Cosmological implications of baryon acoustic oscillation (BAO) measurements

A homogenenous and isotropic cosmological model is specified by the curvature parameter $k$ entering the Friedman-Robertson-Walker metric

which governs conversion between radial and transverse distances, and by the evolution of $a(t)=(1+z)^{-1}$. In General Relativity (GR), this evolution is governed by the Friedmann equation Friedman22 , which can be written in the form

where $H\equiv\dot{a}/a$ is the Hubble parameter, $\rho(a)$ is the total energy density (radiation + matter + dark energy), and the subscript 0 denotes the present day ($a=1$). We define the density parameter of component $x$ by the ratio

and the curvature parameter

where the sum is over all matter and energy components and $\Omega_{k}=0$ for a flat ($k=0$) universe. Density parameters and $\rho_{\rm crit}$ always refer to values at $z=0$ unless a dependence on $a$ or $z$ is written explicitly, e.g., $\Omega_{x}(z)$. We will frequently refer to the Hubble constant $H_{0}$ through the dimensionless ratio $h\equiv H_{0}/100\,{\rm km}\,{\rm s}^{-1}\,{\rm Mpc}^{-1}$. The dimensionless quantity $\omega_{x}\equiv\Omega_{x}h^{2}$ is proportional to the physical density of component $x$ at the present day.

Given the curvature parameter and $H(z)$ from the Friedmann equation, the comoving angular diameter distance can be computed as

where the line-of-sight comoving distance is

where

Positive $k$ corresponds to negative $\Omega_{k}$. We do not use the small $\Omega_{k}$ approximation of equation (5) in our calculations, but we provide it here to illustrate that for small non-zero curvature the change in distance is linear in $\Omega_{k}$ and quadratic in $D_{C}(z)$.

Curvature affects $D_{M}(z)$ both through its influence on $H(z)$ and through the geometrical factor in equation (5). The luminosity distance (relevant to supernovae) is $D_{L}=D_{M}(1+z)$.

The energy components considered in our models are pressureless (cold) dark matter, baryons, radiation, neutrinos, and dark energy. The densities of CDM and baryons scale as $a^{-3}$; we refer to the density parameter of these two components together as $\Omega_{cb}$. The energy density of neutrinos with non-zero mass scales like radiation at early times and like matter at late times, with

where both CMB temperature $T_{\rm CMB}$ and neutrino temperature scale inversely with scale factor, and the neutrino temperature is given by $T_{\nu}=T_{\rm CMB}\left(\frac{4}{11}\right)^{\nicefrac{{1}}{{3}}}g_{c}$, where $g_{c}=\left(\nicefrac{{3.046}}{{3}}\right)^{\nicefrac{{1}}{{4}}}$ accounts for small amount of heating of neutrinos due to electron-positron annihilation. The sum in the above expression is over neutrino species with masses $m_{i}$. The integral $I$ is given by

and must be evaluated numerically. For massless neutrinos $I(0)=\nicefrac{{7}}{{8}}$, while in the limit of very massive neutrinos $I(r)\sim 45\zeta(3)(2\pi^{4})^{-1}r$ (for $r\gg 1$; here $\zeta(3)$ is the Riemann function), i.e., scaling proportionally with $a$ so that neutrinos behave like pressureless matter. When we refer to the $z=0$ matter density parameter $\Omega_{m}$, we include the contributions of radiation (which is small compared to the uncertainties in $\Omega_{m}$) and neutrinos (which are non-relativistic at $z=0$), so that $\Omega_{m}+\Omega_{\rm de}+\Omega_{k}\equiv 1$. Following the Planck Collaboration PlanckXVI , we adopt $\sum m_{\nu}=0.06\,$ eV with one massive and two massless neutrino species in all models except the one referred to as $\nu$CDM, where it is a free parameter. The default implies $\omega_{\nu}=6.57\times 10^{-4}$ including massless species and $\omega_{\nu}=6.45\times 10^{-4}$ excluding them. The effect of finite neutrino temperature at $z=0$ is a very small $10^{-4}$ relative effect. The adopted values are close to the minimum value allowed by neutrino oscillation experiments.

We consider a variety of models for the evolution of the energy density or equation-of-state parameter $w=p_{\rm de}/\rho_{\rm de}$. Table 1 summarizes the primary models discussed in the paper, though we consider some additional special cases in Section VI. $\Lambda$CDM represents a flat universe with a cosmological constant ($w=-1$). $o\Lambda$CDM extends this model to allow non-zero $\Omega_{k}$. $w$CDM adopts a flat universe and constant $w$, and $ow$CDM generalizes to non-zero $\Omega_{k}$. $w_{0}w_{a}$CDM and $ow_{0}w_{a}$CDM allow $w(a)$ to evolve linearly with $a(t)$, $w(a)=w_{0}+w_{a}(1-a)$. PolyCDM adopts a quadratic polynomial form for $\rho_{\rm de}(a)$ and allows non-zero space curvature, to provide a highly flexible description of the effects of dark energy at low redshift. Finally, Slow Roll Dark Energy is an example of a one-parameter evolving-$w$ model, based on a quadratic dark energy potential.

We focus in this paper on parameter constraints and model tests from measurements of cosmic distances and expansion rates, which we refer to collectively as “expansion history” or “geometric” constraints. We briefly consider comparisons to measurements of low-redshift matter clustering in Section VII. In this framework, the crucial roles of CMB anisotropy measurements are to constrain the parameters (mainly $\omega_{m}$ and $\omega_{b}$) that determine the BAO scale and to determine the angular diameter distance to the redshift of recombination. For most of our analyses, this approach allows us to use a highly compressed summary of CMB constraints, discussed in Section II.3 below, and to compute parameter constraints with a simple and fast Markov Chain Monte Carlo (MCMC) code that computes expansion rates and distances from the Friedmann equation. The code is publicly available with data used in this paper at https://github.com/slosar/april.

The BAO data in this work are summarized in Table 2 and more extensively discussed below.

The robustness of BAO measurements arises from the fact that a sharp feature in the correlation function (or an oscillatory feature in the power spectrum) cannot be readily mimicked by systematics, whether observational or astrophysical, as these should be agnostic about the BAO scale and hence smooth over the relevant part of the correlation function (or power spectrum). In most current analyses, the BAO scale is determined by adopting a fiducial cosmological model that translates angular and redshift separations to comoving distances but allowing the location of the BAO feature itself to shift relative to the fiducial model expectation. One then determines the likelihood of obtaining the observed two-point correlation function or power spectrum as a function of the BAO offsets, while marginalizing over nuisance parameters. These nuisance parameters characterize “broad-band” physical or observational effects that smoothly change the shape or amplitude of the underlying correlation function or power spectrum, such as scale-dependent bias of galaxies or the LyaF, or distortions caused by continuum fitting or by variations in star-galaxy separation. In an isotropic fit, the measurement is encoded in the $\alpha$ parameter, the ratio of the measured BAO scale to that predicted by the fiducial model. In an anisotropic analysis, one separately constrains $\alpha_{\perp}$ and $\alpha_{\parallel}$, the ratios perpendicular and parallel to the line of sight. In real surveys the errors on $\alpha_{\perp}$ and $\alpha_{\parallel}$ are significantly correlated for a given redshift slice, but they are typically uncorrelated across different redshift slices. While the values of $\alpha$ are referred to a specified fiducial model, the corresponding physical BAO scales are insensitive to the choice of fiducial model within a reasonable range.

The BAO scale is set by the radius of the sound horizon at the drag epoch $z_{d}$ when photons and baryons decouple,

where the sound speed in the photon-baryon fluid is $c_{s}(z)=3^{-1/2}c\left[1+\frac{3}{4}\rho_{b}(z)/\rho_{\gamma}(z)\right]^{-1/2}$. A precise prediction of the BAO signal requires a full Boltzmann code computation, but for reasonable variations about a fiducial model the ratio of BAO scales is given accurately by the ratio of $r_{d}$ values computed from the integral (10). Thus, a measurement of $\alpha_{\perp}$ from clustering at redshift $z$ constrains the ratio of the comoving angular diameter distance to the sound horizon:

A measurement of $\alpha_{\parallel}$ constrains the Hubble parameter $H(z)$, which we convert to an analogous quantity:

with

An isotropic BAO analysis measures some effective combination of these two distances. If redshift-space distortions are weak, which is a good approximation for luminous galaxy surveys after reconstruction but not for the LyaF, then the constrained quantity is the volume averaged distance

with

There are different conventions in use for defining $r_{d}$, which differ at the 1-2% level, but ratios of $r_{d}$ for different cosmologies are independent of the convention provided one is consistent throughout. In this work we adopt the CAMB convention for $r_{d}$, i.e., the value that is reported by the linear perturbations code CAMBLewis00 . In practice, we use the numerically calibrated approximation

This approximation is accurate to 0.021% for a standard radiation background with $N_{\rm eff}=3.046$, $\sum m_{\nu}<0.6\,{\rm eV}$, and values of $\omega_{b}$ and $\omega_{cb}$ within $3\sigma$ of values derived by Planck. It supersedes a somewhat less accurate (but still sufficiently accurate) approximation from Anderson14 (their eq. 55). Note that $\omega_{\nu}=0.0107(\sum m_{\nu}/1.0\,{\rm eV})$, and a 0.5 (1.0) eV neutrino mass changes $r_{d}$ by $-0.26\%$ ($-0.92\%$) for fixed $\omega_{cb}$. For neutrino masses in the range allowed by current cosmological constraints, the CMB constrains $\omega_{cb}$ rather than $\omega_{cb}+\omega_{\nu}$ because neutrinos remain relativistic at recombination, even though they are non-relativistic at $z=0$. For the case of extra relativistic species, a useful fitting formula is

which is accurate to 0.119% if we restrict to neutrino masses in the range $0<\sum m_{\nu}<0.6\,{\rm eV}$ and $3<N_{\rm eff}<5$. Increasing $N_{\rm eff}$ by unity decreases $r_{d}$ by about 3.2%.

For $\Lambda$CDM models (with $\sum m_{\nu}=0.06\,{\rm eV}$, $N_{\rm eff}=3.046$) constrained by Planck, $r_{d}=147.49\pm 0.59$ Mpc. This 0.4% uncertainty is only slightly larger for $o\Lambda$CDM, $ow$CDM, or even $ow_{0}w_{a}$CDM (see Table 1 for cosmological model definitions), because the relevant quantities $\omega_{cb}$ and $\omega_{b}$ are constrained by the relative heights of the acoustic peaks, not by their angular locations. The inference of matter energy densities from peak heights thus depends on correct understanding of physics in the pre-recombination epoch, where curvature and dark energy are negligible in any of these models.

BAO measurements constrain cosmological parameters through their influence on $r_{d}$, their influence on $D_{H}(z)$ via the Friedmann equation, and their influence on $D_{M}(z)$ via equation (5). For standard models, the 0.4% error on $r_{d}$ from Planck is small compared to current BAO measurement errors, so the constraints come mainly through $D_{H}$ and $D_{M}$. From the Friedmann equation, we see that $D_{H}(z)$ is directly sensitive to the total energy density at redshift $z$, while $D_{M}(z)$ constrains an effective average of the energy density and is also sensitive to curvature.

Measurements in Table 2 are reported in terms of $D_{V}/r_{d}$, $D_{M}/r_{d}$, and $D_{H}/r_{d}$, using the $r_{d}$ convention of equation (10). Expressed in these terms, the results are independent of the fiducial cosmologies assumed in the individual analysis papers. Note that some of the referenced papers quote values of $D_{A}/r_{d}$ rather than $D_{M}/r_{d}$, differing by a factor $(1+z)$. An anisotropic analysis yields anti-correlated errors on $D_{M}$ and $D_{H}$, and the correlation coefficients are reported in Table 2. Each sample spans a range of redshift, and the quoted effective redshift is usually weighted by statistical contribution to the BAO measurement. Because redshift-space positions are scaled to comoving coordinates based on a fiducial cosmological model, and BAO measurements are obtained as ratios relative to that fiducial model, the values of the effective redshift in Table 2 can be treated as exact, e.g., one should compare the BOSS CMASS numbers to model predictions computed at $z=0.5700.$

In this paper we focus on constraints on the expansion history of the homogeneous cosmological model. For this purpose, we compress the Cosmic Microwave Background (CMB) measurements to the variables governing this expansion history. This approach greatly simplifies the required computations, allowing us to fit complex models that have a simple solution to the Friedmann equation without the need to numerically solve for the evolution of perturbations. It is also physically illuminating, making clear what relevant quantities the CMB determines and distinguishing expansion history constraints from those that depend on the evolution of clustering. For some models or special cases we use more complete CMB results obtained by running the industry-standard cosmomc 2002PhRvD..66j3511L or by relying on the publicly available Planck MCMC chains.⁵⁵5http://wiki.cosmos.esa.int/planckpla/index.php/ Cosmological_Parameters

The CMB plays two distinct but important roles in our analysis. First, we treat the CMB as a “BAO experiment” at redshift $z_{\star}=1090$, measuring the angular scale of the sound horizon at very high redshift. Here we ignore the small dependence of the last-scattering redshift $z_{\star}$ on cosmological parameters and the fact that the relevant scale for the CMB is $r_{\star}$ rather than the drag redshift $r_{d}$ that sets the BAO scale in low-redshift structure. We have checked that both approximations are valid to around $0.1\sigma$ for the case of BAO and Planck data and the $\Lambda$CDM model. In its second important role, the CMB calibrates the absolute length of the BAO ruler through its determination of $\omega_{b}$ and $\omega_{cb}$.

Inspired by the existence of well-known degeneracies in CMB data 1999MNRAS.304…75E ; 2002PhRvD..66f3007K ; Wang13 , we compress the CMB measurements into three variables: $\omega_{b}$, $\omega_{cb}$ and $D_{M}(1090)/r_{d}$. The mean vector and the $3\times 3$ covariance matrix are used to describe the CMB constraints by a simple Gaussian likelihood. In order to calibrate these variables, we rely on the publicly available Planck chains. In particular, we use the base_Alens chains with the planck_lowl_lowLike dataset corresponding to the Planck dataset with low-$\ell$ WMAP polarization (referred to in this paper as Planck+WP). We find that the data vector

can be described by a Gaussian likelihood with mean

and covariance

The fractional diagonal errors on $\omega_{b}$, $\omega_{cb}$, and $D_{M}(1090)/r_{d}$ are 1.5%, 1.9%, and 0.06%, respectively. We similarly compress the WMAP 9-yr data into

and covariance

For reference in interpreting the cosmological constraints from CMB+BAO data, especially in Section VI below, note that the contributions to $D_{M}(1090)$ accumulate over a wide range of redshift, with 14%, 25%, 38%, 47%, 69%, 88%, and 99% of the integral in equation (6) coming from redshifts $z<0.5$, 1.0, 2.0, 3.0, 10, 50, and 640, respectively.

The base_Alens model corresponds to the basic flat $\Lambda$CDM cosmology with explicit marginalisation over the foreground lensing potential. Our decision to use the flat model was intentional, since we found that in curved models there is significant non-Gaussian correlation of $\omega_{b}$ and $\omega_{m}$ with curvature. Because our BAO data inevitably collapse more complex models to nearly flat ones, use of the flat data is more appropriate. We have tested the data compression in a couple of simple cases by comparing results of BAO+CMB data to cosmomc chains and found less than $0.5\sigma$ differences in best-fit parameter values between using compressed and full chains. The residual differences are driven by the fact that our compressed likelihood attempts to extract purely geometric information from the CMB data (for example, values of $\omega_{b}$ and $\omega_{m}$ are different at roughly the same level between chains that marginalise over lensing potential and those that do not). For BAO-only data combinations the results are completely consistent.

Throughout the paper, we refer to the constraints represented by equations (18)-(20) simply as “Planck” (although they also include information from WMAP polarization measurements). In Section III we treat the CMB as a BAO experiment measuring $D_{M}(1090)/r_{d}$, but we eliminate its calibration of the absolute BAO scale by artificially blowing up the errors on $\omega_{b}$ and $\omega_{cb}$; we denote this case as “$+$Planck $D_{M}$”. Conversely, in Section IV we use the CMB information on $\omega_{b}$ and $\omega_{cb}$ to set the size of our standard ruler $r_{d}$ but omit the $D_{M}(1090)/r_{d}$ information by artificially inflating its errors; we denote this case as “$+r_{d}$”. When we use a full Planck chain instead of the compressed information, we adopt the notation “Planck (full)” and specify what additional parameters (such as $A_{\rm lens}$, $N_{\rm eff}$, or tensor-to-scalar ratio $r$) are being varied in the chain.

If one assumes a flat universe, a cosmological constant ($w=-1$), standard relativistic background ($N_{\rm eff}=3.046$), and minimal neutrino mass ($\sum m_{\nu}=0.06\,{\rm eV}$), then the CMB data summarized by equations (18)-(20) also provide a precise constraint on the Hubble parameter $h$, and thus on $\Omega_{cb}$, $\Omega_{b}$, and $\Omega_{\Lambda}$. At various points in the paper we refer to a “fiducial” Planck $\Lambda$CDM model for which we adopt $\omega_{b}h^{2}=0.022032$, $\Omega_{m}=0.3183$, and $h=0.6704$, which are the best fit parameters for “Planck+WP” combination as cited in the Table 2 of PlanckXVI . The CMB constraints on $h$ and $\Omega_{m}$ become much weaker if one allows $w\neq-1$ or $\Omega_{k}\neq 0$, so for more general models BAO data or other constraints are needed to restore high precision on cosmological parameters.

A comprehensive set of relative luminosity distances of 740 SNIa was presented in Betoule14 , based on a joint calibration and training set of the SDSS-II Supernova Survey Sako14 and the Supernova Legacy Survey (SNLS) 3-year data set Conley11 . The 374 supernovae from SDSS-II and 239 from SNLS were combined with 118 nearby supernovae from Contreras10 ; Hicken09 ; Jha06 ; Altavilla04 ; Riess99 ; Hamuy96 and nine high-redshift supernovae discovered and studied by HST Riess07 . We use this set, dubbed Joint Light-curve Analysis (JLA), rather than the Union 2.1 compilation of Suzuki2012 because of the demonstrated improvement in calibration and corresponding reduction in systematic uncertainties presented in Betoule14 .

While Betoule14 also provide a full cosmomc module and a covariance matrix in relevant parameters, we here instead use their compressed representation of relative distance constraints due to conceptual simplicity and a drastic increase in computational speed when combining with other cosmology probes. The compressed information consists of a piece-wise linear function fit over 30 bins (leading to 31 nodes) spaced evenly in $\log{z}$ (to minimum $z\sim 0.01$) with a $31\times 31$ covariance matrix that includes all of the systematics from the original analysis. SNIa constrain relative distances, so the remaining marginalization required to use this compressed respresentation in a comological analysis is over the fiducial absolute magnitude of a SNIa, $M_{B}$. In Section IV we also utilize a similar compression of the Union 2.1 SN data set, which we have constructed in analogous fashion.

Figure 1 shows the “Hubble diagram” (distance vs. redshift) from a variety of recent BAO measurements of $D_{V}/r_{d}$, $D_{M}/r_{d}$, or $zD_{H}/r_{d}$; these three quantities converge at low redshift. In addition to the data listed in Table 2, we show measurements from the DR7 data set of SDSS-II by Percival10 and from the WiggleZ survey by Blake11 , which are not included in our cosmological analysis because they are not independent of the (more precise) BOSS measurements in similar redshift ranges. Curves represent the predictions of the fiducial Planck $\Lambda$CDM model, whose parameters are determined independently of the BAO measurements but depend on the assumptions of a flat universe and a cosmological constant. Overall, there is impressively good agreement between the CMB-constrained $\Lambda$CDM model and the BAO measurements, especially as no parameters have been adjusted in light of the BAO data. However, there is noticeable tension between the Planck $\Lambda$CDM model and the LyaF BAO measurements.

Figure 2 displays a subset of these BAO measurements with scalings that elucidate their physical content. In the upper panel, we plot $H(z)/(1+z)$, which is the proper velocity between two objects with a constant comoving separation of 1 Mpc. This quantity is declining in a decelerating universe and increasing in an accelerating universe. We set the $x$-axis to be $\sqrt{1+z}$, which makes $H(z)/(1+z)$ a straight line of slope $H_{0}$ in an Einstein-de Sitter ($\Omega_{m}=1$) model. For the transverse BAO measurements in the lower panel, we plot $c\ln(1+z)/D_{M}(z)$, chosen so that a constant (horizontal) line in the $H(z)/(1+z)$ plot would produce the same constant line in this panel, assuming a flat Universe. This quantity would decrease monotonically in a non-accelerating flat cosmology. The quantities in both the upper and lower panels approach $H_{0}$ as $z$ approaches zero, independent of other cosmological parameters. We convert the BOSS LOWZ and MGS measurements of $D_{V}(z)$ to $D_{M}(z)$ in the lower panel assuming the fiducial Planck $\Lambda$CDM parameters; this is a robust approximation because all acceptable cosmologies produce similar scaling at these low redshifts. Note that the $H(z)$ and $D_{M}(z)$ measurements from a given data set (i.e., at a particular redshift) are covariant, in the sense that the points on these panels are anti-correlated (see Table 2). For example, if $H(z)$ at $z=2.34$ were scattered upward by a statistical fluctuation, then the $z=2.34$ point in the lower panel would be scattered downward.

As discussed below in Section IV, the galaxy BAO and JLA supernova data can be combined to yield an “inverse distance ladder” measurement of $H_{0}$, which utilizes the CMB measurements of $\omega_{cb}$ and $\omega_{b}$ but no other CMB information. This value of $H_{0}$ is robust to a wide range of assumptions about dark energy evolution and space curvature, although it does assume a standard radiation background for the calculation of $r_{d}$. We plot the resulting determination of $H_{0}=67.3\pm 1.1\,{\rm km}\,{\rm s}^{-1}\,{\rm Mpc}^{-1}$ as the open square in both panels.

The grey swath in both panels of Figure 2 represents the 1$\sigma$ region for the fiducial Planck $\Lambda$CDM model, with the top panel clearly showing the transition from deceleration to acceleration at $z\approx 0.6$. Formally, we are scaling both panels by $(r_{d}/r_{d,{\rm fid}})$, so that the comparison of the BAO data points to the CMB prediction is invariant to changes in the sound horizon. The galaxy BAO measurements of $D_{M}(z)$ from BOSS and MGS are in excellent agreement with the predictions of this model (as are the other measurements shown previously in Fig. 1), and the combination of BAO and SNe yields an $H_{0}$ value in excellent agreement with this model’s prediction. The expansion rate $H(0.57)$ from CMASS is high compared to the model prediction, at moderate significance. Compared to Planck, the best-fit value of $\Omega_{m}h^{2}$ from the 9-year WMAP analysis WMAP9 is lower, 0.143 vs. 0.137, implying lower $\Omega_{m}$ and slightly higher $h$ for a $\Lambda$CDM model. The model using these best-fit parameters, shown by the dashed lines, agrees better with the CMASS $H(z)$ measurement but is in tension with the distance data, especially the CMASS value of $D_{M}(0.57)$.

The Ly$\alpha$ forest measurements are much more difficult to reconcile with the $\Lambda$CDM model: compared to the Planck curve, the LyaF BAO $H(z)$ is low and $[D_{M}(z)]^{-1}$ is high. It is important to keep the error anti-correlation in mind when assessing significance — if $H(z)$ fluctuates up then $1/D_{M}(z)$ will fluctuate down, which tends to reduce the tension relative to the CMB. However, our subseqent analyses (and those already reported by Delubac14 ) will show that the discrepancy is significant at the $2-2.5\sigma$ level. The dotted curves show predictions of cosmological models with $\Omega_{k}=0.01$ or $1+w=\pm 0.3$. While changing curvature or the dark energy equation-of-state can improve agreement with some of the data points, it worsens agreement with other data points, and on the whole (as demonstrated quantitatively in Section V) such variations do not noticeably improve the fit to the combined CMB, BAO, and SN data.

Not plotted in Figure 2 is the value of $D_{M}(1090)$ that comes from the angular acoustic scale in the CMB. Connecting the acoustic scale measured in CMB anisotropy to that measured in large-scale structure does require model assumptions about structure formation at the recombination epoch. However, it would be difficult to move the relative calibration significantly without making substantial changes to the CMB damping tail, which is already well constrained by observations. Using the ratio of $D_{M}(1090)/r_{d}$ in equation (19) and $r_{d}=147.49$ Mpc, we find $c\ln(1+z)/D_{M}(z)=151\,{\rm km}\,{\rm s}^{-1}\,{\rm Mpc}^{-1}$ at $z=1090$ with percent level accuracy, a factor of two larger than any of the low-redshift values in Figure 2. On their own, the BAO data in Figure 2 clearly favor a universe that transitions from deceleration at $z>1$ to acceleration at low redshifts, and this evidence becomes overwhelming if one imagines the corresponding CMB measurements off the far left of the plot. We quantify these points in the following section.

It is tempting to consider a flat cosmology with a constant $H/(1+z)$ as an alternative model of these data Melia12 . Note that although this form of $H(z)$ occurs in coasting (empty) cosmologies in general relativity, those models have open curvature and hence a sharply different $D_{M}(z)$. But even for the flat model, the data are not consistent with a constant $H(z)/(1+z)$, first because the increase in $c\ln(1+z)/D_{M}(z)$ from $z=0.57$ to $z=0.0$ is statistically significant, and second because of the factor of two change of this quantity relative to that inferred from the CMB angular acoustic scale. The change from $z=0.57$ to $z=0$ is more significant than the plot indicates because the data points are correlated; this occurs because the $H_{0}$ value results from normalizing the SNe distances with the BAO measurements. We measure the ratio of the values, $H_{0}D_{M}(0.57)/c\ln(1.57)$, to be $1.080\pm 0.014$ from the combination of BAO and SNe datasets, a 5.5$\sigma$ rejection of a constant hypothesis and an indication of the strength of the SNe data in detecting the low-redshift accelerating expansion.

For quantitative contraints, we start by considering BAO data alone with the simple assumption that the BAO scale is a standard comoving ruler, whose length is independent of redshift and orientation but is not necessarily the value computed using CMB parameter constraints. A similar analysis has been presented in 2013MNRAS.436.1674A . In this case, a simple dimensional analysis shows that in addition to fractional densities in cosmic components, one can constrain the dimensionless quantity $P=c/(H_{0}r_{d})$.

Figure 3 presents constraints on relevant quantities in $o\Lambda$CDM models, which assume that dark energy is a cosmological constant but allow $\Omega_{\Lambda}=0$ and arbitrary $\Omega_{k}$. The combination of galaxy and LyaF BAO measurements yields a marginalised constraint of $\Omega_{\Lambda}=0.73^{+0.25}_{-0.68}$ at 99.7% confidence, implying a $>3\sigma$ detection of dark energy from BAO alone without CMB data.

These constraints become much tighter if we assume that the CMB is measuring the same acoustic scale, functioning as an additional BAO experiment at a much higher redshift. As discussed in Section II.3, we implement this case by retaining the high-precision CMB measurement of $D_{M}(1090)/r_{d}$ but drastically inflating (by a factor of 100) the CMB errors on $\omega_{cb}$ and $\omega_{b}$, so that the value of $r_{d}$ itself remains effectively unknown. Combining the CMB measurement with galaxy or LyaF BAO alone yields a strong detection of non-zero $\Omega_{\Lambda}$, but with different central values reflecting the tensions already discussed in Section II.5 and examined further in Sections V–VI. Combining all three measurements yields a marginalised $\Omega_{\Lambda}=0.72^{+0.030}_{-0.034}$ (at 68% confidence, with reasonably Gaussian errors), implying $>20\sigma$ preference for a low-density universe dominated by dark energy. The dimensionless quantity $P=c/(H_{0}r_{d})=29.63^{+0.48}_{-0.45}$ is determined with 1.6% precision. Most importantly, this data combination also requires a nearly flat universe, with a total density $\Omega_{m}+\Omega_{\Lambda}=1.011^{+0.014}_{-0.016}$ determined to 1.5% and consistent with the critical density. Thus, with the minimal assumption that the BAO scale is a standard ruler, these data provide strong support for the standard cosmological model.

We proceed further by computing the sound horizon scale $r_{d}$ from the standard physics of the pre-recombination universe but adopting empirical constraints external to the CMB. In particular, we adopt a prior on the baryon density of $\omega_{b}=0.02202\pm 0.00046$ determined from big bang nucleosynthesis (BBN) and the observed primordial deuterium abundance Cooke14 , and we assume a standard relativistic background ($N_{\rm eff}=3.046$, $\omega_{\nu}\approx 0$). For any values of $\Omega_{m}$ and the Hubble parameter $h$ that arise in our MCMC chain, we can then compute the value of $r_{d}$ from equation (16). Compared to the previous section, the addition of the physical scale allows us to convert the measured value of $c/(H_{0}r_{d})$ into a measurement of the dimensional parameter $H_{0}$. In practice, we derive constraints in a separate MCMC run where, instead of a flat prior on $P$, we have a flat prior on $h$ and the above prior on $\omega_{b}$. We also fix the curvature parameter $\Omega_{k}$ to zero. Results are presented in Figure 4. The red (galaxy BAO) and blue (LyaF BAO) contours in this figure use no CMB information at all, but they do assume a spatially flat universe in contrast to Figure 3.

The point of this exercise is the following. The homogeneous part of the minimal $\Lambda$CDM model has just two adjustable parameters, $\Omega_{m}$ and $h$, which matches the two degrees of freedom offered by a measurement of anisotropic BAO at a single redshift. (The weak BBN prior is required to fix the magnitude of $r_{d}$, but it does not affect the expansion history.) One can therefore get meaningful constraints from either galaxy BAO or LyaF BAO alone, though this is no longer true if one allows non-zero curvature and therefore introduces a third parameter. There is substantial $\Omega_{m}-h$ degeneracy for either measurement individually, but both are generally compatible with standard values of these parameters. The tension of the LyaF BAO with the Planck $\Lambda$CDM model manifests itself here as a best fit at relatively low matter density and high Hubble parameter. Combining the galaxy and LyaF measurements produces a precise measurement of both $\Omega_{m}$ and the Hubble parameter coming from BAO alone, independent of CMB data. In combination, we find $h=0.67\pm 0.013$ and $\Omega_{m}=0.29\pm 0.02$ (68% confidence). The small black ellipse in Figure 4 shows the Planck constraints for $\Lambda$CDM, computed from full Planck chains, which are in excellent agreement with the region allowed by the joint BAO measurements.