Hubble constant difference between CMB lensing and BAO measurements
HHubble constant tension between CMB lensing and BAO measurements
W. L. Kimmy Wu, Pavel Motloch, Wayne Hu, and Marco Raveri Kavli Institute for Cosmological Physics, University of Chicago, Chicago, Illinois 60637, U.S.A Canadian Institute for Theoretical Astrophysics, University of Toronto, M5S 3H8, ON, Canada Kavli Institute for Cosmological Physics, Department of Astronomy & Astrophysics,Enrico Fermi Institute, University of Chicago, Chicago, Illinois 60637, U.S.A Center for Particle Cosmology, Department of Physics and Astronomy,University of Pennsylvania, Philadelphia, PA 19104, USA
We apply a tension metric Q UDM , the update difference in mean parameters, to understandthe source of the difference in the measured Hubble constant H inferred with cosmic microwavebackground lensing measurements from the Planck satellite ( H = 67 . +1 . − . km / s / Mpc) and fromthe South Pole Telescope ( H = 72 . +2 . − . km / s / Mpc) when both are combined with baryon acousticoscillation (BAO) measurements with priors on the baryon density (BBN). Q UDM isolates the relevantparameter directions for tension or concordance where the two data sets are both informative, andaids in the identification of subsets of data that source the observed tension. With Q UDM , weuncover that the difference in H is driven by the tension between Planck lensing and BAO+BBN, atprobability-to-exceed of 6.6%. Most of this mild tension comes from the galaxy BAO measurementsparallel to the line of sight. The redshift dependence of the parallel BAOs pulls both the matterdensity Ω m and H high in ΛCDM, but these parameter anomalies are usually hidden when the BAOmeasurements are combined with other cosmological data sets with much stronger Ω m constraints. I. INTRODUCTION
The standard cosmological model ΛCDM is extremelysuccessful in describing observations over a wide rangeof scales and redshifts: from the cosmic microwave back-ground (CMB) to the expansion of the universe today.However, increasingly precise measurements of cosmo-logical parameters obtained in the past several years un-covered mild to strong tensions between different datasets. Most notably,
Planck infers the Hubble constant tobe H = 67 . ± .
54 km / s / Mpc under ΛCDM [1] whilethe Cepheid-calibrated Type Ia Supernovae from SH0ESgives H = 74 . ± .
42 km / s / Mpc [2] (see [3–7] forother measurements). Such tensions between differentdata sets could suggest need to extend the ΛCDM modelto accommodate the observations or, alternatively, exis-tence of unmodeled systematics in the data sets. It isthus very important that we find independent measure-ments that can clarify the source(s) of the current H tension.One such example has been reported recently for thedata sets that result from measurements of the weightedgravitational potential integrated along the line of sight(CMB lensing) from the Planck satellite and the SouthPole Telescope (SPTpol), baryon acoustic oscillation(BAO) parallel and perpendicular to the line of sightin galaxy surveys, and the baryon density inferred fromthe deuterium abundance (
D/H ) measurements (de-noted by BBN). In [8], the values of H inferred fromthe BAO+BBN+ Planck lensing and the BAO+BBN+SPTpol lensing data sets are 67 . +1 . − . km / s / Mpc and72 . +2 . − . km / s / Mpc respectively. This mild discrepancyis intriguing since it is reminiscent of the tension betweenthe SH0ES vs.
Planck
CMB power spectra measurementsabove.In addition, the constraints on the lensing amplitudeas captured by the parameter combination σ Ω . m be- tween the two lensing data sets are completely consis-tent. This presents somewhat of a puzzle since the mildtension in H appears through adding BAO+BBN to theotherwise consistent lensing data sets. Therefore, we setout to investigate the underlying driver(s) of the differ-ences in the inferred H .To do this, we apply a tension metric developed in [9]to quantify tension between BAO+BBN+ Planck lensingand BAO+BBN+SPTpol lensing. We find that the dif-ference in H between the two is driven by the differentinferences from the shape of the Planck lensing spec-trum and BAO+BBN. Specifically, it is the line-of-sightBAO measurements that pull H high. This preferenceis ordinarily hidden when the line-of-sight BAO measure-ments are combined with other cosmological data sets,since it requires a high matter density in ΛCDM whichis strongly ruled out by these other data sets.This paper is organized as follows: In Sec. II we sum-marize the tension metric we use, before presenting thedata sets used in this work in Sec. III. In Sec. IV, we iso-late and quantify the tension between the BAO+BBN+SPTpol lensing and the BAO+BBN+ Planck lensingdata sets and show that it originates from the
Planck lensing and the parallel BAO measurements. In Sec. V,we show that values of cosmological parameters preferredby the parallel BAO measurements are strongly ruled outby other cosmological data sets. We discuss the resultsand conclude in Sec. VI.
II. QUANTIFYING TENSIONS
To quantify tensions between uncorrelated data sets,we use the update difference-in-mean (UDM) statisticdefined in [9]. This statistic compares the mean pa-rameter values from a data set A alone, θ Aα , with their“updated” values after adding another data set B to A , a r X i v : . [ a s t r o - ph . C O ] A p r θ A + Bα . The index α here enumerates the individual pa-rameters. Specifically, this statistic computes the squareof the difference between the mean parameter values ofthe two sets ∆¯ θ α = ¯ θ Aα − ¯ θ A + Bα , (1)in units of its covariance C ∆ , Q tot UDM = (cid:88) α,β ∆¯ θ α (cid:0) C − (cid:1) αβ ∆¯ θ β . (2) C ∆ is inferred from the covariances of A and A + B as C ∆ = C A − C A + B . (3)To quantify the tension, we use the fact that if the A and A + B parameter posteriors are Gaussian distributed anddrawn from a self-consistent model, Q tot UDM is chi-squareddistributed with the number of parameters measured byboth A and B as the degrees of freedom.There are two advantages of using Q tot UDM as opposedto the simpler difference-in-mean statistic. The first ispractical. If the parameter posterior for B is highlynon-Gaussian due to weak constraints or degeneracies,then the squared difference-in-mean is also far from chi-squared distributed, making its significance difficult toquantify. However, if the parameter posterior for A ismore Gaussian, then the posterior of the combined dataset A + B is also more Gaussian and Q tot UDM is closer tochi-squared distributed. The second advantage is theability to pre-identify parameter directions in which thecombination of A and B improves the errors over A or B individually and hence can exhibit interesting tensionor confirmation bias. This is in contrast to defining theparameter space of investigation by inspecting the pa-rameter differences in mean a posteriori , e.g. by pickingthe most discrepant directions.An effective method to isolate these directions is toidentify the Karhunen-Lo`eve (KL) eigenmodes of the co-variance matrices [9]. These eigenmodes φ aα are the so-lutions to the generalized eigenvalue problem (cid:88) β C Aαβ φ aβ = λ a (cid:88) β C A + Bαβ φ aβ , (4)and are normalized so that (cid:88) αβ φ aα C A + Bαβ φ bβ = δ ab . (5)The parameters in the KL basis p a = (cid:88) α φ aα θ α (6)are uncorrelated for both A and A + B with variance λ a and 1, respectively. From Eqs. (2) and (3), Q UDM = (cid:88) a (∆ p a ) λ a − . (7) If the sum is over all KL modes then Q UDM = Q tot UDM .However, to isolate the directions of interest, we restrictthis sum to eigenvalues0 . < λ a − < . (8)Notice that this selection does not involve the actual val-ues of the difference in means, only the expected abilityof B to update A and vice versa. For cases where λ a ≈ B does not update the constraints of A appre-ciably whereas for λ a (cid:29)
1, data set A itself becomes ir-relevant and cannot update B . In the former case, thereare also numerical problems due to the MCMC samplingof the posteriors. This selection also covers cases wherethere are nuisance parameters that are constrained byonly one of A or B .With only the interesting directions in the parameterspace retained, we can then determine the significance oftheir associated difference in means by noting that Q UDM is chi-squared distributed with the number of remainingparameter directions as the degrees of freedom. Fromthis point forward we refer to Q UDM as defined by Eqs. (7)and (8) as the update difference-in-mean statistic.We will also be interested in how much informationeach KL eigenmode contributes to constraining individ-ual cosmological parameters θ α . Recall that the Fisherinformation matrix is the inverse of the parameter co-variance matrix F αβ = ( C αβ ) − and each diagonal entrycorresponds to the inverse variance of the parameter ifall other parameters are held fixed. Using Eq. (5), wecan express this Fisher information of data set A for theparameter θ α as F αα = (cid:88) a F aαα = (cid:88) a φ aα φ aα /λ a . (9)The Fisher information of data set A + B is the sameexpression with λ a →
1, but this will not be needed inour analysis below.The fractional Fisher information F aαα /F αα ∈ [0 ,
1] pa-rameterizes how important KL mode a is in constrainingthe cosmological parameter θ α , where low values meanthat dropping this mode does not significantly affect itsconstraints.When considering correlated data sets, in particularthe internal consistency of parallel and perpendicularBAO measurements, we use the generalization of theabove discussion as in [10]. Specifically, we duplicate theparameter space of the model, and fit the joint data setwith one copy of parameters controlling the theory pre-diction for the first part of the joint data set (e.g. parallelBAOs) and the other copy controlling the theory predic-tion for the second part (e.g. perpendicular BAOs). Wethen assess the confidence intervals of the difference inthese two parameter sets by sampling its posterior. Be-cause we fit the joint data set, the correlations are prop-erly accounted for. This technique also has the benefitof applying to non-Gaussian posterior distributions. III. DATA
The data sets we investigate in Sec. IV include
Planck
D/H measurements. In Sec. V we compare BAO con-straints with parameter constraints from the
Planck tem-perature and polarization power spectra [17] and fromthe Pantheon supernova sample [18].The applicability of the Q UDM statistic hinges uponthe data sets being uncorrelated. For the data sets weare considering, the BAO measurements are uncorre-lated with the lensing measurements. While the
Planck and the SPTpol measurements have partial sky overlap,the overlap is very small ( ∼ Planck ’s L = [8 , L = [100 , Q UDM is appropriate for quantifying tension between our datasets. We only require the parameter duplication gen-eralization discussed in the previous section to quantifytension between correlated subsets of a given data set(e.g. BAO).In all cases, we use
CosmoMC [19] to sample the pos-teriors of these data sets. We impose the following priorsfor ΛCDM parameters when sampling: uniform priorsfor the cold dark matter density Ω c h = [0 . , . A s ) =[1 . , . θ MC = [0 . , σ )for the initial spectrum tilt n s : (0 . , .
02) and thebaryon density Ω b h : (0 . , . D/H measurements/BBN data) and wefix the optical depth to recombination τ to 0.055. Todraw the contour plots, we use GetDist [20]. IV. SPTPOL AND PLANCK LENSING VS.BAO+BBN
In this section, we start with identifying the key pa-rameter directions that contribute to the apparent ten-sion between SPTpol lensing and
Planck lensing whenboth data sets are combined with BAO+BBN using theupdate difference in mean statistic of Sec. II. Upon find-ing that the apparent tension is not between SPTpoland
Planck lensing, but rather between
Planck lensingand the parallel BAO measurements, we then focus onand quantify tension between those measurements.
A. Parameters
The BAO+BBN+lensing data sets depend on 5 out ofthe 6 ΛCDM parameters (they are not sensitive to τ ). Inorder to obtain more Gaussian covariance matrices, weperform our investigations in the parameter space that is native to the BAO and CMB lensing measurements.This way, it is also easier to interpret the influence ofeach. Specifically, we work in the parameter basis θ = (cid:104) θ ⊥ , θ (cid:107) , σ Ω / m , Ω b h , n s (cid:105) . (10)Here θ ⊥ = D M ( z BAO ) r fid r d ,θ (cid:107) = H ( z BAO ) r d r fid , (11)where D M ( z ) is the comoving angular diameter distanceto redshift z , H ( z ) is the expansion rate at this redshift, r d is the comoving BAO scale (the sound horizon at theend of the Compton drag epoch), r fid ≡ .
78 Mpc isthe fiducial r d , and σ is the root mean square of the lin-ear matter density fluctuations at the 8 h − Mpc scale.We choose z BAO = 0 .
61, one of the DR12 points, butother choices of z within the DR12 range do not quali-tatively affect our results.The first three parameters are the most relevant to thiswork and may be interpreted as a perpendicular BAO,parallel BAO, and CMB lensing amplitude parameter.Given that under ΛCDM H D M ( z ) and H ( z ) /H arefunctions of Ω m alone, the first two parameters span thesame space as do Ω m and H r d . Ω b h is constrainedmainly by the BBN data and n s is a nuisance parameterthat is constrained by the prior given in Sec. III.The BAO measurements do not depend on the lens-ing amplitude parameter σ Ω / m , while the shape of thelensing power spectrum within ΛCDM does depend onthe BAO parameters, in this case mainly supplying ex-tra information on θ (cid:107) . The difference between SPTpoland Planck lensing can be attributed to the fact thatthe former mainly constrains the amplitude of the lens-ing power spectrum whereas the latter constrains boththe amplitude and shape [8].We can see some of these properties in the posteriordistributions of these three parameters shown in Fig. 1.In this space, the tension between BAO+BBN+SPTpollensing and BAO+BBN+
Planck lensing is confined tothe parallel BAO parameter θ (cid:107) . From the BAO+BBNresult, we can see the shift in this parameter is alreadypresent without the SPTpol lensing data, albeit withlarger uncertainties. Another notable observation is thatthe lensing amplitude parameter constrained by SPT-pol lensing and Planck lensing appears to agree too wellwith each other. Finally, the perpendicular BAO param-eter θ ⊥ is mainly constrained by the BAO data them-selves and the addition of either lensing data set doesnot change its posterior appreciably.To tie the tension between BAO+BBN+SPTpol lens-ing and BAO+BBN+ Planck lensing seen in θ (cid:107) to thetension originally identified in H , we show the poste-riors of these two parameters in Fig. 2. Note that thedistributions and shifts in means follow each other dueto the high correlation between the two parameters. Wewill hereafter use tension in θ (cid:107) as a proxy for this tension .
55 0 .
60 0 . σ Ω / m θ k [ k m / s / M p c ] θ ⊥ [Mpc] . . . σ Ω / m
92 96 100 104 θ k [km / s / Mpc]
BAO+BBNBAO+BBN+SPTpol lensBAO+BBN+
P lanck lens
FIG. 1. The posterior distributions of the data setsBAO+BBN, BAO+BBN+SPTpol lensing, BAO+BBN+
Planck lensing in the parameter basis [ θ ⊥ , θ (cid:107) , σ Ω / m ] whichis native to the BAO and CMB lensing measurements.The posteriors for the combined BAO+BBN+ Planck lens-ing+SPTpol lensing data set are not shown for clarity asthey are qualitatively close to those from BAO+BBN+
Planck lensing. in H . In the following sections, we use the Q UDM anal-ysis to quantify these tensions and further isolate theirorigin in the various data sets.
B. BAO+BBN+SPTpol lensing vs
Planck lensing
We first analyze tension between BAO+BBN+SPTpollensing and its update BAO+BBN+SPTpol lensing+
Planck lensing through the Q UDM statistic. In this case,two directions satisfy the KL update criteria on eigenval-ues (Eq. 8). These two KL eigenmodes are a = 4 , σ Ω / m and theparallel BAO parameter θ (cid:107) respectively. We will referto them below as the amplitude mode and the parallelmode. With these two degrees of freedom, Q UDM = 4 . ∼ z BAO .Taken at face value, the PTE signals that the param-eters of BAO+BBN+SPTpol lensing and
Planck lensingare not particularly in tension. However, as can be seenin Fig. 1 and also [8], the best-fit σ Ω / m between SPT-pol lensing and Planck lensing are almost too consis-tent with each other despite being nearly uncorrelatedin their lensing information. It is therefore interesting
92 96 100 104 θ k [km / s / Mpc]
66 72 78 84 H θ k [ k m / s / M p c ] BAO+BBNBAO+BBN+SPTpol lensBAO+BBN+
P lanck lens
FIG. 2. The posterior distributions of the data setsBAO+BBN, BAO+BBN+SPTpol lensing, BAO+BBN+
Planck lensing for H and θ (cid:107) . The differences observed in H are highly correlated with the differences in θ (cid:107) , whichsuggests a common origin in the parallel BAO measurements. to examine the individual contributions to Q UDM of theamplitude and parallel modes. We find that the am-plitude mode contributes only 0 .
11 whereas the paral-lel mode contributes 3.9 to the total Q UDM . Consideredseparately, these correspond to a PTE of 74% and 4 . Q UDM from theamplitude mode is smaller than expected, the total sig-nificance downplays the tension in the parallel mode.This parallel mode reflects the tension originally identi-fied in H . Indeed if we computed the update differencein mean for only the marginal H distributions, we wouldobtain a 6.8% PTE or effectively a 1.8 σ tension.As discussed in the previous section, the BAO+BBNdata do not contribute to the σ Ω / m constraint and theSPTpol lensing measurements contribute little to the θ (cid:107) constraint. The Q UDM contributions imply that the σ Ω / m measurements from SPTpol lensing and Planck lensing are slightly too consistent, while the θ (cid:107) param-eter from the BAO data set and Planck lensing are inmild tension of ∼ σ . With this information, we now seethat the difference between SPTpol lensing and Planck lensing when both are combined with BAO+BBN, firstidentified in H , is actually driven by the mild tension in θ (cid:107) between Planck lensing and the BAO measurements.We next focus on quantifying this tension between
Planck lensing and BAO+BBN.
KL modes, a Ω b h n s σ Ω / m θ ⊥ θ k p a r a m e t e r s , α . . . . . . FIG. 3. Fractional Fisher information F aαα /F αα of theBAO+BBN+SPTpol lensing data set computed using theKL eigenmodes from updating it with Planck lensing. Thenumbers in each row add to one. The KL directions a = 4and a = 5 satisfy the KL update criteria (Eq. 8). They con-tribute the most information to the lensing amplitude param-eter σ Ω / m and the parallel BAO parameter θ (cid:107) respectively. KL modes, a Ω b h n s σ Ω / m θ ⊥ θ k p a r a m e t e r s , α . . . . . . FIG. 4. Fractional Fisher information F aαα /F αα of theBAO+BBN data set computed using the KL eigenmodesfrom updating it with Planck lensing. Only the KL direc-tion a = 4 satisfies the KL update criteria. It contributes themost information to the parallel BAO parameter θ (cid:107) . C. BAO+BBN vs
Planck lensing
In the previous section, we determined that the par-allel BAO parameter θ (cid:107) is the main indicator of tensionin the data sets we consider. Given that SPTpol lensinghas very little information on this parameter, we now fo-cus on the comparison between BAO+BBN and Planck lensing.In this case, we calculate Q UDM between BAO+BBN and its update BAO+BBN+
Planck lensing. From theKL decomposition, only one mode satisfies the KL up-date criteria on eigenvalues. For BAO+BBN, this modeagain dominates the information on the parallel BAOparameter θ (cid:107) , as shown by the fractional Fisher informa-tion of this “parallel” mode a = 4 in Fig. 4. In this case a = 5 dominates the information in the lensing ampli-tude parameter but its constraints come almost entirelyfrom Planck lensing and so do not satisfy Eq. (8).With the parallel mode, Q UDM = 3 .
37 for a single de-gree of freedom and hence the PTE is 6.6%. We obtainsimilar values for all DR12 values of z BAO . If we computethe update difference in means from the marginalizedconstraints on H alone, we obtain a PTE of 10.5%, un-derestimating the significance. To confirm that SPTpollensing does not contribute to θ (cid:107) beyond what Planck lensing does in this context, we calculate the Q UDM be-tween BAO+BBN and its update BAO+BBN+
Planck lensing+SPTpol lensing. The eigenmodes have simi-lar distributions as the BAO+BBN update with
Planck lensing case and the PTE is also 6.6%, concludingthat SPTpol lensing does not affect this result. Again,but now more explicitly, the Q UDM analysis shows thatthe parallel BAO parameter is in mild tension between
Planck lensing and BAO+BBN.For the BAO data we have used in this work, the SDSSDR12 BAO data set have separate parallel BAO andperpendicular BAO measurements. In the next section,we look into the effects on the BAO parameters fromthe parallel and the perpendicular BAO measurementsseparately.
D. BAO DR12 parallel vs
Planck lensing
To identify the origin of tension between BAO+BBNand
Planck lensing, we examine the posterior constraintson the BAO parameters θ ⊥ and θ (cid:107) from various subsetsof the BAO measurements themselves in Fig. 5. Sincethese individual constraints are themselves too weak tohave data-dominated Gaussian posteriors, we do not em-ploy Q UDM here.First, we see that the BAO constraints mainly comefrom the DR12 points with little information from otherBAO data (MGS, 6dF). Next, we see that parallel andperpendicular DR12 measurements map onto these pa-rameters without significant degeneracy, as expectedgiven the design of the parameters. However, the cor-respondence between the measurements and parametersis not entirely one-to-one. There is a small amount ofconstraining power of the perpendicular measurementson the parallel parameter and vice versa due to the com-bination of the three redshift points in each set. Re-call that we take z BAO = 0 .
61 in our fiducial parameterchoice, which is the highest of the three DR12 redshifts.We see in Fig. 5 that the 68% CL regions of the twoposteriors almost overlap, which suggests that the par-allel and perpendicular measurements might be in mildtension. However, the level of tension cannot be inferred θ ⊥ [Mpc] θ k [ k m / s / M p c ] k DR12 BAO ⊥ DR12 BAODR12 BAOall BAO
FIG. 5. θ (cid:107) vs θ ⊥ for various combinations of BAO data. from this observation directly because of the correlationbetween the parallel and perpendicular measurements.To properly account for the correlation, we apply the pa-rameter duplication technique [10] described in Sec. II,and find that the PTE associated with zero parameterdifference is 32%, indicating no significant tension.Comparing the posteriors of the perpendicular andparallel BAO measurements with that of Planck lens-ing in the θ (cid:107) − θ ⊥ plane, we find good overlap betweenperpendicular BAO and Planck lensing but that the 95%CL regions from parallel BAO barely overlaps with the
Planck lensing posterior, as shown in Fig. 6. However,we caution the reader that these two data sets have non-Gaussian posterior probabilities and intuition of tensionbased on Gaussian data-dominated posteriors may notapply. Specifically, due to the weak constraining powerof the parallel BAO data in the θ ⊥ direction, the shapeof the priors (assumed flat in θ MC , Ω c h ) over the con-strained range informs the shape of the posterior.To demonstrate that the prior is informative, we con-sider a goodness-of-fit statistic for the parallel BAO data.We compare χ (cid:107) BAO between the best-fit parameters tothe parallel BAO data (red star in Fig. 6) and a represen-tative test case (black star). The latter sits on the 95%CL line for the parallel BAO data, which for a two di-mensional Gaussian likelihood and flat priors in the BAOparameters would correspond to ∆ χ (cid:107) BAO = 6. However,actually evaluating the difference in χ (cid:107) BAO between thetwo cosmological models highlighted in Fig. 6 leads toonly ∆ χ (cid:107) BAO = 3 .
2. Because of the increase of the priorvolume at high θ ⊥ , which we shall see is associated withthe large range in Ω m that it encompasses, the yellowcontours in Fig. 6 are shifted to the right compared tothe position of the best-fit model.This shows that, in fact, the two models are not asdiscrepant as one would normally infer from a 95% CLexclusion in two dimensions. However, this mild dis- θ ⊥ [Mpc] θ k [ k m / s / M p c ] k DR12 BAO
P lanck lensing
FIG. 6. θ (cid:107) vs θ ⊥ for parallel BAO and Planck lensing. Thered star is the best-fit model to the parallel BAO measure-ments, the black star is a point we choose for illustration. Itlies on the 95% CL line of the parallel BAO posterior and68% CL line of the
Planck lensing posterior. We calculatethe ∆ χ between these two models using the parallel BAOlikelihood. crepancy does account for a large portion of the tensionbetween BAO+BBN and Planck lensing in the previ-ous section. This is in part because the
Planck lensingconstraint, and any tension with it, is effectively one-dimensional in the BAO parameters, where a 95% ex-clusion would correspond to ∆ χ = 3 . Planck lensing data set. On the top panel, we see thatthe test model does not deviate from the BAO+BBN+
Planck lensing best-fit very much. The parallel BAObest-fit model does not fit the perpendicular BAO mea-surements, with χ of 57 for 3 data points. On the lowerpanel, we show the parallel BAO measurements againstthe same sets of models. The measurements are very wellfit by the parallel BAO best-fit model. The test modelreflects the ∆ χ (cid:107) BAO = 3 . m and r d H .As already mentioned, these parameters have a one-to-one mapping with θ (cid:107) and θ ⊥ . From the figure, it isclear that the parallel BAOs pull in the direction of verylarge Ω m . The best-fit parallel BAO model (red star)has Ω m = 0 .
64 and r d H = 84 . / s. With the BBNprior, the best-fit also has a high H = 74 km / s / Mpc, . . . . θ ⊥ / θ ⊥ , r e f k BAO best fitTest modelSDSS DR12 consensus . . . . . . . z . . . . . θ k / θ k , r e f FIG. 7. SDSS DR12 BAO consensus measurements plottedagainst the θ ⊥ ( z ) and θ (cid:107) ( z ) model predictions from the par-allel BAO best-fit and the test model, the red and black starsin Fig. 6 respectively. θ ⊥ and θ (cid:107) are normalized by θ ref , thebest-fit model to the BAO+BBN+ Planck lensing data set. but as we shall show in the next section, there are manyother data sets that would exclude the high Ω m required. V. COMPARISON WITH OTHER DATA SETS
To put things in context, in Fig. 9 we compare con-straints on Ω m and r d H from the parallel BAO and Planck lensing measurements with those from other cos-mological data sets. We plot constraints on Ω m from thePantheon supernova sample [18] assuming flat ΛCDM(dashed lines). While these supernova constraints are ingood agreement with Planck lensing and the combinedBAO constraints, they are in mild tension with the par-allel BAOs. We then show constraints from the
Planck primary CMB power spectra measurements [17] (redcontours), which are compatible with and even strongerthan supernovae.The best-fit Ω m value of the parallel BAO+BBN dataset (red star) is therefore strongly ruled out by both thesupernova sample and the Planck primary CMB mea-surements. This disallowed preference for high Ω m inthe parallel BAO data set is the ultimate origin of thehigh H preferred by the BAO+BBN+SPTpol lensingdata set compared with BAO+BBN+ Planck lensing. . . . . . . . Ω m r d H [ k m / s ] k DR12 BAO
P lanck lensing
FIG. 8. Constraints as in Fig. 6 but here in terms of Ω m and r d H . Note the long degeneracy for the parallel BAOdata out to high Ω m and the high value of Ω m = 0 .
64 for theparallel BAO best fit (red star). .
24 0 .
32 0 .
40 0 .
48 0 .
56 0 . Ω m r d H [ k m / s ] k DR12 BAO
P lanck lensingSN
P lanck
TTTEEE
FIG. 9. Constraints on Ω m and r d H from parallel BAOscompared with those from Planck lensing, the Pantheon su-pernovae sample, and
Planck
CMB power spectra. Note thatthe posteriors extend beyond the part of the parameter planeshown here. All of the other data sets strongly disfavor theparallel BAO best fit (red star) due to its high value of Ω m . VI. CONCLUSION
In this work, we apply the update difference-in-mean statistic Q UDM to quantify tension between twocomposite data sets, BAO+BBN+SPTpol lensing andBAO+BBN+
Planck lensing, and track the origin of thedifferences in H to the individual data sets that are pri-marily responsible. We work in a parameter basis thatis native to the BAO and the CMB lensing measure-ments, replacing the cosmological parameters Ω m , H and A s with θ (cid:107) , θ ⊥ , and σ Ω / m , where the parameterposteriors are nearly Gaussian and their constraints arerelatively easy to map back to the measurements. Withthis setup, we isolate the parameter direction that dom-inates the tension and isolate its origin in the Planck lensing vs. parallel BAO measurements.We arrive at this conclusion through a process of nar-rowing down parameter combinations that matter andremoving data sets that contribute little to the ten-sion. In calculating the update difference in mean ofBAO+BBN+SPTpol lensing updated by
Planck lensing,the parameter direction that dominates the tension is θ (cid:107) with a PTE of 4.7%. This direction is highly cor-related with H , which carries a comparable tension.Knowing that SPTpol lensing contributes little to θ (cid:107) constraints, we next check the Q UDM of BAO+BBN up-dated by
Planck lensing. This test confirms the tensionbetween BAO+BBN and
Planck lensing along the θ (cid:107) di-rection at a PTE of 6.6%.Both update difference in mean statistics point to theparallel BAO parameter as the source of the tension. Wethus divide the BAO measurements into subsets to fur-ther our investigation. While the perpendicular BAOmeasurements are largely compatible with both the par-allel BAO measurements and Planck lensing, there istension between the parallel BAO and
Planck lensingmeasurements around the 95% CL. This exclusion is ex-acerbated by our chosen prior, which in particular allowsa large range in Ω m . Independent of this prior, the ∆ χ between the best-fit model to the parallel BAO data anda representative model that is consistent with Planck lensing is ∆ χ = 3 . Planck lensing and the BAO data set is indeedfrom the parallel BAO measurements. Finally, we tracethe origin of this ∆ χ to a slope in the parallel BAOmeasurements as a function of redshift, which drives itspreference for high Ω m values. In combination with con-straints from BBN, this translates into a preference forhigh H values.We note that the Ω m preferred by the parallel BAOdata under ΛCDM is highly excluded by other data sets,including supernova measurements and Planck primaryCMB measurements. These other measurements tend to have much stronger constraining power on Ω m than theparallel BAO data. For this reason, the mild tension ofthe parallel BAO data with the other data sets is hid-den when analyzed in combination. While this tensionclearly cannot be resolved within ΛCDM, it is useful tobear this in mind when considering alternatives.Finally, we reiterate the importance of not selectingcosmological parameters a posteriori when adjudicatingtension between data sets. Had we calculated the differ-ence in mean of BAO+BBN+SPTpol lensing updatedwith Planck lensing on H alone, the PTE would belower than letting the algorithm reveal that there aretwo relevant parameter combinations. Conversely, hadwe calculated the difference in mean of BAO+BBN up-dated with Planck lensing on H alone, the PTE wouldbe higher than letting the algorithm choose the singlerelevant parameter direction. With parameters selected a posteriori , a trials factor is required to accompanythe resultant PTE for fair interpretation of the statisticand that selection may still not reflect the true sourceof tension. Looking forward, as upcoming surveys pro-vide more precise measurements of our universe, it is ofutmost importance that we identify the origin and sig-nificance of tension accurately to aid the differentiationof the underlying causes of the observed tension—be itunmodeled systematics or new physics. ACKNOWLEDGMENTS
We thank Georgios Zacharegkas for useful discus-sions. WLKW is supported in part by the Kavli In-stitute for Cosmological Physics at the University ofChicago through grant NSF PHY-1125897 and an en-dowment from the Kavli Foundation and its founder FredKavli. WH is supported by by U.S. Dept. of Energycontract DE-FG02-13ER41958 and the Simons Founda-tion. MR is supported in part by NASA ATP Grant No.NNH17ZDA001N, and by funds provided by the Centerfor Particle Cosmology. This work was completed in partwith resources provided by the University of Chicago Re-search Computing Center. [1] N. Aghanim et al. (Planck), (2018), arXiv:1807.06209[astro-ph.CO].[2] A. G. Riess, S. Casertano, W. Yuan, L. M.Macri, and D. Scolnic, Astrophys. J. , 85 (2019),arXiv:1903.07603 [astro-ph.CO].[3] W. L. Freedman et al. , Astrophys. J. , 34 (2019),arXiv:1907.05922 [astro-ph.CO].[4] W. L. Freedman, B. F. Madore, T. Hoyt, I. S. Jang,R. Beaton, M. G. Lee, A. Monson, J. Neeley, andJ. Rich, Astrophys. J. , 57 (2020), arXiv:2002.01550[astro-ph.GA].[5] K. C. Wong et al. , (2019), arXiv:1907.04869 [astro-ph.CO]. [6] W. Yuan, A. G. Riess, L. M. Macri, S. Caser-tano, and D. Scolnic, Astrophys. J. , 61 (2019),arXiv:1908.00993 [astro-ph.GA].[7] D. Pesce et al. , Astrophys. J. , L1 (2020),arXiv:2001.09213 [astro-ph.CO].[8] F. Bianchini et al. (SPT), Astrophys. J. , 119 (2020),arXiv:1910.07157 [astro-ph.CO].[9] M. Raveri and W. Hu, Phys. Rev.
D99 , 043506 (2019),arXiv:1806.04649 [astro-ph.CO].[10] M. Raveri, G. Zacharegkas, and W. Hu, (2019),arXiv:1912.04880 [astro-ph.CO].[11] N. Aghanim et al. (Planck), (2018), arXiv:1807.06210[astro-ph.CO]. [12] W. L. K. Wu et al. , Astrophys. J. , 70 (2019),arXiv:1905.05777 [astro-ph.CO].[13] S. Alam et al. (BOSS), Mon. Not. Roy. Astron. Soc. ,2617 (2017), arXiv:1607.03155 [astro-ph.CO].[14] A. J. Ross, L. Samushia, C. Howlett, W. J. Percival,A. Burden, and M. Manera, Mon. Not. Roy. Astron.Soc. , 835 (2015), arXiv:1409.3242 [astro-ph.CO].[15] F. Beutler, C. Blake, M. Colless, D. H. Jones,L. Staveley-Smith, L. Campbell, Q. Parker, W. Saun-ders, and F. Watson, Mon. Not. Roy. Astron. Soc. , 3017 (2011), arXiv:1106.3366 [astro-ph.CO].[16] R. J. Cooke, M. Pettini, and C. C. Steidel, Astrophys.J. , 102 (2018), arXiv:1710.11129 [astro-ph.CO].[17] N. Aghanim et al. (Planck), (2019), arXiv:1907.12875[astro-ph.CO].[18] D. M. Scolnic et al. , Astrophys. J. , 101 (2018),arXiv:1710.00845 [astro-ph.CO].[19] A. Lewis and S. Bridle, Phys. Rev. D66