cosmology. For to completeness, the we effective have genera Hubble parameter the radiation com luminosity distance initial time, althoughΩ the contribution of radiation is com JCAP12(2014)049 ) a 2.5 6= 1, ) (2.20) )–( D z ] justified 2.3 1 ) is also an ) is different . Comparing ect should be )(1 + 2.2 D 2.1 D w z . Eqs.( ( 0 This leads to the D D a tric ( ), and only to the case many assumptions have it is easy to see that the ole that can be found in o the same definition of e to use in the averaged these constant curvatures e metric of the primordial racteristics on an inhomo- ded to ray bundles should itical: on one side it follows coordinate distance satisfies that it should correctly hold 2.11 ning procedure of spacetime eneities and produces at any ] propose the same metric. ated to the template, clearly te parameter to the backreaction. In terms of rd relation 19 ightly different. The two models are ], where they were first introduced, for . 1 3 + 3 ) = 1 on a large domain 0 ) have the same form, but the time evolution ) and the Ricci scalar of eq. ( n t t ( D ( − a D D ( – 5 – k = k D 6, D H / w 0 ) is implicitly assumed to work in the averaged space- ] in analogy with the FLRW metric, to which it reduces ) is defined in analogy to FLRW, to which it reduces 1 D ) has been used. This has been proposed by [ k 2.6 ], a strong albeit reasonable assumption is that at early 2.10 ) and 1 2.1 = a | ( 0 H D i : -component with the standard DE scaling, we immediately find R D X |h ) w 0 t ( D and k of the n a . This assumption is reasonable because the backreaction eff 3.3 , while other solutions may exist. ], we restrict to the scaling solutions shown in eq. ( p 1 = very small as inhomogeneitiestemplate get smaller. metric, On which the isthe other sound not hand horizon perturbative, if it would is not assumed lead as t the underlying spacetim definition of the soundsection horizon and of the fundamental multip times the spacetime is described by a weakly perturbed FLRW. assumption, again proposed by [ time. This assumption, although reasonable,from the is metricity somewhat of cr spacetimefor only, which the leads template; to on believe thechanges other the side however, lightcone the andchange. coarse-grai the In cross-sectional summary,framework, areas whether is the subten not Etherington certain. relation is saf n when posing when the effective metric becomes FLRW. by the fact that Riccigeneous flow space-time renormalization decreases of the intrinsicgiven average curvature cha time inhomog a constantare curvature not slice, equal. but at Based on different different times considerations, also [ are consequences of these two assumptions. The relation between the curvature As [ The reciprocity relation eq. ( Finally, as again stated in [ The effective redshift of eq. ( The template metric of eq. ( We stress that the backreacted model based on the template me We also stress that in order to build this backreaction model • • • • • • the scaling with relation between which implicitly defines a domain-dependent equation of sta from a standard FLRW cosmology fora two different main differential reasons: equation, first secondkinematically the the the lightcone same, is since sl further detail. of the effective scale factorredshifts, differs the from two the models standard are one due not equivalent: the non standa which is ashows consequence the of difference. the non-standard lightcone associ been made. We summarize them here and refer to [ JCAP12(2014)049 ] is (3.1) , and 30 3 ) , where ] through ] and it is els for the 2 mod T/h 29 σ B 22 – k -model [ + 27 β )( tness and to the 2 2 sys σ et al. [ h/c + state that the effect of is time-dependent, the 2 stat = (2 D σ 0 talog, the ICM is modeled k unted in a different angular I xy clusters [ = ectron density. Instead, the iameter distance of a cluster nd cosmology. This is done . an be detected. The SZ effect brightness of the cluster in the the redshift and the scale factor erse Compton scattering. This 89. The distances are computed m (ICM) shifts the frequency of ! 2 tive redshift. data he template backreaction metric . Owens Valley Radio Observatory 2 σ 0 T Compton scattering, respectively. SnIa and the CMB. (BIMA) interferometric arrays. σ : 2 0 ) and the previous relation does not ψ 2 0 data ) can be safely used in the backreacted 2.8 -model. The spherical I σ 4 β (0) e ) 3.1 4 0 and 22 I 34 − X ) 2 SZ z (0) e N N depends also on the geometry of the cluster through the A = d ) is not constant. Moreover, to be precise, by assuming that A t ( that intrinsically depends on the physics of the ICM and the d ), ] and Laroque [ D A k 3.1 -model (see [ d 31 -model and the spherical β β X-ray data and SZ measures from the ) the factor (1+ are quantities proportional to the bolometric surface brigh ). Since in the template metric the curvature Berkeley-Illinois-Maryland Association 3.1 0 X double t ( N . This depends on the physical properties of the ICM. Many mod X /a ) t Chandra and ( /N a -model of Mohr [ 2 SZ β SZ = the fundamental multipole is as in the standard CMB theory we backreaction upon the observed spectrumsize is completely subtended acco at the decoupling epoch. Universe as soon as N N The SZ clusters catalog we use has been compiled by Bonamente As regards the errors, we follow [ As we see from eq. ( z is the Thompson cross-section. Λ T combining (OVRO) and the 3 Data sets In this section, we describe the data sets used to constrain t illustrated in the previous section. These are3.1 SZ clusters, SZ clusters It is well known that the SZ effect allows the detection of gala usual FLRW, which provides the well-known relation between 1+ There are three different contributions for the error hold anymore. For simplicity we assume that eq. ( composed of 38 clusters in the redshift range 0 an isothermal model baseddouble on the sphericalcopies simmetry of of the isothermal the el model, provided that the redshift is substituted by the effec the following mechanism. Thephotons gas coming of the from intra-cluster thefrequency mediu last shift scattering alters surface theallows CMB through a temperature inv computation fluctuation and of c Here σ SZ intensity fluctuations, respectively, ratio standard ray-tracing equation is substituted by eq. ( function and to theNote that relativistic in corrections eq. to ( the inverse through the combined measurement ofX-ray the band bolometric and surface ofis the given SZ in intensity terms fluctuations. of The these: angular d cluster geometry, whereas it is independent of the backgrou ICM were proposed inusing the past both years. the In the Bonamente et al. ca JCAP12(2014)049 (3.3) (3.2) (3.4) ]. The 36 accounts , which we 0 ), we find it sity distance mod H is the solution σ 3.3 r sample [ redshift, and it is by directly solving nIa absolute magni- cm 4. The SnIa observable r vely, and . , where rror is such, but in our 0 Union2 he CMB shift parameters D = 1 r spectrum. olve the Boltzmann equation z tance r/H r among the two associated to ], we first perform our analysis g a Gaussian prior centered in A full solution has been derived 1 re given in percentage in table I 200). It turns out that this does = , , . ′ 0 t contributions to the marginalized . (0 µ 015 to cm dz . r 2 ) ∈ cm 2 ′ a 0 )+ z ]. This sample is composed by 115 SnIa are intrinsically small, regardless of how kr = 0 1 ) ( depends parametrically on L H a 0 d z 20 − ( H ]. We test the dependence of our results on A 0 H 1 d z ]. Also, instead of using eq. ( H H 35 0 ( p Z 39 10 ) , – 7 – z = ) through 38 ) a ) catalog [ ( z,θ catalog, in our work we neglect the covariance contri- ( ]), which is an update of the da cm L 01. We then update the analysis by using a second set ) = 5 log = (1+ . 23 d z dr 1 L SNLS ( ( d µ = 8 Km/s/Mpc [ 0 Union2.1 are listed in table III. 219 419 537 813 ∗ ∗ WMAP-9 is determined through the fitting formulae Planck z # z ( ( ∗ r 9 7 9 8 eff m z . . . . Ω 0 1 2 9 N Ω and WMAP-3 WMAP dip – 9 – l 3 ± ± ± ± / = 4 − 9 4 8 5 . . . . ∗ we list the position of the first three peaks and of ] r to the position of the peaks are given by the fitting : we fit the peaks and dips with exponential functions 4 1 WMAP-9 24 11 ), where 220 415 537 813 m ∗ ] and depend on the ratio Planck dip φ z l WMAP-9 7 8 40 2 3 1 l l l dip is computed in terms of the corresponding redshift l 2. The reason for this can be found in our fitting procedure. . The position of the first dip, on the other hand, shows only ∗ (1 + data, are used to determine its position (see bottom panel of / 1+ ) a / WMAP-3 " . Positions of the CMB peaks and related errors. . In table γ ρ is given by [ = 1 A Planck Planck r ∗ ] for = ρ σ a Planck r 21 ρ + Table 1 ] and references therein. WMAP 43 σ , is the effective number of neutrino species, which takes into 42 ). If we had treated the boundaries of the dip as free paramete =( 04, for simplicity, and because observations are consisten eff . σ N 10 ], used also by [ Finally, the correction terms ] the positions of the CMB peaks from = 3 1 44 eff the position of the peaksat of the the time power of spectrumof preparing of [ this the CMB work, were we had to compute them. Our The radiation density treating them as freebecause parameters. a full We fitting usedthe procedure this shape would approach require of for atprocedure the s least spectrum see 14 up par appendix to the third peak. For a detailed e and parabolas, but differently from it, we fix the boundaries d In [ with those provided by marginal consistency among the two data sets. A rough estima analysis, we use the more recent the first dip ofpositions the of CMB the spectrum, CMB together peaks with the obtained correspon by fitting scale factor at recombination formulae of Doran and Lilley in [ existence of extraN relativistic species. In this work we ass the standard relation where where ¯ figure given in [ As said previously, we arbitrarily fix the boundaries delimi dip, this is problematic becausepoints, it in is the a narrow case feature of with r JCAP12(2014)049 0 R H sive (3.15) ] state, has been 40 is defined WMAP-9 WMAP-3 R R utrino masses ], and over , like non-zero ] and [ 45 y directly solving ), thus obtaining 49 puting the integral is the fundamental ted from . The parameter 2.9 0 assuming a Gaussian a LRW models (see for l sing a different set of D 2 h The latter case is indeed b en we compare the results r/H W, whose most important and models with late-time based on the position of the = x, is much stronger. Under action template model and a ]. Here es not significantly change the e 1, which shows that changing cm 47 is almost the same for many DE r measurement compatible with the ). , ) R z 46 3.1 ( dz ) [ H . ∗ everywhere (instead of using the more ∗ A Planck z 0 ] the authors suggest a way for extracting ]. The model dependency of 0 D m Z 42 52 ,R,z 0 ] instead, a likelihood analysis with a l H 49 is the redshift of recombination and – 10 – 0 m Ω ∗ z p ), = . 3.6 R 1 has a wider field of application, as both [ 3.3.1 parameter should be stable, since no tensorial modes nor mas 0020 coming from Big Bang Nucleosynthesis data [ . 0 R ± , related to the comoving distance by ]) and in local voids ones [ is carried out involving extra parameters, like positive ne r a 51 l – 0214 . 49 for the FLRW case). , and ), that corresponds to the comoving distance, we compute it b m = 0 ]. Again, as done for the luminosity distance, instead of com one. For more details see the appendix . On the other hand, the dependency on more exotic parameters 46 2 R R 24 h ) for the FLRW case, and, for backreaction, by solving eq. ( 3.15 b ] When analyzing the data, we marginalize numerically over Ω In the following, we used the notation Ω 3.4 48 data in [ discussed by several authors. For example, in [ position would likely increase enough to make the has been usedinstance to analyze [ CMB data many times in the context of F for a flat FLRW Universe. We use the CMB shift parameters compu neutrino masses, tensorialour modes assumptions, or the a running spectral inde by [ showing that it stillgeometry works or well photon both dynamicsrelevant for departing for our from models work. a with standard early FLRW. DE 4 Data analysis andWe results analyze the differentFLRW Universe data with sets DE separatelyobtained with using for constant the the equation two backe of different state, cosmologies. and th 3.3.2 CMB shiftThe parameters shift ofobservables, the called acoustic CMB peaks shift of parameters: the ( CMB can be quantified by u in eq. ( eq. ( the dimensionless model-independent constraints from it.data In fot [ WMAP-9 prior Ω multipole of the CMB defined in ( with the same prior used for clusters data (see section standard Ω and tensor modes. Theycosmic summarize curvature their or results slightly invalue modifying their of the tabl DE parameters do aspects are summarized in figure 4.1 FLRW cosmology Here we present the analysis of the constraints on the flat FLR neutrinos are considered.models, On this the parameter other must hand, bepeaks used although described with in caution. section The method JCAP12(2014)049 Union2.1 0.6 0.6 for the FLRW constraints from w 0.5 0.5 and 0.4 0.4 Planck 0 D m ers data. These are un- 0 0 D D m m 0.3 0.3 W W ling the cluster geometry. By 0.2 0.2 constraints from SnIa, it is clear that present cluster data cannot m CMB data: 0.1 0.1 hown in yellow. 0.8 1.0 1.2 1.4 1.6 1.8 2.0 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 0.6 constraints from the CMB shift on the lower ------ w w constraints are shown in the left panels, – 11 – 0.6 0.6 WMAP-9 SNLS 0.5 0.5 0.4 0.4 0 0 D D m m 0.3 0.3 W W 0.2 0.2 likelihood contours for the effective parameters Ω σ 0.1 0.1 and 2 1.4 1.6 1.8 2.0 2.0 0.6 0.8 1.0 1.2 1.0 1.2 1.4 1.6 1.8 0.6 0.8 ------ w w σ . 1 The red regions correspond to constraints coming from clust model. Red fields are given by clusters. Green fields comes fro Figure 1 fortunately very wide duecomparing the to red the fields with hugethat the uncertainties the blue in latter ones, corresponding are model fully to contained in the former. This means constraints are in the right panels. Joint constraints are s ones. The blue fields correspond to SnIa: CMB peaks are given in the upper panels, JCAP12(2014)049 . , 0 1, he D m − , but = w - Planck 0 w D m SnIa data, ]. Here the od based on rom htly towards 1 f [ SNLS description of the data are more precise. er spectrum, the green he position of the peaks. r panels show constraints smology since the coarse- g to the produces worse constraints tfully combined with CMB f the CMB power spectrum when combining them with 3. meters in the right panel of data, we notice that the im- Planck / ed by a constant line ers. The purple fields are very on average ch larger number of data of the ully consistent with each other el based on the template metric tion about the curvature of the is that all combinations of data . It is nevertheless interesting to nces based on the SZ-effect may ts do not present any significant ] and the positions of the CMB + 3) 1 data. Since the procedure to fit the 20 n Planck s from ( data, extracted with the same method − Union2.1 , but it seems that the shift parameters a l = , while we show constraints on Ω constraints towards a larger value of Ω n w - 0 for more details). We do not show constraints WMAP-9 D m A WMAP-9 – 12 – Planck ] and references therein). 53 and to our fitting procedure. The method based on the we treat two interesting aspects. The left panel shows shows a comparison between constraints coming from two 2 2 if backreaction is assumed. This effect can be considered the WMAP-9 data. The wide purple fields correspond to constraints from t 0 D m -3. The main feature is clear: likelihood contours move slig data, because the errors we obtained were larger than those f data. The bottom ones show constraints derived from the meth WMAP-9 Planck W MAP ). Our results are compared with those summarized by figure 2 o Planck are completely contained into the constraints coming from t 2.1 2 For completeness, in figure If we compare the blue fields of the left panels, correspondin The green fields indicate constraints from CMB data. The uppe Overall, the most important result shown in figure Universe can be separatedgraining procedure from does the not correspondingbare completely metric. flat destroy Results FLRW the are co shown informa in the plane Ω it is easy to relate the two planes, knowing that as found in most recent work (see e.g. [ shift parameters gives tighter constraints and prediction a Universe withsignature larger Ω of curvature: the bare metric is curved and any significantly improve the constraints wenotice can that infer these from two SnIa in completely the independent context of datasets our are FLRW f cosmology. with those on the right panel, corresponding to the position of thecontours first are three constraints peaks derivedwide from and due the to the CMB larger first shift errors of paramet dip of the CMB pow We also note thatfigure the constraints derived from the shift para This is because both sets of parameters depend on authors show combined constraints from the SnIa of [ the improvement induced on the constraints given by different sets of peaks from derived from the position of the peaks from provement in the SnIa constraintlatter is sample. significant, due to the mu are fully consistent with the ΛCDM model, which is represent constraints from clusters data.improvement, clusters Even data if slightly combined move constrain used for the given by the positionfrom of the the first three peaks and the first dip o encode more information included in the CMB spectrum. 4.2 Backreaction cosmology Here we discuss our resultsof concerning eq. the ( backreaction mod the CMB shift parameters computed from the suggesting that more precisebe measurements used of as cluster andata. dista independent The probe right of panel cosmology of which figure can be frui position of the CMBwith peaks respect to is the not second one very (see refined, appendix the first dataset JCAP12(2014)049 n Union2.1 for the FLRW rs in the upper CMB shift pa- w catalog (see the and 0 D m , so that constraints are . We verified that this is t expected in general for 0 w WMAP-9 D m Union2.1 fitting procedure give worse RW models and backreacted parameters (green fields); only tice that the left edges of the and 0 nding to the backreacted scenario ction have the expected behavior D m is data set are tighter than those stant Ω . The that move towards lower values of 0 D m ints provided by the position of the CMB , which suggests that SnIa observations σ , constraints derived from the positions of ’s peak positions and – 13 – constraints (empty contours) combining clusters data shows constraints from clusters data. The big errors WMAP-9 , and it happens both for FLRW and for backreaction. 0 3 Planck H we show constraints derived from our sets of CMB data. Planck , as happens for SnIa. data and show the behavior we expected: we recover the 3 0 D m ]. Indeed, filled regions, corresponding to the backreactio 1 SNLS likelihood contours for the effective parameters Ω σ . This is mainly due to the larger number of data points of the and 3 we show results coming from each dataset separately. Contou σ 3 SNLS , 2 σ data were used. . 1 In the lower panels of figure In figure More importantly, we see that The upper right panel of figure , contrarily to what happens for SnIa and clusters, and to wha 0 D m Figure 2 results shown in figure 2 of [ We note that the positions of the CMB peaks extracted with our constraints with respect toprobability CMB contours shift are sharply parameters. cut We in also the no direction of con on the cluster geometry project onto very large errors on Ω model, are slightly pushed towards higher values of Ω left panel comes from the very loose, but we observeof that preferring contours a relative higher to value backrea of Ω model. Left panel: improvement of (filled contours). Right panel:peaks comparison and between dips constra WMAP-9 (purple fields) and those given by the CMB shift upper central panel) behavesgiven likewise. by Constraints the from th due to the choice of the flat prior on rameters, that give relativelyΩ small errors, give contours backreaction models. (In the case of (580 vs 117). We note foroverlap this those dataset that for contours correspo the flat FLRW model only at 3 may provide in theones, future if a combined way with for other distinguishing observations. between FL JCAP12(2014)049 ] 40 0.6 0.6 0.5 0.5 0.4 0.4 , but the fact 0 0 D D 0 m m 0.3 0.3 W W D m 0.2 0.2 0.1 0.1 regions corresponding to CMB peaks position. Lower ogy, since it depends only his effect regards both the data. Upper right: clusters. s) and a flat FLRW with DE σ 1.2 1.4 1.6 1.8 2.0 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 0.6 0.8 1.0 ------ w w likelihood contours are shown for model dependent, ensures that d have ascribed to this problem sidering backreaction. Since the aller values of Ω xplanation. Let us suppose that 0.6 0.6 σ WMAP-9 Union2.1 0.5 0.5 and 3 σ 0.4 0.4 , 2 0 0 D D σ m m 0.3 0.3 W W from Doran and Lilley’s fitting formulae [ ∗ z 0.2 0.2 – 14 – 0.1 0.1 data. Upper central: 1.8 2.0 0.6 0.8 1.0 1.2 1.4 1.6 1.0 1.2 1.4 1.6 1.8 2.0 0.6 0.8 ------ w w SNLS 0.6 0.6 0.5 0.5 0.4 0.4 CMB shift parameters. Lower central: 0 0 D D m m 0.3 0.3 W W 0.2 0.2 CMB peaks position. WMAP-9 0.1 0.1 . Comparison between the backreaction model (filled contour Planck 1.4 1.6 1.8 2.0 0.8 1.0 1.2 2.0 0.6 1.4 1.6 1.8 0.8 1.0 1.2 0.6 ------ w w the peaks on the CMB power spectrum are very loose and the 1 the backreacted model and toCMB shift the constraints FLRW and overlap.) the peaks WeCMB note positions shift ones, that parameters when t are con the slightly model-dependent, strange behavior we coul of such constraints, moving towards sm right: that this also happenssuch hypothesis for is the wrong. peaks Wethe positions, propose estimate instead that the of are following the e less recombination redshift Figure 3 with constant equation of state (blank contours). 1 is valid both in the case of a FLRW and of a backreaction cosmol Lower left: different data sets. Upper left: JCAP12(2014)049 . e a ], ∗ l ns 1 a (4.1) over, , from )= ∗ z ) does not ( rmines the z viously the D ] found that a dependence of 54 (1+ / 4 . To do so in our and consequently − D : the scale factor ∗ a z a = 1 a such that from 0 andard relation between the difficulties explained e Doran & Lilley’s fitting dx D ]. The D be questioned, because we a omputing the fundamental ing very roughly consistent ause a theoretical problem, )) a me problem affects also the ue to a wrong prediction of 17 0 ) ed in the CMB, is projected ndamental parameter should the same in the two different ckreaction, since the path of 0 ed that, since the Universe is D y between our template model the authors of [ m is then understanding which growing inhomogeneities. This D ugh to be completely negligible . Finally, the assumption that oretical inconsistency: together imate of the sound horizon and tion quation ties [ x,a ( x,a hen radiation is no more negligible, minated Universe has extra effects, 2 ( 2 r 0 r ), as the relation ) D 0 . Once this problem is solved, one should D a 2.5 l x,a ( in the standard FLRW cosmology, assuming x,a D ( ∗ k . This is in contrast to what happens in [ z ′ D k − , one has to compute – 15 – ). The underlying idea is simple: the multipole 1 D a 0 2.5 both in FLRW and in backreaction. If none of the D Planck a ∗ 0 z Z and . The latter is used for the computation of the positions of )= 1+ a 0 / l , rather than the redshift, because in the Boltzmann equatio D is built in analogy with standard cosmology and its effects ar ∗ depends on a a a 2 D . Unfortunately, we were unable to solve this equation. More l 0 . This would explain why both the CMB shift parameters and the H is the fundamental parameter for the recombination physics ∗ D WMAP-9 2log( z a ∗ ), instead of eq. ( z of the species present in the baryon-photon plasma that dete z ). As 3 D , that also depends on the recombination scale factor, and ob a ( and R na = 1, we are assuming a further boundary condition for (1 + a l ∗ / 0 a = 1. This reflects into a wrong prediction of the sound horizon a = 0 = 1 a l a = a 0 , D z a is indeed the fundamental parameter, we are not allowed to us ∗ a and it is the density time of recombination. We tried hence to find an initial condi This boundary-value problem translates into the integral e formulae, because they give the expression of implicitly in an error in predicting the fundamental multipole negligible on the likelihood contours,the but at effect early of times, coarse w a graining perturbed may, FLRW metric incannot holds principle, be up be sure to whether important the recombinationand coarse could to graining also leave effect the is small CMB eno theory unaffected. To conclude, given all at recombination is fixed to 1 if should be affected onlyto by an how observer the today.photons early from The time the projection physics, last instead imprint argument scattering seems is surface to affected be is by reasonablewith perturbed ba but by it the actually implies a the on the early timemodels. cosmology, which to This a seems goodas to approximation be we is a will reasonable explainmultipole assumption, in but the it following. can c The problem arises when c which the other one hasbe to the be size determined. of the In our Universe, opinion the fu two boundary conditions isconsequently relaxed, of this leads the to multipole a wrong est also deal with thedue issue to that coarse-graining coarse-graining radiation-do the radiation pressure inhomogenei backreaction model, in principle we have to use eq. ( the radiation density in their model of backreactionthe size was of inconsistent the with soundand CMB horizon. the data We CMB found d thewhere data same the of inconsistenc CMB iswith consistent their with the result, model. wealmost followed To FLRW recover a up someth different toa approach: the time we of assum recombination, one can use the st hold in the backreaction model. Using such correspondence, the peaks and isshift also parameter one of the three shift parameters. The sa third shift parameter, where the unknown is one between peaks positions are affected by the same behavior. The proble JCAP12(2014)049 data, models ned by Union2.1 region of the Planck and 1 erstand whether data, while lower and the averaged one , but differs from it in d, or, as we believe, a D ities only, and in terms data are less precise SNLS a data likelihoods. There , while in the latter case, vables at the time of decoupling e time of recombination the he different datasets become SNLS WMAP-9 ing to in the direction of a Universe ards the second explanation. ur work suggests that future ction of rovements induced by adding ft parameters or r, the backreaction context, since data, respectively), green fields rison, also the improvement on as explained above, it is critical ] finds model-independent parameters emplate backreaction metric of omes narrower for backreaction aints from the shift parameters, and to check that our results are a backreaction cosmology, brings = 1 starts producing noticeable ears, as will be the case, although om the position of the peaks and rs in non-FLRW models are given 56 data provides tighter constraints. Union2.1 onstraints. to the model being excluded by the 0 rfectly model-independent variables at the a data, respectively), red fields indicate the bare scale factor with = . Ref. [ ity. However while in the case of the void model D esent-day parameters, because two incompatible σ 0 Union2.1 z ) redshift. For this reason it would be arbitrary to D . Here blue fields indicate SnIa constraints a Union2.1 or 4 Planck , which enters directly the calculation of the and D ∗ a z – 16 – and we show how CMB peaks constraints improve when 5 SNLS WMAP-9 . Temperatures should be more appropriate in inhomogeneous z or a data. As can be seen in the left panels, the improvement obtai while those from CMB data move in the opposite direction, the 0 ], but unfortunately none of them helps us solve our problem. D m Planck 56 , on which CMB constraints are based. It is interesting to und ] it is possible to determine the exact relation between obser a ] shows how to write some CMB parameters in terms of local quant l 55 55 ) is only marginally compatible with the data that we have use ] and [ Combined constraints are shown in figure Since constraints from clusters and SnIa data move slightly 2.1 Ref. [ 55 is made so that our metric looks like a FLRW model if seen as a fun 1 D adding clusters data isSnIa not data very (shown strong. in On the the contrary, central imp and right panels, correspond above, our approach instandard this cosmological work model is is toconsistent still simply with a assume the good that other approximation, data atit sets. will th be If impossible an in inconsistencydata principle app or to tell by whether a thisInteresting wrong is proposals analysis due for of using the consistentlyin data, CMB [ we paramete will be inclined tow the relation between (averaged)decide scale how factor the and measured (averaged temperatures are related to the inconsistency on the boundary conditions of temperatures rather than and observables today, in our case a correspondence between from the CMB, buttime this does of not decoupling, help we us could because not even correctly if relate we had these pe to pr boundary conditions are chosen. (upper and lower panels correspond to combined with constraints derivedCMB from constraints clusters. from For SnIapanels compa is show shown. The upper panels show indicate CMB constraints (leftwhile panels central correspond and right todips panels constr of correspond the to CMB spectrum constraints from fr a multipole are two possibleeq. interpretations ( of the latter. Either the t cluster constraints, and yellow fields correspond to joint c with higher Ω which produce likelihoods that do not overlap at 2 effects and cannot bedata neglected may anymore. be Inmore the able care former to will case, be rule o constraints needed out may in be the correctly strongly defining backreaction affectedto observables by template understand in them. well metric In the particular, meaning and use of parameter space where thewith respect likelihood to functions what happens overlapless for bec compatible. the standard FLRW The case, most and t interesting cases regards CMB shi the clusters dataset used hererelevant improvements. for the Hence, in first figure time to constrain and combined constraints are looser, while the data, respectively) are much more significant. In particula because they can be derived through the local radiation dens treated in [ JCAP12(2014)049 0.6 0.6 0.5 0.5 ints from constraints constraints 0.4 0.4 0 0 D D m m 0.3 0.3 W W data, but when for the backreaction 0.2 0.2 Union2.1 WMAP-9 w SNLS 0.1 0.1 and 0 constraints from the position D m 1.2 1.4 1.6 1.8 2.0 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 0.6 0.8 1.0 peaks caused by our fitting ------ w w with these data sets. This is e until now. ve, this may either imply that r that observations are precise are shown in the right panels. The 0.6 0.6 rom CMB data: WMAP-9 nts. 0.5 0.5 Planck 0.4 0.4 0 0 D D m m 0.3 0.3 W W regions of the joint constraints do not overlap σ 0.2 0.2 – 17 – 0.1 0.1 1.8 2.0 0.6 0.8 1.0 1.2 1.4 1.6 1.0 1.2 1.4 1.6 1.8 2.0 0.6 0.8 constraints are shown in the upper panels, ------ w w data, the 1 data, they are still compatible with 0.6 0.6 SNLS 0.5 0.5 Planck data, we note that all combined constraints overlap constra Union2.1 0.4 0.4 likelihood contours for the effective parameters Ω 0 0 D D m m σ 0.3 0.3 W W 0.2 0.2 WMAP-9 0.1 0.1 . 1 and 2 2.0 1.4 1.6 1.8 0.8 1.0 1.2 2.0 0.6 1.4 1.6 1.8 0.8 1.0 1.2 0.6 ------ w w Figure 4 model. Red fields correspond to clusters. Green fields comes f are in the lower ones. The yellow contours are joint constrai As regards of the peaks are shown in the central ones, constraints from CMB data alone,also indicating that due backreaction to isprocedure. the compatible large Regarding errors associated to thethose corresponding position to of CMB datathe the alone, model and, is as not explainedenough abo compatible to with necessitate these a observations refinement or of rathe the approximations mad from the CMB shift parameters are shown in the left panels, blue fields correspond to SnIa: combining them with JCAP12(2014)049 . 0.6 0.6 Planck ’s peak is used 0.5 0.5 a l data. Right 0.4 0.4 data provide 0 0 Planck D D m m W W 0.3 0.3 SNLS 0.2 0.2 , but for CMB shift 5 d combined ones, for the Union2.1 0.1 0.1 ork, since the two sets of ce the multipole 1.2 1.4 1.6 1.8 2.0 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 0.6 0.8 1.0 ------ w w se the . Finally, from the right panel of 0.6 0.6 g backreaction, the two methods he likelihoods are not independent ne. Filled contours are joint constraints. : combination with 0.5 0.5 0.4 0.4 0 0 D D data, those in the lower ones comes from m m W W 0.3 0.3 – 18 – 0.2 0.2 0.1 0.1 WMAP-9 data that there is concordance between the CMB peaks 1.8 2.0 0.6 0.8 1.0 1.2 1.4 1.6 1.0 1.2 1.4 1.6 1.8 2.0 0.6 0.8 ------ w w data. 0.6 0.6 WMAP . The latter constraints are tighter than those from 0.5 0.5 Union2.1 0.4 0.4 0 0 D D m m WMAP-9 W W 0.3 0.3 we show the same as in the bottom panels of figure 6 0.2 0.2 . Comparison between constraints from the CMB peaks alone an 0.1 0.1 we had noted from 2 In figure 1.6 1.8 2.0 0.8 1.0 1.2 1.4 2.0 0.6 1.4 1.6 1.8 0.8 1.0 1.2 0.6 ------ w w Left panels: combination with cluster data. Central panels Figure 5 backreacted Universe. Blank contours refersContours to in CMB the data upper alo panels comes from positions. The right panel shows clearly that also inand this CMB shift ca parametersconstraints methods overlap in very the well. standard Thissince FLRW framew is they natural, are given based that t on the same data set and in particular sin in both procedures. Surprisingly instead, when considerin panels: combination with parameters from constraints only marginally compatible withfigure CMB shift data JCAP12(2014)049 ] 1 0.6 are and data. 0 R D m 0.5 and 0.4 ∗ proximately z 0 D m W 0.3 WMAP-3 data. Right panel: ) suggested in [ the position of the 0.2 2.1 regions relative to the SNLS distinguish among the 0.1 gical parameter Ω data improve noticeably σ we have described earlier SNLS ned using ers and combined ones for the esponding region from the ontours given by the shift verse and testing it against 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 ------w c of eq. ( catalog and by employing for ondence between an averaged e and better approximations. uncertainties on modeling the r diameter distances of galaxy . Indeed if only the theoretical ical predictions of ∗ slices and a standard flat FLRW z 0.6 . The 1 7 Union2.1 ne. Filled contours are joint constraints. 0.5 ombination with Union2.1 0.4 0 D m W 0.3 too, we are led to guess that the evaluation of ∗ – 19 – 0.2 z by combining SnIa of the 0.1 n 1.0 1.2 1.4 1.6 1.8 2.0 0.6 0.8 ------w due to the higher number of SnIa, no useful information 0.6 ]. We found that while the 22 SNLS data. 0.5 0.4 implicitly depends on were affected by an inconsistency both constraints should ap 0 D m R W 0.3 a Union2.1 l 0.2 ] the authors published constraints on the effective cosmolo 0.1 . Comparison between constraints from the CMB shift paramet 1 In [ 2.0 1.4 1.6 1.8 0.8 1.0 1.2 0.6 ------w CMB shift parametersCMB do peaks. not overlap Weand claim completely that that with the this the main is problem corr the consists mark in of correctly the predicting inconsistency computation of the scaling index of the backreaction the constraints from the backreacted Universe. Blank contours refersLeft to panel: CMB combination data with alo cluster data. Central panel: c Figure 6 give results that are not concordant, as shown in figure first two peaks and the first dip of the CMB power spectrum obtai against recent data sets.Universe based The on morphon Buchert’s field averagingspacetime procedure creates on filled a spatial with corresp two DE models by of assuming constantpresent a data. equation template metric of for state. the averaged Uni We may corrupted. Since modify likewise.parameters The can strong be explained modification by which stating that affects also only the theoret c combination with We updated their analysis by using the SnIa of the the first time inclusters the from context Bonamente of [ backreaction the set of angula the decoupling epoch in the averaged model requires more car 5 Conclusions We tested the backreaction model based on the template metri is added by clusters, since the errors are heavily affected by JCAP12(2014)049 data would data improves han the former. ], and used the 1 Planck e CMB. We used two WMAP-3 idering backreaction, we f the CMB peaks and dips data, extracting two sets lysis of the CMB provides 0.6 o mperature power spectrum. re we plan to improve our dataset. gested in [ and the first dip of the CMB MB shift parameters provided ocedure we used for extracting ve solution of the Einstein field or constraining the cosmological ration points between peaks and reen fields) for the averaged model. 0.5 not very refined fitting procedure. Planck gly that clusters data confirm the ], suggesting that future surveys of 1 and 0.4 0 D m W 0.3 WMAP-9 – 20 – 0.2 0.1 1.8 2.0 0.8 1.0 1.2 1.4 1.6 0.6 ------w data were used. . We found that the latter dataset gives better constraints t . Comparison between constraints provided by the position o WMAP-9 We also updated the likelihood analysis based on data from th WMAP-9 different sets of observables. First we followed the path sug Figure 7 geometry of the clusters. Nevertheless we found interestin (purple fields) and thoseOnly given by the CMB shift parameters (g behavior followed by the constraints shown in figure 2 of [ be sufficiently precise tothe give positions good results, ofestimate but the by the using peaks fitting additional pr by produces parameters using related large a to refined the errors.good Monte sepa Carlo constraints Markov In Chain for the technique. the The futu FLRW ana spacetime. However, when cons galaxy clusters, and refinedparameters modeling and errors, may may help be inequation useful distinguishing f and between the an standard effecti homogeneous and isotropic FLRW. positions of the firstWe three found peaks that and adding the first a dip point of corresponding the to CMB the te third peak t the constraints despite its bigWe error, which applied was derived the by same a fitting procedure to of data corresponding tospectrum. the As position an of alternativeby to the this first observable, three we peaks used the C This is partly due to the poor estimate of errors of the former JCAP12(2014)049 ) is at ∗ D 0 a 1 D m − ) in the = ∗ ∗ z D a CMB shift )=(1+ z e extension of each ( a data and the overlap WMAP-9 ed to demonstrate that er curve. The first peak rst dip are modeled with er best fit value of Ω dard flat FLRW filled with sition of the peaks and dips SNLS = 1 together with day. We noted that, in the ing the size of the Universe, rder to avoid the complication nce it has a thermodynamical 0 aks, and we checked that the cy should affect the likelihood ll posed. This led us to suggest ft of recombination, caused by at the inconsistency arises due rlapping regions of neighboring ced by this approach are now of e other hand, clearly tell us that D tely, our attempts of solving the on a spacetime described by the a ased on current data are strongly ], we tried to explain this behavior d by = 1200. We arbitrarily divide the ) and the corresponding statistical nd the effective redshift before and = 54 l π raw data and to the combined power 0 (2 a / l C ) cannot correspond to + 1) D =2 up to z l ]. As a consequence, the WMAP-9 ( ( l 1 have failed, as we ran into an integral which we l D – 21 – a a , from l ] of a Friedmannian evolution for any domain 1 data are used. Following [ . 700] . , ∗ 950]. z , 350] 550] , , Planck [400 ∈ [50 Union2.1 [650 data consist of a list of l [300 ∈ ∈ l l ∈ l 1200] into subsets corresponding to a rough estimation of th , [2 WMAP-9 ∈ l First peak: First dip: Second peak: Third peak: SnIa and clusters provide compatible constraints and a high The • • • • find a behavior opposite to what shownreduces in further [ when for both thethe FLRW spacetime approximation and suggested the in backreacted [ model. We tri likelihood contours only marginally overlap those produce in terms of inconsistenciesthe non in trivial the relation between computationafter the of the effective scale the recombination factor redshi epoch.to a the In assumption particular, of standard we boundary believe conditions th ( backreacted Universe up to thethat recombination epoch the is correspondence not between we DE an should averaged be modelstandard formulated and cosmology, a for fixing stan suitable theand boundary recombination we suggested conditions epoch that to meansmeaning the fix related main to role the is numberboundary played density conditions of by species. problem the latter, in Unfortuna si terms of were unable to solve.analysis because Finally the we effective argued scale that factor the inconsisten the recombination epoch spectrum measured by found if backreaction is considered.the Current inconsistency CMB related data, to on how th of we rewriting treat the all CMB the physicstemplate microphysics in metric of o cannot the be photon-baryon neglectedthe fluid anymore. same order Mistakes of indu theaffected experimental by errors, them. and predictions b A Fitting procedure Here we discuss theof fitting the procedure CMB. we We applied used the for same finding scheme the to po error as a function of the multipole peak or dip: range We decided to takedifferent overlapping fitting intervals parabolas for alwaysranges. cross the each different After other pe that, inis the we modeled ove model with each a peak Gaussian, while or the dip other with two a peaks 3-paramet and the fi JCAP12(2014)049 9 ectra. and the act that . In figure m fitting the l ing that it is well , corresponding to l data, but no error is corresponding to the σ st dip this contribution e vertex, once we have boundaries of the fitting 700 and the prediction of order of magnitude of the range in the following way. ues of truly statistical. The final e is arbitrary, but it should ent random number to each of each peak as the standard realizations. We find that the the fitting range is significant ollow a Gaussian distribution, he first peak and to the first re too distant in l > an and 5 We assume that each boundary s of the position of each vertex fit. Once the best fitting values nce their variance contains the WMAP-9 andard deviation corresponding iation. nding Gaussian is superimposed the binning procedure employed efitting procedure is done at least holding the extrema fixed. Hence n 10 . a data. Here we note that the statistical 2 shows that the position of the third peak b/ is held fixed and the random numbers are 9 l − = data are used. The predicted positions of the Planck v – 22 – Planck and figure data distribute quite asymmetrically, while the position 8 . For the first peak, we fit the logarithm of the spectrum, in c raw data increase drastically for + bx WMAP-9 we show the fitting parabolas superimposed on the measured sp + 10 2 data shows a more symmetric distribution. This is due to the f ax WMAP-9 -th peak (or dip) is computed by taking the mean of the sample, we show the histograms of the values of the vertices found fro n )= 8 x Planck are found for each peak or dip, we estimate its position assum ( p c . We assumed the same ranges we used for fitting and b We applied the same procedure to the A comparison between figure Finally, in figure In order to achieve a statistical sample of the position of th In figure We tried to estimate the error introduced by fixing the fitting times. This number of iterations ensures that the vertices f , Planck a 5 three peaks and the firstfor dip. For an each histogram easy thedip correspo visualization. are slightly The widercontribution Gaussians than coming corresponding the from to envelope fixing the of t fitting the ranges. histogram, si uncertainty of each data point is drastically reduced due to by approximated by the position of the vertex computed relative to fixing the range, since the data points a derived from uncertainties on of the fitting parabolas and the corresponding Gaussian. we show the histograms corresponding to the predicted value parabolas: the position of the third peak may be biased towards lower val third peak derived from lower errors. order to transform the Gaussian fit into a standard parabolic of is estimated with better precision if extracted from a Gaussian distributionto with the null statistical mean error and st associated10 to each data point. The r estimated the position ofdata the point and vertex, we we refit randomly the add data. a The differ range in hence that theposition main of uncertainty in the the position of the peak is corresponding error is estimated as the sample standard dev uncertainty on the positiononly of for each the peak first introducedstatistical peak by uncertainty fixing and predicted the bywe first the decided dip fitting to because procedure add itin to is quadrature, the assuming of final that the error the same on errors are the uncorrelated. first peak and on the fir 10% of the widthnot of affect the too interval. muchrange our Again are estimate. we changed. note We Wedeviation that then refit of this estimate the the corresponding the choic data sample, error each which on contains time agai the the position We let the boundaries of thepoint fitting can range fluctuate fluctuate with randomly. a Gaussian distribution with null me JCAP12(2014)049 l l l l 420 818 422 950 816 418 420 814 900 416 418 812 414 850 810 Lower left: second peak. Lower right: Lower left: second peak. Lower right: 416 412 808 800 L l L L L H l l l H H H N N N N 800 600 400 200 500 1000 600 500 400 300 200 100 800 600 400 200 3000 2500 2000 1500 1000 l l l l – 23 – 223 550 223 data: distribution of the estimator of the position of the 542.5 data: distribution of the estimator of the position of the 222 545 540 222 Planck 221 WMAP-9 221 537.5 540 220 535 220 535 219 532.5 219 530 218 530 218 217 525 L L L l l l . Fitting procedure of L . Fitting procedure of H H H l H N N N N 500 400 300 200 100 300 200 100 600 700 600 500 400 200 800 600 400 400 200 800 600 Figure 9 Figure 8 peaks and dips. Upperthird peak. left: first peak. Upper right: first dip. peaks and dips. Upperthird peak. left: first peak. Upper right: first dip. JCAP12(2014)049 l l d fitting curves. Upper 1- 289442). E. M. also 46 and by the European I-2010-08112), by CICYT Excelencia Severo Ochoa” 800 iscussions. E. M. was sup- 800 Government (CAM) through aniti, Thomas Buchert, Marco 600 600 – 24 – data. 400 400 Planck 200 200 l l C C L L 1 Π 1 Π 2 + 2 + l data. Lower panel: l H H l l 2000 1000 5000 4000 3000 5000 4000 3000 2000 1000 . Fitting procedure of CMB data: superposition of raw data an WMAP-9 Bersanelli, Davide Bianchiported and by the Alida Spanish MICINNsthrough Marchetti Juan the de for project la Cierva FPA-2012-31880, usefulthe programme by (JC d project the Madrid HEPHACOS Regional S2009/ESP-1473Union under FP7 ITN grant P-ESP-003 INVISIBLESacknowledges (Marie the Curie support Actions, ofProgramme PITN- the under Spanish GA-201 Grant MINECO’s No. “Centro SEV-2012-0249. de Acknowledgments We acknowledge Luigi Guzzo, Martin Kunz, Pierstefano Coras panel: Figure 10 JCAP12(2014)049 and ]. α ] Ω , ] ] ng SPIRE , , IN , (2011) 525 ] [ ]. ]. 56 ]. (2008) 467 Astron. Astrophys. , ]. 40 SPIRE SPIRE ativity: perfect fluid ativity. 1. Dust Measurements of ]. 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