Elucidating ΛCDM: Impact of Baryon Acoustic Oscillation Measurements on the Hubble Constant Discrepancy

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 ${r}_{d}/{D}_{V}(z)$ or ${D}_{V}(z)/{r}_{d}$. 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 ${D}_{M}(z)/{r}_{d}$ and $H(z){r}_{d}$ 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	${z}_{\mathrm{eff}}$	Measurementa	Constrainta	References
6dFGS	galaxies	0.106	${r}_{d}/{D}_{V}({z}_{\mathrm{eff}})$	0.336 ± 0.015	Beutler et al. (2011)
SDSS MGS	galaxies	0.15	${D}_{V}({z}_{\mathrm{eff}}){r}_{d,\mathrm{fid}.}/{r}_{d}$ [Mpc]	664 ± 25	Ross et al. (2015)
BOSS DR12	galaxies	0.38	${D}_{M}({z}_{\mathrm{eff}}){r}_{d,\mathrm{fid}.}/{r}_{d}$ [Mpc]	1512 ± 25	Alam et al. (2017)
			$H({z}_{\mathrm{eff}}){r}_{d}/{r}_{d,\mathrm{fid}.}$ [km s−1 Mpc−1]	81.2 ± 2.4
		0.51	${D}_{M}({z}_{\mathrm{eff}}){r}_{d,\mathrm{fid}.}/{r}_{d}$ [Mpc]	1975 ± 30
			$H({z}_{\mathrm{eff}}){r}_{d}/{r}_{d,\mathrm{fid}.}$ [km s−1 Mpc−1]	90.9 ± 2.3
		0.61	${D}_{M}({z}_{\mathrm{eff}}){r}_{d,\mathrm{fid}.}/{r}_{d}$ [Mpc]	2307 ± 37
			$H({z}_{\mathrm{eff}}){r}_{d}/{r}_{d,\mathrm{fid}.}$ [km s−1 Mpc−1]	99.0 ± 2.5
BOSS DR11 Lyα	Lyα absorbersb	2.34	${D}_{A}({z}_{\mathrm{eff}})/{r}_{d}$	11.28 ± 0.65	Delubac et al. (2015)
			$c/[H({z}_{\mathrm{eff}}){r}_{d}]$	9.18 ± 0.28
BOSS DR11 QSO × Lyα	QSO, Lyαb	2.36	${D}_{A}({z}_{\mathrm{eff}})/{r}_{d}$	10.8 ± 0.4	Font-Ribera et al. (2014)
			$c/[H({z}_{\mathrm{eff}}){r}_{d}]$	9.0 ± 0.3

Notes.

^aNote that the fiducial sound horizon, ${r}_{d,\mathrm{fid}.}$, differs across different analyses. We provide constraints here only to show relative precision. For parameter fitting we use full-likelihood surfaces, including correlations across the BOSS redshift bins or between D_M and H. ^bFor brevity we refer to the Lyα and QSO × Lyα measurements collectively as Lyα.

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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 $\sim 2.5\sigma$ tension with Planck predictions, assuming a standard flat ${\rm{\Lambda }}\mathrm{CDM}$ 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 $\sim 2.5\sigma$ 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 $0.8\leqslant z\leqslant 2.2$ (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.

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, $\sim 2.5\sigma$ tension exists between determination of some parameters from splitting the Planck power spectrum into multipoles ${\ell }\lt 800$ and ${\ell }\gt 800$, where the choice of 800 corresponds to a roughly even division of overall constraining power (Addison et al. 2016). In the full ${\rm{\Lambda }}\mathrm{CDM}$ model space, the tension is not significant ($1.8\sigma$ 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 ${\rm{\Lambda }}\mathrm{CDM}$ parameter space, but they do provide independent and precise constraints on a subset of parameters, including H₀, ${{\rm{\Omega }}}_{m}$, and ${\sigma }_{8}$. These parameters are therefore of particular interest when it comes to assessing the performance of the ${\rm{\Lambda }}\mathrm{CDM}$ 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).

In Table 2 we show H₀ constraints within the ${\rm{\Lambda }}\mathrm{CDM}$ 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 $\tau =0.07\pm 0.02$ prior used by Planck Collaboration et al. (2016c) and Addison et al. (2016), or the $\tau =0.055\pm 0.009$ 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 $R-1\lt 0.01$ (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 "$\phi \phi$" 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 ${\ell }=800$ (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 $(1.3\sigma )$	gal+Lyα	68.30 ± 0.72 $(2.6\sigma )$
	''	''	''	gal	68.19 ± 0.72 $(2.7\sigma )$
	''	''	''	Lyα	71.01 ± 2.10 $(0.8\sigma )$
ACTPol Two-Season	$\tau =0.07\pm 0.02$	TT, TE, EE, $\phi \phi$	67.12 ± 2.67 $(1.9\sigma )$	gal+Lyα	67.23 ± 0.80 $(3.1\sigma )$
	''	''	''	gal	66.94 ± 0.77 $(3.3\sigma )$
	''	''	''	Lyα	69.59 ± 2.61 $(1.3\sigma )$
	''	TT, TE, EE	67.60 ± 3.56 $(1.4\sigma )$	gal+Lyα	67.29 ± 0.83 $(3.1\sigma )$
	$\tau =0.055\pm 0.009$	TT, TE, EE, $\phi \phi$	66.55 ± 2.59 $(2.1\sigma )$	gal+Lyα	67.21 ± 0.83 $(3.1\sigma )$
SPT-SZ	$\tau =0.07\pm 0.02$	TT, $\phi \phi$	71.38 ± 3.09 $(0.5\sigma )$	gal+Lyα	68.52 ± 0.90 $(2.4\sigma )$
	''	''	''	gal	68.25 ± 0.91 $(2.5\sigma )$
	''	''	''	Lyα	73.74 ± 2.84 $(0.2\sigma )$
	''	TT	73.20 ± 3.54 $(0.0\sigma )$	gal+Lyα	68.49 ± 0.92 $(2.4\sigma )$
	$\tau =0.055\pm 0.009$	TT, $\phi \phi$	70.67 ± 3.06 $(0.7\sigma )$	gal+Lyα	68.46 ± 0.88 $(2.5\sigma$)
Planck	lowTEBd	plikHM_TT 2015, $\phi \phi$	67.86 ± 0.92 $(2.7\sigma )$	gal+Lyα	68.06 ± 0.56 $(2.8\sigma )$
	''	''	''	gal	67.95 ± 0.54 $(2.9\sigma )$
	''	''	''	Lyα	68.17 ± 0.93 $(2.6\sigma )$
	lowTEB	plikHM_TT 2015	67.81 ± 0.92 $(2.8\sigma )$	gal+Lyα	67.97 ± 0.56 $(2.9\sigma )$
	$\tau =0.055\pm 0.009$, lowle	plikHM_TT 2015	66.88 ± 0.91 ($3.2\sigma$)	gal+Lyα	67.72 ± 0.54 $(3.0\sigma )$
	$\tau =0.055\pm 0.009$, lowl	plikHM_TTTEEE 2015	66.93 ± 0.62 ($3.4\sigma$)	gal+Lyα	67.53 ± 0.45 $(3.2\sigma )$
	$\tau =0.07\pm 0.02$, lowl	plikHM_TT ${\ell }\lt 800$	70.08 ± 1.96 $(1.2\sigma )$	gal+Lyα	68.34 ± 0.67 $(2.6\sigma )$
	$\tau =0.055\pm 0.009$, lowl	plikHM_TT ${\ell }\lt 800$	69.78 ± 1.86 $(1.4\sigma )$	gal+Lyα	68.29 ± 0.66 $(2.7\sigma )$
	$\tau =0.07\pm 0.02$	plikHM_TT ${\ell }\gt 800$	65.12 ± 1.45 $(3.6\sigma )$	gal+Lyα	67.91 ± 0.66 $(2.9\sigma )$
	$\tau =0.055\pm 0.009$	plikHM_TT ${\ell }\gt 800$	64.30 ± 1.31 $(4.1\sigma )$	gal+Lyα	67.55 ± 0.62 $(3.1\sigma )$

Notes.

^aPixel-based and other likelihoods used at multipoles ${\ell }\lesssim 30$. For some fits, particularly with the ACTPol and SPT experiments that do not probe these scales, we indicate the Gaussian prior adopted on τ instead. ^bTemperature, E-mode polarization, temperature-polarization cross-spectrum, and lensing potential power spectra are denoted TT, EE, TE, and $\phi \phi$, respectively. ^c"Gal" refers to galaxy BAO; "Lyα" refers to Lyα forest and QSO × Lyα BAO (see Table 1). ^dlowTEB is the combined temperature-plus-polarization Planck likelihood for ${\ell }\lt 30$. ^eSince the Planck Collaboration et al. (2017) low-multipole polarization likelihood is not publicly available, we approximate its inclusion with a prior $\tau =0.055\pm 0.009$, which produces constraints in very good agreement with their Table 8. lowl is the Planck temperature-only likelihood for ${\ell }\lt 30$ (no polarization).

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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 ${\rm{\Lambda }}\mathrm{CDM}$ is illustrated in Figure 1 using the BOSS DR12 anisotropic BAO measurements at ${z}_{\mathrm{eff}}=0.61$ 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.

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In conjunction with the CMB, and in the context of ${\rm{\Lambda }}\mathrm{CDM}$, 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 ${\ell }\gt 800$. 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 $\gt 3\sigma$ 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 ${\ell }\gt 800$ Planck temperature power spectrum is larger than expected at the 2.2σ and $2.8\sigma$ level for the $\tau =0.07\pm 0.02$ 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 $\phi \phi$ 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 $\sim 0.2\sigma$ 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.

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

$$\begin{eqnarray}{H}_{0} & = & 69.3\pm 0.7\,{\rm{km}}\,{{\rm{s}}}^{-1}\,{{\rm{Mpc}}}^{-1}\\ & & ({WMAP}+\mathrm{ACT}+\mathrm{SPT}+\mathrm{BAO}14).\end{eqnarray} \tag{ 2 }$$

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

$$\begin{eqnarray}{H}_{0} & = & 68.34\pm 0.61\,{\rm{km}}\,{{\rm{s}}}^{-1}\,{{\rm{Mpc}}}^{-1}\\ & & ({WMAP}+\mathrm{ACTPol}+\mathrm{SPT}+\mathrm{BAO}).\end{eqnarray} \tag{ 3 }$$

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 ${D}_{V}(z)/{r}_{d}$ 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 ${H}_{0}=69.07\pm 0.70$ 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 ${\chi }^{2}$ 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 ${z}_{\mathrm{eff}}=0.15$ 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 ${\rm{\Lambda }}\mathrm{CDM}$ model, all the information from any BAO measurement is contained in contours in the two-dimensional ${{\rm{\Omega }}}_{m}-{H}_{0}{r}_{d}$ space (Addison et al. 2013). The relative late-time expansion history is determined by ${{\rm{\Omega }}}_{m}$, with ${{\rm{\Omega }}}_{{\rm{\Lambda }}}$ 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 ${{\rm{\Omega }}}_{r}{h}^{2}$, or in converting to the fractional density, ${{\rm{\Omega }}}_{r}$, can be neglected. The combination ${H}_{0}{r}_{d}$ 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 ${{\rm{\Omega }}}_{m}-{H}_{0}{r}_{d}$ plane for the DR11 BOSS CMASS sample at ${z}_{\mathrm{eff}}=0.57$ (Anderson et al. 2014). We show the transverse (${D}_{M}/{r}_{d}$) and line-of-sight (Hr_d) contours separately, as well as the contour from combining both, and the angle-averaged ${D}_{V}(z)/{r}_{d}$ contour. While there is substantial overlap between the combined anisotropic contour and the ${D}_{V}(z)/{r}_{d}$ contour, a portion of the parameter space is allowed by ${D}_{V}(z)/{r}_{d}$ 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 ${D}_{M}(z)/{r}_{d}+H(z){r}_{d}$ constraint pulling down more strongly on ${H}_{0}{r}_{d}$, and hence H₀, since H₀ and r_d are only partially degenerate in the CMB. The same effect is apparent in Figure 8 of Cuesta et al. (2016).

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We conclude that the shift in H₀ from using the angle-averaged ${D}_{V}(z)/{r}_{d}$ 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 ${D}_{V}(z)/{r}_{d}$. It is therefore preferable to use the anisotropic constraints where possible.

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 ${\rm{\Lambda }}\mathrm{CDM}$ model, BAO measurements provide contours in the ${{\rm{\Omega }}}_{m}-{H}_{0}{r}_{d}$ plane. Combining the galaxy and Lyα BAO provides a tight constraint on ${{\rm{\Omega }}}_{m}$ from the late-time expansion history, even when marginalizing over the normalization ${H}_{0}{r}_{d}$. For the BAO listed in Table 1 we find constraints of

$$\begin{eqnarray}{{\rm{\Omega }}}_{m} & = & 0.292\pm 0.020\\ {H}_{0}{r}_{d} & = & (10119\pm 138){\text{km s}}^{-1}.\end{eqnarray} \tag{ 4 }$$

The left panel of Figure 3 shows constraints from the galaxy and Lyα BAO in the ${{\rm{\Omega }}}_{m}-{H}_{0}{r}_{d}$ plane. The orientation of these contours can be approximately understood from considering the redshift dependence of H(z). Similar arguments hold for D_A(z). At the Lyα BAO redshifts the universe is matter-dominated, and $H{(z)\simeq {H}_{0}{{\rm{\Omega }}}_{m}^{1/2}(1+z)}^{3/2}$, so that $H(z){r}_{d}$ constraints produce contours along the direction with ${H}_{0}{r}_{d}\cdot {{\rm{\Omega }}}_{m}^{1/2}$ roughly constant. At lower redshifts, where dark energy becomes dominant, H(z) depends less strongly on ${{\rm{\Omega }}}_{m}$, 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 ${\rm{\Delta }}{\chi }^{2}={\chi }_{X+Y}^{2}-{\chi }_{X}^{2}-{\chi }_{Y}^{2}$, where in this case X and Y are the galaxy and Lyα BAO data, respectively, ${\chi }_{X+Y}^{2}$ denotes the best-fit ${\chi }^{2}$ from the joint fit, and ${\chi }_{X}^{2}$ and ${\chi }_{Y}^{2}$ are the best-fit ${\chi }^{2}$ from the fits to just the galaxy or just the Lyα data. For Gaussian-distributed data,¹⁴ if X and Y are independent and ${\rm{\Lambda }}\mathrm{CDM}$ is the correct model, then ${\rm{\Delta }}{\chi }^{2}$ is drawn from a ${\chi }^{2}$ distribution with ${N}_{{\rm{\Delta }}{\chi }^{2}}={N}_{X+Y}-{N}_{X}-{N}_{Y}$ degrees of freedom (dof). We find

$$\begin{eqnarray*}{\chi }_{\mathrm{gal}}^{2} & = & 2.98\quad {N}_{\mathrm{gal}}=8\mbox{--}2=6\\ {\chi }_{\mathrm{Ly}\alpha }^{2} & = & 0.92\quad {N}_{\mathrm{Ly}\alpha }=4\mbox{--}2=2\\ {\chi }_{\mathrm{gal}+\mathrm{Ly}\alpha }^{2} & = & 13.63\quad {N}_{\mathrm{gal}+\mathrm{Ly}\alpha }=12\mbox{--}2=10\\ {\rm{\Delta }}{\chi }^{2} & = & 9.73\quad {N}_{{\rm{\Delta }}{\chi }^{2}}=10\mbox{--}6\mbox{--}2=2.\end{eqnarray*}$$

The probability of exceeding (PTE) for ${\chi }^{2}=9.73$ with ${N}_{\mathrm{dof}}=2$ is $7.71\times {10}^{-3}$, which corresponds to a $2.4\sigma$ disagreement. This is comparable to the $2.5\sigma$ 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 $0\lt z\lt 1$, 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 ${{\rm{\Omega }}}_{m}$ for the flat ${\rm{\Lambda }}\mathrm{CDM}$ model.

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Obtaining a constraint on H₀ from the BAO requires adding information to break the ${H}_{0}-{r}_{d}$ 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 ${H}_{0}{r}_{d}$ product, some residual H₀ dependence still exists because the decoupling redshift and the sound speed depend on the physical matter and radiation densities, ${{\rm{\Omega }}}_{m}{h}^{2}$ and ${{\rm{\Omega }}}_{r}{h}^{2}$, respectively, while the expansion rate H(z) depends on the fractional densities ${{\rm{\Omega }}}_{m}$ and ${{\rm{\Omega }}}_{r}$.

In the BAO fit with an external baryon density prior, ${{\rm{\Omega }}}_{m}$ performs double duty. It not only goes into determining the late-time expansion (D_M 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 ${N}_{\mathrm{eff}}=3.046$) are held fixed. As well as providing an indirect H₀ constraint, the BAO+${{\rm{\Omega }}}_{b}{h}^{2}$ fit also serves as something of a self-consistency test of early and late-time expansion.

The most precise constraints on ${{\rm{\Omega }}}_{b}{h}^{2}$ 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 ${{\rm{\Omega }}}_{b}{h}^{2}$. 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

$$\begin{eqnarray}&&{10}^{5}{{\rm{D}}}_{{\rm{I}}}/{{\rm{H}}}_{{\rm{I}}}=2.547\pm 0.033\end{eqnarray} \tag{ 5 }$$

by combining six such systems. The $d{(p,\gamma )}^{3}\mathrm{He}$ reaction rate plays a key role in the conversion from D/H to ${{\rm{\Omega }}}_{b}{h}^{2}$. Using the theoretical calculation for this rate from Marcucci et al. (2016), C16 find

$$\begin{eqnarray}&&100{{\rm{\Omega }}}_{b}{h}^{2}=2.156\pm 0.020\\ &&\quad ({\rm{D}}/{\rm{H}},\mathrm{theoretical}\,\mathrm{rate}),\end{eqnarray} \tag{ 6 }$$

which is $\gt 2\sigma$ lower than the Planck value (assuming a standard ${\rm{\Lambda }}\mathrm{CDM}$ model throughout). Using instead an empirically derived $d{(p,\gamma )}^{3}\mathrm{He}$ rate, C16 find

$$\begin{eqnarray}&&100{{\rm{\Omega }}}_{b}{h}^{2}=2.260\pm 0.034\\ &&\quad ({\rm{D}}/{\rm{H}},\mathrm{empirical}\,\mathrm{rate}),\end{eqnarray} \tag{ 7 }$$

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 ${{\rm{\Omega }}}_{b}{h}^{2}$ 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.  ${\rm{\Lambda }}\mathrm{CDM}$ Constraints from the BAO+D/H Fits, Using either the Theoretical or Empirical $d{(p,\gamma )}^{3}{\rm{He}}$ 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)
$100{{\rm{\Omega }}}_{b}{h}^{2}$	2.156 ± 0.020	2.257 ± 0.034	2.265 ± 0.049	2.215 ± 0.021
$100{{\rm{\Omega }}}_{c}{h}^{2}$	10.94 ± 1.20	11.19 ± 1.29	11.37 ± 0.46	12.07 ± 0.21
$100{\theta }_{\mathrm{MC}}$	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
${{\rm{\Omega }}}_{m}$	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

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In the BAO+D/H fits, ${{\rm{\Omega }}}_{b}{h}^{2}$ is driven solely by the D/H prior, as expected, and ${{\rm{\Omega }}}_{m}$ matches the BAO-only value. While the choice of the $d{(p,\gamma )}^{3}\mathrm{He}$ reaction rate significantly impacts the value of ${{\rm{\Omega }}}_{b}{h}^{2}$, it has a reduced impact on the inferred H₀, because r_d only depends weakly on ${{\rm{\Omega }}}_{b}{h}^{2}$ (Eisenstein & Hu 1998; Addison et al. 2013). Specifically, replacing the theoretical rate with the empirical one shifts the center of the ${{\rm{\Omega }}}_{b}{h}^{2}$ 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 ${{\rm{\Omega }}}_{b}{h}^{2}$ 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 $2.5\sigma$ lower than the R16 distance ladder value of 73.24 ± 1.74 km s⁻¹ Mpc⁻¹ for the theoretical and empirical $d{(p,\gamma )}^{3}\mathrm{He}$ rates, respectively. The combination of precise BAO and D/H measurements enables determinations of H₀ within the context of the flat ${\rm{\Lambda }}\mathrm{CDM}$ 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 ${{\rm{\Omega }}}_{m}-{H}_{0}$ plane for the BAO+D/H fits with the theoretical $d{(p,\gamma )}^{3}\mathrm{He}$ 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.

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The 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 r_d depends on ${{\rm{\Omega }}}_{m}$ and H₀. For a given value of ${{\rm{\Omega }}}_{b}{h}^{2}$, r_d depends approximately on the combination ${H}_{0}\cdot {{\rm{\Omega }}}_{m}^{1/2}$ (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 ${{\rm{\Omega }}}_{m}$ at the galaxy BAO redshifts is weaker, and the direction of the galaxy BAO contour in Figure 4 is approximately determined by requiring ${H}_{0}{r}_{d}$ to be roughly constant as ${{\rm{\Omega }}}_{m}$ varies. This produces a positive correlation between H₀ and ${{\rm{\Omega }}}_{m}$ because r_d decreases as ${H}_{0}{{\rm{\Omega }}}_{m}^{1/2}$ increases.

For the BAO+D/H fits, we ran CosmoMC as one would when fitting to the CMB: the fitted parameters are ${{\rm{\Omega }}}_{b}{h}^{2}$, the physical cold dark matter density, ${{\rm{\Omega }}}_{c}{h}^{2}$, and the angular sound horizon, ${\theta }_{\mathrm{MC}}$, and H₀, ${{\rm{\Omega }}}_{m}$, and r_d 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 ${\rm{\Lambda }}\mathrm{CDM}$ 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.

In the ${\rm{\Lambda }}\mathrm{CDM}$+${N}_{\mathrm{eff}}$ model, there is a perfect degeneracy between ${{\rm{\Omega }}}_{b}{h}^{2}$ and ${N}_{\mathrm{eff}}$ from D/H measurements (Figure 6 of C16). Closed contours in the ${{\rm{\Omega }}}_{b}{h}^{2}-{N}_{\mathrm{eff}}$ 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), ${Y}_{p}=0.2551\pm 0.0022$, 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 ${Y}_{p}=0.2449\pm 0.0040$, while Peimbert et al. (2016) found ${Y}_{p}=0.2446\pm 0.0029$. The different Y_p values lead to significantly different inferences for ${N}_{\mathrm{eff}}$ 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 ${N}_{\mathrm{eff}}=2.85\pm 0.28$, and Peimbert et al. (2016) found ${N}_{\mathrm{eff}}=2.90\pm 0.22$, 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 ${N}_{\mathrm{eff}}\gt 3$ is effective at resolving ${\rm{\Lambda }}\mathrm{CDM}$ tensions. Given the spread in Y_p values discussed above, and the impact of the choice of $d{(p,\gamma )}^{3}\mathrm{He}$ rate when ${N}_{\mathrm{eff}}$ 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 ${\rm{\Lambda }}\mathrm{CDM}$+${N}_{\mathrm{eff}}$. Instead, we note that combining BAO measurements with D/H and ⁴He constraints on ${N}_{\mathrm{eff}}$ 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 ${N}_{\mathrm{eff}}$ would produce a higher H₀, improving agreement with the distance ladder. The BAO measurements, only being sensitive to ${H}_{0}{r}_{d}$, and not to H₀ or ${N}_{\mathrm{eff}}$ directly, are unable to discriminate between these possibilities.