We present cosmological results from the final galaxy clustering data set of the Baryon Oscillation Spectroscopic Survey, part of the Sloan Digital Sky Survey III. Our combined galaxy sample comprises 1.2 million massive galaxies over an effective area of 9329 deg² and volume of 18.7 Gpc³, divided into three partially overlapping redshift slices centred at effective redshifts 0.38, 0.51 and 0.61. We measure the angular diameter distance D_M and Hubble parameter H from the baryon acoustic oscillation (BAO) method, in combination with a cosmic microwave background prior on the sound horizon scale, after applying reconstruction to reduce non-linear effects on the BAO feature. Using the anisotropic clustering of the pre-reconstruction density field, we measure the product D_MH from the Alcock–Paczynski (AP) effect and the growth of structure, quantified by fσ₈(z), from redshift-space distortions (RSD). We combine individual measurements presented in seven companion papers into a set of consensus values and likelihoods, obtaining constraints that are tighter and more robust than those from any one method; in particular, the AP measurement from sub-BAO scales sharpens constraints from post-reconstruction BAOs by breaking degeneracy between D_M and H. Combined with Planck 2016 cosmic microwave background measurements, our distance scale measurements simultaneously imply curvature Ω_K = 0.0003 ± 0.0026 and a dark energy equation-of-state parameter w = −1.01 ± 0.06, in strong affirmation of the spatially flat cold dark matter (CDM) model with a cosmological constant (ΛCDM). Our RSD measurements of fσ₈, at 6 per cent precision, are similarly consistent with this model. When combined with supernova Ia data, we find H₀ = 67.3 ± 1.0 km s⁻¹ Mpc⁻¹ even for our most general dark energy model, in tension with some direct measurements. Adding extra relativistic species as a degree of freedom loosens the constraint only slightly, to H₀ = 67.8 ± 1.2 km s⁻¹ Mpc⁻¹. Assuming flat ΛCDM, we find Ω_m = 0.310 ± 0.005 and H₀ = 67.6 ± 0.5 km s⁻¹ Mpc⁻¹, and we find a 95 per cent upper limit of 0.16 eV c⁻² on the neutrino mass sum.

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-$\alpha$ 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 $H_0=67.3\pm 1.1\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$, 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 $\mathrm{\Lambda }\mathrm{CDM}$ cosmology is an important corroboration of this minimal cosmological model. For constant dark energy ($\mathrm{\Lambda }$), our $\mathrm{BAO}+\mathrm{SN}+\mathrm{CMB}$ combination yields matter density ${\mathrm{\Omega }}_m=0.301\pm 0.008$ and curvature ${\mathrm{\Omega }}_k=-0.003\pm 0.003$. When we allow more general forms of evolving dark energy, the $\mathrm{BAO}+\mathrm{SN}+\mathrm{CMB}$ parameter constraints are always consistent with flat $\mathrm{\Lambda }\mathrm{CDM}$ values at $\approx 1\sigma$. While the overall ${\chi }^2$ of model fits is satisfactory, the LyaF BAO measurements are in moderate ($2–2.5\sigma$) 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, $\sum m_{\nu }<0.56\mathrm{eV}$ (95% confidence), improving to $\sum m_{\nu }<0.25\mathrm{eV}$ if we include the lensing signal in the Planck CMB power spectrum. In a flat $\mathrm{\Lambda }\mathrm{CDM}$ model that allows extra relativistic species, our data combination yields $N_{\mathrm{eff}}=3.43\pm 0.26$; while the LyaF BAO data prefer higher $N_{\mathrm{eff}}$ when excluding galaxy BAO, the galaxy BAO alone favor $N_{\mathrm{eff}}\approx 3$. 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.

The fourth generation of the Sloan Digital Sky Survey (SDSS-IV) began observations in 2014 July. It pursues three core programs: the Apache Point Observatory Galactic Evolution Experiment 2 (APOGEE-2), Mapping Nearby Galaxies at APO (MaNGA), and the Extended Baryon Oscillation Spectroscopic Survey (eBOSS). As well as its core program, eBOSS contains two major subprograms: the Time Domain Spectroscopic Survey (TDSS) and the SPectroscopic IDentification of ERosita Sources (SPIDERS). This paper describes the first data release from SDSS-IV, Data Release 13 (DR13). DR13 makes publicly available the first 1390 spatially resolved integral field unit observations of nearby galaxies from MaNGA. It includes new observations from eBOSS, completing the Sloan Extended QUasar, Emission-line galaxy, Luminous red galaxy Survey (SEQUELS), which also targeted variability-selected objects and X-ray-selected objects. DR13 includes new reductions of the SDSS-III BOSS data, improving the spectrophotometric calibration and redshift classification, and new reductions of the SDSS-III APOGEE-1 data, improving stellar parameters for dwarf stars and cooler stars. DR13 provides more robust and precise photometric calibrations. Value-added target catalogs relevant for eBOSS, TDSS, and SPIDERS and an updated red-clump catalog for APOGEE are also available. This paper describes the location and format of the data and provides references to important technical papers. The SDSS web site, http://www.sdss.org, provides links to the data, tutorials, examples of data access, and extensive documentation of the reduction and analysis procedures. DR13 is the first of a scheduled set that will contain new data and analyses from the planned ∼6 yr operations of SDSS-IV.

We analyse the baryon acoustic oscillation (BAO) signal of the final Baryon Oscillation Spectroscopic Survey (BOSS) data release (DR12). Our analysis is performed in the Fourier space, using the power spectrum monopole and quadrupole. The data set includes 1198 006 galaxies over the redshift range 0.2 < z < 0.75. We divide this data set into three (overlapping) redshift bins with the effective redshifts z_eff = 0.38, 0.51 and 0.61. We demonstrate the reliability of our analysis pipeline using N-body simulations as well as ∼1000 MultiDark-Patchy mock catalogues that mimic the BOSS-DR12 target selection. We apply density field reconstruction to enhance the BAO signal-to-noise ratio. By including the power spectrum quadrupole we can separate the line of sight and angular modes, which allows us to constrain the angular diameter distance D_A(z) and the Hubble parameter H(z) separately. We obtain two independent 1.6 and 1.5 per cent constraints on D_A(z) and 2.9 and 2.3 per cent constraints on H(z) for the low (z_eff = 0.38) and high (z_eff = 0.61) redshift bin, respectively. We obtain two independent 1 and 0.9 per cent constraints on the angular averaged distance D_V(z), when ignoring the Alcock–Paczynski effect. The detection significance of the BAO signal is of the order of 8σ (post-reconstruction) for each of the three redshift bins. Our results are in good agreement with the Planck prediction within Λ cold dark matter. This paper is part of a set that analyses the final galaxy clustering data set from BOSS. The measurements and likelihoods presented here are combined with others in Alam et al. to produce the final cosmological constraints from BOSS.

We study the cosmological constant ($\mathrm{\Lambda }$) in the standard $\mathrm{\Lambda }$ cold dark matter model by introducing the graduated dark energy (gDE) characterized by a minimal dynamical deviation from the null inertial mass density of the $\mathrm{\Lambda }$ in the form ${\rho }_{\text{inert}}\propto {\rho }^{\lambda }<0$ with $\lambda <1$ being a ratio of two odd integers, for which its energy density $\rho$ dynamically takes negative values in the finite past. For large negative values of $\lambda$, it creates a phenomenological model described by a smooth function that approximately describes the $\mathrm{\Lambda }$ spontaneously switching sign in the late Universe to become positive today. We confront the model with the latest combined observational datasets of $\mathrm{Planck}+\mathrm{baryon}\mathrm{acoustic}\mathrm{oscillations}+\mathrm{supernova}+H$. It is striking that the data predict bimodal posterior probability distributions for the parameters of the model along with large negative $\lambda$ values; the new maximum significantly excludes the $\mathrm{\Lambda }$, and the old maximum contains the $\mathrm{\Lambda }$. The improvement in the goodness of fit for the $\mathrm{\Lambda }$ reaches highly significant levels, $\mathrm{\Delta }{\chi }_{\min }^2=6.4$, for the new maxima, while it remains at insignificant levels, $\mathrm{\Delta }{\chi }_{\min }^2\lesssim 0.02$, for the old maxima. We show that, in contrast to the old maxima, which do not distinguish from the $\mathrm{\Lambda }$, the new maxima agree with the model-independent $H_0$ measurements, high-precision $\mathrm{Ly}\text{−}\alpha$ data, and model-independent $Om{h}^2$ diagnostic estimates. Our results provide strong hints of a spontaneous sign switch in the cosmological constant and lead us to conjecture that the Universe has transitioned from anti-de Sitter vacua to de Sitter vacua, at a redshift $z\approx 2.32$, and triggered the late-time acceleration, and suggests looking for such mechanisms in string theory constructions.