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A 2% Distance to z = 0.35 by Reconstructing Baryon Acoustic Oscillations - III : Cosmological Measurements and Interpretation

We use the 2% distance measurement from our reconstructed baryon acoustic oscillations (BAOs) signature using the Sloan Digital Sky Survey (SDSS) Data Release 7 (DR7) Luminous Red Galaxies (LRGs) from Padmanabhan et al. (2012) and Xu et al. (2012) combined with cosmic microwave background (CMB) data from Wilkinson Microwave Anisotropy Probe (WMAP7) to measure parameters for various cosmological models. We find a 1.7% measurement of H_0 = 69.8 +/- 1.2 km/s/Mpc and a 5.0% measurement of Omega_m = 0.280 +/- 0.014 for a flat Universe with a cosmological constant. These measurements of H_0 and Omega_m are robust against a range of underlying models for the expansion history. We measure the dark energy equation of state parameter w = -0.97 +/- 0.17, which is consistent with a cosmological constant. If curvature is allowed to vary, we find that the Universe is consistent with a flat geometry (Omega_K = -0.004 +/- 0.005). We also use a combination of the 6 Degree Field Galaxy Survey BAO data, WiggleZ Dark Energy Survey data, Type Ia supernovae (SN) data, and a local measurement of the Hubble constant to explore cosmological models with more parameters. Finally, we explore the effect of varying the energy density of relativistic particles on the measurement of H_0.


INTRODUCTION
Since the discovery of the accelerated expansion of the Universe (Riess et al. 1998;Perlmutter et al. 1999), there has been a growing interest to understand the nature of dark energy and measure various cosmological parameters. This understanding requires improved measurements of the expansion history of the Universe via the distance-redshift relation. In particular, baryon acoustic oscillations (BAO) have been widely used to study this relation by measuring cosmic distances. The physics behind these oscillations is well understood (Sakharov 1967;Peebles & Yu 1970;Sunyaev & Zeldovich 1970;Bond & Efstathiou 1984, 1987Hu & Sugiyama 1996;Hu et al. 1997;Hu & Dodelson 2002). In the pre-recombination era of the Universe, the baryons were coupled to the photons in a hot plasma. Small overdensities in the underlying dark matter distribution caused the baryons and photons to fall into the overdensities due to gravity. As the plasma density grows, the radiation pressure from the photons drive an acoustic wave of baryons and photons around the original dark matter overdensity. As the Universe cools, the electrons and protons combine to form atoms, and the photons decouple from the baryons causing the sound speed in the plasma to drop dramatically. This leaves the baryons in a spherical shell around the initial overdensity. This shell has a characteristic scale of about 150 Mpc, defined by the distance traveled by the acoustic wave in the pre-recombination era, and its angular scale has been measured in the cosmic microwave background (CMB) to be about 1 • (Bennett et al. 2003;Jarosik et al. 2011). Under the influence of gravity, these overdensities grow and form galaxies imprinting the characteristic acoustic scale into the distribution of galaxies (Hu & Sugiyama 1996;Meiksin et al. 1999). Thus, the BAO scale can be used as a robust standard ruler in large galaxy surveys (Tegmark 1997; Goldberg & Strauss 1998;Efstathiou & Bond 1999) with an important application to the study of dark energy (Eisenstein 2002;Blake & Glazebrook 2003;Hu & Haiman 2003;Linder 2003;Seo & Eisenstein 2003). The large physical size of this acoustic scale causes the standard ruler to be highly accurate (Eisenstein et al. 2007b;Seo et al. 2008;Seo et al. 2010;Mehta et al. 2011).
The BAO signal was first measured in the SDSS LRG survey and the 2dF Galaxy survey (Eisenstein et al. 2005;Cole et al. 2005) and has since been observed in multiple surveys SDSS (Tegmark et al. 2006;Percival et al. 2007Percival et al. , 2010Kazin et al. 2010;Chuang et al. 2010), 6dF Galaxy Survey (Beutler et al. 2011), and the WiggleZ Dark Energy Survey (Blake et al. 2010(Blake et al. , 2011a. Weinberg et al. (2012) provides an overall review of observational cosmology and discusses the current state of the field in depth.
The acoustic scale hence gives us a measurement of the distance to a given redshift. Padmanabhan et al. (2012) (hereafter Paper I) presents the BAO measurements via the correlation function in the SDSS Luminous Red Galaxies (LRG) Data Release 7 (DR7) dataset using the reconstruction technique first introduced by Eisenstein et al. (2007a). Xu et al. (2012) (hereafter PaperII) describes a robust methodology to measure the acoustic scale, which is heavily tested against the LasDamas mock catalogs. Also shown in PaperII are the results of the testing performance of the reconstruction technique to improve the BAO measurement. The reconstruction technique improves the distance measurement to z = 0.35 to a 1.9% measurement compared to a 3.5% measurement before reconstruction. We show in this paper how this new measurement of the acoustic scale in SDSS helps improve our measurements of the cosmological parameters over a wide range of cosmological models.
The CMB angular acoustic scale gives us a distance measurement to the redshift at recombination that helps us break the degeneracy between Ωm and H0, therefore precisely measuring the parameters in the flat ΛCDM or "vanilla" cosmological model. However, with higher dimensional models, we need to have more distance measurements to break degeneracies between various cosmological parameters. We show how BAO data helps break these degeneracies by providing a second distance measurement at low redshift. We extend our redshift range to lower redshifts by adding SN data.
The combination of degree-scale CMB anisotropy, large-scale structure, and SN Ia data offers powerful constraints on cosmology and dark energy. Notable early papers include Efstathiou et al. (2002), Percival et al. (2002), Spergel et al. (2003), and Tegmark et al. (2004). With the discovery of the acoustic peak in the large-scale clustering of galaxies, the results from the WMAP satellite, and the construction of yet-larger supernova samples, these constraints have gotten increasingly precise. Many papers have combined these data sets; some recent examples include Komatsu et al. (2009) In this paper, we use the results of this reconstructed SDSS data in conjunction with CMB measurements from the Seven-year Wilkinson Microwave Anisotropy Probe (WMAP7) (Jarosik et al. 2011;Komatsu et al. 2011), Type Ia supernovae (SN) measurements from the 3 year Supernovae Legacy Survey (SNLS) (Conley et al. 2011), and direct measurement of the Hubble constant from the SH0ES project (Riess et al. 2011). To break degeneracies between different parameters, we use additional BAO data from the 6dF Galaxy Survey (Beutler et al. 2011), the WiggleZ Dark Energy Survey (Blake et al. 2011b), and SN data from the 3 year Supernova Legacy Survey (SNLS3) by Conley et al. (2011).
We start with the concordance cosmological model, ΛCDM (which here denotes a flat Universe), and add other cosmological parameters to explore higher dimensional models. We vary the curvature of the Universe, ΩK , and the constant dark energy equation of state, w0 independently for the oCDM and wCDM models respectively. For higher dimensionality, we vary both ΩK and w0 simultaneously in the owCDM model and in the w0waCDM model we assume a flat Universe but allow the dark energy equation of state parameter to vary in time. We allow all three parameters: ΩK , w0, wa to vary in our most general model, ow0waCDM.
In Section 2, we describe the Markov Chain Monte Carlo fitting and the various datasets we used in this study. We introduce the results of the BAO data and describe the cosmological implications in Section 3.1. In Sections 3.2 to 3.7, we show our results for various cosmological models. In Section 3.8 we show the robustness of our measurements of Hubble constant and the matter density over different models for the expansion history. Section 3.9 explores a possibility to solve an apparent tension in the H0 measurement by varying the energy density of relativistic species. We conclude with a summary of our results in Section 4.

METHODOLOGY
Here, in the third paper of this series, we use the reconstructed SDSS DR7 LRG results presented in PaperI and PaperII to measure cosmological parameters. We use a reconstruction technique first introduced in Eisenstein et al. (2007a), to model and remove effects of non-linear evolution of large scale structure and large scale velocity flows (redshift space distortions). As shown in Mehta et al. (2011), this method can also be applied to biased tracers of the matter density distribution, such as LRG. This effect was tested using N-body simulations (Noh et al. 2009;Mehta et al. 2011). As shown in PaperI and PaperII, reconstruction improves the measurement on the acoustic scale by about 40%. Therefore, in this paper we use the reconstructed BAO data from SDSS DR7 (hereafter BAO) unless otherwise specified. In practice, the BAO data measures the acoustic scale relative to some fiducial cosmology. This ratio of the acoustic scales is defined to be α, which is given by where DV is the spherically averaged distance scale to the pivot redshift, DV (z) = (D 2 A (z)cz/H(z)) 1/3 , rs is the sound  , for a given combination of (Ωmh 2 , Ω b h 2 ). Both definitions agree to within 0.2% level even for cosmologies 5σ away from the current WMAP7 constraints. Our fiducial cosmology with the WMAP7 1-sigma errors are shown as the grey cross. We use the Markov-Chain Monte Carlo code CosmoMC (http://cosmologist.info/cosmomc/) to compute the constraints on the cosmological parameters (Lewis & Bridle 2002). The original BAO routine in CosmoMC was replaced in order to use information from the probability distribution function of the BAO peak location p(α) from the χ 2 fitting described in PaperII. Using p(α), we estimate the likelihood of the acoustic scale α that corresponds to the cosmological parameters at a given step in the Markov chain. After the Markov chains converge, CosmoMC outputs the posterior probability distribution for each of the cosmological parameters, given the observations. Table 1 gives the mean values and the RMS errors of the cosmological parameters for various cosmological models.
A relevant issue that is often overlooked is that the sound horizon rs has several definitions in the literature. Its value depends on the definition of parameters such as z drag , the redshift at which the electrons are no longer dragged by the photons due to Compton scattering. In this paper, we use the definition of z drag proposed in   (hereafter EH98, to compute the sound horizon, rs. The difference between the definition of rs used in this paper and the implementation used in CAMB, is only a few percent for a wide range in Ωm and Ω b . In Figure 1 we show the relative difference of rs/r fid s between the EH98 and CAMB definitions as a function of Ωmh 2 and Ω b h 2 , and find that the differences are negligible for our analysis. Thus the definition dependence cancels out when computing α except for a small residual (≃ 0.1%) difference.
In our CosmoMC chains, we use the WMAP7 data (Komatsu et al. 2011) as our base dataset defined as "CMB". We then add our SDSS DR7 LRG reconstructed BAO data (PaperI and PaperII) to get the "CMB+BAO" dataset. We also include the other two latest BAO measurements from the 6-degree Field Galaxy Survey (6dFGS) (Beutler et al. 2011) and the WiggleZ Dark Energy Survey (Blake et al. 2011a). The combination of WMAP7 and all BAO datasets is denoted by "CMB+AllBAO". While the SDSS and 6dFGS provide single redshift points, the WiggleZ survey measures BAO in three correlated redshift slices. We use all three redshift slices in our code and use their covariance matrix to account for the covariant points. Conley et al. (2011) provide a covariance matrix analysis of Type Ia supernova cosmology from the 3 year Supernovae Legacy Survey (SNLS3) accompanied by a CosmoMC module. We use their dataset and module in conjunction with the WMAP7 to create the "CMB+SN" dataset and add to our SDSS BAO data to create the "CMB+BAO+SN" dataset. Finally, we also use the direct H0 measurement by Riess et al. (2011) and combine it with WMAP7, our SDSS BAO, and SNLS3 dataset into the "CMB+BAO+H0+SN" dataset. In the next section, we discuss the various cosmological parameters we measure using these datasets and how adding various datasets help measure and constrain various parameters in high dimensional cosmological models.

Cosmology with BAO Data
PaperI applies reconstruction to the SDSS DR7 LRG BAO dataset and PaperII shows the robustness of our BAO measurements. After using reconstruction, we measure DV (z = 0.35)(r fid s /rs) = 1356 ± 25 Mpc and DV (z = 0.35)/rs = 8.88 ± 0.17 giving us a 1.9% measurement of the distance to z = 0.35. We can combine our BAO measurement with the measurements from 6dFGS (Beutler et al. 2011) (DV = 456 ± 27 Mpc to z = 0.106) and WiggleZ Dark Energy Survey (Blake et al. 2011b) (DV = 2.23 ± 0.11 Gpc to z = 0.6) to make this BAO Hubble diagram. We have combined the three correlated WiggleZ redshift slices into one data point in order to show only uncorrelated points. WMAP has measured the angular acoustic scale to about 0.1% and has measured the baryon density with enough precision that its contribution to the sound horizon uncertainty is subdominant. Therefore, for any given value of Ωmh 2 , we have a precise prediction for the sound horizon. Given an exact statement of the spatial curvature and w(z), here flat ΛCDM, only one value of Ωm (and hence H0) will satisfy the angular acoustic scale. Hence, each value of Ωmh 2 makes a unique prediction for DV (z)/rs. We plot this prediction in Figure 2 for the best-fit value of Ωmh 2 = 0.1351 as the solid black line, and the 1-sigma range of Ωmh 2 = 0.1351 ± 0.0051 as the shaded region. This line is not a fit to the BAO data.
We see that the BAO data are a remarkable match to the WMAP7 ΛCDM prediction. To focus on the residuals, in Figure 3, we normalize the data to the WMAP7 best-fit model. Also plotted as the open square is the Percival et al. (2010) BAO data point at z = 0.275. Again, we see that the BAO data are consistent with ΛCDM. As in Figure 2, Figure 2. 6dFGS, reconstructed SDSS DR7, and WiggleZ BAO data points. The black line represents the ΛCDM prediction using WMAP7 data only (Komatsu et al. 2011). The shaded gray region is the effect of varying Ωmh 2 within the 1σ measurement errors of WMAP7. We see that the BAO data is consistent with the ΛCDM cosmological model.
for any assumption of ΩK and w(z), WMAP7 predicts a region on this plot with the width set by the uncertainty in Ωmh 2 . In this figure, we explore the effects of varying the equation of state parameter, w and the curvature of the Universe ΩK respectively. The red region corresponds to a flat Universe with w = −0.7, while the blue region corresponds to a Universe with a cosmological constant and ΩK = 0.01. Ωm is adjusted to keep the sound horizon constant. From this figure, we see that changing w mostly changes the slope of the line on this plot while a non-zero ΩK mostly changes the vertical offset. The relative distance measure from comparing the flux of SN constrain only the slope of the lines, while the BAO data can measure an absolute distance and hence the vertical offset. This explains why SN data is more effective at constraining w, while the BAO data is more effective at constraining ΩK . The Riess et al. (2011) direct H0 measurement is also plotted in this figure assuming the fiducial sound horizon value. While the sound horizon varies by about 1% within the WMAP7 results, this effect is subdominant to the quoted errors on H0. We explore the apparent tension between the BAO measurement and the direct measurement of H0 in Section 3.9.
Conventionally, the Hubble constant has been measured by building a distance ladder from local measurements out to measuring the cosmological Hubble flow. Conversely, the CMB and BAO data build an inverse distance ladder starting from a distance measurement at the recombination epoch. The CMB data provides an accurate measurement of the distance to the recombination redshift and our BAO data provides a measurement of distance to z = 0.35, thereby building an inverse distance ladder. The combination of these two datasets has the power to distinguish between different cosmological models. The supernovae data extrapolate the distance measurements to lower redshift and, therefore, precisely measure the expansion of the Universe at z = 0, which is the Hubble constant, H0. In the following sections we use a combination of these datasets to explore a variety of cosmological models, and we use the The shaded regions around these lines correspond to 1σ uncertainty in Ωmh 2 around the WMAP7 measurement. We see that the BAO data has the power to distinguish between various cosmological models. The H 0 point is the direct H 0 measurement from Riess et al. (2011).
CMB+BAO+SN dataset to obtain robust measurements of H0 and Ωm.

ΛCDM: The Vanilla Model
The WMAP7 measurements of the CMB give us very good measurements of the various parameters in the "vanilla cosmology" model, also known as the ΛCDM model. In the CosmoMC code, we vary the standard CDM parameters of matter and baryon densities (Ωm, Ω b ), the primordial spectrum amplitude and slope (ns), matter clustering amplitude (σ8), and the optical depth to reionization (τ ). Adding BAO measurement to the WMAP7 results improves the measurement of Ωm by about 40% and H0 by almost 30%. With reconstruction, we measure Ωm = 0.280 ± 0.014 and H0 = 69.8 ± 1.2 km/s/Mpc giving us a 1.7% measurement of the Hubble constant. Figure 4 shows the 68% and 95% confidence level contours for H0 vs Ωm and we can see the improvement in these parameters by adding the BAO data. Table 1 shows the values for Ωmh 2 , Ωm, and H0 for various cosmological models and the corresponding datasets used.
The acoustic standard ruler is calibrated by the WMAP measurement of Ωmh 2 . Komatsu et al. (2011) shows that allowing for a running spectral index, dns/d ln k increases the errors on Ωmh 2 . Thus, we explore the effects of varying the running spectral index, dns/d ln k with the CMB and CMB+BAO datasets. We note that the nuisance parameters used in our BAO fitting techniques (PaperII) make our measurement of DV /rs insensitive to the running spectral index. Table 2 shows the effect of varying the running spectral index on cosmological parameters. We see that the running spectral index is consistent with 0: dns/d ln k = −0.024 ± 0.020 using the CMB+BAO dataset. We find that including this parameter in the case of CMB data only, the Ωmh 2 measurements are degraded by a factor of 1.4 from Table 1. From top to bottom, the blocks correspond to the flat ΛCDM model (Section 3.2), oCDM (Section 3.3), wCDM (Section 3.4), owCDM (Section 3.5), w 0 waCDM (Section 3.6), and ow 0 waCDM (Section 3.7) respectively. The first column shows the data set used in each case, whereas the rest of the columns show the cosmological parameter values, with their uncertainties indicated in parenthesis. Empty values correspond to the cases in which the parameter is kept fixed to its fiducial value, i.e. Ω K = 0, w 0 = −1, wa = 0  Ωmh 2 = 0.1341 ± 0.0056 to 0.1393 ± 0.0080. This corresponds to an increased uncertainty in the measurements of Ωm, H0, and the spectral index ns. Adding the BAO data improves the measurement of Ωmh 2 , Ωm, H0, and ns and are consistent with the values with no running spectral index. We also note that the value of ns is less than 1.0 in all cases which is expected by typical inflation models.

oCDM: Varying Spatial Curvature
The BAO measurements calibrate the acoustic scale at low redshifts to the high redshift measurement from the CMB data. Therefore, the BAO accurately measures the curvature of the Universe (see Figure 3). As shown in the previous section, in the ΛCDM model, the CMB breaks the Ωm − H0 degeneracy with a distance measurement to the recombination redshift. However, when we vary the curvature parameter ΩK in the oCDM model, the CMB data has a degeneracy between Ωm, ΩK , and H0. The BAO measurement adds a second distance measurement in the inverse distance ladder, breaking this degeneracy and significantly improving these measurements. Using the BAO data, we find that the Universe is consistent with a flat geometry: ΩK = −0.004 ± 0.005. Figure 5 shows the improvements in the H0 vs Ωm, ΩK vs Ωm and ΩK vs H0 contours. From these different panels and Table 1, we clearly see that the CMB degeneracies between these three parameters are greatly reduced by adding the BAO data.
The "AllBAO" dataset gives us BAO measurements at various redshifts: z = 0.106 (6dF Galaxy Survey), 0.35 (reconstructed SDSS DR7), 0.6 (WiggleZ Dark Energy Survey). Figure 6 shows the 68% confidence level contours for various data sets. Table 1 provides the values for the cosmological parameters. From the table and Fig. 6, we see that the SN and additional BAO data add little to constrain the parameters over the CMB+BAO dataset.

wCDM: Varying the Constant Dark Energy Equation of State Parameter
In this section, we allow the dark energy equation of state parameter w to vary and we measure its value. Using the CMB+BAO dataset, we measure the equation of state parameter w = −0.97 ± 0.17, which consistent with a cosmological constant (w = −1). Figure 7 shows the 68% and 95% confidence level contour plots for H0, Ωm and w. The combination of low redshift (SDSS DR7) and high redshift (WMAP7) measurement of the acoustic scale measures the expansion of the Universe and helps measure the equation of state parameter for dark energy. We see that the combined CMB and BAO data precisely measures H0, Ωm and w as listed in Table 1. Similar to the oCDM case (section 3.3), we see that the CMB alone provides a robust measurement of Ωmh 2 but adding the BAO data breaks the degeneracy between the H0, Ωm, and w. We compare these measurements of H0, Ωm, and w with values for different datasets. The CMB+AllBAO dataset slightly improves our measurements on w and H0 as shown in Table 1. Figure 8 shows the 68% confidence level contours for H0, Ωm and w. In the oCDM case, the BAO data precisely measured ΩK , Ωm, and H0 and we see no additional improvement by adding the SN data. In this case, however, we see that adding the SN data improves our measurements of w, Ωm, and H0. This is expected from Figure 3 as the BAO data constrains the vertical offset rather than the slope of the lines. We see that while all the measured values are consistent with each other within 1σ, the CMB+BAO dataset results tend to favor lower H0 values and therefore higher values of Ωm and w compared to the CMB+BAO+SN dataset. However, as Figure 8 shows, the different data sets are consistent with the ΛCDM value of w = −1.

owCDM: Varying Curvature and the Constant Dark Energy Equation of State Parameter
Next, we move onto models where we vary two extra parameters in addition to the ΛCDM model parameters. In this case, we choose to vary both the dark energy equation of state parameter w, and the curvature parameter ΩK as free parameters. Figure 9 shows the 68% and 95% confidence level contours for w vs ΩK for the CMB+AllBAO, CMB+SN, and CMB+BAO+SN datasets. We measure w = −1.08 ± 0.08, and H0 = 70.3 ± 1.9 km/s/Mpc giving us an 4.4% and 2.7% measurement of w and H0. We precisely measure the curvature of the Universe to be consistent with being flat ΩK = −0.005 ± 0.005. We note that the CMB+AllBAO results are consistent with the CMB+SN measurements. Adding the low redshift H0 measurement by Riess et al. (2011) gives us consistent measurements with CMB+BAO+SN as shown in Table 1.

w0waCDM: Varying the Time Dependent Dark Energy Equation of State Parameter
As we probe higher redshifts, we can measure the evolution in w, the dark energy equation of state parameter. The most popular way to parametrize an evolving w, introduced by Chevallier & Polarski (2001) and Linder (2003), is: where a = 1/(1 + z) is the scale factor. In this section, we assume a flat Universe and measure w0 and wa. For cosmological models that use this parameterization of dark energy equation of state (w0, wa), we use the parameterized Post-Friedmann prescription for dark energy perturbations as implemented in a CAMB module (Lewis et al. 2000) by Wenjuan Fang (http://camb.info/ppf/). This modified code generalizes it to support a time-dependent equation of state w(a). Figure 10 shows the 68% and 95% contours for wa vs w0 using the CMB+BAO+SN, CMB+BAO, and CMB+AllBAO datasets. From Table 1, we see that we find very similar constraints on H0 and Ωm as the previously presented cosmological models: Ωm = 0.272±0.15 and H0 = 71.2 ± 1.9 km/s/Mpc giving us a 2.7% measurement

ow0waCDM: Varying Curvature and Time Dependent Dark Energy Equation of State Parameter
In the most general cosmological model we analyze, we vary the curvature of the Universe, ΩK , and both the dark energy parameters w0 and wa as free parameters. Figure 11 shows the 68% and 95% contour levels for H0 vs Ωm, w0 vs ΩK , wa vs ΩK , and wa vs w0 using the CMB+BAO+SN dataset. It is noteworthy that even though we have curvature and both dark energy parameters as free parameters, the data is still consistent with a flat Universe with a cosmological constant. We see the precision in the measurements of H0 and Ωm in the upper left panel of Figure 11. We obtain a 2.7% measurement of H0 = 69.9 ± 1.9 km/s/Mpc and w0 = −0.90 ± 0.16. Our measurement of wa = −1.30 ± 0.99 is consistent with no evolution in w(z). We measure ΩK = −0.010 ± 0.007, which is still consistent with a flat Universe. We find that even with the high dimensionality of the cosmological model, we are able to measure and constrain various cosmological parameters.
The Dark Energy Task Force (DETF) compares various cosmology missions and defines their Figure of Merit (FoM) in the context of this cosmological model (Albrecht et al. 2006). The DETF FoM is defined as the inverse square-root of the determinant of the w0-wa covariance matrix. The 68% and 95% contours for wa vs w0 are shown in the bottom right panel in Figure 11. Using the CMB+BAO+SN dataset, we compute the DETF FoM to be 11.5. However, we note that this is an upper limit since the dataset allows wa outside our prior of −3.0 wa 2.0. Table 1 provides the values for CMB+BAO+SN, CMB+AllBAO, and CMB+BAO+SN+H0 datasets. We see that all the measured values are consistent with each other at the 1σ level. In order to prevent the Markov chains from exploring very extended and remote parameter spaces, we use a prior of −3.0 wa 2.0. However, the chains run with CMB+BAO+SN and CMB+AllBAO datasets tend to allow values beyond wa < −3.0 in their 95% confidence level contours.

Robust measurement of H0 and Ωm
From previous sections, we have found that our measurements of H0 and Ωm remain unchanged as we increase the dimensionality of our cosmological models. In this subsection, we explore this result and explain why our measurements of H0 and Ωm are robust regardless of the model for the late-time behavior of dark energy. In Figure 12, we show the measurements of H0 from the CMB+BAO+SN datasets while varying the dark energy parameterization and the inclusion of spatial curvature. Figure 13 shows the same set of results as 1σ contours in H0 and Ωm. One can see that regardless of the cosmological model, we obtain highly consistent values and error bars for these quantities. This robustness is due to the inverse distance ladder discussed in section 3.1. The CMB data provides a measurement of Ωmh 2 and the sound horizon rs. The BAO data uses the measurement of rs to provide a distance measurement to z = 0.35. The SN data then provide precise measures of the relative distance between z = 0.35 and the local distance scale. Hence, we have an empirical measure of the local distance and hence H0, independent of spatial curvature or the model parameterization of the dark energy equation of state. Combining this measurement of H0 with the CMB measurement of Ωmh 2 yields the value of Ωm. We note that while this result is independent of the parameterization of latetime dark energy and the presence of spatial curvature, it would be sensitive to new cosmological physics at z 1000 that alters the inference of Ωmh 2 and the sound horizon from the CMB.
It is of course important to compare our result for H0 to direct measurements of the local distance scale. The Hubble constant has long been measured using distance ladders that build from local calibrations out to more distant galaxies situated in the Hubble flow (Freedman et Figure 13. 1σ contours of H 0 in km/s/Mpc vs Ωm for various cosmological models using the CMB+BAO+SN dataset. Also plotted is the H 0 value measured by Riess et al. (2011). We see that we get a robust measurement of H 0 and Ωm regardless of the cosmological model.

2009; Freedman & Madore 2010
). A precise value of H0 = 73.8 ± 2.4 km/s/Mpc was recently obtained by the SHOES project (Riess et al. 2011) using the NGC 4258 water maser (Argon et al. 2007;Humphreys et al. 2008) and Cepheid variable stars measured in the near-infrared. We plot this measurement in Figure 12 and 13. One sees that the direct measurement lies about 5% higher than our inference from CMB+BAO+SN. However, this discrepancy only has a statistical significance of 1.5σ and hence is not unusual. Nevertheless, we will return to this in the next subsection.
We note that the CMB+BAO+SN combination consistent favors H0 values around 71, while CMB+BAO alone give slightly lower best-fit values of 69. The latter is not independent of the model for the expansion history; without SN, we are extrapolating the z = 0.35 distance to z ≈ 0 using the cosmological model rather than an empirical measurement. Similarly, Beutler et al. (2011) measure H0 = 67.2 ± 3.2 km/s/Mpc using a BAO detection at z = 0.1. While this is all well within statistical uncertainties, apparently there is a small difference between the SN distance-redshift relation and that predicted from the combination of CMB and BAO data.

Energy Density of Relativistic Species
The measurements of H0 and Ωm discussed in the previous section depend on knowledge of cosmological physics at z 1000. Further, we found a small tension between the CMB+BAO+SN measurement of H0 and the direct measurement by Riess et al. (2011). Hence, we are motivated to consider altering the standard cosmological model by adding additional relativistic particles with negligible interaction cross-section. These would be in addition to the usual cosmic background of the three neutrino species, and hence the new energy density is parameterized by altering the number of neutrino species from 3 to a new value NREL. We note that the particles need not actually be neutrinos, simply highly relativistic and negligibly interacting at late times. This possibility has a long history in cosmology, including constraints from Big Bang nucleosynthesis (Steigman et al. 1977;Hansen et al. 2002;Dolgov 2002;Bowen et al. 2002). Eisenstein & White (2004) pointed out that extra density in relativistic particles would cause CMB and BAO measurements to underestimate the value of Ωmh 2 and H0. Numerous recent papers have constrained the density of relativistic particles with modern cosmology data (Seljak et al. 2006;Ichikawa et al. 2007;Mangano et al. 2007;Hamann et al. 2010;Reid et al. 2010b;Riess et al. 2011;Giusarma et al. 2011;Komatsu et al. 2011;Calabrese et al. 2011;Archidiacono et al. 2011).
We therefore consider cosmological models that vary the relativistic density. In our MCMC chains, we use a prior of NREL 3. Figure 14 shows the 68% and 95% confidence level contours for NREL vs H0 using the CMB+BAO+H0+SN dataset with a ΛCDM + NREL cosmology model. Table 3 gives the values of NREL and other cosmological parameters for three different models of the expansion history of the Universe. From Figure 14 and Table 3, we see that the best-fit value for NREL is around 4. Models with extra relativistic particle density increase the values of Ωmh 2 and H0, allowing a better fit to the Riess et al. (2011) measurement of H0 = 73.8 ± 2.4 km/s/Mpc. In terms of the inverse distance ladder, the added relativistic species affects Ωmh 2 , which moves the acoustic scale, and therefore changes the calibration of the distance ladder to larger values of H0. Hence, it is not surprising to find that the other cosmological parameters such as Ωm, w, and ΩK remain unaffected by the addition of a new relativistic species.
From Figure 14, we see that the shift away from NREL = 3 is not statistically significant. Table 3 shows this shift to be about 2σ. This is larger than the 1.5σ tension between the H0 measurements; this is likely due to the NREL 3  This increase in the relativistic particle density also causes the model to predict a higher value of σ8 as shown in the lower panel of Figure 14 and in Table 3. The bestfit value shifts from σ8 = 0.81 ± 0.02 for NREL = 3 to σ8 = 0.86 ± 0.03 for NREL ≈ 4. For comparison, Allen et al. (2011) (Table 2) gives a comparison of σ8 measurements from galaxy cluster studies. X-ray (Henry et al. 2009) and optical (Rozo et al. 2010) studies of cluster abundances measure σ8 = 0.88±0.04 and 0.80±0.07, respectively. Thus, our σ8 measurements for NREL ≈ 4 are consistent with galaxy cluster measurements.

CONCLUSIONS
In this series of papers, we have used the reconstructed SDSS DR7 LRG dataset to measure the BAO acoustic scale at the median redshift of z = 0.35. The reconstruction technique that provided this measurement has been discussed to great detail in Paper I, and the measurement itself has been extensively studied and tested in PaperII. In this paper, we use this BAO measurement of DV (z = 0.35)/rs = 8.88 ± 0.17, which is a 1.9% measurement of the distance to z = 0.35. To measure various cosmological parameters in a variety of cosmological models, we use our BAO data in combination with the CMB data from WMAP7 (Komatsu et al. 2011) and the type Ia supernovae data from SNLS3 (Conley et al. 2011) to extend the CMB+BAO inverse distance ladder to z = 0. With this CMB+BAO+SN dataset, we explore higher dimensional cosmological models and robustly measure the Hubble constant and the matter density of the Universe. We also use the BAO data from the 6dFGS and WiggleZ surveys in combination with our BAO data and the CMB data to measure cosmological parameters. In particular: • We find that our BAO dataset is consistent with ΛCDM as shown in Figure 2. We improve on the WMAP7 measurements in ΛCDM and obtain a 1.7% measurement of the Hubble constant: H0 = 69.8 ± 1.2 km/s/Mpc.
• As shown in Figure 3, the distance measured from our BAO result is in good agreement with past work. It is therefore unsurprising that the cosmological parameters resulting from our chains are similar to those in recent works combining BAO with other data sets, e.g., Komatsu et al. (2009), Percival et al. (2010, Reid et al. (2010a), Blake et al. (2010), and Beutler et al. (2011).
• Paper I and PaperII show that reconstruction improves the BAO distance measurement by a factor of 1.8. We see this improvement as a reduction in the errors around H0 and Ωm by a factor of 1.5.
• Under the ΛCDM model, we explore the effect of allowing a running spectral index, dns/d ln k and find that only using the CMB data degrades the measurements of Ωmh 2 by a factor of 1.4. This translates into a larger uncertainty in measurements of Ωm, H0, and ns. Adding BAO data decreases the uncertainty to 1.1. We also find that with both datasets that the value of dns/d ln k is still consistent with 0.
• The CMB+BAO dataset breaks the degeneracy between H0, Ωm, and w or ΩK in the wCDM and oCDM models respectively. We measure w = 0.97 ± 0.17 and ΩK = −0.003 ± 0.005, both consistent with ΛCDM. We find that adding the other BAO data slightly improves the measurements on these parameters.
• For the higher dimensional cosmological models (owCDM, w0waCDM, and ow0waCDM), we use the combined CMB+BAO+SN dataset to measure cosmological parameters. We find that even in these high dimensional models, the data is consistent with a flat Universe with a cosmological constant, i.e. consistent with ΛCDM. The Dark Energy Task Force (DETF) Figure of Merit (FoM) is 11.5 using the CMB+BAO+SN dataset and using a prior on wa.
• Using the inverse distance ladder built from the CMB+BAO+SN dataset, we show that we obtain robust and precise measurements of both the Hubble constant and the matter density of the Universe despite varying the underlying model for the expansion history of the Universe. Even in our most general case, we measure H0 = 69.9 ± 1.9 km/s/Mpc and Ωm = 0.276 ± 0.015.
• Our value of the Hubble constant is in mild tension (1.5 σ) with the direct measurement of 73.8 ± 2.4 km/s/Mpc by Riess et al. (2011). We explore the possibility that this tension could be resolved by increasing the density of relativistic particles beyond the usual background of three species of neutrino. We find that such a model can fit the H0 value better if one adds density equivalent to 1 extra species of neutrinos. However, we stress that the conventional model is not rejected by our data.
Looking towards the future, measurements of the distance-redshift relation with baryon acoustic oscillations will improve considerably. The SDSS-III Baryon Oscillation Spectroscopic Survey (BOSS) is underway and will extend the galaxy sample out to z = 0.7 . We expect that the methods used in this SDSS DR7 analysis will be applicable to the BOSS sample. Yet larger surveys probing higher redshifts, such as Euclid and the Wide-Field Infrared Survey Telescope (WFIRST) missions, will use reconstruction to approach the cosmic variance statistical limit available to the acoustic peak method. We expect that baryon acoustic oscillations will play a major role in the precision mapping of the cosmic distance scale and expansion history of the Universe.