Cosmological parameters from SDSS and WMAP

TL;DR

Combining the SDSS three-dimensional power spectrum with WMAP and ancillary data yields a largely concordant vanilla flat ΛCDM cosmology, with SDSS sharply tightening key parameters like the Hubble parameter $h$ and the matter density $oldsymbol{ extOmega}_m$. The analysis demonstrates that inflationary predictions of near-flatness and approximate scale invariance are well supported, while no compelling evidence for tensor modes or a running spectral index is found. Neutrino masses are constrained to $M_ u<1.7$ eV (95%), and the dark energy equation of state remains consistent with a cosmological constant ($w=-1$) at the ~20% level. The results underscore the power of linking CMB measurements with large-scale structure data to break degeneracies and to test the robustness of the standard cosmological model, while highlighting remaining uncertainties and avenues for future progress.

Abstract

We measure cosmological parameters using the three-dimensional power spectrum P(k) from over 200,000 galaxies in the Sloan Digital Sky Survey (SDSS) in combination with WMAP and other data. Our results are consistent with a ``vanilla'' flat adiabatic Lambda-CDM model without tilt (n=1), running tilt, tensor modes or massive neutrinos. Adding SDSS information more than halves the WMAP-only error bars on some parameters, tightening 1 sigma constraints on the Hubble parameter from h~0.74+0.18-0.07 to h~0.70+0.04-0.03, on the matter density from Omega_m~0.25+/-0.10 to Omega_m~0.30+/-0.04 (1 sigma) and on neutrino masses from <11 eV to <0.6 eV (95%). SDSS helps even more when dropping prior assumptions about curvature, neutrinos, tensor modes and the equation of state. Our results are in substantial agreement with the joint analysis of WMAP and the 2dF Galaxy Redshift Survey, which is an impressive consistency check with independent redshift survey data and analysis techniques. In this paper, we place particular emphasis on clarifying the physical origin of the constraints, i.e., what we do and do not know when using different data sets and prior assumptions. For instance, dropping the assumption that space is perfectly flat, the WMAP-only constraint on the measured age of the Universe tightens from t0~16.3+2.3-1.8 Gyr to t0~14.1+1.0-0.9 Gyr by adding SDSS and SN Ia data. Including tensors, running tilt, neutrino mass and equation of state in the list of free parameters, many constraints are still quite weak, but future cosmological measurements from SDSS and other sources should allow these to be substantially tightened.

Figures (18)

• Figure 1: Summary of observations and cosmological models. Data points are for unpolarized CMB experiments combined (top; Appendix A.3 details data used) cross-polarized CMB from WMAP (middle) and Galaxy power from SDSS (bottom). Shaded bands show the 1-sigma range of theoretical models from the Monte-Carlo Markov chains, both for cosmological parameters (right) and for the corresponding power spectra (left). From outside in, these bands correspond to WMAP with no priors, adding the prior $f_\nu=0$, $w=-1$, further adding the priors $\Omega_k=r=\alpha=0$, and further adding the SDSS information, respectively. These four bands essentially coincide in the top two panels, since the CMB constraints were included in the fits. Note that the $\ell$-axis in the upper two panels goes from logarithmic on the left to linear on the right, to show important features at both ends, whereas the $k$-axis of the bottom panel is simply logarithmic.
• Figure 2: Constraints on individual cosmological quantities using WMAP alone (shaded yellow/light grey distributions) and including SDSS information (narrower red/dark grey distributions). Each distribution shown has been marginalized over all other quantities in the class of 6-parameter $(\tau,\Omega_\Lambda,\omega_d,\omega_b,A_s,{n_s})$ "vanilla" models as well as over a galaxy bias parameter $b$ for the SDSS case. The $\alpha$-distributions are also marginalized over $r$ and $\Omega_k$. The parameter measurements and error bars quoted in the tables correspond to the median and the central 68% of the distributions, indicated by three vertical lines for the WMAP+SDSS case above. When the distribution peaks near zero (like for $r$), we instead quote an upper limit at the 95th percentile (single vertical line). The horizontal dashed lines indicate $e^{-x^2/2}$ for $x=1$ and $2$, respectively, so if the distribution were Gaussian, its intersections with these lines would correspond to $1\sigma$ and $2\sigma$ limits, respectively.
• Figure 3: 95% constraints in the $({n_s},\omega_b)$ plane. The shaded dark red/grey region is ruled out by WMAP alone for 6-parameter "vanilla" models, leaving the long degeneracy banana discussed in the text. The shaded light red/grey region is ruled out when adding SDSS information. The hatched band is required by Big Bang Nucleosynthesis (BBN). From right to left, the three vertical bands correspond to a scale-invariant Harrison-Zel'dovich spectrum and to the common inflationary predictions ${n_s}=1-2/N\sim 0.96$ and ${n_s}=1-3/N\sim 0.94$ (Table 6), assuming that the number of e-foldings between horizon exit of the observed fluctuations and the end of inflation is $50<N<60$.
• Figure 4: 95% constraints in the $(\omega_d,\omega_b)$ plane. Shaded dark red/grey region is ruled out by WMAP alone for 6-parameter "vanilla" models. The shaded light red/grey region is ruled out when adding SDSS information. The hatched band is required by Big Bang Nucleosynthesis (BBN).
• Figure 5: 95% constraints in the $(\Omega_m,h)$ plane. Shaded dark red/grey region is ruled out by WMAP alone for 6-parameter "vanilla" models, leaving the long degeneracy banana discussed in the text. The shaded light red/grey region is ruled out when adding SDSS information, which can be understood as SDSS accurately measuring the $P(k)$ "shape parameter" $h\Omega_m=0.21\pm 0.03$ at $2\sigma$ (sloping hatched band). The horizontal hatched band is required by the HST key project Freedman01. The dotted line shows the fit $h = 0.7(\Omega_m/0.3)^{-0.35}$, explaining the origin of the accurate constraint $h(\Omega_m/0.3)^{0.35} = 0.70\pm 0.01$$(1\sigma)$.