Testing the consistency between cosmological data: the impact of spatial curvatureand the dark energy EoS

Javier E. Gonzalez,1, 2, ∗ Micol Benetti,3, 4, 5, † Rodrigo von Marttens,6, ‡ and Jailson Alcaniz.6, §

1Facultad de Ciencias e Ingeniería, Universidad de Manizales, 170002, Manizales, Colombia2Departamento de Física, Universidade Federal do Rio Grande do Norte, 59078-970, Natal, RN, Brasil

3Dipartimento di Fisica "E. Pancini", Università di Napoli “Federico II”,Complesso Universitario di Monte Sant’Angelo, Edificio G, Via Cinthia, I-80126, Napoli, Italy

4Istituto Nazionale di Fisica Nucleare (INFN) Sezione di Napoli,Complesso Universitario di Monte Sant’Angelo, Edificio G, Via Cinthia, I-80126, Napoli, Italy

5Scuola Superiore Meridionale, Università di Napoli “Federico II”, Largo San Marcellino 10, 80138 Napoli, Italy6Observatório Nacional, 20921-400, Rio de Janeiro, RJ, Brasil

(Dated: April 29, 2021)

The results of joint analyses of available cosmological data have motivated an important debateabout a possible detection of a non-zero spatial curvature. If confirmed, such a result would implya change in our present understanding of cosmic evolution with important theoretical and observa-tional consequences. In this paper we discuss the legitimacy of carrying out joint analyses with thecurrently available data sets and explore their implications for a non-flat universe and extensionsof the standard cosmological model. We use a robust tension estimator to perform a quantitativeanalysis of the physical consistency between the latest data of Cosmic Microwave Background, typeIa supernovae, Baryonic Acoustic Oscillations and Cosmic Chronometers. We consider the flat andnon-flat cases of the ΛCDM cosmology and of two dark energy models with a constant and varyingdark energy EoS parameter. The present study allows us to better understand if possible incon-sistencies between these data sets are significant enough to make the results of their joint analysesmisleading, as well as the actual dependence of such results with the spatial curvature and darkenergy parameterizations.

I. INTRODUCTION

Over the last two decades, the standard Λ - Cold DarkMatter (ΛCDM) cosmology has been consolidated as thebest description of cosmological observations [1]. Thismodel has shown great success in explaining the dynam-ics of the Universe up to first-order perturbations requir-ing only half a dozen parameters, namely: the physicalbaryon and CDM densities, the optical depth, the an-gular acoustic scale, the scalar spectral index, and theprimordial curvature amplitude.

In the concordance ΛCDM model, the mechanism be-hind the late-time cosmic acceleration is the cosmologicalconstant Λ, characterized by a constant equation-of-state(EoS) parameter w = −1, and the Universe is assumed tobe spatially flat. The latter assumption has been stronglysupported by observational results obtained from jointanalyses of many different cosmological observables (seee.g. [2–10]). From a statistical point of view, the addition

∗ gonzalezsjavier@gmail.com† micol.benetti@unina.it‡ rodrigovonmarttens@gmail.com§ alcaniz@on.br

of a non-zero curvature in such analyses does not seem tosignificantly improve the goodness-of-fit when comparedwith the flat scenario [11] whereas from the theoreticalpoint of view, the flatness of the Universe is well moti-vated by the simplest models of inflation [12, 13], whichsuggests that the case for a zero spatial curvature is morethan a mere simplification.

An important aspect worth considering in this discus-sion concerns the actual consistency among the availablecosmological data and the need to introduce new degreesof freedom in the cosmological description to capture un-known features presented in the data. Recently, the latterpossibility has been explored in several ways. In particu-lar, given the latest results of the Planck Collaboration,a debate about a possible evidence for a positive curva-ture (Ωk < 0) in the power spectrum of the temperaturefluctuations and polarization of the Cosmic MicrowaveBackground (CMB) has emerged, which could imply acosmic discordance between the current CMB data andother cosmological probes such as Baryon Acoustic Oscil-lations (BAO), Type Ia Supernovae (SNe Ia) and CosmicChronometers (CC) observations.

This apparent inconsistency has been pointed outby the Planck Collaboration [2] and interpreted by DiValentino et al. [14], Handley [15], and Park & Ratra

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[16] as a cosmic discordance in the framework of theΛCDM cosmology. In Ref. [17] the authors exploreda four-parameter extension of the standard model andpresented cosmological constraints from CMB and SNeIa data which exclude a flat universe at 99% (CL). Adifferent interpretation of the results obtained from thecurrently available sets of observations was given by thePlanck Collaboration [2] and more recently by Efstathiou& Gratton [11]. In the latter work, the authors performeda joint analysis of the main cosmological data to break thegeometrical degeneracy of the parameters, showing thata flat universe is favoured.

The tension level among the current data sets or, morespecifically, if it allows performing a legitimate joint anal-ysis is an important question not completely explored inthese studies. Currently, the main inconsistency seem toarise between BAO and Planck data when the Plik like-lihood is used, while the use of CamSpec likelihood pro-vides a better consistency. Furthermore, the geometricaldegeneracy is partially broken when the CMB tempera-ture and polarization data are combined with the CMBlensing information, which alleviates the CMB/BAO ten-sion. It is also worth mentioning that the Planck evidencefor a non-zero spatial curvature is not present in the theACT Collaboration CMB data, which shows a good agree-ment with a flat universe [9]. Another relevant question iswhether these inconsistencies also appear when one con-siders more general models e.g. a ΛCDM scenario withnon-zero spatial curvature or even a flat universe domi-nated by a dark energy component with a more generalEoS parameter (w 6= −1).

Currently, there is not a commonly accepted tensionmeasure and a large number of estimators have been pro-posed in the literature e.g. the Bayes Evidence ratio T[18], the Bayes Evidence ratio R [19], the Suspiciousness[20], the parameter difference [21], the Surprise [22], theIndex of Inconsistency (IOI) [23, 24], among others. TheBayes Evidence ratio is a commonly used tension estima-tor, however, like any quantity based on the evidence, itdepends on a prior, which may affect the tension estimate.Since our purpose is to use very large uninformative priorsto allow data to provide the posterior distribution with-out biases from previous information, the Evidence Ratiois not a good choice.

In our analysis we use the moment-based tension es-timator IOI to quantify the (in)consistency among themain cosmological data sets. It is worth mentioning thatthe IOI, as well as other discordance estimators proposedin the literature, quantifies a mix of discrepancies causedby incomplete or incorrect models, systematic errors anddata scatter. Therefore, in order to explore the consis-tency of different theoretical models from the compatibil-

ity of the main cosmological data sets (rather than thedata scatter), we adopt the novel approach presented inRef [25], the so-called level of physical inconsistency βthat quantifies only discrepancies of physical origin, i.e.,due to wrong models or systematic errors. In our anal-ysis, we use the latest data of CMB, SNe Ia, BAO andCC, and consider the flat and non-flat cases of the ΛCDMcosmology and of two dark energy models with a constantand varying dark energy EoS parameter.

This paper is organized as follows: in Sec. II we presentthe primary data sets used in our analysis and the formal-ism adopted to estimate tensions among them. In Sec.III, we show the confidence levels of the statistical anal-yses with the aim to expose graphically the data incon-sistencies and, then, analyse the quantitative estimates ofthe tensions in Secs. IV and V. Finally, in Sec. VI, wepresent the main conclusions of this work.

II. OBSERVATIONS AND DATA TENSIONS

In this section, we shall present the observational dataused in our analysis and the method adopted to computethe consistency between data sets.

A. Data

We use the most commonly used data to perform pa-rameter selection in the literature. They are the following:

• CMB(Plik/CamSpec): Cosmic MicrowaveBackground measurements, through the Planck(2018) data [26], using “TT,TE,EE+lowE" databy combination of temperature TT, polarizationEE and their cross-correlation TE power spectraover the range ` ∈ [30, 2508], the low-` temperatureCommander likelihood, and the low-` SimAll EElikelihood. Regarding the high-`, we analyze boththe likelihood “Plik" and “CamSpec". The formeris considered the more robust high-` data of thePlanck Collaboration, while the latter uses thesame data of Plik with a variation on some keydata and model choices (e.g., polarization maskand polarization efficiency). In polarization, themain differences are the use of a single mask toreduce the amount of computation required tocalculate covariance; Galactic dust subtraction inpolarization; effective calibration handling for TEand EE; the coaddition process; and the absence ofpolarized dust nuisance parameters.

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• Lens: The CMB lensing reconstruction power spec-trum composed by 9 correlated data points from thelatest Planck satellite data release (2018) [2, 26];

• Baryon Acoustic Oscillation (BAO): Followingthe analysis of the latest Planck release [26], we usethe BAO measurements from 6dFGS [27], SDSS-MGS [28], and BOSS DR12 [29] surveys;

• Type Ia SNe (Pantheon): The Pantheon compi-lation [5] contains 1048 SNe Ia in the redshift range0.01 < z < 2.3, which provides accurate measure-ments of the peak magnitudes in the rest frame ofthe B band mB . The SNe Ia absolute magnitudeM is considered as a nuisance parameter.

B. Discordance and physical inconsistencyestimators

As mentioned earlier, recent analyses combining theabove data sets have led to different results on the con-straints of the main cosmological parameters. In orderto explore the consistency among these sets of obser-vations, we adopt the Index of Inconsistency (IOI) toquantify possible tensions among them, as proposed inRefs. [23, 24].

Given the parameter joint distributions of a model con-strained by two different data sets, the IOI is defined as

IOI ≡ 1

2(µ1 − µ2)(C1 +C2)−1(µ1 − µ2)T , (1)

where µi and Ci correspond to the mean parameter vec-tor and its covariance matrix, respectively, constrainedby the i-th data set. This tension estimator considers thedifference between the parameter mean vectors of the pos-teriors and also the effective size of the parameter spaceencoded in the covariance matrix.

It is important to highlight that the tension measuredoes not depend only on the data sets, but also on themodel adopted in the analysis. Other crucial point is thatthe analysis of tensions must not be carried out parameterby parameter (marginalizing over the other ones), but ajoint calculation that takes into account the complete setof parameters and their correlations, i.e., the joint pos-terior distribution of the model parameters. The mainreason for the joint approach is that the tension anal-ysis considering marginalized distributions of separatedparameters can hide disagreements due to the parametercorrelations, as shown in Ref. [23, 30].

As any other discordance estimator in literature, theIOI measures a combination of physical inconsistency,

which may be caused by incomplete/incorrect model orsystematic errors, and data scatter. In order to quantifyuniquely the level of physical inconsistency, β, we adoptthe novel approach proposed in Ref [25]. The main idea ofthis novel approach is to calculate the conditional prob-ability of the physical inconsistency, given a discordanceestimator value, and to infer from this distribution thediscordance estimators values that probabilistically implylevels of physical inconsistency that cannot be neglected.For this purpose it is used the Bayes’ theorem,

P (β|√

2IOI) =P (√

2IOI|β)P (β)

P (√

2IOI), (2)

with its usual interpretation.The first step is to estimate if the discordance measure

(in our case the IOI) implies that the probability P (β >

1|√

2IOI) exceeds a specific level P , i.e.,∫ ∞1

P (β|√

2IOI)dβ > P. (3)

The criterion adopted to define the probability thresholdis the α significance level with α = 0.15 (P = 1 − α).This choice is due to the fact that in an one-parameterdistribution the α value corresponds to 2σ confidence level(See Table 1 of Ref. [25]). The second step is to obtain themost representative value of β and its credible interval.For this purpose it is used the median statistics, with themedian value of β (βmed) defined as∫ βmed

0

P (β|√

2IOI)dβ = 0.5 , (4)

and its 68%-percentile lower and upper limits∫ βlower(∞)

0(βupper)

P (β|√

2IOI)dβ = 0.16. (5)

The next step is to choose an adequate scale to interpretthe β results. The scale proposed in Ref. [25] is motivatedby the classification performed in Ref. [31] based on then-σ and it is shown in Table I1. See also Ref. [25] for acomplete discussion about the level of physical inconsis-tency.2

1 An available code to quantify β can be found inhttps://github.com/WeikangLin/IOI.

2 As noted in Ref. [24], the original term "moderate tension" shouldnot be interpreted as an irrelevant inconsistency, mainly if thevalue is close to the upper limit. In order to avoid an underestima-tion of this tension level, we replace the original term "moderate"to "substantial".

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No evidence of physical inconsistency exceeding the thresholdP (β > 1|

√2IOI) < (1− α) (Ranking number: 1)

Evidence of physical inconsistency exceeding the thresholdP (β > 1|

√2IOI) > (1− α)

Single β value Level inconsistency β range[βlower, βupper)

Ranking inconsistency(Ranking number)

β<3.5 SubstantialInside the

substantial levelSubstantial

(2)Spans substantialand strong level

Substantial-to-strong(3)

3.5<β<5 StrongSpans substantial, strongand very strong level

Strong(4)

Spans strong andvery strong level

Strong-to-verystrong (5)

5>β Very strong Inside the strong level Very strong(6)

Table I. Guiding interpretation of the level of physical inconsistency. The inconsistency levels are motivated by the interpretationof the 3σ, 4σ and 5σ tension in 1D distribution considering a specific value of β. The ranking of inconsistency is a more suitablescale that takes into account that β is characterized by a range of values rather than a single number, i.e., the β distribution.The first rank corresponds to probability of β > 1 lower than the threshold of significant tension, P = 1− α = 0.85 [25].

In addition to the IOI, it is also useful to define aquantity that allows identifying outliers in a compositionof multiple data sets. For this purpose, we use the Out-lier Index (Oj) that consists in the calculation of the IOIbetween the j-th data set, the one considered as a pos-sible outlier, and a joint multiple data sets, which mustnot include the j-th data set. Thus, the Oj value is ob-tained by considering in Eq. (1) the mean and covariancematrix of the parameters constrained by the j-th dataset and the ones constrained by the joint analysis of theother data sets (without considering the j-th data set)and subtracting the term (Np − 1)/2, where Np is thenumber of common parameters of the mean vectors andmatrices. The interpretation of the outlier index is madein relative terms by comparing the Oj value of each dataset. An outlier data is identified when its own Oj valueis significantly higher than the other ones [32].

III. MODEL ANALYSIS

We shall now perform the statistical analysis that willbe used to compute the consistency between the datasets and to identify possible outliers. Here, in order togive a first qualitative idea of the problem, we show somecontour planes to illustrate our main results discussed inSec. VI. However, we emphasize that a correct consis-tency test cannot be made regarding only 2D contours

separately, but must take into account all the parameterspace simultaneously.

The models we consider are the following:

• ΛCDM(+Ωk): The standard cosmological modelwith zero and non-zero spatial curvature as afree parameter, which we denote as “ΛCDM” and“oΛCDM”, respectively.

• wCDM(+Ωk): A simple one-parameter extensionof the previous model, where the DE component isno longer described by the cosmological constant,but by a dark component with constant EoS w. Aswell as in the first case, we also consider the modelwith spatial curvature as a free parameter. Fromnow on, the flat and non-flat model of this kind aredenoted by “wCDM” and “owCDM”, respectively;

• CPL(+Ωk): A two-parameter extension of theoΛCDM, where the DE component is described bythe CPL parameterization, i.e., a time-dependentDE EoS given by w (a) = w0 + wa (1 + a) [33, 34].This parameterization is of particular interest be-cause it is widely used for parameter selection [26]and forecasts [35] (see also [36–40] for other DE EoSparameterizations). From now on, the flat and non-flat model of this kind is denoted by “CPL” and“oCPL”, respectively.

We divide our analysis in two parts. First, in order

5

0.1

0.0

0.1

ΩK

BAOSNe IaPlikPlik+Lens

50 60 70 80H0

0.1

0.0

0.1

ΩK

50 60 70 80H0

0.2 0.3 0.4 0.5 0.6Ωm

0.2 0.3 0.4 0.5 0.6Ωm

BAOSNe IaCamSpecCamSpec+Lens

Figure 1. Constraints on the H0 − Ωk and Ωm − Ωk planes from SNe, BAO and CMB (+Lens) data considering the oΛCDMmodel. In the upper panels we present the confidence contours for CMB (+Lens) data using the Plik likelihood whereas in thelower panels we use the CampSpec CMB (+Lens) likelihood.

to compute the physical consistency among the individ-ual probes, we perform a parameter estimation analysiswith the individual probes presented in Sec. II A. Thesecond analysis performed is for the outlier diagnostic. Inthis case we combine the data sets in pairs to be com-pared with the remaining data set. Even though CMBand CMB+Lens are considered single data sets sepa-rately, they are not combined. Thus, the possible groupsare: CMB+BAO, (CMB+Lens)+BAO, CMB+SNe Ia,(CMB+Lens)+SNe Ia, and BAO+SNe Ia. The lastgroup, BAO+SNe Ia, is compared with CMB and withCMB+Lens. In all cases, CMB stands for both Plik andCamSpec likelihoods.

In Fig. 1, we show the results of the H0 − Ωk andΩm − Ωk planes for the oΛCDM model using the inde-pendent data sets separately. Since type Ia SNe can-not constraint H0, all panels that contain this quantitydo not show results from SNe Ia. Note that both BAOand SNe data seem to be in tension with CMB (with-out Lens) data, being more evident in the H0 −Ωk planefor BAO data and in the Ωm − Ωk for SNe Ia observa-

tions. In the latter case, the tension arises because ofthe difference in the Ωm estimates, as SNe data do notprovide any information about H0 and do not constraineffectively the curvature. This disagreement is reducedif one uses the CamSpec likelihood instead of the Pliklikelihood, the primary Planck Collaboration likelihood.Clearly, the inclusion of the CMB lensing breaks the geo-metrical degeneracy for both CMB likelihoods, as pointedout in [2], and the principal source of tension betweenBAO and CMB+Lens data remains in the H0 estimate.As shown in the right panels of Fig. 1, the inclusionof CMB lensing data to CMB data reduces considerablythe SNe/CMB tension, mainly when it is used the Pliklikelihood, which will be quantified in the next section.On the other hand, BAO and SNe Ia data show a goodagreement. As it was discussed in previous works, theCamSpec allows better compatibility with a flat universethan the Plik likelihood, whilst the CMB+Lens, SNe, andBAO data sets are consistent with Ωk = 0 [11, 20].

In order to identify possible outlier data sets, we showin Fig. 2 and 3 the constraints from a single observable

6

0.2 0.3 0.4 0.5 0.6Ωm

0.1

0.0

0.1

ΩK

SNe IaPlik+BAOPlik+Lens+BAO

0.2 0.3 0.4 0.5 0.6Ωm

50 60 70 80H0

BAOPlik+SNe Ia

Plik+Lens+SNe Ia

Figure 2. The 1σ and 2σ constraints on the H0 − Ωk and Ωm − Ωk planes from single data sets.

50 60 70 80H0

0.1

0.0

0.1

ΩK

0.2 0.3 0.4 0.5 0.6Ωm

BAO+SNe Ia PlikPlik+Lens

50 60 70 80H0

0.1

0.0

0.1

ΩK

0.2 0.3 0.4 0.5 0.6Ωm

BAO+SNe Ia

CamSpec CamSpec+Lens

Figure 3. Current 1σ and 2σ constraints on the H0 − Ωk and Ωm − Ωk planes from combination of data sets. The search foroutliers is performed for CMB data (Plik and CamSpec likelihoods).

and from the joint analyses of the remaining data sets,respectively. In Fig. 2, we show the results for the out-lier analysis considering the CMB(+Lens) combined withBAO and SNe Ia. We choose to show only the Plik likeli-hood case because the results using both CMB likelihoodsare very similar. As one may see, the SNe constraints arein a very good agreement with the CMB(+Lens)+BAO

constraints. In the second and third panels, we explorethe possibility of the BAO data being an outlier. In theΩm − Ωk plane we note a good compatibility betweenBAO and CMB(+Lens)+SNe whereas in the H0 − Ωkplane a certain difference in the contours is evident. Inthese cases the addition of SNe Ia or BAO observations tothe CMB(+Lens) data is enough to break the parameter

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Figure 4. Values of the level of physical tension β between independent data sets assuming a flat universe (blue) and a universewith curvature (green). The left, middle and right panels correspond to the matrix tensions in the ΛCDM (oΛCDM), wCDM(owCDM) and CPL (oCPL) models, respectively. The color scale represents the ranking number introduced in Table I. Notethat the highest inconsistency rank obtained is substantial-to-strong (3).

degeneracy, making the CMB lensing not very relevant.In Fig. 3, we present the SNe+BAO and CMB con-

straints. In both planes, Ωm − Ωk and H0 − Ωk, it ispossible to visualize the strong inconsistency between thelate and early-time data when the Plik CMB likelihoodis considered. However, this inconsistency is considerablyreduced if the CMB data is analysed with the CamSpeclikelihood or if the CMB lensing is also included in theanalysis (CMB+Lens).

We also explore the observational constraints on cos-mological parameters in the framework of the owCDMmodel. Contrary to the results for the oΛCDMmodel, theanalysis for the owCDM model shows that a non-null cur-vature is not confirmed at 2σ confidence level using onlyCMB data, and when lensing data are also considered, thespatial geometry is compatible with a flat universe at 1σconfidence level. However, for Plik CMB likelihood thecurvature parameter Ωk = 0 needs high H0 values andan EoS crossing the phantom line, while for the Cam-Spec likelihood we obtain more compatible results withthe standard cosmology. Apparently, the owCDM modelcan alleviate the inconsistencies between all data sets in-dependently of the CMB likelihood. However, such re-sult needs to be confirmed by a consistency estimator asthe marginalized distributions can hide possible tensions,which we perform in the next section. In the case ofthe oCPL model, the constraints are not very restrictivewhen a single observable is considered and its analysis oftensions from the marginalized confidence contours is notvery useful.

IV. QUANTITATIVE TENSION ESTIMATES

In what follows, we quantify the results of the previ-ous section and estimate possible tensions between theanalysed data sets taking into account the full posteriorparameter distributions and using the Index of Inconsis-tency as a discordance estimator, and the level of physicalinconsistency as a tension measure caused by physical ef-fects [23–25] (see also Sec. II).

Fig. 4 presents heatmaps with the β values of pairsfor data sets. The left, middle and right maps corre-spond to the ΛCDM, wCDM and CPL models, respec-tively, where blue maps represent spatially flat modelswhile green maps represent models with non-zero curva-ture. For the ΛCDMmodel, we note that all data are con-sistent, i.e., no tension exceeds the threshold of evidenceof physical inconsistency P = 0.85 (see Eq. 3), ranking(1) of the Table I, and the final result is a high precisionestimate of the six model parameters. For the wCDMand oΛCDM models, we note that the tensions increase,being higher for BAO and CMB(Plik) data, which showssubstantial and substantial-to-strong inconsistencies, re-spectively. The use of the CamSpec likelihood alleviatesthe inconsistencies, but it still remains in the substantialrank for the oΛCDM model. In this heatmap it is alsoseen that the addition of the lensing power spectrum tothe CMB data alleviates the physical inconsistencies, andwe obtain non-substantial tensions.

Extending our analysis to the owCDM and CPL mod-els, we find that the inconsistencies decrease when a two-

8

SNe (Plik)

SNe (CamSpec)

BAO (Plik)

BAO (CamSpec)

CMB (Plik)

CMB (Plik+lens)

CMB (CamSpec)

CMB (CamSpec+lens)

oCPL

owCDM

o CDM

-0.7 -0.5 -0.4 -0.4 -1.4 -1.3 -1.3 -1.5

-1.0 -1.0 0.3 0.3 -0.2 -0.8 -0.4 -0.8

-0.4 -0.4 1.9 2.4 5.8 2.6 2.9 2.6

10123456

Figure 5. Values of the Oj values as a measure of the compatibility between the constraints from the j-th data set and the jointconstraints from the rest of the data sets. For SNe and BAO data, the Oj value is calculated considering CMB constraints fromthe Plik and CamSpec likelihoods. The color scale represents the Oj value.

parameter extension of the ΛCDMmodel is considered. Itis worth noticing that for the owCDM model, substantial-to-strong inconsistency appears between CMB(Plik) andBAO data, although such inconsistencies become insignif-icant when the CMB lensing data are also considered.Therefore, assuming the owCDM model and taking intoaccount the lensing data, all discrepancies are in an ac-ceptable level to perform joint analyses involving thesethree observables. The last model analysed, the oCPLmodel, does not exhibit considerable improvements interms of the tension analysis. We also note that inde-pendently of the cosmological model adopted, the highesttensions are always found for the BAO data3.

In Fig. 5, we show the calculations of the Outlier In-

3 It is worth mentioning that the usual approach to measure BAOin galaxy surveys make use of a fiducial cosmology (usually theflat ΛCDM model) to transform observed redshifts and angles tothe estimated angular and radial BAO peak positions. In [41],the impact of the fiducial model on the inferred BAO scale wasdiscussed by considering flat wCDM models. The influence of anon-zero curvature is unknown.

dex Oj considering SNe, BAO and CMB+Lens data andmodels with non-zero curvature (oΛCDM, owCDM andoCPL). The heatmap evaluates if one specific data setconstitutes an outlier with respect to the other two datasets, and in parenthesis we specify which CMB likelihoodis used in the analysis. In the case of the Outlier Indexfor the CMB data, we also calculate it without consider-ing the CMB lensing (fifth and seventh column). As canbe seen, the unique outlier found with high significanceis the CMB when the high multipoles of the temperaturepower spectrum are analyzed with the Plik likelihood.The same significance is not obtained for CamSpec likeli-hood or when the lensing power spectrum measurementsare taken into account.

V. COSMIC CHRONOMETERS

Recently, the authors of [42] argued that measurementsof the expansion rate H(z) from the relative ages of pas-sively evolving galaxies [43–49] – cosmic chronometers(CC) – can break the geometrical degeneracy discussedabove without presenting tensions with the remaining

9

Figure 6. Values of the level of physical inconsistency β be-tween CC and the other independent data sets assuming a flatuniverse (blue) and a universe with curvature (green). Thecolor scale represents the ranking number introduced in Ta-ble I. Note that the highest inconsistency rank obtained issubstantial-to-strong (3).

data sets.By comparing the constraints obtained on the

cosmological parameters from the joint analysis ofCMB(Plik)+BAO in the framework of the oΛCDMmodel [2, 50],

H0 = 67.88± 0.66 km.s−1.Mpc−1

Ωm = 0.310± 0.007

ΩK = 0.0008± 0.0019

with the constraints obtained from the joint analysis ofCMB(Plik)+CC [42],

H0 = 65.23± 2.14 km.s−1.Mpc−1

Ωm = 0.336± 0.022

ΩK = −0.0054± 0.0055,

it is possible to infer that (i) the results from CC andBAO data are not inconsistent and, (ii) given the resultspresented in Fig. (1), there seems to be a tension betweenCMB (Plik) and CC data.

In Fig. 6, we show the heatmaps of the level of physicalinconsistency β between the CC and the other data sets,considering non-flat (green) and flat (blue) cosmologies.

As seen in the upper panel of Fig. 6, we find a substantial-to-strong inconsistency between CMB(Plik) and CC data,with β = 3.0+1.1

−1.1. This result changes substantially whenthe lensing power spectrum measurements are taken intoaccount (β = 1.1+1.0

−0.7), and differs from the mild disagree-ment found in Ref. [42], which uses the I diagnostic basedon the deviance information criterion (DIC). Finally, wealso note that the tension between CC and CMB obser-vations is significantly reduced when the CamSpec like-lihood is used. Moreover, we find no significant incon-sistency involving the CC data when a flat geometry isassumed in the analysis (lower panel of Fig. 6).

VI. CONCLUSIONS

Some of the most important aspects of our present un-derstanding of the cosmic evolution are based on the re-sults of joint analyses involving the currently available ob-servational data. In this paper we assessed the reliabilityof such analyses by quantifying possible inconsistenciesbetween the data sets through the Index of Inconsistency,the Outlier Index and the level of physical inconsistencyβ, defined in Sec. IIb. We used the latest observationsof CMB, SNe Ia, BAO and CC and discussed the actualdependence of the results of their combined analysis withthe spatial curvature parameter and dark energy param-eterizations.

We presented the calculations of the level of physi-cal inconsistency β between all independent data setsconsidering six cosmological models mentioned in Sec.III. We found that a model extension that considers anon zero curvature is necessary to describe all the fea-tures present in the data sets (separately) considered inthis work, specially for the Planck CMB data (temper-ature+polarisation). This is evident when the ΛCDMmodel is extended to the oΛCDM and substantial-to-strong tensions appear between BAO and CMB data sets.However, our analysis also confirms that by consideringjointly all data sets or even only CMB Planck informa-tion (temperature+polarisation+lensing), there is no sig-nificant evidence for a non-flat Universe.

As shown in Figs. 4 and 5, the owCDM is thesimplest model showing only weak inconsistencies be-tween all cosmological data including the CMB lens-ing information. Nevertheless, when the joint analy-sis of CMB+Lens+SNe+BAO is performed, the curva-ture and EoS parameters are tightly constrained, i.e.,Ωk = −0.0001+0.0023

−0.0021 and w = −1.026+0.039−0.032, and the

10

model is reduced to the flat ΛCDM cosmology4. On theother hand, we found no considerable improvement of thedata tensions when an evolving dark energy EoS is al-lowed, which means that considering higher order exten-sions of the standard cosmology (such as the oCPL model)do not seem to be the way to alleviate data inconsisten-cies.

Finally, our analysis did not find significant tensionsthat a prevent joint analysis of the currently available cos-mological data in non zero curvature models when lensingdata is considered. Therefore, the most relevant tensionfound is the one between the CMB(Plik) and BAO datasets, which can possibly be solved with the data from thenext generation of galaxy surveys such as J-PAS, Euclidand DESI.

ACKNOWLEDGMENTS

We thank Gabriela Antunes, Weikang Lin andMustapha Ishak for very useful discussions. JEG ac-knowledges the Federal University of Rio Grande doNorte for financial support during my lockdown in Na-tal. MB acknowledges the Istituto Nazionale di FisicaNucleare (INFN), sezione di Napoli, iniziativa specificaQGSKY. RvM acknowledges financial support from thePrograma de Capacitação Institucional (PCI) do Obser-vatório Nacional/MCTI. JA is supported by the ConselhoNacional de Desenvolvimento Científico e TecnológicoCNPq (Grants no. 310790/2014-0 and 400471/2014-0)and Fundação de Amparo à Pesquisa do Estado do Riode Janeiro FAPERJ (grant no. 233906). The computa-tional analyses of this work were developed thanks to theHigh Performance Computing Center at the Federal Uni-versity of Rio Grande do Norte (NPAD/UFRN) and theNational Observatory Data Center (DCON).

[1] D. H. Weinberg, M. J. Mortonson, D. J. Eisenstein, C. Hi-rata, A. G. Riess, and E. Rozo, Phys. Rept. 530, 87(2013), arXiv:1201.2434 [astro-ph.CO].

[2] N. Aghanim et al. (Planck), Astron. Astrophys. 641, A8(2020), arXiv:1807.06210 [astro-ph.CO].

[3] G. Hinshaw et al. (WMAP), Astrophys. J. Suppl. 208, 19(2013), arXiv:1212.5226 [astro-ph.CO].

[4] S. Alam et al. (eBOSS), (2020), arXiv:2007.08991 [astro-ph.CO].

[5] D. Scolnic et al., Astrophys. J. 859, 101 (2018),arXiv:1710.00845 [astro-ph.CO].

[6] M. Betoule et al. (SDSS), Astron. Astrophys. 568, A22(2014), arXiv:1401.4064 [astro-ph.CO].

[7] T. Abbott et al. (DES), Astrophys. J. Lett. 872, L30(2019), arXiv:1811.02374 [astro-ph.CO].

[8] V. Bonvin et al., Mon. Not. Roy. Astron. Soc. 465, 4914(2017), arXiv:1607.01790 [astro-ph.CO].

[9] S. Aiola et al. (ACT), (2020), arXiv:2007.07288 [astro-ph.CO].

[10] M. Costanzi et al. (DES, SPT), (2020), arXiv:2010.13800[astro-ph.CO].

[11] G. Efstathiou and S. Gratton, Mon. Not. Roy. Astron.Soc. 496, L91 (2020), arXiv:2002.06892 [astro-ph.CO].

[12] A. H. Guth, Phys. Rev. D 23, 347 (1981).

4 As shown in Ref. [4], an analogous result is also obtained for theoCPL model where the flat ΛCDM cosmology is recovered within∼ 1σ CL by considering Pantheon, CMB(Plik), SDSS BAO+RSDand DES cosmic shear, galaxy clustering, and galaxy-galaxy lens-ing data.

[13] A. D. Linde, Phys. Lett. B 116, 335 (1982).[14] E. Di Valentino, A. Melchiorri, and J. Silk, Nature As-

tron. 4, 196 (2019), arXiv:1911.02087 [astro-ph.CO].[15] W. Handley, (2019), arXiv:1908.09139 [astro-ph.CO].[16] C.-G. Park and B. Ratra, Astrophys. J. 882, 158 (2019),

arXiv:1801.00213 [astro-ph.CO].[17] E. Di Valentino, A. Melchiorri, and J. Silk, ApJL 909,

L9 (2021), arXiv:2003.04935 [astro-ph.CO].[18] L. Verde, P. Protopapas, and R. Jimenez, Phys. Dark

Univ. 2, 166 (2013), arXiv:1306.6766 [astro-ph.CO].[19] P. Marshall, N. Rajguru, and A. Slosar, Phys. Rev. D

73, 067302 (2006), arXiv:astro-ph/0412535.[20] W. Handley and P. Lemos, Phys. Rev. D 100, 043504

(2019), arXiv:1902.04029 [astro-ph.CO].[21] R. A. Battye, T. Charnock, and A. Moss, Phys. Rev. D

91, 103508 (2015), arXiv:1409.2769 [astro-ph.CO].[22] S. Seehars, A. Amara, A. Refregier, A. Paranjape,

and J. Akeret, Phys. Rev. D 90, 023533 (2014),arXiv:1402.3593 [astro-ph.CO].

[23] W. Lin and M. Ishak, Phys. Rev. D 96, 023532 (2017),arXiv:1705.05303 [astro-ph.CO].

[24] W. Lin and M. Ishak, Phys. Rev. D 96, 083532 (2017),arXiv:1708.09813 [astro-ph.CO].

[25] W. Lin and M. Ishak, (2019), arXiv:1909.10991 [astro-ph.CO].

[26] N. Aghanim et al. (Planck), Astron. Astrophys. 641, A5(2020), arXiv:1907.12875 [astro-ph.CO].

[27] F. Beutler, C. Blake, M. Colless, D. Jones, L. Staveley-Smith, L. Campbell, Q. Parker, W. Saunders, andF. Watson, Mon. Not. Roy. Astron. Soc. 416, 3017 (2011),arXiv:1106.3366 [astro-ph.CO].

11

[28] A. J. Ross, L. Samushia, C. Howlett, W. J. Percival,A. Burden, and M. Manera, Mon. Not. Roy. Astron. Soc.449, 835 (2015), arXiv:1409.3242 [astro-ph.CO].

[29] S. Alam et al. (BOSS), Mon. Not. Roy. Astron. Soc. 470,2617 (2017), arXiv:1607.03155 [astro-ph.CO].

[30] P. Lemos et al. (DES), (2020), arXiv:2012.09554 [astro-ph.CO].

[31] L. Verde, T. Treu, and A. G. Riess, Nature Astron. 3,891 (2019), arXiv:1907.10625 [astro-ph.CO].

[32] W. Lin, K. J. Mack, and L. Hou, Astrophys. J. Lett. 904,L22 (2020), arXiv:1910.02978 [astro-ph.CO].

[33] M. Chevallier and D. Polarski, Int.J.Mod.Phys. D10, 213(2001), arXiv:gr-qc/0009008 [gr-qc].

[34] E. V. Linder, Phys. Rev. Lett. 90, 091301 (2003),arXiv:astro-ph/0208512 [astro-ph].

[35] M. Aparicio Resco et al., Mon. Not. Roy. Astron. Soc.493, 3616 (2020), arXiv:1910.02694 [astro-ph.CO].

[36] E. M. Barboza Jr. and J. S. Alcaniz, Phys.Lett.B. 666,415 (2008), arXiv:0805.1713 [astro-ph].

[37] E. Barboza, J. Alcaniz, Z.-H. Zhu, and R. Silva,Phys.Rev. D80, 043521 (2009), arXiv:0905.4052 [astro-ph.CO].

[38] R. Lazkoz, V. Salzano, and I. Sendra, Phys.Lett. B694,198 (2010), arXiv:1003.6084 [astro-ph.CO].

[39] H. K. Jassal, J. S. Bagla, and T. Padmanabhan, Mon.Not. Roy. Astron. Soc. 356, L11 (2005), arXiv:astro-ph/0404378 [astro-ph].

[40] D. Adak, D. Majumdar, and S. Pal, (2012), 10.1093/mn-ras/stt1941, arXiv:1210.2565 [astro-ph.CO].

[41] P. Carter, F. Beutler, W. J. Percival, J. DeRose, R. H.Wechsler, and C. Zhao, Mon. Not. Roy. Astron. Soc. 494,2076 (2020), arXiv:1906.03035 [astro-ph.CO].

[42] S. Vagnozzi, A. Loeb, and M. Moresco, (2020),arXiv:2011.11645 [astro-ph.CO].

[43] R. Jimenez and A. Loeb, Astrophys. J. 573, 37 (2002),arXiv:astro-ph/0106145.

[44] J. Simon, L. Verde, and R. Jimenez, Phys. Rev. D 71,123001 (2005), arXiv:astro-ph/0412269.

[45] D. Stern, R. Jimenez, L. Verde, M. Kamionkowski, andS. Stanford, JCAP 02, 008 (2010), arXiv:0907.3149 [astro-ph.CO].

[46] M. Moresco, Mon. Not. Roy. Astron. Soc. 450, L16(2015), arXiv:1503.01116 [astro-ph.CO].

[47] C. Zhang, H. Zhang, S. Yuan, T.-J. Zhang, andY.-C. Sun, Res. Astron. Astrophys. 14, 1221 (2014),arXiv:1207.4541 [astro-ph.CO].

[48] M. Moresco, L. Pozzetti, A. Cimatti, R. Jimenez,C. Maraston, L. Verde, D. Thomas, A. Citro, R. To-jeiro, and D. Wilkinson, JCAP 05, 014 (2016),arXiv:1601.01701 [astro-ph.CO].

[49] A. Ratsimbazafy, S. Loubser, S. Crawford, C. Cress,B. Bassett, R. Nichol, and P. Väisänen, Mon. Not. Roy.Astron. Soc. 467, 3239 (2017), arXiv:1702.00418 [astro-ph.CO].

[50] S. Vagnozzi, E. Di Valentino, S. Gariazzo, A. Melchiorri,O. Mena, and J. Silk, (2020), arXiv:2010.02230 [astro-ph.CO].