Quantifying discordance in the 2015 Planck CMB spectrum
G. E. Addison, Y. Huang, D. J. Watts, C. L. Bennett, M. Halpern, G. Hinshaw, J. L. Weiland
SSubmitted to the Astrophysical Journal, 30 October 2015
Preprint typeset using L A TEX style emulateapj v. 01/23/15
QUANTIFYING DISCORDANCE IN THE 2015
PLANCK
CMB SPECTRUM
G. E. Addison , Y. Huang , D. J. Watts , C. L. Bennett , M. Halpern , G. Hinshaw , and J. L. Weiland Submitted to the Astrophysical Journal, 30 October 2015
ABSTRACTWe examine the internal consistency of the
Planck (cid:96) <
Wilkinson Microwave Anisotropy Probe ), and from (cid:96) ≥ c h ,which is discrepant at 2 . σ for a Planck -motivated prior on the optical depth, τ = 0 . ± .
02. Wefind some parameter tensions to be larger than previously reported because of inaccuracy in the codeused by the
Planck
Collaboration to generate model spectra. The
Planck (cid:96) ≥ Planck ’s own measurement of the CMB lensingpower spectrum (2 . σ ), and the most precise baryon acoustic oscillation (BAO) scale determination(2 . σ ). The Hubble constant predicted by Planck from (cid:96) ≥ H = 64 . ± . − Mpc − ,disagrees with the most precise local distance ladder measurement of 73 . ± . − Mpc − atthe 3 . σ level, while the Planck value from (cid:96) < . ± . − Mpc − , is consistent within1 σ . A discrepancy between the Planck and South Pole Telescope (SPT) high-multipole CMB spectradisfavors interpreting these tensions as evidence for new physics. We conclude that the parametersfrom the
Planck high-multipole spectrum probably differ from the underlying values due to either anunlikely statistical fluctuation or unaccounted-for systematics persisting in the
Planck data.
Keywords: cosmic background radiation – cosmological parameters – cosmology: observations INTRODUCTIONMeasurements of the power spectrum of cosmic mi-crowave background (CMB) temperature fluctuations(hereafter ‘TT spectrum’) are a cornerstone of mod-ern cosmology. The most precise constraints are cur-rently provided by the final 9-year
Wilkinson MicrowaveAnisotropy Probe ( WMAP ) analysis (Bennett et al.2013; Hinshaw et al. 2013), high-resolution ground-basedinstruments including the Atacama Cosmology Telescope(ACT; Sievers et al. 2013) and the South Pole Tele-scope (SPT; Story et al. 2013), and most recently
Planck (Planck Collaboration XIII 2015). Significant improve-ments in both CMB polarization and low-redshift, late-time observations are anticipated in the near future andwill be used to measure or tightly constrain key cosmo-logical quantities including the total neutrino mass, de-viations of dark energy from a cosmological constant andthe amplitude of primordial gravitational waves (e.g.,Abazajian et al. 2013a,b; Kim et al. 2013). Many ofthese future results will rely on having precise and accu-rate TT constraints. Assessing consistency both betweenand internally within each TT measurement is thereforeextremely important.While the
Planck data from the first data releasein 2013 (Planck Collaboration XVI 2014) were qualita-tively in agreement with
WMAP , supporting the mini-mal ΛCDM model, there were small but highly significantquantitative differences between the cosmological param-eters inferred. For example, Larson et al. (2015) found a [email protected] Dept. of Physics & Astronomy, The John Hopkins Univer-sity, 3400 N. Charles St., Baltimore, MD 21218-2686 Department of Physics and Astronomy, University of BritishColumbia, 6224 Agricultural Road, Vancouver, BC V6T 1Z1,Canada ∼ σ overall parameter discrepancy after accounting forthe cosmic variance common to both experiments.Several systematic effects were corrected in the Planck . σ in units of the 2015 uncertainty (Ta-ble 1 of Planck Collaboration XIII 2015), and an artifactwith a statistical significance of 2 . − . σ near multipole (cid:96) (cid:39) WMAP and
Planck
Planck parameters pull away from
WMAP (Section4.1.6 of Planck Collaboration XI 2015), leading to ten-sion between
Planck and several low-redshift cosmologi-cal measurements if ΛCDM is assumed, including a 2 . σ tension with the Riess et al. (2011) determination of theHubble constant, H , 2 − σ tension with weak lensingmeasurements of the CFHTLens survey (Heymans et al.2012), and tension with the abundance of massive galaxyclusters (e.g., Planck Collaboration XXIV 2015).In this paper we examine the internal consistency ofthe Planck
TT spectrum. We show that tension ex-ists between ΛCDM parameters inferred from the
Planck
TT spectrum at the multipoles accessible to
WMAP ( (cid:96) (cid:46) (cid:96) (cid:38) Planck ’s own lensing potential powerspectrum measurement and baryon acoustic oscillation a r X i v : . [ a s t r o - ph . C O ] F e b G. E. Addison et al. (BAO) from galaxy surveys, while the low-multipole
Planck
TT,
Planck lensing,
WMAP , BAO, and distanceladder H data are all in reasonable agreement.We describe the data sets used and parameter fittingmethodology in Section 2 and present results in Section 3.Discussions and conclusions follow in Sections 4 and 5. DATA AND PARAMETER FITTINGWe use
CAMB (Lewis et al. 2000) to calculate tem-perature and lensing potential power spectra as a func-tion of cosmological parameters and CosmoMC (Lewis& Bridle 2002) to perform Monte-Carlo Markov Chain(MCMC) parameter fitting and obtain marginalized pa-rameter distributions, adopting the default Planck set-tings, including a neutrino mass of 0.06 eV (Planck Col-laboration XVI 2014). We use the public temperature-only
Planck lowl likelihood for 2 ≤ (cid:96) ≤
29, thebinned plik likelihood for 30 ≤ (cid:96) ≤ Planck C φφL covering 40 ≤ L ≤
400 (Planck Collaboration XI 2015;Planck Collaboration XV 2015). We fit for six ΛCDM pa-rameters: the physical baryon and CDM densities, Ω b h and Ω c h , the angular acoustic scale, parametrized by θ MC , the optical depth, τ , the primordial scalar fluctu-ation amplitude, A s , and the scalar spectral index, n s .Other parameters, including H , the total matter den-sity, Ω m , and the present-day mass fluctuation ampli-tude, σ , are derived from these six. Additional fore-ground and calibration parameters used in the fits aredescribed by Planck Collaboration XI (2015).At the present time the analysis of Planck ’s polariza-tion data is only partially complete. At high multipoles,significant systematic errors remain in the TE and EEspectra, putatively due to beam mismatch, which leadsto temperature-polarization leakage (Sec. 3.3.2 of PlanckCollaboration XIII 2015). At low multipoles ( (cid:96) < . The LFI 70 GHzdata, in conjunction with the 30 and 353 GHz mapsas Galactic foreground tracers, are used to constrain τ .Using the polarized 353 GHz map as a dust tracer re-sults in a value of τ lower than constraints from WMAP (0 . ± .
016 compared to 0 . ± . τ is only weakly con-strained, but it does couple to other cosmological param-eters. We considered two approaches for setting priors on τ . First we adopted a Gaussian prior of τ = 0 . ± . σ with the range of values inferred from WMAP and
Planck data (Hinshaw et al. 2013; Planck Collab-oration XIII 2015). Second, to gain more insight intoexactly how τ does or does not affect our conclusions camb.info http://cosmologist.info/cosmomc/ According to the
Planck http://wiki.cosmos.esa.int/planckpla2015/index.php/Frequency_Maps about TT consistency, we also ran chains with τ fixed tospecific values: 0.06, 0.07, 0.08, and 0.09.When assessing consistency between parameter con-straints from two data sets that can be considered inde-pendent we use the difference of mean parameter values,which we treat as multivariate Gaussian with zero meanand covariance given by the sum of the covariance ma-trices from the individual data sets. The mean and co-variance for each data set are estimated from the MCMCchains. We then quote equivalent Gaussian ‘sigma’ levelsfor the significance of the parameter differences.We also considered using the difference of best-fit pa-rameters, rather than difference of means, for these com-parisons. For Gaussian posterior distributions this choiceshould make little difference. We find that this is gen-erally true, with significance levels for parameter differ-ences changing only at the 0 . − . σ level. In a fewcases, however, we found a significant shift, due to anoffset between the mean and best-fit parameters. In allcases the Gaussian distribution specified by the mean andcovariance matrix from the chains provided an excellentmatch to the distribution of the actual MCMC samples,and for this reason we quote results based on the differ-ences of the mean rather than best-fit parameters. It ispossible that the mismatches are caused by problems inthe algorithm used to determine the best-fit parameters .Note that simply taking the maximum-likelihood param-eters directly from the MCMC chains is unreliable dueto the large parameter volume sampled (typically around20 parameters, including nuisance parameters, e.g., forforegrounds). The overall posterior distribution is wellmapped out by a converged chain but the tiny region ofparameter space close to the likelihood peak is not. RESULTSFigure 1 shows the two-dimensional ΛCDM parame-ter constraints for the
Planck ≤ (cid:96) < ≤ (cid:96) ≤ τ = 0 . ± . τ . Twodifferences in our fit act to pull some of the low and highmultipole parameter constraints away from one another.Firstly, the constraints in the Planck figure only extenddown to (cid:96) = 30 because the intention was to test ro-bustness of the plik likelihood only. We use the fullrange 2 ≤ (cid:96) < Planck fit uses the
PICO (Fendt & Wandelt 2007) code rather than CAMB to gen-erate TT spectra. We find that the
PICO and
CAMB results are noticeably different for the 1000 ≤ (cid:96) ≤ PICO requires only a fraction of the computationtime and provides a good approximation to
CAMB , butonly within a limited volume of parameter space. Someparameter combinations outside this volume are allowedby the 1000 ≤ (cid:96) ≤ PICO output deviates from the
CAMB spectrum and a poor like-lihood is returned, leading to artificial truncation of thecontours, particularly for Ω b h and n s .From Figure 1 it is clear that some tension existsbetween parameters inferred from the (cid:96) < (cid:96) ≥ Planck
TT spectra. Assuming the two sets of See http://cosmologist.info/cosmomc/readme.html https://pypi.python.org/pypi/pypico Planck
Discordance .
88 0 .
96 1 .
04 1 . n s . . . . Ω c h . . . θ M C . . . . τ . . . . A s e − τ . . . Ω b h . . . . n s .
11 0 .
12 0 .
13 0 . Ω c h .
039 1 .
042 1 . θ MC .
00 0 .
04 0 .
08 0 . τ . . . . A s e − τ Planck ≤ ℓ < Planck ≤ ℓ ≤ Planck ≤ ℓ ≤ Figure 1.
Contours enclosing 68.3% and 95.5% of MCMC sample points from fits to the
Planck
TT spectrum. Results are shown for2 ≤ (cid:96) < WMAP , and higher multipoles, 1000 ≤ (cid:96) ≤ c h differs by 2 . σ . Results are also shown for the 1000 ≤ (cid:96) ≤ PICO code isused to estimate the theoretical TT spectra instead of the more accurate
CAMB . Using
PICO leads to an artificial truncation of the contoursand diminishes the discrepancy between the high and low multipole fits for some parameters. We adopt a Gaussian prior of τ = 0 . ± . constraints are independent, the values of Ω c h differ by2 . σ . Independence is a valid assumption because eventhe bins on either side of the (cid:96) = 1000 split point are onlycorrelated at the 4% level and the degree of correlationfalls off with increasing bin separation. Taken togetherthe five free ΛCDM parameters differ by 1 . σ , howeverit should be noted that Ω c h plays a far more signifi-cant role in comparisons with low-redshift cosmologicalconstraints (Section 3.3) than, for example, θ MC . For fixed τ we find differences in Ω c h of 3 .
0, 2 .
7, 2 . . σ for τ values of 0.06, 0.07, 0.08 and 0.09, respec-tively. Constraints on each parameter for these casesare shown in Figure 2. Apart from the expected strongcorrelation with A s (the TT power spectrum amplitudescales as A s e − τ ) there is relatively little variation with τ .Note that while increasing τ reduces the tension in Ω c h ,higher values of τ are mildly disfavored by Planck ’s ownpolarization analysis (Planck Collaboration XIII 2015).
G. E. Addison et al. . . Ω b h . . . Ω c h . . . θ M C +1 . . . l og ( A s ) . . . n s H .
06 0 .
07 0 .
08 0 . τ . . Ω m .
06 0 .
07 0 .
08 0 . τ . . σ .
06 0 .
07 0 .
08 0 . τ . . . . . A s e − τ Planck TT 2015 ≤ ℓ < Planck TT 2015 ≤ ℓ ≤ Figure 2.
Marginalized 68.3% confidence ΛCDM parameter constraints from fits to the (cid:96) < (cid:96) ≥ Planck
TT spectra. Herewe replace the prior on τ with fixed values of 0.06, 0.07, 0.08, and 0.09, to more clearly assess the effect τ has on other parameters in thesefits. Aside from the strong correlation with A s , which arises because the TT spectrum amplitude scales as A s e − τ , dependence on τ isfairly weak. Tension at the > σ level is apparent in Ω c h and derived parameters, including H , Ω m , and σ . We investigated the effect of fixing the foreground pa-rameters to the best-fit values inferred from the fit to thewhole
Planck multipole range rather than allowing themto vary separately in the (cid:96) < (cid:96) ≥ c h decreas-ing to 2 . σ for τ = 0 . ± .
02, for example. The best-fit χ is, however, worse by 3.1 and 4.8 for the (cid:96) < (cid:96) ≥ (cid:96) < (cid:96) ≥ Comparing temperature and lensing spectra
Planck Collaboration XIII (2015) found that allowinga non-physical enhancement of the lensing effect in theTT power spectrum, parametrized by the amplitude pa-rameter A L (Calabrese et al. 2008), was effective at re-lieving the tension between the low and high multipole Planck
TT constraints. For the range of scales coveredby
Planck , the main effect of increasing A L is to slightlysmooth out the acoustic peaks. If ΛCDM parametersare fixed, a 20% change in A L suppresses the fourthand higher peaks by around 0.5% and raises troughs byaround 1%, for example.In Figure 3 we show the effect of fixing A L to valuesother than the physical value of unity on the (cid:96) < (cid:96) ≥ τ = 0 . ± . A L > (cid:96) ≥ (cid:96) < c h andhigher values of H . Planck Collaboration XIII (2015)found A L = 1 . ± .
10 for plik combined with the low- (cid:96)
Planck joint temperature and polarization likelihood,although note that this fit was performed using
PICO rather than
CAMB , which uses a somewhat different A L definition.Lensing also induces specific non-Gaussian signaturesin CMB maps that can be used to recover the lens-ing potential power spectrum (hereafter ‘ φφ spectrum’).Planck Collaboration XV (2015) report a measurementof the φφ spectrum using temperature and polarizationdata with a combined significance of ∼ σ . The φφ spectrum constrains σ Ω . m = 0 . ± . b h = 0 . ± . n s = 0 . ± .
02, and0 . < H /
100 km s − Mpc − < . Planck
TT data using a τ = 0 . ± .
02 prior: σ Ω . m = 0 . ± .
021 (
Planck φφ ),= 0 . ± .
019 (
Planck (cid:96) < . ± .
020 (
Planck (cid:96) ≥ . (1)The (cid:96) < (cid:96) ≥ . σ ,consistent with the difference in Ω c h discussed above.The (cid:96) ≥ φφ values are in tension at the 2 . σ level (for fixed values of τ in the range 0 . − .
09 wefind a 2 . − . σ difference). The (cid:96) < φφ values are consistent within 0 . σ .It is worth noting that while allowing A L > (cid:96) and high- (cid:96) TT results,it does not alleviate the high- (cid:96)
TT tension with φφ . For A L = 1 . σ Ω . m =0 . ± .
019 from (cid:96) ≤ φφ spectrumrequires σ Ω . m = 0 . ± . φφ power roughly scales as A L ( σ Ω . m ) , so, for fixed φφ , increasing A L by 20% requires a ∼
10% decrease in
Planck
Discordance . . . . . Ω b h . . . Ω c h . . θ M C +1 . . . . . τ . . l og ( A s ) . . . n s H . . . . . Lensing Amplitude A L . . . . . Ω m . . . . . Lensing Amplitude A L . . . . σ . . . . . Lensing Amplitude A L . . . . . A s e − τ Planck TT 2015 ≤ ℓ < Planck TT 2015 ≤ ℓ ≤ Figure 3.
Marginalized 68.3% parameter constraints from fits to the (cid:96) < (cid:96) ≥ Planck
TT spectra with different values ofthe phenomenological lensing amplitude parameter, A L , which has a physical value of unity (dashed line). Increasing A L smooths out thehigh order acoustic peaks, which improves agreement between the two multipole ranges. Note that a high value of A L is not favored bythe direct measurement of the φφ lensing potential power spectrum (see text). . . . . . . Lensing Amplitude A L . . . . . σ Ω . m Planck
TT 2015 2 ≤ ‘ < Planck
TT 2015 1000 ≤ ‘ ≤ Planck
Figure 4.
Constraints on σ Ω . m from fits to the (cid:96) < (cid:96) ≥ φφ lensing spec-trum. Results are shown as a function of the phenomenologicallensing amplitude parameter A L . The φφ measurement constrainsthe product A L ( σ Ω . m ) . A similar trend is apparent in the (cid:96) ≥ (cid:96) < A L . The (cid:96) < φφ constraints agree well forthe physical value of A L = 1 (dashed line). Increasing A L helpsreconcile the low- (cid:96) and high- (cid:96) constraints but does not improveagreement between the high- (cid:96) and φφ constraints. σ Ω . m . As shown in Figure 4, there is no value of A L that produces agreement between these data. The φφ spectrum featured prominently in the Planck claim that the true value of τ is lower than the valueinferred by WMAP (Planck Collaboration XIII 2015).While a full investigation into τ is deferred to future workwe note here that the effect of the φφ spectrum on τ is completely dependent on the choice of temperatureand polarization data. The shift to lower τ in the joint Planck φφ fit is partly a reflection of the tensiondiscussed above. Adding the Planck φφ spectrum to the WMAP τ at all, reflecting the fact that the φφ spectrum and WMAP temperature and polarization data (with τ =0 . ± . WMAP alone, the mean values shiftby < . σ . 3.2. Comparison With SPT
Planck Collaboration XVI (2014) reported moderateto strong tension between cosmological parameters fromthe SPT TT spectrum, measured over 2500 square de-grees and covering 650 ≤ (cid:96) ≤ Planck
TT spectrum. Planck CollaborationXIII (2015) comment that this tension has worsenedfor the
Planck
Planck
G. E. Addison et al. .
021 0 .
022 0 .
023 0 . Ω b h .
104 0 .
112 0 .
120 0 . Ω c h .
035 1 .
040 1 . θ MC .
03 0 .
06 0 .
09 0 . τ .
00 3 .
06 3 .
12 3 . log(10 A s ) .
950 0 .
975 1 .
000 1 . n s Planck
WMAP WMAP
Planck
Figure 5.
Marginalized ΛCDM parameter constraints compar-ing results from
Planck
WMAP
WMAP
Planck φφ lensing power spectrum. Adding the φφ spectrumto Planck temperature and polarization data results in a down-ward shift in τ , which reflects internal tension between the high-multipole Planck
TT spectrum and φφ (see text). The WMAP
Planck φφ constraints are in very good agreement. Adding φφ to WMAP leads to a negligible shift in τ and shifts of < . σ inother parameters. from 650 ≤ (cid:96) ≤ WMAP within0 . σ . For the 143 GHz Planck spectrum, most directlycomparable to the 150 GHz SPT channel, the agreementis better than 0 . σ . The disagreement between SPT and Planck therefore cannot be resolved by simply calibrat-ing SPT to
Planck rather than
WMAP in this manner.We note that the high-multipole ACT TT measurementsare consistent with
WMAP and SPT, as well as
Planck
Comparison With BAO and Local H Measurements
Figure 6 shows a comparison of CMB ΛCDM con-straints with the 1% BAO scale measurement fromthe Baryon Oscillation Spectroscopic Survey (BOSS)‘CMASS’ galaxy sample at an effective z = 0 .
57 (An- derson et al. 2014) and the most precise local distanceladder constraint on the Hubble constant, H = 73 . ± . − Mpc − (Riess et al. 2011; Bennett et al. 2014).The BAO scale is parametrized as the ratio of the com-bined radial and transverse dilation scale, D V (Eisensteinet al. 2005), to the sound horizon at the drag epoch, r d ,which has a fiducial value r d ,fid = 149 .
28 Mpc (Andersonet al. 2014).The BOSS BAO D V /r d constraint is at the higher endof the range preferred by WMAP and
Planck (cid:96) < σ . The Planck (cid:96) ≥ D V /r d , and lower values of H ,than the BOSS BAO and distance ladder measurementsat the 2 . σ and 3 . σ level, respectively, for τ = 0 . ± .
02. The difference between the
Planck high-multipoleconstraint and the Riess et al. H constraint is extremelyunlikely to be explained by statistical fluctuation alone.The SPT-only values provided by Story et al. (2013) arealso shown. The SPT predictions for D V /r d and H arediscrepant with those from Planck (cid:96) ≥ . σ and 2 . σ levels. Note that SPT used a WMAP -based τ prior but that τ couples very weakly to the inferred BAOscale.The consistency between the Planck and BAO con-straints has been repeatedly highlighted (Planck Collab-oration XVI 2014; Planck Collaboration XIII 2015). Wefind that this agreement arises more in spite of than be-cause of the high-multipole TT spectrum that
WMAP did not measure. Figure 7 shows constraints in theΩ m − H plane for the BAO constraint from combiningBOSS CMASS with the BOSS ‘LOWZ’ sample (Ander-son et al. 2014), Sloan Digital Sky Survey Main GalaxySample (Ross et al. 2015, SDSS MGS), and Six-degree-Field Galaxy Survey (Beutler et al. 2011, 6dFGS) mea-surements. This is the same combination utilized in the Planck b h = 0 . > σ tension between the Planck (cid:96) ≥ WMAP
9, ACT, SPT,BAO, and distance ladder measurements and found thatthese measurements are consistent and together con-strain H = 69 . ± . − Mpc − . This concordancevalue differs from the Planck (cid:96) ≥ . ± . − Mpc − at 3 . σ but agrees well with the Planck (cid:96) < . ± . − Mpc − .3.4. Choice of multipole split
The choice of (cid:96) = 1000 as the split point for param-eter comparisons matches the tests described by PlanckCollaboration XI (2015) and roughly corresponds to themaximum multipoles accessible to
WMAP , but the exactchoice is arbitrary. To test the robustness of our findingswe also considered the effect of splitting the
Planck
TTspectrum at (cid:96) = 800. This choice achieves an almost-even division of the
Planck
TT spectrum constrainingpower as assessed by the determinants of the ΛCDM pa-rameter covariance matrices from fits to 2 ≤ (cid:96) ≤
799 and http://pole.uchicago.edu/public/data/story12/chains/ Planck
Discordance
60 64 68 72 76 80 H [km s − Mpc − ] D V ( z = . ) r d , fid / r d [ M p c ] BAO+ H WMAP
Planck
TT 2015 2 ≤ ‘ < Planck
TT 2015 1000 ≤ ‘ ≤ Figure 6.
BOSS BAO scale and local distance ladder H mea-surements (Riess et al. 2011; Anderson et al. 2014; Bennett et al.2014) with ΛCDM CMB 68.3 and 94.5% confidence contours over-plotted. The Planck (cid:96) ≥ . σ and 3 . σ levels,respectively, while the WMAP
Planck (cid:96) < σ . Constraints from SPT (cov-ering 650 ≤ (cid:96) ≤ Planck and SPT currentlyprovide the most precise measurements of the CMB damping tailand their predictions for the z = 0 .
57 BAO scale and H differ atthe 2 . σ and 2 . σ level.
56 60 64 68 72 76 800 . . . . . . . .
45 BAO
WMAP
Planck
TT 2015 2 ≤ ‘ < Planck
TT 2015 1000 ≤ ‘ ≤ H [km s − Mpc − ] Ω m Figure 7.
Comparison of CMB, BAO, and distance ladder con-straints in the Ω m − H plane. We show here the BAO constraintsfrom combining the BOSS CMASS, BOSS LOWZ, SDSS MGS,and 6dFGRS measurements, assuming Ω b h = 0 . Planck (cid:96) ≥ ≤ (cid:96) ≤ ≤ (cid:96) < Planck fit has a sig-nificant effect on several parameters, including n s andΩ b h , tightening constraints on these parameters by fac-tors of four and two, respectively. Conversely, the un-certainty on θ MC is increased by 50% for (cid:96) ≤
800 com-pared to (cid:96) ≤ (cid:96) = 1000 remain for asplit at (cid:96) = 800, with the 2 . σ tension in Ω c h for the (cid:96) = 1000 split shifting to 2 . σ for the (cid:96) = 800 case (as-suming a τ = 0 . ± .
02 prior). From (cid:96) ≥
800 we find σ Ω . m = 0 . ± . Planck φφ constraint in equation (1) by 2 . σ , the same differ-ence as for (cid:96) ≥ (cid:96) = 1000 is not driving our results. DISCUSSIONWe have found multiple similar tensions at the > σ level between the Planck
Planck spectrum. A combination of these factors isalso possible.If the tensions were largely due to an unlikely statisti-cal fluctuation, our results suggest that it is parametersfrom the high-multipole
Planck
TT spectrum that havescattered unusually far from the underlying values, onthe basis that the low-multipole
Planck
TT,
WMAP , Planck φφ , BAO and distance ladder H measurementsare all in reasonable agreement with one another (see alsoBennett et al. 2014). One might argue that the (cid:96) < WMAP and
Planck constraints are pulled away fromthe true values by the multipoles at (cid:96) <
30. However,all parameter constraints we have quoted include cosmicvariance uncertainty and thus account for this possibility(assuming Gaussian fluctuations). Furthermore, an un-usual statistical fluctuation in the (cid:96) <
Planck (cid:96) ≥ Planck φφ , BAO, and the distanceladder measurements.Cosmology beyond standard ΛCDM cannot be ruledout as the dominant cause of tension. We do not favorthis explanation because, firstly, none of the physicallymotivated modifications investigated by Planck Collab-oration XIII (2015) were found to be significantly pre-ferred in fits to the full Planck
TT spectrum, and, sec-ondly, the most precise measurements of the CMB damp-ing tail, from
Planck and SPT, disagree, as discussed inSections 3.2 and 3.3.From 2013 to 2015 the
Planck results were revised dueto several significant systematic effects. Without moredetailed reanalysis of the
Planck
Planck high-multipole spectrum. Wedo note that the TT covariance matrices described inPlanck Collaboration XIII (2015) were calculated ana-lytically assuming that sky components are Gaussian.Both foregrounds and the primary CMB have knownnon-Gaussian characteristics (in the latter case due tolensing, see, e.g., Benoit-L´evy et al. 2012) that would re-sult in this approximation underestimating the true TTspectrum uncertainties, particularly at high multipoleswhere the foreground power becomes comparable to theCMB signal and the lensing effect is most important.Finally, we emphasize that, irrespective of what is re-sponsible for these tensions, care must clearly be takenwhen interpreting joint fits including the full range of
Planck multipoles, particularly given
Planck ’s high pre-cision and ability to statistically dominate other mea-surements, regardless of accuracy. CONCLUSIONS
G. E. Addison et al.
We have discussed tensions between the
Planck (cid:96) ≥ WMAP ) and the cosmological mea-surements:(i) the
Planck (cid:96) < c h . σ lower than the high-multipole fit,(ii) the Planck φφ lensing power spectrum, whichhas an amplitude (parametrized by σ Ω . m ) 2 . σ lower than predicted from the (cid:96) ≥ ≤ (cid:96) ≤ . σ higher than Planck (cid:96) ≥ z = 0 .
57, which disagrees at the 2 . σ level,and(v) the most precise local distance ladder determina-tion of H , which is is tension at the 3 . σ level.These differences are quoted assuming τ = 0 . ± . τ but note that this would introducenew tension with Planck polarization data. Definitiveconclusions about τ will require a more detailed under-standing of low- (cid:96) foreground contamination. The Cos-mology Large Angular Scale Surveyor (CLASS) is ex-pected to provide a cosmic variance limited measurementof τ (Essinger-Hileman et al. 2014; Watts et al. 2015).Given these results and the previously reportedtensions with some weak lensing and cluster abundancedata, we suggest that the parameter constraints fromthe high-multipole Planck data appear anomalous dueto either an unlikely statistical fluctuation, remainingsystematic errors, or both. Understanding the origin ofthese discrepancies is important given the role
Planck data might play in future cosmological advances.We are grateful to Adam Riess for reading themanuscript and providing helpful comments. We alsothank Erminia Calabrese and Karim Benabed for helpwith
CosmoMC and the
Planck likelihood code, respec-tively. This research was supported in part by NASAgrant NNX14AF64G and by the Canadian Institute forAdvanced Research (CIFAR). We acknowledge the useof the Legacy Archive for Microwave Background Data Analysis (LAMBDA). This work was based on observa-tions obtained with
Planck ( ), an ESA science mission with instruments andcontributions directly funded by ESA Member States,NASA, and Canada. Part of this research project wasconducted using computational resources at the Mary-land Advanced Research Computing Center (MARCC).REFERENCES), an ESA science mission with instruments andcontributions directly funded by ESA Member States,NASA, and Canada. Part of this research project wasconducted using computational resources at the Mary-land Advanced Research Computing Center (MARCC).REFERENCES