Phys. Rev. D 92, 123516 (2015) - Cosmological implications of baryon acoustic oscillation measurements

Cosmological implications of baryon acoustic oscillation measurements

Éric Aubourg et al. (BOSS Collaboration)

Phys. Rev. D 92, 123516 – Published 14 December 2015

Abstract

We derive constraints on cosmological parameters and tests of dark energy models from the combination of baryon acoustic oscillation (BAO) measurements with cosmic microwave background (CMB) data and a recent reanalysis of Type Ia supernova (SN) data. In particular, we take advantage of high-precision BAO measurements from galaxy clustering and the Lyman- $α$ forest (LyaF) in the SDSS-III Baryon Oscillation Spectroscopic Survey (BOSS). Treating the BAO scale as an uncalibrated standard ruler, BAO data alone yield a high confidence detection of dark energy; in combination with the CMB angular acoustic scale they further imply a nearly flat universe. Adding the CMB-calibrated physical scale of the sound horizon, the combination of BAO and SN data into an “inverse distance ladder” yields a measurement of $H_{0} = 67.3 \pm 1.1 km s^{- 1} {Mpc}^{- 1}$ , with 1.7% precision. This measurement assumes standard prerecombination physics but is insensitive to assumptions about dark energy or space curvature, so agreement with CMB-based estimates that assume a flat $Λ CDM$ cosmology is an important corroboration of this minimal cosmological model. For constant dark energy ( $Λ$ ), our $BAO + SN + CMB$ combination yields matter density $Ω_{m} = 0.301 \pm 0.008$ and curvature $Ω_{k} = - 0.003 \pm 0.003$ . When we allow more general forms of evolving dark energy, the $BAO + SN + CMB$ parameter constraints are always consistent with flat $Λ CDM$ values at $\approx 1 σ$ . While the overall $χ^{2}$ of model fits is satisfactory, the LyaF BAO measurements are in moderate ( $2 - 2.5 σ$ ) tension with model predictions. Models with early dark energy that tracks the dominant energy component at high redshift remain consistent with our expansion history constraints, and they yield a higher $H_{0}$ and lower matter clustering amplitude, improving agreement with some low redshift observations. Expansion history alone yields an upper limit on the summed mass of neutrino species, $\sum m_{ν} < 0.56 eV$ (95% confidence), improving to $\sum m_{ν} < 0.25 eV$ if we include the lensing signal in the Planck CMB power spectrum. In a flat $Λ CDM$ model that allows extra relativistic species, our data combination yields $N_{eff} = 3.43 \pm 0.26$ ; while the LyaF BAO data prefer higher $N_{eff}$ when excluding galaxy BAO, the galaxy BAO alone favor $N_{eff} \approx 3$ . When structure growth is extrapolated forward from the CMB to low redshift, standard dark energy models constrained by our data predict a level of matter clustering that is high compared to most, but not all, observational estimates.

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Received 18 November 2014

DOI:https://doi.org/10.1103/PhysRevD.92.123516

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Vol. 92, Iss. 12 — 15 December 2015

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Figure 1
The BAO “Hubble diagram” from a world collection of detections. Blue, red, and green points show BAO measurements of $D_{V} / r_{d}$ , $D_{M} / r_{d}$ , and $z D_{H} / r_{d}$ , respectively, from the sources indicated in the legend. These can be compared to the correspondingly colored lines, which represent predictions of the fiducial Planck $Λ CDM$ model (with $Ω_{m} = 0.3183$ , $h = 0.6704$ ; see Sec. 2c). The scaling by $\sqrt{z}$ is arbitrary, chosen to compress the dynamic range sufficiently to make error bars visible on the plot. Filled points represent BOSS data, which yield the most precise BAO measurements at $z < 0.7$ and the only measurements at $z > 2$ . For visual clarity, the $Ly α$ cross-correlation points have been shifted slightly in redshift; autocorrelation points are plotted at the correct effective redshift.
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Figure 2
BAO measurements and model predictions of $H (z)$ and $D_{M} (z)$ as a function of redshift, with physically informative scalings. The top panel shows $H (z) / (1 + z)$ , the proper velocity between two objects 1 comoving Mpc apart. The bottom panel shows $c \ln (1 + z) / D_{M} (z)$ , a scaling that matches a constant line $H (z) = (1 + z) H_{0}$ in the top panel to the same constant line in the bottom panel for a flat universe. Filled circles and squares show the BOSS CMASS and LyaF measurements of $H (z)$ and $D_{M} (z)$ , respectively; we show the LyaF-quasar cross correlation as crosses to distinguish from the LyaF autocorrelation measurements. Filled triangles in the bottom panel show the BOSS LOWZ and MGS measurements of $D_{V} (z)$ converted to $D_{M} (z)$ . Open squares show the value of $H_{0} = 67.3 \pm 1.1 km s^{- 1} {Mpc}^{- 1}$ determined from the combination of BAO and SNIa data described in Sec. 4. The grey swath in both panels is the prediction from the Planck $Λ CDM$ cosmology including $1 σ$ parameter errors; in the top panel, one can easily see the model transition from deceleration to acceleration at $z \approx 0.6$ . The dashed line shows the $Λ CDM$ prediction using the best-fit WMAP parameters, which has lower $Ω_{m} h^{2}$ . Dotted curves show models that match the best-fit Planck values of $ω_{c b}$ , $ω_{b}$ , and $D_{M} (1090) / r_{d}$ but have $Ω_{k} = 0.01$ (blue), $w = - 0.7$ (green), or $w = - 1.3$ (red). The $x$ axis is set to $\sqrt{1 + z}$ both for display purposes and so that a pure matter universe ( $Ω_{m} = 1$ ) appears as a decreasing straight line on the top panel.
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Figure 3
Constraints from BAO on the parameters of $o Λ CDM$ models, treating the BAO scale as a redshift-independent standard ruler of unknown length. Green curves/contours in each panel show the combined constraints from galaxy and LyaF BAO, with no CMB information. Black curves/contours include the measurement of $D_{M} (1090) / r_{d}$ from the CMB acoustic scale, again with no assumption about the value of $r_{d}$ except that it is the same scale as the lower redshift measurements. This combination of BAO measurements yields precise constraints on $Ω_{Λ}$ (top panel) and the dimensionless quantity $c / (H_{0} r_{d})$ (bottom panel), and it requires a low density ( $Ω_{m} \approx 0.29$ ), nearly flat universe (middle panel). Blue and red curves in the top and bottom panels show the result of combining the CMB BAO measurement with either the galaxy or LyaF BAO measurement separately. The dotted line in the middle panel marks $Ω_{m} + Ω_{Λ} = 1$ .
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Figure 4
Constraints on $Ω_{m}$ and $h$ in a flat $Λ CDM$ model from galaxy BAO (red), LyaF BAO (blue), and the combination of the two (green), using a BBN prior on $ω_{b}$ and standard physics to compute the sound horizon $r_{d}$ but incorporating no CMB information. Contours are plotted at 68%, 95%, and 99.7% confidence (the interior white region of the green “donut” is 68%). Black contours show the entirely independent constraints on $Ω_{m}$ and $h$ in $Λ CDM$ from full Planck CMB chains.
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Figure 5
Determination of $H_{0}$ by the “inverse distance ladder” combining BAO absolute distance measurements and SNIa relative distance measurements, with CMB data used to calibrate the sound horizon scale $r_{d}$ . The quantity $c \ln (1 + z) / D_{M} (z)$ converges to $H_{0}$ at $z = 0$ . Filled circles show the four BAO measurements, normalized with $r_{d} = 147.49 Mpc$ ; for the three lower redshift points, $D_{V}$ has been converted to $D_{M}$ assuming $Λ CDM$ . Crosses show the SNIa measurements, with error bars representing diagonal elements of the covariance matrix. Because the absolute luminosity of SNIa is not known a priori, the SNIa points are free to shift vertically by a constant factor, which is chosen here to produce the best joint fit with the BAO data. The red square and error bar shows the value $H_{0} = (67.3 \pm 1.1) km s^{- 1} {Mpc}^{- 1}$ determined by the full inverse distance ladder procedure described in the text. The black curve shows the prediction for a $Λ CDM$ model with $Ω_{m} = 0.3$ and the best-fit $H_{0}$ , and green curves show ten PolyCDM models randomly selected from our MCMC chain that have $Δ χ^{2} < 4$ relative to the best-fit PolyCDM model. This $H_{0}$ determination assumes standard prerecombination physics to evaluate $r_{d}$ . For nonstandard energy backgrounds (e.g., extra relativistic species or early dark energy) the more general result is described by Eq. (23).
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Figure 6
Constraints on the Hubble constant $H_{0}$ from this paper’s inverse distance ladder analysis (blue, at bottom), from three direct distance ladder estimates (red, at top), and from Planck or WMAP CMB data assuming $Λ CDM$ (green, middle). All error bars are $1 σ$ . The inverse distance ladder estimates assume $r_{d} = 147.49 \pm 0.59 Mpc$ , based on Planck constraints for a standard radiation background, while the green points make the much stronger assumptions of a flat universe with a cosmological constant.
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Figure 7
BAO constraints in the $D_{M} - D_{H}$ planes at $z = 0.57$ (left) and $z = 2.34$ (middle) compared to predictions of $o Λ CDM$ (top row) or $w CDM$ (bottom row) models constrained by CMB data. Black curves show 68%, 95%, and 99.7% likelihood contours from the CMASS and LyaF BAO measurements, relative to the best-fit values (black dots). Colored points represent individual models from $Planck + WP + ACT / SPT$ MCMC chains, which are color-coded by the value of $Ω_{k}$ (top row) or $w$ (bottom row) as illustrated in the right panels. Green cross-hairs mark the predictions of the flat $Λ CDM$ model that best fits the CMB data. White curves show 68% and 95% likelihood contours for the CMB data alone.
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Figure 8
Constraints on interesting parameter combinations in a variety of dark energy models: $Λ CDM$ (upper left), $o Λ CDM$ (upper right), $w CDM$ (middle left), $o w CDM$ (middle right), $w_{0} w_{a} CDM$ (bottom left), and $o w_{0} w_{a} CDM$ (bottom right). Curves show 68%, 95%, and 99.7% confidence contours for the data combinations indicated in the legend. In the top panels the red contours are almost fully obscured by the green contours because the $BAO + Planck$ combination is already as constraining as the $BAO + SN + Planck$ combination, but for models with freedom in dark energy the SN and BAO constraints are complementary. The bottom panels, with evolving $w (z)$ , display the value of $w$ at $z = 0.266$ , the “pivot” redshift where $w$ is best constrained by $BAO + SN + Planck$ in the $w_{0} w_{a} CDM$ model. For our $BAO + SN + Planck$ contours, the white zone interior to the dark green annulus marks the 68% confidence region, and the outer edge of the dark annulus is 95%.
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Figure 9
Constraints on $h$ and $δ w_{0} \equiv 1 + w_{0}$ in the slow-roll dark energy model [Eq. (24)], in the same format as Fig. 8.
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Figure 10
$χ^{2}$ values for the best-fit versions of cosmological models considered in the paper. Each bar represents the minimum $χ^{2}$ for the model listed at the left axis, and colors show the $χ^{2}$ contributions of individual data sets. For better visualization, we subtract 30 from the SN $χ^{2}$ . CMB contributions are not included but (with our 3-parameter compression) are always close to zero. The total $χ^{2}$ and model degrees of freedom (d.o.f., 40 data points minus number of fit parameters, which includes the SNIa absolute magnitude normalization as well as cosmological quantities) are listed to the right of each bar. The bottom bar shows the number of d.o.f. associated with each data set. For the $Δ N_{eff}$ model we use cosmomc rather than our compressed CMB description, but we again omit CMB contributions to $χ^{2}$ .
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Figure 11
Constraints on $ρ_{D E} (z)$ assumed to be constant within redshift bins, in units of the present-day critical density $ρ_{c}$ . Shaded areas represent 68% confidence levels. Yellow bands show constraints in the same bins from [67]. Our constraints in the $z = 1.0 - 1.6$ bin are omitted.
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Figure 12
Predicted BAO scales for early dark energy models with CMB observables $Ω_{b} h^{2}$ , $Ω_{m} h^{2}$ , and $D_{M} (1090) / r_{d}$ held fixed to the values of the best-fit $Planck + WP$ $Λ CDM$ model. We adopt Eq. (25) with $w_{0} = - 1$ . Solid lines show the case in which $r_{d}$ is rescaled by $(1 - Ω_{de}^{e})^{1 / 2}$ to represent the effect of early dark energy in the prerecombination era, while dashed lines show the case in which $r_{d}$ is held fixed at the fiducial model value of $r_{d} = 147.49 Mpc$ . We show ratios of $D_{H} (z) / r_{d}$ (top) or $D_{M} (z) / r_{d}$ (bottom) relative to the fiducial ( $Ω_{de}^{e} = 0$ ) model. Points with error bars show the BAO measurements from CMASS galaxies at $z = 0.57$ (filled square) and from LyaF autocorrelation (open circle) and cross correlation (filled circle) at $z = 2.34$ . For visual clarity, the LyaF cross-correlation points have been slightly shifted in redshift.
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Figure 13
Constraints in the $Ω_{de}^{e} - Ω_{m}$ plane from the combination of our compressed CMB description with $galaxy + LyaF$ BAO data. Black contours (68%, 95%, and 99.7%) show the tight constraints on early dark energy for models with fixed $r_{d} = 147.49 Mpc$ . Blue contours show the constraints for models with $r_{d} \propto (1 - Ω_{de}^{e})^{1 / 2}$ as expected if $Ω_{de}^{e}$ is constant into the radiation-dominated era. The red solid line traces the parameter degeneracy $Ω_{m} = Ω_{m, fid} (1 - Ω_{de}^{e})$ predicted by the approximate scaling arguments described in the text.
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Figure 14
Constraints on decaying dark matter from CMB, SN and $galaxy + LyaF$ BAO data. For various choices of the fraction $f_{x}$ of dark matter in the decaying component, we plot posterior probability distributions for the product $λ f_{x}$ , where $λ$ is the decay constant, assuming a flat prior on $λ$ vs value of $Ω_{m}$ today (which, by definition, includes both the nondecaying component and the undecayed fraction of the decaying component). Removing SN data does not significantly relax these constraints.
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Figure 15
BAO constraints in the $D_{M} - D_{H}$ planes at $z = 0.57$ (left) and $z = 2.34$ (middle) compared to predictions of CMB-constrained, flat $Λ CDM$ models in which the neutrino mass $\sum m_{ν}$ or the number of relativistic species $N_{eff}$ is a free parameter. Black curves show 68%, 95%, and 99.7% likelihood contours from the CMASS and LyaF BAO measurements, relative to the best-fit values (black dots). Colored points represent individual models from $Planck + WP + ACT / SPT$ MCMC chains, which are color-coded by the value of $\sum m_{ν}$ (top row) or $N_{eff}$ (bottom row) as illustrated in the right panels. Green cross-hairs mark the predictions of the flat $Λ CDM$ model with $\sum m_{ν} = 0.06 eV$ and $N_{eff} = 3.046$ that best fits the CMB data. White curves show 68% and 95% likelihood contours for the CMB data alone. CMB results in the top row are marginalized over the lensing parameter $A_{L}$ .
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Figure 16
Neutrino mass constraints for several combinations of data and model freedom. In the top panel, the red curve shows the posterior pdf (with a flat prior) on $\sum m_{ν}$ from the expansion history constraint on the neutrino mass, based on the combination of BAO with our compressed CMB description, which yields $\sum m_{ν} < 0.56 eV$ at 95% confidence. The green curve shows the result obtained by replacing our compressed CMB description with the full $Planck + WP$ power spectrum using cosmomc, which strengthens the upper limit to $\sum m_{ν} < 0.25 eV$ . The blue curve adopts the same data combination but additionally marginalizes over the parameter $A_{L}$ , demonstrating that the difference between the green and red curves is driven mainly by the lensing amplitude information in the Planck data. In the lower panel, curves show the posterior pdf of $\sum m_{ν}$ from our usual $BAO + Planck$ (thick) or $BAO + SN + Planck$ (thin) geometrical constraints, assuming $Λ CDM$ , $w CDM$ , or $o Λ CDM$ (red, blue, green, respectively).
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Figure 17
Constraints on the effective number of relativistic species $N_{eff}$ , assuming $w = - 1$ and $Ω_{k} = 0$ . These contours use full $Planck + WP$ CMB constraints computed with cosmomc, combined with LyaF BAO only (red) or with our full set of BAO measurements (green). We marginalize over the tensor-to-scalar ratio $r$ .
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Figure 18
A “tuned oscillation” model in which a Gaussian perturbation of the $Λ CDM$ $D_{M} (z)$ is introduced to allow a good simultaneous fit to the galaxy and LyaF BAO data. Solid lines show the same $Λ CDM$ model plotted in Fig. 1, while the dashed line shows the perturbed model.
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Figure 19
Implied variation in the energy density of the dark energy component for the model shown in Fig. 18. The dotted line corresponds to the density becoming negative. These plots illustrate the difficulty of concurrently fitting the LyaF and galaxy BAO constraints on $D_{M}$ and $D_{H}$ while satisfying the CMB constraint on $D_{M}$ .
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Figure 20
Predictions of matter clustering from our $BAO + SN + CMB$ constrained models compared to observational estimates. The vertical location of the observational estimates (red points) is arbitrary. Labels for the model points (black) in all panels are indicated along the left vertical axis. Panel (a) shows the $z = 0$ parameter combination $σ_{8} (Ω_{m} / 0.3)^{0.4}$ , which approximately describes the quantity best constrained by low-redshift measurements of the cluster mass function or weak lensing. Black points show the mean and $1 σ$ range computed from our model chains, and red points show observational estimates discussed in the text. Panel (b) presents a similar comparison for $σ_{8} (z = 0.57) f (z = 0.57)$ , constrained by redshift-space distortions in CMASS galaxy clustering. Panel (c) compares $σ_{8} (z = 2.5)$ to an estimate from the BOSS LyaF 1-d power spectrum. Observational sources are the cosmic shear measurements of Hey13 [104] and Jee13 [105], the galaxy-galaxy lensing measurement of Man13 [106], the cluster mass function measurements of Vik09 [107], Roz10 [108], Pla13 [109], and Man14 [110], the RSD measurements of Beu14 [111], Sam14 [112], and Rei14 [113], and the LyaF power spectrum measurement of Pal13 [114]. Dotted vertical lines are provided for visual reference.
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Physical Review D

covering particles, fields, gravitation, and cosmology

Cosmological implications of baryon acoustic oscillation measurements

Éric Aubourg et al. (BOSS Collaboration)

Phys. Rev. D 92, 123516 – Published 14 December 2015

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Physical Review D

covering particles, fields, gravitation, and cosmology

Cosmological implications of baryon acoustic oscillation measurements

Éric Aubourg et al. (BOSS Collaboration)

Phys. Rev. D 92, 123516 – Published 14 December 2015

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