A&A 459, 375–389 (2006) Astronomy DOI: 10.1051/0004-6361:20065377 & c ESO 2006 Astrophysics Power spectrum of the SDSS luminous red galaxies: constraints on cosmological parameters G. Hütsi1,2 1 Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str. 1, 86740 Garching bei München, Germany e-mail: [email protected] 2 Tartu Observatory, Tõravere 61602, Estonia Received 6 April 2006 / Accepted 21 August 2006 ABSTRACT In this paper we determine the constraints on cosmological parameters using the CMB data from the Wmap experiment together with the recent power spectrum measurement of the SDSS Luminous Red Galaxies (LRGs). Specifically, we focus on spatially flat, low matter density models with adiabatic Gaussian initial conditions. The spatial flatness is achieved with an additional quintessence component whose equation of state parameter weff is taken to be independent of redshift. Throughout most of the paper we do not allow any massive neutrino contribution and also the influence of the gravitational waves on the CMB is taken to be negligible. The analysis is carried out separately for two cases: (i) using the acoustic scale measurements as presented in Hütsi (2006, A&A, 449, 891), (ii) using the full SDSS LRG power spectrum and its covariance matrix. We are able to obtain a very tight constraint on the = +2.1 −1 −1 Hubble constant: H0 70.8−2.0 km s Mpc , which helps in breaking several degeneracies between the parameters and allows us to determine the low redshift expansion law with much higher accuracy than available from the Wmap + HST data alone. The positive deceleration parameter q0 is found to be ruled out at 5.5σ confidence level. Finally, we extend our analysis by investigating the effects of relaxing the assumption of spatial flatness and also allow for a contribution from massive neutrinos. Key words. cosmology: cosmological parameters – large-scale structure of Universe – cosmic microwave background 1. Introduction information has also been obtained from the Ly-α forest, weak obe1 lensing, galaxy cluster, and large-scale peculiar velocity stud- Since the flight of the C satellite in the beginning of 90’s ies. It is remarkable that this diversity of observational data the field of observational cosmology has witnessed an extremely can be fully explained by a cosmological model that in its sim- rapid development. The data from various Cosmic Microwave plest form has only 5−6 free parameters (Liddle 2004; Tegmark Background (CMB) experiments (Wmap2 (Bennett et al. 2003); obe rcheops3 et al. 2004). As the future data sets will be orders of magnitude C (Smoot et al. 1992); A (Benoît et al. 2003); larger, leading to the extremely small statistical errors, any fur- Boomerang4 (Netterfield et al. 2002); Maxima5 (Hanany et al. bi6 sa7 ther progress is possible only in case we fully understand various 2000); C (Pearson et al. 2003); V (Scott et al. 2003); systematic uncertainties that could potentially bias our conclu- Dasi8 (Halverson et al. 2002) etc.); supernova surveys (Scp9 10 sions about the underlying cosmology. As such, one should try (Perlmutter et al. 1999), High-Z SN Search (Riess et al. 1998)) to use observables that are least sensitive to the theoretical uncer- and large galaxy redshift surveys (SDSS11 (York et al. 2000), 12 tainties, contaminating foregrounds etc. Currently the “cleanest” 2dFGRS (Colless et al. 2001)) has lead us to the cosmolog- constraints on cosmological models are provided by the mea- ical model that is able to accommodate almost all the avail- surements of the angular power spectrum of the CMB. Since the able high quality data – the so-called “concordance” model underlying linear physics is well understood (see e.g. Hu 1995; (Bahcall et al. 1999; Spergel et al. 2003). Useful cosmological Dodelson 2003) we have a good knowledge of how the angular Appendices are only available in electronic form at position and amplitude ratios of the acoustic peaks depend on http://www.aanda.org various cosmological parameters. However, the CMB data alone 1 http://lambda.gsfc.nasa.gov/product/cobe/ is able to provide accurate measurements of only a few com- 2 http://map.gsfc.nasa.gov/ binations of the cosmological parameters. In order to break the 3 http://www.archeops.org/ degeneracies between the parameters one has to complement the 4 http://cmb.phys.cwru.edu/boomerang/ CMB data with additional information from other independent 5 http://cfpa.berkeley.edu/group/cmb/ sources e.g. the data from the type Ia supernovae, large-scale 6 http://www.astro.caltech.edu/∼tjp/CBI/ structure, or the Hubble parameter measurements. In fact, the 7 http://www.mrao.cam.ac.uk/telescopes/vsa/ well understood physical processes responsible for the promi- 8 http://astro.uchicago.edu/dasi/ nent peak structure in the CMB angular power spectrum are also 9 http://supernova.lbl.gov/ predicted to leave imprints on the large-scale matter distribu- 10 http://cfa-www.harvard.edu/oir/Research/supernova/ tion. Recently the analysis of the spatial two-point correlation HighZ.html function of the Sloan Digital Sky Survey (SDSS) Luminous Red 11 http://www.sdss.org/ Galaxy (LRG) sample (Eisenstein et al. 2005), and power spec- 12 http://www.mso.anu.edu.au/2dFGRS/ tra of the 2dF (Cole et al. 2005) and SDSS LRG (Hütsi 2006) Article published by EDP Sciences and available at http://www.aanda.org or http://dx.doi.org/10.1051/0004-6361:20065377 376 G. Hütsi: SDSS LRG cosmological parameters redshift samples, have lead to the detection of these acoustic fea- tures, providing the clearest support for the gravitational insta- 2800 bility picture, where the large-scale structure of the Universe is believed to arise through the gravitational amplification of the 2600 density fluctuations laid down in the very early Universe. ] In the current paper we work out the constraints on cosmo- 2 2400 Mpc logical parameters using the SDSS LRG power spectrum as de- -2 2200 termined by Hütsi (2006). In order to break the degeneracies 2000 between the parameters we complement our analysis with the kP(k) [h data from other cosmological sources: the CMB data from the Wmap, and the measurement of the Hubble parameter by 1800 the HST Key Project13. We focus our attention on simple mod- els with Gaussian adiabatic initial conditions. In the initial phase 1600 of the analysis we further assume spatial flatness, and also neg- ligible massive neutrino and gravitational wave contributions. 0.03 0.1 0.2 This leads us to the models with 6 free parameters: total matter k [hMpc-1] Ω Ω and baryonic matter density parameters: m and b, the Hubble 100000 parameter h, the optical depth to the last-scattering surface τ, the amplitude As and spectral index ns of the scalar perturbation spectrum14. This minimal set is extended with the constant dark energy effective equation of state parameter weff. We carry out our analysis in two parts. In the first part we use only the mea- ] surement of the acoustic scale from the SDSS LRG power spec- 3 trum as given in Hütsi (2006). The analysis in the second part Mpc uses the full power spectrum measurement along with the covari- -3 ance matrix as provided by Hütsi (2006). Here we add two ex- P(k) [h tra parameters: bias parameter b and parameter Q that describes the deformation of the linear power spectrum to the nonlinear 10000 redshift-space spectrum. These extra parameters are treated as nuissance parameters and are marginalized over. Thus the largest parameter space we should cope with is 9-dimensional15.Since 5000 the parameter space is relatively high dimensional it is natural to 0.03 0.1 0.2 use Markov Chain Monte Carlo (MCMC) techniques. For this k [hMpc-1] purpose we use publicly available cosmological MCMC engine osmomc16 Fig. 1. Upper panel: power spectra in somewhat unconventional form. C (Lewis & Bridle 2002). All the CMB spectra and Here the spectra have been multiplied by an extra factor of k to increase matter transfer functions are calculated using the fast Boltzmann the visibility of details. Filled circles with solid errorbars represent the amb17 code C (Lewis et al. 2000). SDSS LRG power spectrum as determined by Hütsi (2006). The upper The paper is organized as follows. In Sect. 2 we describe the data points provide the deconvolved version of the spectrum. The thin observational data used for the parameter estimation. Section 3 solid lines show the best-fitting model spectra. Lower panel:thesame discusses and tests the accuracy of the transformations needed to spectra as above now plotted in the usual form. convert the linear input spectrum to the observed redshift-space galaxy power spectrum. In Sect. 4 we present the main results of the cosmological parameter study and we conclude in Sect. 5. the deconvolution can be done rather “cleanly”. This decon- volved spectrum might be useful for the extra-fast parameter estimation employing analytic approximations for the matter 2. Data transfer functions (Eisenstein & Hu 1998; Novosyadlyj et al. 1999) and fast CMB angular power spectrum generators such as The SDSS LRG power spectrum as determined by Hütsi (2006) 19 20 is shown with filled circles and heavy solid errorbars in Fig. 1. CMBfit (Sandvik et al. 2004), DASh (Kaplinghat et al. 2002) and CMBwarp21 (Jimenez et al. 2004). However, in this pa- There the upper data points correspond to the deconvolved ver- amb sion of the spectrum18. The thin solid lines represent the best- per, as we use an accurate Boltzmann solver C to calculate fitting model spectra, with the lower curve corresponding to CMB power spectra and matter transfer functions, the relative the convolved case.