Abstract
We examine the impact of baryon acoustic oscillation (BAO) scale measurements on the discrepancy between the value of the Hubble constant (H0) inferred from the local distance ladder and that from Planck cosmic microwave background (CMB) data. While the BAO data alone cannot constrain H0, we show that combining the latest BAO results with WMAP, Atacama Cosmology Telescope (ACT), or South Pole Telescope (SPT) CMB data produces values of H0 that are lower than the distance ladder, independent of Planck, and that this downward pull was less apparent in some earlier analyses that used only angle-averaged BAO scale constraints rather than full anisotropic information. At the same time, the combination of BAO and CMB data also disfavors the lower values of H0 preferred by the Planck high-multipole temperature power spectrum. Combining galaxy and Lyα forest BAO with a precise estimate of the primordial deuterium abundance produces
km s−1 Mpc−1 for the flat
model. This value is completely independent of CMB anisotropy constraints and is
lower than the latest distance ladder constraint, although
tension also exists between the galaxy BAO and Lyα BAO. These results show that it is not possible to explain the H0 disagreement solely with a systematic error specific to the Planck data. The fact that tensions remain even after the removal of any single data set makes this intriguing puzzle all the more challenging to resolve.
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1. Introduction
While no single data set currently provides compelling evidence for a deviation from the standard Lambda cold dark matter () cosmological model, the values of some parameters inferred from different measurements now exhibit moderate to severe tension. This is most pronounced in the value of the Hubble constant, H0. Riess et al. (2016; hereafter R16) provided the most recent and most precise local distance ladder constraint, finding
km s−1 Mpc−1 by combining three absolute distance anchors with the empirical period–luminosity relation for Cepheid variable stars and the relationship between observed light curve and intrinsic luminosity of type Ia supernovae (SNe). The most precise H0 prediction from cosmic microwave background (CMB) anisotropy power spectrum measurements is currently provided by the Planck mission. The 2015 Planck temperature and polarization analysis produced
km s−1 Mpc−1 (Planck Collaboration et al. 2016a). An updated analysis with a revised estimate of the optical depth to reionization, τ, found
, or 66.93 ± 0.62 if preliminary small-scale polarization data are also included (Planck Collaboration et al. 2016b). Assuming all uncertainties are Gaussian, these values are, respectively, 3.0σ, 3.2σ, and
lower than the distance ladder determination. Strong lensing timing delay measurements have produced H0 constraints consistent with the distance ladder, and in mild tension with Planck (Bonvin et al. 2017). Tensions also exist between the Planck predictions for the growth of cosmic structure (through the matter density,
, and present-day density fluctuation amplitude,
) and measurements using weak gravitational lensing (e.g., Alsing et al. 2017; Hildebrandt et al. 2017; Joudaki et al. 2017; Köhlinger et al. 2017).
It is not clear whether the problem is with the model or the data. While it is certainly plausible that a failure of the standard model could show up as a discrepancy between the CMB and low-redshift measurements, none of the commonly considered or physically motivated extensions to appear to provide a convincing improvement when considering the full range of data available (e.g., Bernal et al. 2016; Planck Collaboration et al. 2016a). In principle, the CMB prediction for H0 could be significantly increased by modifying the expansion history of the universe post-recombination, for example, by allowing spatial curvature or a dark energy equation of state
. Planck temperature and polarization data alone mildly prefer a non-zero curvature, but H0 goes in the wrong direction. The Planck 2015
constraint is 53.2 ± 5.1 km s−1 Mpc−1 (mean and standard deviation), with 95% of Markov chain Monte Carlo (MCMC) samples lying in
km s−1 Mpc−1 < 63.5.3
Allowing
can shift the Planck prediction to 70 or even 80 km s−1 Mpc−1, however, even leaving aside questions of the physical interpretation of
, resolving the H0 disagreement with evolution in w is strongly disfavored when we include observations of the expansion rate, such as baryon acoustic oscillations (BAO) in the clustering of galaxies, or high-redshift SNe. Alam et al. (2017) combined Planck data with the latest galaxy clustering and SNe data and found
km s−1 Mpc−1 for constant w, or 67.5 ± 1.0 km s−1 Mpc−1 for the
parameterization.
Modifying the early-universe expansion history, for instance, by increasing the number of effective neutrino species, , can increase the CMB H0 prediction. The Planck data do not favor this solution, for example, Alam et al. (2017) reported
(Planck-only), and 3.03 ± 0.18 (Planck plus galaxy clustering), consistent with the standard model value of 3.046, with corresponding H0 constraints of 66.6 ± 1.6 and 67.5 ± 1.2 km s−1 Mpc−1. Adding
in these fits shifted the tension with the distance ladder from
to
(Planck-only) and from
to
(Planck plus galaxy clustering). A fit to the 2015 Planck temperature and polarization data plus BAO fixing
, the value found by R16 to most effectively relieve Planck-distance ladder tension, leads to an increase in the parameter combination best constrained by weak lensing measurements,
, by around 1.5%, 0.8 times the original uncertainty.4
This slightly worsens the tension between Planck and the weak lensing analyses mentioned above, which found
values lower than Planck at the 2σ–3σ level when the standard model was assumed. Brust et al. (2017) found that the Planck-lensing consistency could be improved by also introducing some degree of neutrino or dark radiation self-interaction, but even with a second additional parameter, a joint fit to the Planck, BAO, distance ladder, weak lensing, and galaxy cluster data produced a H0 distribution peaking at 69.95 km s−1 Mpc−1, still almost
lower than the R16 measurement. In short, while a non-standard value of
cannot be ruled out, its inclusion is not justified by the improvements to the fit.
On the other hand, the discrepant data sets have passed a range of systematic checks. The R16 distance ladder analysis used infrared data to greatly reduce the effects of reddening, substituted rungs of the ladder with alternative data, compared different calibrators, corrected for estimated local motion, and constructed a systematic error budget from considering a range of modeling variants (see also, e.g., Cardona et al. 2017; Dhawan et al. 2018; Feeney et al. 2017; Follin & Knox 2017; Zhang et al. 2017). The distance ladder measurements have substantially improved since the analysis by Efstathiou (2014). While the constraints have become tighter, the mean H0 values in recent years have remained fairly constant (e.g., Riess et al. 2009, 2011; Freedman et al. 2012). Likewise, the Planck team has performed an array of robustness checks of their data, investigating the effects of detector nonlinearity, beam shapes and sidelobes, and various other calibration-related issues. Also, the preference for a lower H0 from Planck does not appear to be driven by a particular frequency channel (Planck Collaboration et al. 2017).
Ultimately, it may take additional high-precision measurements to shed light on what is really going on. More precise measurements may bring with them new tensions or disagreements, and the handling of systematic errors will get harder, not easier, as statistical uncertainties are reduced. In the meantime, it is therefore helpful to re-examine existing data and ask whether any extra insight into the discrepancies can be gleaned. To this end, in this paper we investigate in detail the indirect but important role played by BAO measurements in H0 constraints, both with and without CMB anisotropy data. While this topic has been addressed in the literature, we describe several results that have either not previously been discussed, or are not widely appreciated. In Section 2, we review the BAO measurements. In Section 3, we describe results of fitting cosmological parameters to BAO in conjunction with other data sets, focusing on H0. A discussion and conclusions follow in Sections 4 and 5.
2. BAO Measurements
The first convincing detections of the BAO feature in the correlation function or power spectrum of large-scale structure (LSS) tracers were made a little over a decade ago (Cole et al. 2005; Eisenstein et al. 2005). Since that time, deeper surveys with orders of magnitude more galaxies, notably the Baryon Oscillation Spectroscopic Survey (BOSS5
), have led to both improved precision in the BAO scale measurements over a range of redshifts, and improved analysis methodologies (e.g., Anderson et al. 2014; Kazin et al. 2014; Percival et al. 2014; Reid et al. 2016). While current and future BAO surveys are proposed as a means of improving dark energy constraints, BAO measurements also provide significant information about parameters in the standard model, particularly in joint fits with the CMB.
A detailed discussion of BAO physics can be found in Chapter 4 of Weinberg et al. (2013). The BAO scale in the transverse and line-of-sight direction correspond to measurements of and
, where
is the comoving angular diameter distance at the effective redshift of the survey and rd is the sound horizon at the drag epoch where baryons decouple from photons, denoted zd. The sound horizon is defined as6
![Equation (1)](https://content.cld.iop.org/journals/0004-637X/853/2/119/revision1/apjaaa1edeqn1.gif)
where the sound speed, , depends on the ratio of baryon to photon density, with
. The sound horizon is sensitive to the physics of the early universe, including the pre-recombination expansion history and the number of effective neutrino species,
, while DM(z) and H(z) at the effective redshift of the survey depend on the late-time expansion.
In some cases, only a joint constraint, for example, on , where
, is provided, representing an angle-averaged constraint. This can be helpful where the BAO feature is detected at lower significance and the separate line-of-sight and transverse measures are poorly constrained or have distributions with non-Gaussian tails. Whenever possible, we use the joint anisotropic
and
constraints. Quantities like
entail a compression of information that potentially gives a false sense of agreement with other data, as discussed in Section 3.2.
2.1. Current BAO Constraints
The BAO data sets included in fits presented in this paper are listed in Table 1. For the 6dF Galaxy Survey (6dFGS) and Sloan Digital Sky Survey (SDSS) Main Galaxy Sample (MGS), we adopt a simple Gaussian likelihood for or
. Away from the peak of the likelihood, these constraints become non-Gaussian; however, the uncertainties for these measurements are large enough that the preferred model solutions never lie far from the peak in a joint fit with other data. We use the consensus BAO scale measurements from the BOSS Data Release 12 (DR12), including
and
for each of the three redshift bins and the six-by-six covariance matrix described by Alam et al. (2017). We restrict our analysis to the BAO scale as it is the most robust observable from LSS surveys (e.g., Weinberg et al. 2013, and references therein), and do not consider redshift-space distortion constraints or information from the broadband correlation function. We do not include results from the WiggleZ7
survey, which are consistent with BOSS and partially overlap in sky coverage (Beutler et al. 2016).
Table 1. BAO Measurements Used in This Work
Data Set | LSS Tracer |
![]() |
Measurementa | Constrainta | References |
---|---|---|---|---|---|
6dFGS | galaxies | 0.106 |
![]() |
0.336 ± 0.015 | Beutler et al. (2011) |
SDSS MGS | galaxies | 0.15 |
![]() |
664 ± 25 | Ross et al. (2015) |
BOSS DR12 | galaxies | 0.38 |
![]() |
1512 ± 25 | Alam et al. (2017) |
![]() |
81.2 ± 2.4 | ||||
0.51 |
![]() |
1975 ± 30 | |||
![]() |
90.9 ± 2.3 | ||||
0.61 |
![]() |
2307 ± 37 | |||
![]() |
99.0 ± 2.5 | ||||
BOSS DR11 Lyα | Lyα absorbersb | 2.34 |
![]() |
11.28 ± 0.65 | Delubac et al. (2015) |
![]() |
9.18 ± 0.28 | ||||
BOSS DR11 QSO × Lyα | QSO, Lyαb | 2.36 |
![]() |
10.8 ± 0.4 | Font-Ribera et al. (2014) |
![]() |
9.0 ± 0.3 |
Notes.
aNote that the fiducial sound horizon,![${r}_{d,\mathrm{fid}.}$](https://content.cld.iop.org/journals/0004-637X/853/2/119/revision1/apjaaa1edieqn63.gif)
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BAO have been measured in the Lyα forest of BOSS quasars (QSOs), and in the cross-correlation between the QSOs and Lyα absorbers, at effective redshifts of 2.3–2.4 (Busca et al. 2013; Slosar et al. 2013; Font-Ribera et al. 2014; Delubac et al. 2015; Bautista et al. 2017). BAO measurements at these redshifts, when the dark energy contribution to the total energy budget of the universe is small, are a powerful complement to the BAO from lower-redshift galaxy surveys. The analysis methodology and systematic error treatment required to extract the Lyα BAO scale are less mature than those for the galaxy BAO and are an active field of research (e.g., Blomqvist et al. 2015). The anisotropic BAO measurements from the DR11 Lyα and QSO × Lyα analyses are in tension with Planck predictions, assuming a standard flat
model. This tension was reduced slightly in the DR12 Lyα BAO analysis (Bautista et al. 2017). Bautista et al. (2017) found that the shift in the DR12 Lyα constraints was predominantly due to the additional data rather than some different treatment of systematic effects.8
We present results using the DR11 Lyα and QSO × Lyα constraints, and from combining the galaxy and Lyα BAO, noting that
effects can and do arise purely from statistical fluctuations, and that there is currently no known systematic error that explains this tension.
Other BAO measurements have been reported, for example, using galaxy clusters as LSS tracers (e.g., Hong et al. 2016; Veropalumbo et al. 2016). These results are generally less precise than the galaxy BAO, at similar redshifts, and their inclusion would not significantly affect our results. Recently, the first measurement of BAO from the extended Baryon Oscillation Spectroscopic Survey (eBOSS9
) was reported using clustering of quasars at (Ata et al. 2018). BAO constraints from this redshift range are potentially a useful addition to the galaxy and Lyα BAO and upcoming, higher-precision eBOSS measurements will be interesting to include in future analyses.
2.2. Choice of CMB Data for Joint Fits
Joint fits between Planck and BAO have been reported extensively for a range of cosmological models in recent work (e.g., Aubourg et al. 2015; Planck Collaboration et al. 2016a; Alam et al. 2017). While Planck provides the most precise CMB constraints, tension exists between determination of some parameters from splitting the Planck power spectrum into multipoles
and
, where the choice of 800 corresponds to a roughly even division of overall constraining power (Addison et al. 2016). In the full
model space, the tension is not significant (
for the assumptions used by Addison et al. 2016; see also Planck Collaboration et al. 2017). Current low-redshift cosmological observations do not provide strong constraints across the full
parameter space, but they do provide independent and precise constraints on a subset of parameters, including H0,
, and
. These parameters are therefore of particular interest when it comes to assessing the performance of the
model and testing for alternatives. Given the moderate internal Planck tension in these parameters, it is informative to consider other CMB measurements to help understand the extent to which conclusions are driven by Planck data, or are independent of Planck. In this work we therefore also include results from the final WMAP 9-year analysis (Bennett et al. 2013; Hinshaw et al. 2013), the Atacama Cosmology Telescope Polarimeter (ACTPol; Sherwin et al. 2017; Thornton et al. 2016; Louis et al. 2017) two-season survey, covering 548 deg2, and the 2500 deg2 South Pole Telescope Sunyaev–Zel'dovich survey (SPT-SZ; Carlstrom et al. 2011; van Engelen et al. 2012; Story et al. 2013).
3. Results
3.1. Combining BAO and CMB Anisotropy Measurements
In Table 2 we show H0 constraints within the model from CMB data sets with and without the inclusion of the BAO data. ACTPol and SPT use WMAP or Planck data only to provide an absolute calibration, that is, a single scale-independent multiplicative rescaling of the measured power spectrum. Since these experiments do not measure τ, we adopt a Gaussian prior, either the same broader
prior used by Planck Collaboration et al. (2016c) and Addison et al. (2016), or the
constraint from the latest Planck HFI low-ℓ polarization determination (Planck Collaboration Int. XLVI 2016). Here and throughout this paper we quote the mean and standard deviation from MCMC runs using the CosmoMC10
package (Lewis & Bridle 2002), with convergence criterion
(Gelman & Rubin 1992). Since we are not investigating foreground modeling in this work, we use foreground-marginalized CMB likelihood codes for ACTPol and SPT (Calabrese et al. 2013; Dunkley et al. 2013). Uncertainties in foreground and other nuisance parameters propagate to cosmological parameters through an increase in power spectrum uncertainties in these codes. In the Planck rows of Table 2 we include the exact name of the likelihood file for clarity since a range of likelihoods have been provided by the Planck collaboration. These likelihoods include Planck foreground and nuisance parameters as described by Planck Collaboration et al. (2016c). We show results with and without CMB lensing power spectra (denoted "
" in the third column of Table 2), noting that the lensing measurements have a moderate effect on some of the CMB-only constraints but a reduced impact when the BAO are included. In the last four rows of Table 2 we also list constraints from splitting the Planck temperature power spectrum at
(Addison et al. 2016), as mentioned in Section 2.2 and discussed further in Section 4.
Table 2. Constraints on H0 in the ΛCDM Model from the CMB Alone and from Combining CMB with BAO data, with the Significance of the Difference from the Distance Ladder Measurement (73.24 ± 1.74; Riess et al. 2016) in Parentheses, Assuming Uncorrelated Gaussian Errors (All Values Are in km s−1 Mpc−1)
CMB Data Set | Large-scale Likelihooda | Power Spectrum Likelihoodsb | H0 (CMB-only) | BAO datac | H0 (CMB+BAO) |
---|---|---|---|---|---|
WMAP 9-year | WMAP | TT, TE, EE | 69.68 ± 2.17 ![]() |
gal+Lyα | 68.30 ± 0.72 ![]() |
'' | '' | '' | gal | 68.19 ± 0.72 ![]() |
|
'' | '' | '' | Lyα | 71.01 ± 2.10 ![]() |
|
ACTPol Two-Season |
![]() |
TT, TE, EE, ![]() |
67.12 ± 2.67 ![]() |
gal+Lyα | 67.23 ± 0.80 ![]() |
'' | '' | '' | gal | 66.94 ± 0.77 ![]() |
|
'' | '' | '' | Lyα | 69.59 ± 2.61 ![]() |
|
'' | TT, TE, EE | 67.60 ± 3.56 ![]() |
gal+Lyα | 67.29 ± 0.83 ![]() |
|
![]() |
TT, TE, EE, ![]() |
66.55 ± 2.59 ![]() |
gal+Lyα | 67.21 ± 0.83 ![]() |
|
SPT-SZ |
![]() |
TT, ![]() |
71.38 ± 3.09 ![]() |
gal+Lyα | 68.52 ± 0.90 ![]() |
'' | '' | '' | gal | 68.25 ± 0.91 ![]() |
|
'' | '' | '' | Lyα | 73.74 ± 2.84 ![]() |
|
'' | TT | 73.20 ± 3.54 ![]() |
gal+Lyα | 68.49 ± 0.92 ![]() |
|
![]() |
TT, ![]() |
70.67 ± 3.06 ![]() |
gal+Lyα | 68.46 ± 0.88 ![]() |
|
Planck | lowTEBd | plikHM_TT 2015, ![]() |
67.86 ± 0.92 ![]() |
gal+Lyα | 68.06 ± 0.56 ![]() |
'' | '' | '' | gal | 67.95 ± 0.54 ![]() |
|
'' | '' | '' | Lyα | 68.17 ± 0.93 ![]() |
|
lowTEB | plikHM_TT 2015 | 67.81 ± 0.92 ![]() |
gal+Lyα | 67.97 ± 0.56 ![]() |
|
![]() |
plikHM_TT 2015 | 66.88 ± 0.91 (![]() |
gal+Lyα | 67.72 ± 0.54 ![]() |
|
![]() |
plikHM_TTTEEE 2015 | 66.93 ± 0.62 (![]() |
gal+Lyα | 67.53 ± 0.45 ![]() |
|
![]() |
plikHM_TT ![]() |
70.08 ± 1.96 ![]() |
gal+Lyα | 68.34 ± 0.67 ![]() |
|
![]() |
plikHM_TT ![]() |
69.78 ± 1.86 ![]() |
gal+Lyα | 68.29 ± 0.66 ![]() |
|
![]() |
plikHM_TT ![]() |
65.12 ± 1.45 ![]() |
gal+Lyα | 67.91 ± 0.66 ![]() |
|
![]() |
plikHM_TT ![]() |
64.30 ± 1.31 ![]() |
gal+Lyα | 67.55 ± 0.62 ![]() |
Notes.
aPixel-based and other likelihoods used at multipoles![${\ell }\lesssim 30$](https://content.cld.iop.org/journals/0004-637X/853/2/119/revision1/apjaaa1edieqn141.gif)
![$\phi \phi $](https://content.cld.iop.org/journals/0004-637X/853/2/119/revision1/apjaaa1edieqn142.gif)
![${\ell }\lt 30$](https://content.cld.iop.org/journals/0004-637X/853/2/119/revision1/apjaaa1edieqn143.gif)
![$\tau =0.055\pm 0.009$](https://content.cld.iop.org/journals/0004-637X/853/2/119/revision1/apjaaa1edieqn144.gif)
![${\ell }\lt 30$](https://content.cld.iop.org/journals/0004-637X/853/2/119/revision1/apjaaa1edieqn145.gif)
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Adding galaxy BAO to any of the CMB measurements listed in Table 2 substantially tightens the H0 prediction, by more than a factor of three in the case of ACTPol or SPT. While there is still scatter in the CMB + galaxy BAO H0 values, the spread is substantially reduced compared to the CMB-only column. The ACTPol+BAO and SPT+BAO combinations produce H0 constraints of comparable precision to Planck alone. The synergy between the galaxy BAO and CMB measurements for is illustrated in Figure 1 using the BOSS DR12 anisotropic BAO measurements at
as an example. The predictions from the CMB are shown with MCMC samples color-coded by H0, which vary fairly monotonically along the degeneracy line set by the angular acoustic scale, corresponding to the peak spacing in the CMB power spectrum. The MCMC samples shown are drawn from the full chains, and include points from the tails of the distributions in addition to high-likelihood samples. The shaded blue contours correspond to the BOSS measurements, which are precise enough to substantially reduce the range of H0 values allowed by breaking CMB degeneracies. The mixing of colors visible in the ACTPol and SPT panels reflects additional degeneracy between H0 and other parameters arising from the more limited range of angular scales provided by these data.
Figure 1. Including BAO data substantially tightens CMB constraints on H0. The observables corresponding to the transverse and line-of-sight BAO scale, , and
(Section 2 and Table 1), are shown for redshift z = 0.61. The blue shaded contours are the measurements from the final BOSS DR12 analysis (Alam et al. 2017). The different panels contain predictions from different, essentially independent CMB measurements, assuming a flat
model, with MCMC samples color-coded by H0 in km s−1 Mpc−1. The same
prior is used in each case. The addition of BAO tightens the H0 constraint by more than a factor of three in the case of ACTPol or SPT data (Table 2). When combined with any current CMB data set the galaxy BAO disfavors the values of H0 preferred by the distance ladder (73.24 ± 1.74 km s−1 Mpc−1; Riess et al. 2016) at moderate to high significance. The lower values preferred by the high-multipole Planck data (the constraint from the samples shown in the top right panel is 65.12 ± 1.45 km s−1 Mpc−1) are also disfavored.
Download figure:
Standard image High-resolution imageIn conjunction with the CMB, and in the context of , the BAO values have the effect of disfavoring both the higher values of H0 preferred by the distance ladder, and the lower values preferred by the Planck damping tail at
. If we exclude Planck, the CMB + BAO values lie 2.4–3.1σ from the R16 measurement, depending on the choice of CMB data set. While this trend has been reported before using WMAP data (Bernal et al. 2016; Planck Collaboration et al. 2016a), here we show that the measurements of the damping tail from ACTPol and SPT produce the same effect even without information from the larger scales measured by the satellite experiments. The fact that combining ACTPol and BAO data produces an H0 value
lower than R16 provides strong evidence that the H0 discrepancy cannot be explained solely by a systematic specific to the Planck data. On the other hand, using the difference-of-covariance method described in Section 4.1 of Planck Collaboration et al. (2016a), the shift in H0 from adding the BAO to the
Planck temperature power spectrum is larger than expected at the 2.2σ and
level for the
and 0.055 ± 0.009 priors, respectively.
The CMB + Lyα BAO fits yield higher values of H0 than the CMB alone, without significantly smaller uncertainties. This reflects the tension between the CMB and Lyα BAO discussed in earlier work (e.g., Delubac et al. 2015; Planck Collaboration et al. 2016a). In a joint fit with all the BAO data, the Lyα measurements lack the constraining power to overcome the galaxy BAO, and consequently our results are fairly insensitive to whether the Lyα are included along with the galaxy BAO or not. The interaction between the galaxy and Lyα BAO constraints is discussed further in Section 3.3.
We note that the SPT values in Table 2 differ from the value of 75.0 ± 3.5 km s−1 Mpc−1 quoted in Table 3 of the original SPT analysis by Story et al. (2013). This difference is driven by three effects: (i) the inclusion of the SPT lensing power spectrum measurement from van Engelen et al. (2012) in some of our fits; (ii) the difference in τ prior; we used 0.07 ± 0.02 or 0.055 ± 0.009, while Story et al. (2013) used 0.088 ± 0.015; and (iii) different CosmoMC versions or fitting options, including the fact that we set the total neutrino mass to 0.06 eV in our fits, while Story et al. (2013) assumed massless neutrinos, which leads to a
shift in H0. We have verified that if we use the Story et al. (2013) assumptions, we reproduce their 75.0 ± 3.5 constraint. Aylor et al. (2017) recently derived parameters from SPT with an updated Planck-based calibration and improved likelihood, however, the shift they report in H0 is small and would not meaningfully affect our results.
3.2. Angle-averaged versus Anisotropic BAO
Bennett et al. (2014; hereafter B14) used pre-Planck CMB data along with BAO measurements available at the time (6dFGS, BOSS DR11, including the Lyα but not QSO × Lyα cross-correlation; we refer to these data as BAO14) to constrain
![Equation (2)](https://content.cld.iop.org/journals/0004-637X/853/2/119/revision1/apjaaa1edeqn2.gif)
This value is noticeably higher than the CMB+BAO values reported in Table 2. To make a more direct comparison we performed an updated fit using WMAP, ACTPol, SPT, and the latest BAO data, and find
![Equation (3)](https://content.cld.iop.org/journals/0004-637X/853/2/119/revision1/apjaaa1edeqn3.gif)
The difference in these values appears large given the overlap in data sets used, so we investigated this difference in detail. We found that the downward shift in our current fits is due to a combination of several effects:
- (i)The biggest difference comes from using the transverse and line-of-sight BOSS BAO scale measurements now available separately rather than the angle-averaged
used in B14. Using the BOSS DR11 CMASS anisotropic BAO instead of the BOSS DR11 CMASS angle-averaged BAO shifts the WMAP9+ACT+SPT+BAO14 H0 constraint downwards by 0.61 km s−1 Mpc−1, a shift comparable to the total uncertainty. This is discussed in more detail below.
- (ii)A smaller shift of around 0.2 km s−1 Mpc−1 is due to different likelihood codes. We find
km s−1 Mpc−1 using WMAP9+ACT+SPT+BAO14. Our results were obtained with the November 2016 versions of CAMB11 and CosmoMC, while a different MCMC code was used in B14. Furthermore, our implementation of the DR11 Lyα BAO constraint uses the
look-up tables provided by BOSS,12 whereas B14 constructed a likelihood directly from values reported by Delubac et al. (2015).
- (iii)The ACTPol data have a stronger downward pull on H0 than ACT. Both ACT and ACTPol prefer a lower H0 value than WMAP alone (Sievers et al. 2013; Louis et al. 2017). The SPT data prefer a higher H0 value than WMAP, and this preference wins out in the combination with ACT. With ACTPol, however, the downward pull is stronger, and the resulting constraint shifts downward from 69.98 ± 1.58 (WMAP9+ACT+SPT) to 69.08 ± 1.37 km s−1 Mpc−1 (WMAP+ACTPol+SPT). In combination with the BAO the impact of using ACTPol instead of ACT is subdominant to the choice of BAO constraints.
- (iv)The SDSS MGS BAO constraint at
was not used by B14. While the MGS measurement has lower precision than BOSS (4% compared to around 1%), it also has a stronger preference for lower H0 in conjunction with the CMB data.
Why does the choice of anisotropic or angle-averaged BOSS CMASS BAO make such a large difference given the same galaxy sample is used for each? In the flat model, all the information from any BAO measurement is contained in contours in the two-dimensional
space (Addison et al. 2013). The relative late-time expansion history is determined by
, with
determined implicitly by the flatness constraint. The impact of radiation on the late-time expansion is small enough compared to the precision of current BAO measurements that uncertainties in the CMB temperature, which constrains the physical density
, or in converting to the fractional density,
, can be neglected. The combination
provides an overall normalization factor and reflects the fact that the absolute length of the sound horizon, and a change in the normalization of the expansion rate, are completely degenerate when only fitting to measurements of the BAO scale.
The upper left and lower left panels of Figure 2 shows constraints in the plane for the DR11 BOSS CMASS sample at
(Anderson et al. 2014). We show the transverse (
) and line-of-sight (Hrd) contours separately, as well as the contour from combining both, and the angle-averaged
contour. While there is substantial overlap between the combined anisotropic contour and the
contour, a portion of the parameter space is allowed by
but ruled out by the combined anisotropic measurements. This portion is relevant when the BAO and CMB are combined, as shown in the upper right and lower right panels of Figure 2, with the anisotropic
constraint pulling down more strongly on
, and hence H0, since H0 and rd are only partially degenerate in the CMB. The same effect is apparent in Figure 8 of Cuesta et al. (2016).
Figure 2. Use of the angle-averaged BAO constraint, , instead of the full anisotropic information,
plus
, can impact determination of H0 from combined CMB+BAO fits. The upper left panel shows constraints from the same BOSS CMASS DR11 galaxy sample at
(Anderson et al. 2014) but different BAO measures—transverse (
), line of sight (Hrd), and angle-averaged (
, see Section 2.1). The lower left panel shows the anisotropic constraint from combining
and
. While there is significant overlap between the angle-averaged and anisotropic contours, the angle-averaged contour extends to lower values of
, which are not allowed by the anisotropic constraint. The upper right and lower right panels show the effect of adding the BAO information to CMB data (we show the same data sets used by Bennett et al. 2014, WMAP+ACT+SPT). The use of the angle-averaged
constraint diminishes the downward pull on
, and also H0, from the BAO. The vertical dashed lines correspond to the bounds of the contours containing 95% of the CMB+BAO MCMC samples.
Download figure:
Standard image High-resolution imageWe conclude that the shift in H0 from using the angle-averaged instead of the full anisotropic BAO information is not caused by an inconsistency in the BAO measurements, but instead due to the compression of information inherent to
. It is therefore preferable to use the anisotropic constraints where possible.
3.3. Constraints from the BAO Scale Alone
We now consider constraints from the BAO data without the strong additional constraining power of the CMB anisotropy measurements. As discussed above, in the flat model, BAO measurements provide contours in the
plane. Combining the galaxy and Lyα BAO provides a tight constraint on
from the late-time expansion history, even when marginalizing over the normalization
. For the BAO listed in Table 1 we find constraints of
![Equation (4)](https://content.cld.iop.org/journals/0004-637X/853/2/119/revision1/apjaaa1edeqn4.gif)
The left panel of Figure 3 shows constraints from the galaxy and Lyα BAO in the plane. The orientation of these contours can be approximately understood from considering the redshift dependence of H(z). Similar arguments hold for DA(z). At the Lyα BAO redshifts the universe is matter-dominated, and
, so that
constraints produce contours along the direction with
roughly constant. At lower redshifts, where dark energy becomes dominant, H(z) depends less strongly on
, leading to the galaxy BAO contour being oriented more along the direction of the y-axis in Figure 3.13
There is little overlap between the galaxy and Lyα contours. To quantify this difference, we consider the test described in Section 4.1 of Hou et al. (2014). We calculate
, where in this case X and Y are the galaxy and Lyα BAO data, respectively,
denotes the best-fit
from the joint fit, and
and
are the best-fit
from the fits to just the galaxy or just the Lyα data. For Gaussian-distributed data,14
if X and Y are independent and
is the correct model, then
is drawn from a
distribution with
degrees of freedom (dof). We find
![](https://content.cld.iop.org/journals/0004-637X/853/2/119/revision1/apjaaa1edueqn1.gif)
The probability of exceeding (PTE) for with
is
, which corresponds to a
disagreement. This is comparable to the
tension reported between the Lyα BAO and Planck measurements by Delubac et al. (2015). As discussed by Aubourg et al. (2015), modifying the cosmological model to improve three-way agreement between CMB, galaxy BAO, and Lyα BAO appears difficult. Here, we note that the combined contour in Figure 3 lies at the intersection of the main degeneracy directions determined by the redshift coverage of the galaxy and Lyα measurements. If future data shift the galaxy or Lyα BAO constraints along these degeneracy lines (as opposed to perpendicular to them) the main result would be to change the quality of the combined fit rather than changing the parameter values. We further note that the matter density reported in (4) is in agreement with the value of 0.295 ± 0.034 from a joint analysis of type Ia SNe from several surveys covering
, completely independent of LSS clustering (Betoule et al. 2014). This is illustrated in the right panel of Figure 3, which shows a comparison of BAO, WMAP 9-year, Planck 2016,15
and SNe constraints on
for the flat
model.
Figure 3. Left: comparison of BAO-only constraints in the flat model. Contours containing 68 and 95% of MCMC samples are shown for galaxy (
) and Lyα forest (
) BAO separately and in a joint fit using the BAO data listed in Table 1. In flat
the late-time expansion rate is determined only by
, with
acting as an overall expansion normalization. Right: comparison of
constraints from BAO, CMB, and SNe measurements. The SNe constraint is from the "joint light-curve analysis" (JLA) presented by Betoule et al. (2014). While the combined BAO fit produces a tight constraint
, in agreement with the CMB and SNe determinations, there is a
tension between the galaxy and Lyα BAO, which individually prefer higher and lower values of
, respectively.
Download figure:
Standard image High-resolution image3.4. Constraining H0 with BAO plus Deuterium Abundance in ΛCDM
Obtaining a constraint on H0 from the BAO requires adding information to break the degeneracy. One way to do this is to add a constraint on the baryon density (e.g., Addison et al. 2013; Aubourg et al. 2015; Wang et al. 2017). We assume that the photon energy density, or, equivalently, the CMB mean temperature, is also known. The CMB temperature was measured precisely by COBE/FIRAS (Fixsen et al. 1996; Fixsen 2009) and we view this result as independent of the CMB anisotropy measurements performed by more recent experiments. Note that while the H0 in the H(z) in the denominator of (1) cancels in the
product, some residual H0 dependence still exists because the decoupling redshift and the sound speed depend on the physical matter and radiation densities,
and
, respectively, while the expansion rate H(z) depends on the fractional densities
and
.
In the BAO fit with an external baryon density prior, performs double duty. It not only goes into determining the late-time expansion (DM and H at the BAO survey redshifts) but also controls the expansion history in the early universe prior to baryons decoupling from photons, since the photon and neutrino properties (with
) are held fixed. As well as providing an indirect H0 constraint, the BAO+
fit also serves as something of a self-consistency test of early and late-time expansion.
The most precise constraints on independent of the CMB power spectrum come from estimates of the primordial deuterium abundance. In standard Big Bang nucleosynthesis (BBN), the abundance of light nuclei is determined by a single parameter, the baryon-to-photon ratio η (see recent review by Cyburt et al. 2016, and references therein). Taking the photon number density as fixed from the CMB temperature, a measurement of the primordial deuterium abundance in conjunction with knowledge of BBN physics provides a constraint on
. Precise estimates of the primordial deuterium abundance have been made in recent years using extremely metal-poor damped Lyα (DLA) systems along sight lines to high-redshift quasars (e.g., Pettini & Cooke 2012; Cooke et al. 2014, 2016; Riemer-Sørensen et al. 2017). Cooke et al. (2016; hereafter C16) report
![Equation (5)](https://content.cld.iop.org/journals/0004-637X/853/2/119/revision1/apjaaa1edeqn5.gif)
by combining six such systems. The reaction rate plays a key role in the conversion from D/H to
. Using the theoretical calculation for this rate from Marcucci et al. (2016), C16 find
![Equation (6)](https://content.cld.iop.org/journals/0004-637X/853/2/119/revision1/apjaaa1edeqn6.gif)
which is lower than the Planck value (assuming a standard
model throughout). Using instead an empirically derived
rate, C16 find
![Equation (7)](https://content.cld.iop.org/journals/0004-637X/853/2/119/revision1/apjaaa1edeqn7.gif)
which has a larger uncertainty but is in better agreement with CMB-derived values. We performed fits to the galaxy-plus-Lyα BAO data with the addition of each of the Gaussian priors on in (6) and (7) in turn. We show parameter constraints in Table 3, including the WMAP 9-year and Planck 2016 CMB anisotropy constraints for comparison.
Table 3.
Constraints from the BAO+D/H Fits, Using either the Theoretical or Empirical
Reaction Rate, with CMB Anisotropy Constraints from WMAP and Planck Included for Comparison
Parameter | BAO+D/H | BAO+D/H | WMAP 9-year | Planck 2016 |
---|---|---|---|---|
(theoretical) | (empirical) | |||
![]() |
2.156 ± 0.020 | 2.257 ± 0.034 | 2.265 ± 0.049 | 2.215 ± 0.021 |
![]() |
10.94 ± 1.20 | 11.19 ± 1.29 | 11.37 ± 0.46 | 12.07 ± 0.21 |
![]() |
1.0292 ± 0.0168 | 1.0320 ± 0.0173 | 1.04025 ± 0.00223 | 1.04076 ± 0.00047 |
H0 [km s−1 Mpc−1] | 66.98 ± 1.18 | 67.81 ± 1.25 | 69.68 ± 2.17 | 66.89 ± 0.90 |
![]() |
0.293 ± 0.019 | 0.293 ± 0.020 | 0.283 ± 0.026 | 0.321 ± 0.013 |
rd [Mpc] | 151.6 ± 3.4 | 149.2 ± 3.6 | 148.49 ± 1.23 | 147.16 ± 0.48 |
Download table as: ASCIITypeset image
In the BAO+D/H fits, is driven solely by the D/H prior, as expected, and
matches the BAO-only value. While the choice of the
reaction rate significantly impacts the value of
, it has a reduced impact on the inferred H0, because rd only depends weakly on
(Eisenstein & Hu 1998; Addison et al. 2013). Specifically, replacing the theoretical rate with the empirical one shifts the center of the
distribution by 5.2 times the original uncertainty, but shifts the H0 distribution by only 0.7 times the original uncertainty. Our BAO+D/H results for H0 are more robust to the choice of rate than one might expect from the
difference.
The H0 values listed in Table 3 from the BAO+D/H fits have uncertainties of around 1.8% and are 3.0 and lower than the R16 distance ladder value of 73.24 ± 1.74 km s−1 Mpc−1 for the theoretical and empirical
rates, respectively. The combination of precise BAO and D/H measurements enables determinations of H0 within the context of the flat
model that are almost 50% tighter than the distance ladder measurement, and lower at moderate to strong significance. We emphasize that these constraints are completely independent of CMB anisotropy measurements.
Constraints in the plane for the BAO+D/H fits with the theoretical
rate are shown in Figure 4. We show results from the galaxy and Lyα BAO separately and together, as before. Tension between the galaxy and Lyα BAO is again apparent. Adding D/H to these data separately favors higher values of H0, and it is only when galaxy and Lyα BAO are combined that H0 is constrained to the values quoted in Table 3.
Figure 4. Adding an estimate of the baryon density, , in this case from deuterium abundance (D/H) measurements, breaks the BAO
degeneracy in
. The same contours are shown as in Figure 3, with the addition of a Gaussian prior
(Cooke et al. 2016). In contrast to Figure 3, here
determines both the early time expansion, including the absolute sound horizon, rd, as well as the late-time expansion history. The radiation density is fixed from COBE/FIRAS CMB mean temperature measurements. The combined BAO+D/H constraint,
km s−1 Mpc−1 is
lower than the Riess et al. (2016) distance ladder determination and is independent of CMB anisotropy data.
Download figure:
Standard image High-resolution imageThe direction of the Lyα BAO contour is roughly the same in the left panel of Figure 3 and in Figure 4, while that of the galaxy BAO contour changes. This behavior can be understood by considering how rd depends on and H0. For a given value of
, rd depends approximately on the combination
(Equation (26) of Eisenstein & Hu 1998). This is the same dependence as H(z) at the Lyα redshifts (Section 3.3) and is related to the fact that the universe is largely matter-dominated in both cases. The dependence of H(z) on
at the galaxy BAO redshifts is weaker, and the direction of the galaxy BAO contour in Figure 4 is approximately determined by requiring
to be roughly constant as
varies. This produces a positive correlation between H0 and
because rd decreases as
increases.
For the BAO+D/H fits, we ran CosmoMC as one would when fitting to the CMB: the fitted parameters are , the physical cold dark matter density,
, and the angular sound horizon,
, and H0,
, and rd are derived from these three. Since the BAO+D/H data are insensitive to the amplitude and tilt of the primordial power spectrum, and the optical depth to reionization, these other
parameters are held fixed. Consistent results were obtained using earlier BAO and D/H data by Addison et al. (2013) and Aubourg et al. (2015). We note that Riemer-Sorensen & Sem Jenssen (2017) recently obtained a tighter constraint on D/H than we have used here by combining the DLAs used by C16 with a number of additional measurements. Using this tighter constraint would not impact our conclusions.
3.5. BAO and Light Element Abundance Constraints with Varying
In the +
model, there is a perfect degeneracy between
and
from D/H measurements (Figure 6 of C16). Closed contours in the
plane can be obtained from combining estimates of the primordial D/H and 4He abundance (e.g., review by Cyburt et al. 2016, and references therein). The primordial 4He abundance is estimated from He and H emission lines in extragalactic H ii regions. Obtaining accurate constraints is challenging due to dependence on environmental parameters such as temperature, electron density, and metallicity, which must be modeled. An important recent development is the use of the HeI line at 10830 Å to help break modeling degeneracies (Izotov et al. 2014). The value of the primordial helium fraction reported by Izotov et al. (2014),
, is, however, significantly higher than values found in some subsequent analyses of the same H ii sample using different selection criteria and fitting methodology. For example, Aver et al. (2015) found
, while Peimbert et al. (2016) found
. The different Yp values lead to significantly different inferences for
when used in combination with D/H or CMB power spectra measurements. Izotov et al. (2014) found evidence for additional neutrino species at 99% confidence, while, for instance, Cyburt et al. (2016) reported
, and Peimbert et al. (2016) found
, consistent with the standard model value of 3.046.
Current D/H and 4He constraints clearly have the precision to weigh in significantly on the question of whether allowing is effective at resolving
tensions. Given the spread in Yp values discussed above, and the impact of the choice of
rate when
is allowed to vary (Section 5.2 of C16), we do not present a full set of results including BAO and light element abundance data for
+
. Instead, we note that combining BAO measurements with D/H and 4He constraints on
that are consistent with the standard model value would produce H0 values consistent with the values in Table 3, although with larger uncertainties, while higher values of
would produce a higher H0, improving agreement with the distance ladder. The BAO measurements, only being sensitive to
, and not to H0 or
directly, are unable to discriminate between these possibilities.
4. Discussion
We have presented evidence for a lower H0 value than measured by the local distance ladder that is independent of Planck, both from combining BAO with other CMB data sets (WMAP, ACTPol, and SPT), and from joint fits to BAO and D/H measurements, within the context of the standard model. In light of this analysis it is clear that the H0 tension cannot be resolved solely through a systematic error specific to the Planck data. It should be noted, however, that it is not simply a case of having a "high" H0 from the distance ladder, and a "low" H0 from Planck and the joint BAO fits. The high-multipole Planck temperature data prefer H0 values that are even lower than the CMB+BAO or BAO+D/H values (bottom two rows of Table 2 and top right panel of Figure 1). Restricting the Planck temperature power spectrum to multipoles
produces
![Equation (8)](https://content.cld.iop.org/journals/0004-637X/853/2/119/revision1/apjaaa1edeqn8.gif)
or
![Equation (9)](https://content.cld.iop.org/journals/0004-637X/853/2/119/revision1/apjaaa1edeqn9.gif)
depending on the choice of τ prior. These values are not only in strong tension with R16, but are in moderate tension with some of the CMB+BAO values reported in Table 2. For example, for , the SPT+BAO value is lower than R16 by
, but the Planck
value is
lower again than SPT+BAO. The shift in H0 from adding the BAO to the
Planck constraints is also larger than expected given the improvement in precision, as discussed in Section 3.1. Some H0 tension remains even if we do not consider the distance ladder constraints. In fact, concordance cannot be achieved through the removal of any single data set (e.g., BAO, CMB, distance ladder, or D/H). This is part of the reason the H0 discrepancy is challenging to resolve: a convincing solution must simultaneously address multiple avenues of disagreement.
A wide range of fits to expanded cosmological models, with various combinations of data, have been presented in recent years to try to reconcile H0 and other parameter tensions. Our fits in this paper have been restricted to the standard flat model, partly because our results for expanded models would be similar to those already presented by Planck Collaboration et al. (2016a), Alam et al. (2017), Heavens et al. (2017), and others. The BAO, CMB, and light element abundance measurements have some common dependence on the early-universe expansion history, which makes allowing freedom in, for example,
, seem attractive. As discussed in Section 1, varying
does not sufficiently relieve tensions and is not statistically favored over standard
for the current BAO, CMB, and distance ladder data. There are good prospects for tightening
constraints through improved measurements of the high-ℓ CMB damping tail in E-mode polarization (e.g., Abazajian et al. 2015, 2016). Future BAO data, for example, from the Dark Energy Spectroscopic Instrument (DESI16
), Euclid,17
and WFIRST,18
will also provide significant improvements in precision over current measurements (for BAO+
forecasts for H0, see Wang et al. 2017).
5. Conclusions
We have examined the role of BAO measurements in determining H0. While the BAO data alone are unable to distinguish between a change in H0 and a change in the absolute sound horizon at decoupling, rd, this degeneracy is broken, and a precise H0 value obtained, when BAO are combined with either CMB power spectra or deuterium abundance measurements. Overall we find convincing evidence for a lower H0 in than what was obtained from the latest local distance ladder measurement (73.24 ± 1.74 km s−1 Mpc−1; Riess et al. 2016), even without using data from Planck. The motivation and results of this study are summarized as follows:
- (i)Tension at the
level exists between determinations of H0 from the distance ladder and the CMB anisotropy measurements from Planck, within the context of the standard flat
model. Other tensions also exist, for example between Planck data and constraints on the growth of structure from some weak lensing surveys.
- (ii)None of the cosmological modifications commonly proposed appear to provide a statistically compelling solution to these tensions, although some, such as allowing freedom in the number of effective relativistic species,
, do reduce the H0 disagreement.
- (iii)Combining BAO measurements with CMB power spectrum data from WMAP, ACTPol, or SPT, produces H0 values that are lower than the distance ladder by
, independent of Planck (Table 2). The difference was less pronounced in some earlier analyses because of using the angle-averaged BOSS CMASS BAO measurement. The angle-averaged
constraint is a compression of information and allows a region of parameter space that is ruled out by the full anisotropic BAO constraints (Figure 2). Adding the BAO improves the H0 constraint from ACTPol or SPT by more than a factor of three, making their precision comparable to the Planck-only results.
- (iv)Combining BAO data with primordial deuterium (D/H) abundance estimates from metal-poor DLA systems produces precise H0 values that are lower than the distance ladder by
, depending on assumptions about the
reaction rate (e.g., 66.98 ± 1.18 km s−1 Mpc−1 for the theoretical rate, see Table 3). This result is independent of any CMB anisotropy measurement and relies only on the CMB mean temperature from COBE/FIRAS.
- (v)The two previous results taken together indicate that it is not possible to resolve the H0 disagreement solely through some systematic error specific to the Planck data set.
- (vi)The Planck high-multipole (
) damping tail measurements prefer lower values of H0 than the combined BAO fits, such as, for example, 65.12 ± 1.45 and 64.30 ± 1.31 km s−1 Mpc−1, for
and
, respectively. The shift in H0 from adding the BAO to these data is larger than expected at the 2.2σ and
levels for these τ priors. The H0 disagreement is not as simple as the distance ladder value being "high" and other constraints coming out "low," and cannot be resolved through the removal of any single data set.
- (vii)We note that a
tension exists between the galaxy (
) and Lyα (
) BAO, as previously discussed by Aubourg et al. (2015). The BAO+D/H constraints rely on combining these measurements, thus it is important to review their consistency with future data.
In recent years new precise measurements have led to multiple tensions, particularly in H0, that are uncomfortably too large to be explained by statistical scatter within the context of the standard model. Whether this is the sign of new physics or underestimated uncertainties, or some combination of effects, remains unclear, and no straightforward explanation has yet presented itself. Near-term improvements in CMB, LSS, and distance ladder data are expected; however, continuing to scrutinize existing measurements, as we have in this work, could also prove important in moving toward an eventual resolution.
This research was supported in part by NASA grants NNX15AJ57G, NNX16AF28G, and NNX17AF34G, and JPL grant 1563692, and by the Canadian Institute for Advanced Research (CIFAR). We acknowledge the use of the Legacy Archive for Microwave Background Data Analysis (LAMBDA), part of the High Energy Astrophysics Science Archive Center (HEASARC). HEASARC/LAMBDA is a service of the Astrophysics Science Division at the NASA Goddard Space Flight Center. This work was partly based on observations obtained with Planck (http://www.esa.int/Planck), an ESA science mission with instruments and contributions directly funded by ESA Member States, NASA, and Canada. Part of this research project was conducted using computational resources at the Maryland Advanced Research Computing Center (MARCC).
The authors would like to thank Erminia Calabrese, Antony Lewis, James Rich, Adam Riess, and Ashley Ross for helpful discussions and clarifications regarding the data sets and software used in this work, and we are also grateful to Adam Riess for reading a draft of the manuscript and providing useful suggestions. We acknowledge the use of the GetDist plotting package.19
Footnotes
- 3
The public Planck MCMC chains can be downloaded from the Planck legacy archive: http://pla.esac.esa.int/pla/.
- 4
This result is taken from the Planck 2015 base_plikHM_TTTEEE_lowTEB_post_BAO and base_nnu_plikHM_TTTEEE_lowTEB_nnup39_BAO chains.
- 5
- 6
The sound horizon was referred to as rs by Addison et al. (2013); we have adopted the rd notation here for consistency with other work.
- 7
- 8
The DR12 QSO × Lyα analysis, released while this work was in review, produced results consistent with DR11, in tension with Planck predictions at the
level (du Mas des Bourboux et al. 2017).
- 9
- 10
- 11
- 12
- 13
If BAO measurements at z = 0 were possible, they would produce exactly vertical contours in Figure 3.
- 14
This is a reasonable approximation when the Lyα and QSO × Lyα BAO are combined (Delubac et al. 2015).
- 15
We refer to the combination of the 2015 TT constraints with updated
prior from Planck Collaboration et al. (2017) as "Planck 2016."
- 16
- 17
- 18
- 19