The Completed SDSS-IV extended Baryon Oscillation Spectroscopic Survey: Cosmological Implications from two Decades of Spectroscopic Surveys at the Apache Point observatory

TL;DR

This work consolidates final SDSS BAO and RSD measurements across eight tracers with Planck CMB, Pantheon SNe, and DES weak lensing to deliver high-precision constraints on the expansion history and growth of structure. It demonstrates that curvature is constrained to be essentially zero and that the dark energy equation of state is consistent with a cosmological constant, while neutrino masses are tightly bounded. The results show that combining multiple probes breaks key degeneracies present in CMB data alone and that GR remains an excellent description of gravity on cosmological scales within the explored parameter space. Despite strong consistency with ΛCDM, the analysis highlights the persistent H0 tension between early- and late-Universe measurements, a central puzzle guiding future work with DESI and beyond.

Abstract

We present the cosmological implications from final measurements of clustering using galaxies, quasars, and Ly$α$ forests from the completed Sloan Digital Sky Survey (SDSS) lineage of experiments in large-scale structure. These experiments, composed of data from SDSS, SDSS-II, BOSS, and eBOSS, offer independent measurements of baryon acoustic oscillation (BAO) measurements of angular-diameter distances and Hubble distances relative to the sound horizon, $r_d$, from eight different samples and six measurements of the growth rate parameter, $fσ_8$, from redshift-space distortions (RSD). This composite sample is the most constraining of its kind and allows us to perform a comprehensive assessment of the cosmological model after two decades of dedicated spectroscopic observation. We show that the BAO data alone are able to rule out dark-energy-free models at more than eight standard deviations in an extension to the flat, $Λ$CDM model that allows for curvature. When combined with Planck Cosmic Microwave Background (CMB) measurements of temperature and polarization the BAO data provide nearly an order of magnitude improvement on curvature constraints. The RSD measurements indicate a growth rate that is consistent with predictions from Planck primary data and with General Relativity. When combining the results of SDSS BAO and RSD with external data, all multiple-parameter extensions remain consistent with a $Λ$CDM model. Regardless of cosmological model, the precision on $Ω_Λ$, $H_0$, and $σ_8$, remains at roughly 1\%, showing changes of less than 0.6\% in the central values between models. The inverse distance ladder measurement under a o$w_0w_a$CDM yields $H_0= 68.20 \pm 0.81 \, \rm km\, s^{-1} Mpc^{-1}$, remaining in tension with several direct determination methods. (abridged)

Figures (15)

• Figure 1: Top: Distance measurements from the SDSS lineage of BAO measurements presented as a function of redshift. Measurements include those from SDSS MGS ross15ahowlett15a, BOSS galaxies alam17a, eBOSS LRGs LRG_corrgil-marin19a, eBOSS ELGs tamone19ademattia19a, eBOSS quasars hou19aneveux19a, the BOSS+eBOSS $\hbox{Ly$\alpha$}$ auto-correlation, and the BOSS+eBOSS $\hbox{Ly$\alpha$}$-quasar cross-correlation measurements 2019duMasdesBourbouxH. Red points correspond to transverse BAO, while green points to radial BAO. The MGS $D_V$ measurement is plotted in orange with a translation to $D_M$ assuming a $\Lambda$CDM model for illustrative purposes. The red and green theory curves are not fit to the BAO data; they are the Planck bestfit predictions for a flat $\Lambda$CDM model. Bottom: Growth rate measurements from the SDSS lineage of $f\sigma_8$ measurements as a function of redshift. The measurements match the BAO samples except for $z>2$, where we do not report a measurement of the growth rate. As for the upper panel, theory curve is not a fit, but a bestfit Planck model.
• Figure 2: Demonstration of BAO, SN, and CMB constraining power as a function of redshift. To construct alternative models, we have fixed to their best-fit $\Lambda$CDM values the quantities that are best measured by the CMB: $\Omega_b h^2$, $\Omega_c h^2$ and the angular acoustic scale $D_M(z=1150)/r_d$. Because the sound horizon at decoupling is a function of $\Omega_b h^2$, $\Omega_c h^2$ and $N_{\rm eff}$ only, the models have the same value of $r_d=147.16$ Mpc. Top: The Hubble diagram residuals of BAO $D_M(z)$ measurements, presented as the ratio of the measured value of $D_M(z)/r_d$ relative to the prediction for that value based on the best-fit $\Lambda$CDM model from CMB alone. $D_V(z)$ measurements are shown as open circles. We display the CMB determination of the angular position of the acoustic peak as a measurement of transverse BAO, and we split the redshift scale to include this data point. Center: The Hubble diagram residuals of BAO $D_H(z)=c/H(z)$ measurements, normalized in the same manner as the $D_M(z)$ measurements. Bottom: The Hubble diagram residuals of SNe Ia measurements, with relative normalization of the luminosity distance estimates. In order to increase the signal-to-noise, the supernovae data were binned into 11 bins between redshifts 0.1 and 2.5. Spacing was chosen to maintain a relatively constant signal-to-noise ratio. Since the distance modulus varies significantly across the bin at low redshift, we have averaged the signal by averaging the inverse covariance weighted deviations from the $\Lambda$CDM model after the absolute normalization has been fitted. The covariance matrix was taken from the Pantheon dataset. In each case, the residuals are computed relative to the best-fit $\Lambda$CDM model from CMB alone. The curves represent the difference between the $\Lambda$CDM model and single-parameter extensions allowed by the CMB data. The o$\Lambda$CDM model favored by Planck ($\Omega_k=-0.044$) is presented in dashed red lines, the $w$CDM model favored by Planck ($w = -1.585$) is presented in dot-dashed green lines, and a $\Lambda$CDM model with non-zero neutrino mass is presented in solid blue lines. The model with massive neutrinos assumes a summed mass equal to $0.268$ eV, corresponding to the Planck 95% upper limit.
• Figure 3: Cosmological constraints under the assumption of a model with a $w=-1$ cosmological constant with free curvature (o$\Lambda$CDM, as in Table \ref{['tab:main']}). Left: 68% and 95% constraints on $\Omega_m$--$\Omega_\Lambda$ from the Planck CMB temperature and polarization data (gray), Pantheon SNe Ia sample (red), and SDSS BAO-only measurements (blue). The dashed line represents a model with zero curvature. Right: The $\Omega_m$--$\Omega_k$ constraints for the combination of CMB (gray), CMB + SN (red), and CMB + BAO (blue).
• Figure 4: Constraints on the $w$CDM and $\nu\Lambda$CDM models, as in Table \ref{['tab:main']}. Left: $w$--$\Omega_m$ constraints under the assumption of a flat $w$CDM cosmology from the Planck CMB temperature and polarization data (gray), Pantheon SNe Ia sample (red), and SDSS BAO-only measurements (blue). Right: $\sum m_\nu$--$\Omega_m$ constraints under the assumption of a flat $\Lambda$CDM cosmology where the summed neutrino mass is allowed as a free parameter, for the combination of CMB (grey), CMB + SN (red), and CMB + BAO (blue).
• Figure 5: Left:$H_0$ versus $\Omega_m$ from the inverse distance ladder (CMB+BAO+SN) under two different cosmological models. Right:$H_0$ versus $\Omega_m$ from the combination of BAO and BBN, in a $\Lambda$CDM model (blue). The red (gray) contours show the results when using only BAO measurements below (above) $z=1$. The horizontal shaded area shows the (68%, 95%) measurement of $H_0$ from the distance ladder technique (riess19a).