The clustering of galaxies in the SDSS-III Baryon Oscillation Spectroscopic Survey: cosmological implications of the full shape of the clustering wedges in the data release 10 and 11 galaxy samples

We explore the cosmological implications of the angle-averaged correlation function, xi(s), and the clustering wedges, xi_perp(s) and xi_para(s), of the LOWZ and CMASS galaxy samples from Data Release 10 and 11 of the SDSS-III Baryon Oscillation Spectroscopic Survey. Our results show no significant evidence for a deviation from the standard LCDM model. The combination of the information from our clustering measurements with recent data from the cosmic microwave background is sufficient to constrain the curvature of the Universe to Omega_k = 0.0010 +- 0.0029, the total neutrino mass to Sum m_nu<0.23 eV (95% confidence level), the effective number of relativistic species to N_eff=3.31 +- 0.27, and the dark energy equation of state to w_DE = -1.051 +- 0.076. These limits are further improved by adding information from type Ia supernovae and baryon acoustic oscillations from other samples. In particular, this data set combination is completely consistent with a time-independent dark energy equation of state, in which case we find w_DE=-1.024 +- 0.052. We explore the constraints on the growth-rate of cosmic structures assuming f(z)=Omega_m(z)^gamma and obtain gamma=0.69 +- 0.15, in agreement with the predictions from general relativity of gamma=0.55.


INTRODUCTION
The large-scale distribution of galaxies contains the signature of acoustic waves that propagated through the Universe prior to the epoch of recombination.This signal, referred to as baryon acoustic oscillations (BAO), appears as a modulation in the amplitude of the galaxy power spectrum, P (k), and a broad peak in the large-scale two-point correlation function, ξ(s) (Eisenstein & Hu 1998;Meiksin et al. 1999;Matsubara 2004).The wavelength of the oscillations in P (k) and the location of the peak in ξ(s) can be associated with the maximum distance that these acoustic waves can travel before the decoupling of matter and radiation, that is, the sound horizon at the drag redshift, r d .As this scale can be constrained with high accuracy from observations of the cosmic microwave background (CMB), the acoustic scale inferred from the clustering of galaxy samples at different redshifts can be used as a standard ruler to measure the distance-redshift relation, providing a powerful and robust probe of the expansion history of the Universe (Blake & Glazebrook 2003;Linder 2003).
The BAO signal was first detected in the clustering of the Two-degree Field Galaxy Redshift survey (2dFGRS, Colless et al. 2001Colless et al. , 2003) ) by Cole et al. (2005) and the luminous red galaxy (LRG, Eisenstein et al. 2001) sample of the Sloan Digital Sky Survey (SDSS, York et al. 2000) by Eisenstein et al. (2005).Since then, subsequent analyses on various galaxy samples have provided BAO measurements with increasing precision (Padmanabhan et al. 2007;Beutler et al. 2011;Blake et al. 2011;Xu et al. 2012;Seo et al. 2012;Anderson et al. 2012Anderson et al. , 2013a)).Using these results it is now possible to construct a Hubble diagram based entirely on BAO distance measurements.It has become standard practice to use this information, in combination with additional data sets, when deriving constraints on cosmological parameters.
Separate measurements of the acoustic scale in the directions parallel and perpendicular to the line of sight ⋆ E-mail: arielsan@mpe.mpg.decan be used to obtain constraints on the Hubble parameter, H(z), and the angular diameter distance, DA(z), through the Alcock-Paczynski test (Alcock & Paczynski 1979;Hu & Haiman 2003).However, the BAO signal on angle-averaged clustering measurements such as P (k) or ξ(s) provide estimates of the average distance DV(z) ∝ DA(z) 2 /H(z).Although most analyses have focused on angle-averaged quantities, the large volumes probed by present-day galaxy samples make it possible to extend these analyses to anisotropic clustering measurements (Cabré & Gaztañaga 2009;Blake et al. 2012;Xu et al. 2012;Anderson et al. 2013a;Kazin et al. 2013) using the full power of the BAO test.
The clustering of galaxies encodes additional information beyond that contained in the BAO signal that can significantly improve the cosmological constraints derived from large-scale structure (LSS) data sets.This extra information is particularly important for anisotropic clustering measurements, where the signature of the so-called redshiftspace distortions (RSD) can be used to constrain the growth rate of cosmic structures (Guzzo et al. 2008).In this way, anisotropic clustering measurements can provide information of the expansion history of the Universe and the growth of density fluctuations, which can be used to distinguish between the dark energy and modified gravity scenarios for the origin of cosmic acceleration.
The most accurate BAO measurements to date have been obtained from the Baryon Oscillation Spectroscopic Survey (BOSS, Dawson et al. 2013), which is one of the four component surveys of SDSS-III (Eisenstein et al. 2011).After applying a modified version of the reconstruction technique of Eisenstein et al. (2007), the BAO signal in the galaxy clustering of BOSS SDSS Data Release 9 (DR9, Ahn et al. 2012) provided a 1.7 per cent accuracy measurement of the average distance DV(z) at z = 0.57 (Anderson et al. 2012), as well as separate constraints on DA(z) and H(z) at the same redshift with 3 and 8 per cent accuracy, respectively (Anderson et al. 2013a).These measurements have been complemented by analyses of the full shape of isotropic and anisotropic clustering measurements c 0000 RAS, MNRAS 000, 000-000 (Reid et al. 2012;Sánchez et al. 2012Sánchez et al. , 2013;;Samushia et al. 2013a;Chuang et al. 2013a).Besides galaxy clustering analyses, a sample of high-redshift quasars from BOSS has been used to detect for the first time the signature of the BAO in the fluctuations of the Lyman-α forest at z ≃ 2.4 (Busca et al. 2013;Slosar et al. 2013).
In this paper we use information from the full-shape of the two-point correlation function and the clustering wedges statistic (Kazin, Sánchez, & Blanton 2012) measured from BOSS data to derive constraints on cosmological parameters.We extend the analyses of Sánchez et al. (2012Sánchez et al. ( , 2013) ) based on a high-redshift galaxy sample from BOSS DR9 to the data corresponding to DR10 (Ahn et al. 2013) and DR11 (internal data-release), including results from the lowredshift BOSS galaxy sample.
As the statistical uncertainties characterizing different cosmological observations become smaller, it is important to explore potential systematics that can be introduced by the analysis techniques and models applied to the data.The comparison of the results obtained by applying multiple methods to the same data can be used to identify the presence of systematics errors.Our analysis is part of a series of papers examining the clustering properties of the BOSS DR10 and DR11 galaxy samples with different methodologies.Tojeiro et al. (in preparation) and Anderson et al. (2013b) analyse the isotropic and anisotropic BAO signal in these samples and explore their cosmological implications.Ross et al. (2013) study the sensitivity of these BAO measurements to the properties of the galaxy population being analysed.Vargas-Magaña et al. (2013) investigate the potential systematic errors affecting anisotropic BAO measurements.Percival et al. (2013) perform a detailed analysis of the effect of the uncertainties in the covariance matrices determined from mock catalogues on the obtained constraints.These analyses are complemented by those of Chuang et al. (2013b), Samushia et al. (2013b) and Beutler et al. (2013), who analyse the full shape of the monopole-quadrupole pair in configuration and Fourier space.These studies attempt to condense the information of the clustering measurements into a few numbers reflecting the geometric constraints and the measurements of the growth of structures, that are then compared with the predictions from different cosmological models.We follow an alternative approach in which we perform the comparison with cosmological models at the level of the galaxy clustering measurements themselves.The consistency of the results presented here and those of our companion papers is a reassuring indication of the robustness of our results.
The outline of this paper is as follows.In Section 2 we describe our galaxy sample, the procedure followed to obtain our clustering measurements and their respective covariance matrices, as well as the additional data sets included in our analysis.In Section 3 we review our model of the full shape of the correlation function and the clustering wedges and our methodology to obtain cosmological constraints.In Section 4 we present the constraints on cosmological parameters obtained from different combinations of data sets and parameter spaces.Finally, Section 5 contains our main conclusions.

Galaxy clustering measurements from BOSS
We use the LOWZ and CMASS samples of BOSS corresponding to SDSS DR10 (Ahn et al. 2013) and DR11, which will become publicly available with the final data release of the survey.These galaxy samples were selected on the basis of the SDSS multicolour photometric observations (Gunn et al. 1998(Gunn et al. , 2006) ) to cover the redshift range 0.15 < z < 0.7 with a roughly uniform comoving number density n ≃ 3 × 10 −4 h 3 Mpc −3 (Eisenstein et al. 2011;Dawson et al. 2013, Padmanabhan et al. in preparation).Up to ∼ 30 and 2 per cent of LOWZ and CMASS targets, respectively, were observed during the SDSS I/II surveys (York et al. 2000) and thus already have a redshift.The remaining redshifts were measured from the spectra obtained with the double-armed BOSS spectrographs (Smee et al. 2013) by applying the minimum-χ 2 template-fitting procedure described in Aihara et al. (2011) and Bolton et al. (2012).
The LOWZ sample consists primarily of red galaxies that lie in massive haloes, with a satellite fraction of 12 per cent (Parejko et al. 2013).The CMASS sample is approximately complete down to a limiting stellar mass of M ≃ 10 11.3 M⊙ (Maraston et al. 2013), and has a ∼10 per cent satellite fraction (White et al. 2011;Nuza et al. 2013).Although this sample is dominated by early type galaxies, it contains a significant fraction of massive spirals (∼26 per cent, Masters et al. 2011).Anderson et al. (2013b) describes the construction of catalogues for LSS analyses based on these samples.We use these samples separately, restricting our analysis to the redshift ranges 0.15 < z < 0.43 for the LOWZ sample, and 0.43 < z < 0.7 for the CMASS galaxies.
We study the clustering properties of these galaxy samples by means of the angle-averaged correlation function, ξ(s), and the clustering wedges statistic (Kazin et al. 2012), ξ∆µ(s), which corresponds to the average of the full twodimensional correlation function ξ(µ, s) over the interval ∆µ = µmax − µmin, that is We use two wide clustering wedges, ξ ⊥ (s) and ξ (s), defined for the intervals 0 µ 0.5 and 0.5 µ 1, respectively.The basic procedure implemented to obtain these measurements from the LOWZ and CMASS samples is analogous to that of Anderson et al. (2013b) and Sánchez et al. (2013).
Here we summarize the most important points and refer the reader to these studies for more details.We convert the observed redshifts into distances assuming a flat ΛCDM fiducial cosmology characterized by a matter density parameter of Ωm = 0.274.We use the estimator of Landy & Szalay (1993) to compute the full correlation function ξ(µ, s) of the LOWZ and CMASS samples, with random samples following the same selection function as the original catalogues but containing 50 times more objects.The value of µ of a given pair is defined as the cosine of the angle between the separation vector, s, and the line-of-sight direction at the midpoint of s.We infer the correlation function ξ(s) and the clustering wedges ξ ⊥ (s) Figure 1.Angle-averaged correlation functions ξ(s), (left panels) and clustering wedges ξ ⊥ (s) and ξ (s) (right panels) of the LOWZ and CMASS DR10 galaxy samples.The error bars were derived from the diagonal entries of the full covariance matrices obtained as described in Sec.2.1.2.The dashed lines correspond to the best-fitting ΛCDM model obtained from the combination of information from the full shape of the LOWZ and CMASS DR11 clustering wedges with the CMB temperature fluctuation measurements from Planck and the nine-year polarization measurements from WMAP (see Section 4.1).and ξ (s) by averaging the full ξ(µ, s) over the corresponding µ intervals.As discussed in Kazin et al. (2012), this procedure correctly accounts for the µ dependence of the random-random counts, which is ignored when the estimator of Landy & Szalay (1993) is applied to the averaged counts directly, leading to a bias in the recovered clustering measurements.
When computing the pair counts, we assign a series of weights to each object in our catalogue.First, we apply a radial weight designed to minimize the variance of our measurements (Feldman, Kaiser, & Peacock 1994) given by where n(z) is the expected number density of the catalogue at the given redshift and Pw is a scale-independent parameter, which we set to Pw = 2 × 10 4 h −3 Mpc 3 .We also in-clude angular weights to account for redshift failures and fibre collisions.For the CMASS sample we apply additional weights to correct for the systematic effect introduced by the local stellar density and the seeing of the observations, as described in detail in Anderson et al. (2013b).
The left panels of Figs. 1 and 2 show the resulting angle-averaged correlation function ξ(s) of the SDSS-DR10 and DR11 LOWZ (upper) and CMASS (bottom) samples, respectively, while the left panels show the corresponding clustering wedges ξ ⊥ (s) (circles) and ξ (s) (squares).The anisotropic clustering pattern generated by redshiftspace distortions leads to significant differences in the amplitude and shape of the two clustering wedges, with ξ (s) showing a lower amplitude and a stronger damping of the BAO peak than ξ ⊥ (s).The dashed lines correspond to the best-fitting ΛCDM model obtained from c 0000 RAS, MNRAS 000, 000-000 the combination of the LOWZ and CMASS DR11 clustering wedges with CMB observations from the Planck satellite (Planck Collaboration I 2013) and the CMB polarization measurements from WMAP (Bennett et al. 2013) as described in Section 4.1, which provide an excellent description of all our measurements.

Covariance matrix estimation
When comparing our BOSS clustering measurements with theoretical predictions we assume a Gaussian likelihood function of the form L ∝ exp(−χ 2 /2).The calculation of the χ 2 value of a given model requires the knowledge of the inverse covariance matrix of our measurements, which we estimate using mock catalogues matching the selection functions of the LOWZ and CMASS samples.These mocks were constructed from two sets of PTHalos realizations (Scoccimarro & Sheth 2002), corresponding to our fiducial cosmology, as described in Manera et al. (2013) and Manera et al. (in preparation) 1 .Our CMASS mocks are based on 600 independent simulations with a box size of L box = 2.4 h −1 Gpc, while those of the LOWZ sample were constructed from a separate set of 500 boxes with the same volume.In the construction of these mocks, the Northern Galactic Cap (NGC) and Southern Galactic Cap (SGC) components of the survey were considered as being independent, and sampled separately from the same PTHalos realizations.The volume of the LOWZ sample allowed us to obtain two separate NGC and SGC mocks per PTHalos realization, leading to 1000 independent combined NGC+SGC LOWZ mock catalogues.The larger volume of the CMASS sample makes it more difficult to construct mocks of the NGC and SGC components from the boxes without overlap.This means that the NGC and SGC CMASS mocks drawn from the same box are not independent.For DR10 the overlap between the NGC and SGC mocks is approxi- mately 75 per cent of the area covered by the SGC, while for DR11 the whole of the Southern component is also covered by the NGC.To account for this overlap in our covariance matrix estimations we construct two sets of 300 independent NGC+SGC CMASS mocks, drawing the matched components from different boxes.
We measured the angle-averaged correlation function and the clustering wedges of each LOWZ and CMASS mock catalogue using the same binning and weighting schemes as for the real data.These measurements were used to obtain an estimate of the full covariance matrix C of our clustering measurements.For the CMASS sample we define our covariance matrix as the average of the results obtained in the two sets of independent mocks.The error bars in Figs. 1 and 2 correspond to the square root of the diagonal entries in C.
Our estimations of C are affected by noise, as they are inferred from a finite number of mock catalogues.This uncertainty has important implications on the derived constraints.The distribution of covariance matrices recovered from multiple, independent sets of simulations follows a Wishart distribution, and its inverse, C −1 , an inverse-Wishart distribution (Wishart 1928).As the inverse Wishart distribution is asymmetric, C −1 provides a biased estimate of the true inverse covariance matrix (see e.g.Hartlap et al. 2007;Taylor et al. 2013;Percival et al. 2013).This bias can be corrected for by rescaling the inverse covariance matrix as with where N mocks is to the total number of mocks used to estimate C and N bins corresponds to the total number of bins in our measurements.We restrict our analysis to 40 h −1 Mpc < s < 160 h −1 Mpc with a bin-width of ds = 5 h −1 Mpc, leading to N bins = 30 for the angle-averaged correlation function and N bins = 60 for the clustering wedges.Equation (4) shows that an accurate estimate of the inverse covariance matrix requires a large number of independent realizations.While for the LOWZ sample our covariance matrix estimates are based on N mocks = 1000, for the CMASS sample we use two sets of 300 independent mocks.As these two sets are correlated, their combination does not lead to the same noise that would correspond to using 600 independent estimates.To account for this fact we follow Percival et al. (2013) and compute the correction term D in equation (4) using N mocks = 300, multiplied by (1 + r 2 )/2, where r corresponds to the correlation coefficient between the mock clustering measurements.The volume overlap between our mock catalogues implies r = 0.33 for DR10 and r = 0.49 for DR11 (for more details see Percival et al. 2013).The resulting correction factors for the clustering measurements used in our analysis are listed in Table 1.
Although the correction factor of equation ( 4) leads to an unbiased estimation of the inverse covariance matrix, it does not correct for the effect of the uncertainties in this estimate, which should be propagated into the obtained cosmological constraints.Percival et al. (2013) present a detailed description of the effect of the noise in the covariance matrix estimated from a set of mock realizations and derive formulae for their impact on the errors of the cosmological constraints measured by integrating over the likelihood function.They demonstrated that, to account for this extra uncertainty, the recovered parameter covariance constraints must be rescaled by a factor that depends on N bins , N mocks and the number of parameters included in the analysis, Npar (see equation 18 in Percival et al. 2013).Depending on the parameter space, our choice of range of scales and binning leads to a modest correction factor of at most 2.4 per cent for the results inferred from the clustering wedges, which we include in our constraints, and a negligible correction (less than 0.3 per cent) for the results obtained from the angleaveraged correlation function.

Additional data-sets
We combine the information encoded in the full shape of our clustering measurements with additional observations in order to improve the obtained cosmological constraints.Here we give a brief description of each additional data set.
We use the low-ℓ and high-ℓ CMB temperature power spectrum from the one-year data release of the Planck satellite (Planck Collaboration I 2013) with the low-ℓ polarization measurements from the nine-year of observations of the WMAP satellite (Bennett et al. 2013;Hinshaw et al. 2013).This data set corresponds to the 'Planck+WP' case considered in Planck Collaboration XVI (2013).For simplicity we refer to this combination simply as 'Planck'.We extend this data set using the high-ℓ CMB measurements from the Atacama Cosmology Telescope (ACT, Das et al. 2013) and the South Pole Telescope (SPT, Keisler et al. 2011;Story et al. 2013;Reichardt et al. 2012).We refer to this combination as 'ePlanck'.We also explore the constraints obtained by replacing the Planck CMB data by the final nineyear results from the WMAP satellite (Bennett et al. 2013;Hinshaw et al. 2013) to test the consistency between these data sets.We refer to these measurements as 'WMAP9'.
We also include information from distance measurements inferred from the angle-averaged BAO signal from independent samples.We use the results of Beutler et al. (2011) from the large-scale correlation function of the 6dF Galaxy Survey (6dFGS, Jones et al. 2009) corresponding to z = 0.106, and the distance measurements inferred from the Lyman-α forest in BOSS (Busca et al. 2013;Slosar et al. 2013;Kirkby et al. 2013), corresponding to z = 2.4.These data sets constrain the parameter combination DV(z)/r d , where c 0000 RAS, MNRAS 000, 000-000 We do not include the results of Xu et al. (2012) based on the final SDSS-II LRG sample as this catalogue is partially contained in the LOWZ sample used here, or the results from Blake et al. (2011) from the final WiggleZ Dark Energy Survey (Drinkwater et al. 2010) at z = 0.44, 0.6 and 0.73 due to the overlap of these data with the CMASS sample.Finally, we also use the information from the Union2.1 type Ia supernovae (SN) compilation (Suzuki et al. 2012).This sample combines 833 SN drawn from 19 different data sets using the scheme of the original Union sample of Kowalski et al. (2008).For comparison, in some cases we also present results obtained using the SN compilation of Conley et al. (2011), which includes the high-redshift SN from the first three years of the Supernova Legacy Survey (SNLS).When using these data we follow the recipe of Conley et al. (2011) to take into account systematic errors in our cosmological constraints, which requires the introduction of two additional nuisance parameters, α and β, related to the stretch-luminosity and colour-luminosity relationships.When quoting cosmological constraints based on this sample, the values of these parameters are marginalized over.
With the exception of Section 3.3, we use our BOSS clustering measurements in combination with CMB data.Unless stated otherwise, we use the information from our clustering measurements in the LOWZ and CMASS samples in combination, and refer to them as 'BOSS ξ(s)' for the angle-averaged correlation functions and 'BOSS ξ∆µ(s)' for the clustering wedges.Although the bulk of our analysis is based on our DR11 BOSS clustering measurements as they posses smaller statistical uncertainties, we test the consistency of these results with the constraints inferred using their DR10 counterparts.Our tightest constraints are obtained when the additional BAO and Union2.1 SN data are also included in our analysis.We refer to this case as our 'Full' combination.

Modelling of our clustering measurements
We follow the recipe of Sánchez et al. (2013) to model the full shape of the angle-averaged correlation function and the clustering wedges.This description takes into account the effects of non-linear evolution, redshiftspace distortions and bias which, if unaccounted for, could introduce systematic errors in the derived cosmological constraints (Smith et al. 2008;Crocce & Scoccimarro 2008;Angulo et al. 2008;Sánchez et al. 2008).Here we summarize the main details of our modelling and refer the reader to section 3 of Sánchez et al. (2013) for more details.
Both the angle-averaged correlation function and the clustering wedges can be obtained by integrating ξ(µ, s) over different µ-intervals.This means that a theoretical description of these measurements requires a model of the anisotropic correlation function.To obtain this model, it is convenient to decompose ξ(µ, s) in terms of Legendre polynomials as where the multipoles ξ ℓ (s) are given by These multipoles are related to those of the two-dimensional power spectrum, P (µ, k), by where j ℓ (x) is the spherical Bessel function of order ℓ (Hamilton 1997).We describe P (µ, k) with a simple parametrization as where f ≡ d ln D d ln a is the logarithmic structure growth-rate parameter, β = f /b, and PNL(k) represents the non-linear real-space power spectrum, given by with b, σv, and AMC treated as free parameters.Here PMC(k) is given by where F2(k, q) is the standard second order kernel of perturbation theory.The parametrization of equation ( 10) is motivated by renormalized perturbation theory (RPT, Crocce & Scoccimarro 2006) and is the basis of the parametrization of the non-linear correlation function proposed by Crocce & Scoccimarro (2008).This simple recipe provides an accurate description of the power spectra and correlation functions measured from N-body simulations (e.g.Sánchez et al. 2008;Montesano et al. 2010) and has been applied to the analysis of numerous galaxy samples (Sánchez et al. 2009(Sánchez et al. , 2012(Sánchez et al. , 2013;;Montesano et al. 2012;Beutler et al. 2011;Blake et al. 2011).The Lorentzian prefactor in equation ( 9) accounts for the Finger-of-God effect (Jackson 1972) under the assumption of an exponential galaxy velocity distribution function (Park et al. 1994;Cole et al. 1995).
Only a small number of multipoles of ξ(µ, s) have nonnegligible values on large scales.We base our description of the full ξ(µ, s) on the multipoles ξ ℓ (s) with ℓ < 4 of the parametrization of equation ( 9).Sánchez et al. (2013) showed that discarding contributions from multipoles with ℓ 4, this simple recipe provides an accurate description of the full shape of the angle-averaged correlation function and the clustering wedges ξ ⊥ (s) and ξ (s), leading to unbiased cosmological constraints.
One additional ingredient must be added to our model before it can be compared with real clustering measurements.As described in Section 2.1.1,these measurements require the assumption of a fiducial cosmology to convert the observed redshifts into distances.This choice must be taken into account in our models.The relation between the true values of s and µ characterizing a given galaxy pair and those measured in the fiducial cosmology can be written as (Ballinger et al. 1996) c 0000 RAS, MNRAS 000, 000-000 where the primes denote the quantities in the fiducial cosmology and the scaling factors are given by that is, the ratios of the angular diameter distance and the Hubble parameter evaluated at the mean redshift of the sample being considered, zm.These relations encode the effect of the fiducial cosmology on our clustering measurements, as they can be used to transform the integral in equation ( 1) from the fiducial cosmology space to the true cosmology as We use this relation to transform our theoretical predictions of ξ(s) and ξ∆µ(s) to the fiducial cosmology assumed in our BOSS clustering measurements.

Cosmological parameter spaces
We assume that primordial fluctuations are adiabatic, Gaussian, and have a power-law spectra of Fourier amplitudes, with a negligible tensor component.Table 2 lists the parameters that specify a given cosmological model under these assumptions.We use the data sets described in Section 2 to obtain constraints on these parameters.We start our analysis with the basic ΛCDM parameter space, which corresponds to a flat universe where the energy budget contains contributions from baryons, cold dark matter (CDM) and dark energy, described by an equation of state wDE = pDE/ρDE = −1.We follow Planck Collaboration XVI ( 2013) and assume a non-zero fraction of massive neutrinos with a fixed total mass mν = 0.06 eV.The free parameters required to characterize this model are listed in the upper part of Table 2.We also explore a number of possible extensions of this parameter space by allowing for variations on the additional parameters presented in the middle section of Table 2.These extensions include more general dark energy models, non-flat universes, different contributions from massive neutrinos, additional relativistic species and possible deviations from the predictions of general relativity (GR).The final part of Table 2 lists a number of important quantities whose values can be derived from the remaining parameters.
When studying the properties of the dark energy component we explore the cases of a time-independent dark energy equation of state wDE, and when this parameter is allowed to vary with time, in which case we assume the standard parametrization of Chevallier & Polarski (2001) and Linder (2003) given by When exploring the constraints on other potential extensions of the ΛCDM model, we investigate the impact of allowing also for variations on wDE.
We explore these parameter spaces by means of the Markov Chain Monte Carlo (MCMC) technique.We use the June 2013 version of CosmoMC (Lewis & Bridle 2002), modified to include our BOSS clustering measurements as additional data sets.This code uses camb to compute power spectra for the CMB and matter fluctuations (Lewis et al. 2000), which implements the parametrized post-Friedman framework (Hu & Sawicki 2007) to account for models with wDE < −1 and dynamical dark energy models, as described in Fang et al. (2008).Besides the cosmological parameters described here, the analysis of the CMB data requires the inclusion of a number of nuisance parameters that are included in our MCMC and marginalized over, as described in Planck Collaboration XVI (2013).When including clustering measurements from BOSS in our analysis, the parameters b, σv and AMC for each data set are included as additional free parameters in our MCMC and marginalized over.

The cosmological information in the correlation function and the clustering wedges
In this section we analyse the information on geometrical quantities encoded in our measurements of ξ(s) and the clustering wedges ξ ⊥ (s) and ξ (s).As discussed in detail by Kazin et al. (2012), while angle-averaged mea- plane at zm = 0.32 (left panel) and zm = 0.57 (right panel) derived from the LOWZ and CMASS DR11 samples, respectively.The grey long-dashed contours show the results obtained using information from the angle-averaged correlation function while the red solid lines correspond to those inferred from the clustering wedges ξ ⊥ (s) and ξ (s) assuming that f (z) follows the predictions of GR.The short-dashed contours correspond to the prediction for these parameters derived from the ePlanck data set (see Section 2.2) under the assumption of a ΛCDM model.
surements such as ξ(s) provide constraints on the ratio DV(zm)/r d , anisotropic clustering measurement such as the clustering wedges constrain the combinations DA(zm)/r d and H(zm)r d .In this way, the BAO signal in the clustering wedges can be used to break the degeneracy between DA(zm) and H(zm) obtained from ξ(s).When the full shape of these measurements is taken into account, the extra information on f (zm) provided by the amplitude difference between ξ ⊥ (s) and ξ (s) improves the constraints from those recovered when only the BAO signal is considered (Sánchez et al. 2013).
We now investigate the constraints on these quantities that can be derived from our clustering measurements.To do this, we explore our most general parameter space treating these quantities as derived parameters, with their values computed in the context of the cosmological model being tested.This approach differs from the one applied in our companion papers (Anderson et al. 2013b;Chuang et al. 2013b;Samushia et al. 2013b;Beutler et al. 2013, Tojeiro et al. in preparation) where the values of DA(z) and H(z) are treated as free parameters (i.e.without adopting a specific relation between their values).While this approach will lead to more general constraints on these parameters than the ones derived here, our results can be used as an indication of the information content in our clustering measurements and a consistency test with the results of these analyses.For this exercise, we apply flat priors on the parameters Φ = (ω b , ωc, ns), which determine the shape of the linear-theory matter power spectrum, centred on the values corresponding to the best-fitting ΛCDM model to the Planck CMB data, with a width equivalent to six times their 68 per cent confidence levels (CL).
Table 3 lists the geometrical constraints obtained from the full shape of our clustering measurements alone, assum-ing that f (zm) follows the predictions of GR.Here we have rescaled our results by the sound horizon at the drag redshift for our fiducial cosmology, r fid d = 149.31Mpc, to express them in units of Mpc and km s −1 Mpc −1 .In all cases the results recovered from the DR10 and DR11 samples are in good agreement.For the CMASS sample, the extra volume of DR11 leads to a reduction of approximately 20 per cent in the allowed regions.For the DR11 LOWZ sample, although the limits on H(zm) and DA(zm) inferred from the clustering wedges exhibit an improvement of about 10 per cent with respect to the corresponding DR10 results, the constraints on DV(zm) obtained from ξ(s) show the opposite behaviour.This is consistent with the results of the BAO-only analysis of Tojeiro et al. (in preparation) and might be related to the higher amplitude of the acoustic peak in the DR10 LOWZ ξ(s), which leads to a more accurate determination of its centroid.Fig. 3  We also test the effect of relaxing the assumption of GR to compute f (zm) by treating its value as a free parameter.Table 4 summarizes the results obtained from the clustering wedges of our galaxy samples, which are in good agreement with the ones obtained from the BAO-only analysis of the same galaxy samples in Anderson et al. (2013b) and Tojeiro et al. (in preparation).Exploring this parameter space we can constrain the combination f σ8(z), for which we ob-Table 3. Marginalized 68 per cent geometrical constraints derived from the full shape of our clustering measurements alone, under the assumption that f (zm) follows the predictions of GR.We have rescaled our results by the sound horizon at the drag redshift for our fiducial cosmology, r fid d = 149.31Mpc, to express them in units of Mpc and km s  Although these values are lower than the ones reported here, Chuang et al. (2013b) apply a significantly wider prior on the parameters Φ and find evidence for an increase in the recovered value for this quantity when smaller scales are included in the analysis.The dashed lines in Fig. 3 correspond to the constraints obtained from the ePlanck CMB measurements under the assumption of a ΛCDM model.The results obtained from the clustering wedges are in good agreement with the predictions of the ΛCDM model that best describes these CMB data, indicating the consistency between these data sets and their agreement with the ΛCDM model.For more gen- eral parameter spaces, the region in the DA(zm) r fid d /r d -H(zm) r d /r fid d plane allowed by the CMB data increases substantially.In these cases the combination of the CMB data with the information provided by our clustering measurements improves the constraints over those recovered from the CMB information alone.In Section 4 we explore the cosmological implications of the information contained in our clustering measurements.

COSMOLOGICAL CONSTRAINTS
Here we describe the cosmological implications of our BOSS clustering measurements.Section 4.1 presents the constraints on the parameters of the standard ΛCDM model, while Sections 4.2-4.5 explore the results obtained in more general parameter spaces.We pay particular attention to the constraints on the properties of the dark energy component and study how the limits in other parameters are changed when more general dark energy models are considered.Appendix A gives a complete list of the cosmologic 0000 RAS, MNRAS 000, 000-000 cal constraints derived from different data set combinations, while Table 5 summarizes the results on the most important parameters for the various cases we consider.

The ΛCDM parameter space
The simple ΛCDM model is able to describe an ever increasing amount of precise cosmological observations, with the CMB measurements from the Planck satellite being perhaps the most striking example (Planck Collaboration XVI 2013).However, as the statistical uncertainties of these measurements improve, the careful analysis of the consistency of the results derived from different data sets becomes crucial as it can be used to detect the presence of systematic errors.
Here we review the constraints on the parameters of the ΛCDM model obtained by combining our BOSS clustering measurements with different data sets.
The blue dashed lines in Fig. 4 correspond to the constraints in the Ωm − h plane derived from the Planck (upper panel) and WMAP9 (lower panel) CMB measurements.Both sets of constraints are elongated along the same degeneracy, which approximately corresponds to a constant value of Ωmh 3 (Percival et al. 2002;Planck Collaboration XVI 2013), indicated by the dotted lines.Along this degeneracy, the constraints from WMAP9 extend towards lower values of Ωm and higher values of h than those derived from the Planck data.This behaviour leads to different, but consistent, marginalized constraints on these parameters.As shown by the red solid lines in Fig. 4, when these CMB measurements are combined with the informa- tion from the LOWZ and CMASS DR11 clustering wedges the obtained constraints are significantly improved, leading to Ωm = 0.283 ± 0.010 and h = 0.6947 ± 0.0097 for the WMAP9+BOSS ξ∆µ(s) combination and Ωm = 0.2974 ± 0.0098 and h = 0.6859±0.0076for the Planck+BOSS ξ∆µ(s) data set.Although the differences in the constraints from Planck and WMAP9 are propagated to these results, our BOSS DR11 clustering measurements select the lowest values of Ωm allowed by Planck, leading to final constraints which are consistent within one σ with those derived using WMAP9.As shown in Table 5, using the angle-averaged correlation function of the LOWZ and CMASS DR11 samples leads to similar constraints.
The Dashed lines in Figs. 1 and 2 correspond to the best-fitting ΛCDM model to the Planck+BOSS ξ∆µ(s) combination.This model provides an excellent description of the broad band shape and the location of the BAO peak in these measurements, with χ 2 values of 49.7 and 48.3 over 52 degrees of freedom for the DR11 LOWZ and CMASS clustering wedges, respectively.Despite the fact that this model was obtained by fitting the clustering wedges of the DR11 galaxy samples it also provides an excellent description of all our clustering measurements, including our DR10 results.This illustrates the consistency between these data sets, which can also be seen in the cosmological constraints obtained when the Planck CMB data is combined with the DR10 LOWZ and CMASS clustering wedges, in which case we find Ωm = 0.294 ± 0.010 and h = 0.6882 ± 0.0079, in good agreement with the results obtained using DR11 information.
Our tightest constraints on the parameters of the ΛCDM model are obtained by combining the ePlanck CMB data set with the information from our DR11 BOSS ξ∆µ(s), SN and BAO data sets, leading to Ωm = 0.2924 ± 0.0086 and h = 0.6899 ± 0.0070

The dark energy equation of state
In the standard ΛCDM model, the current phase of accelerated cosmic expansion is due to a dark energy component characterized by a constant equation of state wDE = −1.As this hypothesis is consistent with all current cosmological observations, it has become the standard model for dark energy.However, a variety of alternative models have been proposed (for a review see e.g., Peebles & Ratra 2003;Frieman et al. 2008).Here we explore the constraints on more general dark energy models by allowing for variations in wDE and its possible evolution with time.
We start our analysis by extending the ΛCDM parameter space including wDE, assumed constant in time, as a free parameter.The blue long-dashed contours in Fig. 5 correspond to the 68 and 95 per cent confidence levels in the Ωm-wDE plane obtained in this case from the WMAP9 (right panel) and Planck (left panel) CMB data.The constraints derived from both of these data sets exhibit a degeneracy between these parameters.However, the strong degeneracy seen in the WMAP9 constraints is somewhat reduced in the results derived from Planck, as the information from the higher acoustic peaks restricts the region of the parameter space with wDE > −1.When these CMB data sets are combined with the information from the DR11 LOWZ and CMASS angle-averaged correlation functions, the allowed region for these parameters is reduced to a narrow degeneracy that is mostly driven by the DV/r d constraint provided by the CMASS sample (grey long-dashed lines in Fig. 5).In these cases the dark energy equation of state is only weakly constrained, with wDE = −1.05+0.29  −0.13 and wDE = −1.28+0.24 −0.16 c 0000 RAS, MNRAS 000, 000-000 Figure 6.Marginalized 68 and 95 per cent CL in the w 0 -wa plane when we explore the redshift dependence of the dark energy equation of state, parametrized as in equation ( 17).The contours show the results obtained using the ePlanck CMB data alone (blue dashed lines), the ePlanck+BOSS ξ ∆µ (s) combination (red solid lines), and when this information is combined with our BAO and SN data sets (green dot-dashed lines).The dotted lines correspond to the fiducial values of these parameters in the ΛCDM model, w 0 = −1 and wa = 0.
for the WMAP9+BOSS ξ(s) and Planck+BOSS ξ(s) combinations, respectively.The red solid lines in Fig. 5 show the constraints obtained when the WMAP9 and Planck CMB data sets are combined with the information from the full shape of the DR11 LOWZ and CMASS clustering wedges.The additional information provided by ξ ⊥ (s) and ξ (s) can break the degeneracy present in the CMB results much more efficiently than the angle-averaged correlation function, leading to broadly similar results for both CMB data sets.In particular, the marginalized constraints on the matter density parameter are almost identical, with Ωm = 0.288 ± 0.015 for the WMAP9+BOSS ξ∆µ(s) case and Ωm = 0.289 ± 0.016 for the Planck+BOSS ξ∆µ(s) combination.However, the differences in the CMB data sets lead to slightly different constraints on the dark energy equation of state of wDE = −0.964±0.077(WMAP9+BOSS ξ∆µ(s)) and wDE = −1.049± 0.078 (Planck+BOSS ξ∆µ(s)).These results show that the combination of current CMB and LSS data sets can constrain the dark energy equation of state with an accuracy of 8 per cent, leading to results in good agreement with a cosmological constant, indicated by the dotted line in Fig. 5. Our results are consistent with those reported in our companion papers (Anderson et al. 2013b;Samushia et al. 2013a;Chuang et al. 2013b), who find similar constraints on wDE from the combination of CMB data with various types of anisotropic clustering information from the DR11 CMASS and LOWZ samples.This agreement illustrates the robustness of these limits with respect to the methodology implemented to obtain them.
Although the constraints obtained using the WMAP9 and Planck data sets are consistent at the one σ level, the difference between these results highlights the importance of understanding the origin of the discrepancies between these data sets.The same behaviour is seen in other parameter spaces, once combined with our measurements of the LOWZ and CMASS clustering wedges, the WMAP9 and Planck CMB data sets give similar results, although the mean values are shifted by up to one σ.In the following sections we focus on the Planck CMB measurements and derive constraints using the ePlanck combination, but we compare the results with those derived using WMAP9 in some particular cases.
Combining the Planck CMB data with our DR10 clustering measurements leads to consistent results.The combination of Planck and the DR10 LOWZ and CMASS ξ∆µ(s) provides the constraints Ωm = 0.279 ± 0.019 and wDE = −1.092+0.092  −0.088 .The smaller statistical uncertainties associated with the DR11 LOWZ and CMASS clustering measurements lead to a reduction of ∼ 15 per cent in the constraints on the dark energy equation of state.As the same agreement is seen in all cosmological parameter spaces, from now on we focus on the results obtained using the DR11 galaxy samples.
Including the information from the high-ℓ CMB experiments improves the constraints only marginally.Using the ePlanck + BOSS ξ∆µ(s) combination we find wDE = −1.051± 0.076.Our final constraints are obtained when the information from the additional BAO and Union2.1 SN measurements are added to this data combination, leading to wDE = −1.024± 0.052, in good agreement with the ΛCDM model value of wDE = −1, and Ωm = 0.293 ± 0.011.Replacing the information from the Union2.1 SN sample by the SNLS data leads to a change in the recovered values of about one σ, with wDE = −1.071± 0.055 and Ωm = 0.283 ± 0.011, showing a preference for values of wDE < −1.As pointed out by Planck Collaboration XVI (2013), the difference between the results obtained using these samples might indicate that the treatment of the systematic errors affecting these SN data sets is incomplete.
Although exploring the constraints on a constant wDE could indicate a deviation from the standard ΛCDM paradigm, more general dark energy models, such as those based on a scalar field, will be characterized by a timedependent equation of state (e.g.Wetterich 1988).We explore the constraints on the time-dependence of wDE, parametrized as in equation ( 17).The blue dashed lines in Fig. 6 correspond to the two-dimensional marginalized constraints in the w0-wa plane obtained from the ePlanck CMB, covering a large region of the parameter space.The red solid lines in Fig. 6 correspond to the results obtained by combining the ePlanck CMB measurements with our BOSS ξ∆µ(s) data set, showing a significant reduction of the allowed region for these parameters.In this case we find w0 = −0.83+0.38  −0.34 and wa = −0.61+0.89 −0.96 .As shown by the green dot-dashed lines in the same figure, the information from our Full data set combination tightens the constraints, leading to w0 = −0.95± 0.14 and wa = −0.29 ± 0.47, in agreement with the standard ΛCDM model values indicated by the dotted lines.

Non-flat universes
The standard ΛCDM model assumes a flat universe.Here we test this assumption by adding Ω k to the list of free parameters of our base model.The blue dashed contours in Fig. 7 show the 68 and 95 per cent marginalized constraints in the Ωm-Ω k plane derived by means of the ePlanck CMB data combination, which exhibit the so-called geometrical degeneracy (Efstathiou & Bond 1999) relating models with the same angular scale of the acoustic peaks in the CMB.This degeneracy extends over a wide range of values of Ω k , leading to weak constraints on this parameter.In this case we find 100Ω k = −4.2+2.7  −1.7 .As shown by Planck Collaboration XVI (2013), including information from the lensing signal inferred from the CMB partially reduces this degeneracy, but to obtain significantly tighter constraints it is necessary to combine these measurements with additional data sets.
When the ePlanck CMB data are combined with the monopole correlation functions of the DR11 LOWZ and CMASS samples, the constraints on DV/r d provided by these measurements are sufficient to break the geometrical degeneracy, leading to a constraint of 100Ω k = 0.07±0.31, in excellent agreement with a flat Universe.The red solid lines in Fig. 7 show the constraints obtained when the ePlanck data are combined with the clustering wedges of these galaxy samples, leading to a similar result of 100Ω k = 0.10 ± 0.29.Including our SN and BAO data sets leads only to a small shift in the recovered mean value for this parameter, with 100Ω k = 0.15 ± 0.29, showing no evidence for a deviation from the flat Universe hypothesis, indicated by a dotted line in Fig. 7.
Both Ω k and wDE are involved in the geometrical degeneracy, as they change the distance to the last-scattering surface.This means that when both of these parameters are varied simultaneously, the geometric degeneracy gains an extra degree of freedom, leading to a significant degradation of the obtained constraints.This effect is shown by the blue short-dashed contours in Fig. 8, which correspond to the constraints in the wDE -Ω k plane obtained from the ePlanck CMB data.In this case, the information from the LOWZ and CMASS angle-averaged correlation functions is not enough to break the geometric degeneracy completely (as shown by the grey long-dashed lines in Fig. 8).Although this information can constrain the curvature of the Universe to 100Ω k = −0.38 +0.24  −0.28 , it leaves a wide range of allowed values for the dark energy equation of state in the region where wDE < −1.
The red solid contours in Fig. 8 correspond to constraints obtained after combining the ePlanck CMB data with the full shape of the clustering wedges of the DR11 LOWZ and CMASS samples.The additional information in the clustering wedges reduces the allowed region of this parameter space significantly, leading to the constraints 100Ω k = 0.02 ± 0.43 and wDE = −1.05± 0.11.As shown by the green dot-dashed lines in Fig. 8, including the information from the SN and additional BAO measurements can improve the constraints even further, leading to 100Ω k = −0.14 ± 0.33 and wDE = −1.009+0.062 −0.060 , in excellent agreement with the ΛCDM model.In particular, the constraints on the dark energy equation of state obtained in this case have a similar accuracy as those presented in Sec.4.2 under the assumption of a flat universe.
c 0000 RAS, MNRAS 000, 000-000 Figure 9. Marginalized constraints in the Ωm-Σmν plane obtained when ΛCDM parameter set is extended by treating the neutrino mass as a free parameter.The short-and lon-dashed lines correspond to the 68 and 95 per cent CL derived by the Planck and ePlanck CMB data, respectively.The solid lines show the results obtained from the combination of the ePlanck CMB measurements with the full shape of the LOWZ and CMASS clustering wedges (red solid lines).

Massive neutrinos and relativistic species
In recent years, neutrino oscillation experiments have measured non-zero mass-squared differences between neutrino flavours, implying that they are massive and contribute to the total energy budget of the Universe.However, absolute neutrino mass measurements are more difficult to perform in the laboratory (Lobashev 2003;Eitel 2005;Otten & Weinheimer 2008) and current constraints are weaker than those imposed by cosmological observations such as CMB and LSS measurements (Lesgourgues & Pastor 2012).
Following the analysis of Planck Collaboration XVI (2013), our base ΛCDM model includes a non-zero contribution from massive neutrinos with mν = 0.06 eV to the total energy budget of the Universe.In this section we allow this parameter to vary freely assuming three neutrino species of equal mass and explore the constraints that can be imposed on this quantity by means of our DR11 clustering measurements.
The grey short-dashed lines in Fig. 9 correspond to the 68 and 95 per cent CL in the Ωm-mν plane derived from the Planck CMB data.These constraints are elongated along a line that corresponds to models with a constant redshift of matter-radiation equality, zeq, which is accurately measured from CMB observations (Komatsu et al. 2009).As indicated by the blue long-dashed lines in the same figure, extending these data with the high-ℓ CMB measurements from ACT and SPT improves the results significantly, but leaves a residual degeneracy that limits the constraints on mν.This is shown by the blue short-dashed line in Fig. 10, which corresponds to the one-dimensional marginalized constraints on mν obtained from the ePlanck CMB data, corresponding to mν < 0.66 eV (95 per cent CL).The constraints on the total neutrino mass can be improved by combining the CMB information with our isotropic and anisotropic galaxy clustering measurements.As this information improves the constraints on Ωm, it can break the degeneracy present in the CMB-only constraints.This effect is illustrated by the red solid lines in Fig. 9, which correspond to the results derived from our ePlanck + BOSS ξ∆µ(s) combination.The effect of the extra information from BOSS on the constraints on the neutrino mass is shown by the red solid line in Fig. 10.In this case we obtain the limit mν < 0.24 eV (95 per cent CL).As shown in Table 5, combining the CMB information with the LOWZ and CMASS angle-averaged correlation functions leads to a similar constraint.This limit is not improved by including the additional BAO and Union2.1 SN information in the analysis, in which case we obtain mν < 0.23 eV (95 per cent CL).When the SNLS SN compilation is used instead of the Union2.1 sample we find a slightly tighter constrain, with mν < 0.21 eV (95 per cent CL).The grey long-dashed lines in Fig. 10 correspond to the marginalized constraints on Σmν obtained using WMAP9 data alone, which extend over a wide range of allowed values.The green dot-dashed line shows the result obtained when the WMAP9 data is combined with the LOWZ and CMASS DR11 clustering wedges.Interestingly, in this case the constraints show a preference for a non-zero value of Σmν ≃ 0.2 eV.This is due to a slight difference in the values of zeq preferred by the Planck and WMAP9 data.When using WMAP9 data, the degeneracy seen in Fig. 9 is shifted towards lower values of Ωm.Adding the information from the clustering wedges measured from BOSS helps to tighten the constraints on the matter density, breaking the degeneracy obtained from the CMB at a region that shows a slight preference for non-zero values of Σmν .
The constraints on the total neutrino mass derived from cosmological observations are model dependent, as they vary depending on the parameter space being studied (Zhao et al. 2013).For example, if the dark energy equation of state is allowed to vary, the degeneracy in the CMB constraints gains an extra degree of freedom.This behaviour can be seen in the blue dashed contours of Fig. 11, which correspond to the two-dimensional marginalized constraints in the wDE-mν plane derived from the ePlanck data set.As shown by the grey long-dashed lines, in this parameter space the information provided by our BOSS ξ(s) measurements cannot break the CMB degeneracy efficiently, leading to poor marginalized constraints.The additional information in the full shape of the clustering wedges reduces the allowed region for these parameters significantly, leading to the limit mν < 0.47 eV (95 per cent CL) and wDE = −1.13± 0.12.After also including the BAO and SN information the constraints on the neutrino mass are improved to mν < 0.33 eV (95 per cent CL).This demonstrates that, when the uncertainties in the exact value of wDE are taken into account, the upper bound on mν is increased by 50 per cent with respect to the one obtained under the assumption of a ΛCDM model.
It is also interesting to explore for potential deviations on the effective number of relativistic species from its standard value N eff = 3.046.For this analysis we extend the ΛCDM parameter space including N eff as a free parameter, while assuming that the additional relativistic species are massless.As discussed in Planck Collaboration XVI (2013), the ePlanck CMB combination can place tight constraints on this parameter.These constraints are shown by the blue dashed lines in Fig. 12 which correspond to our ePlanck constraints in the Ωm-N eff plane, leading to N eff = 3.35 ± 0.33.Adding the information from the LOWZ and CMASS angle-averaged correlation functions leads to an improvement of this limit, with N eff = 3.31 ± 0.27.Using the information from our BOSS ξ∆µ(s) or adding the information from the BAO and SN data leaves this result essentially unchanged.

Constraining deviations from general relativity
In Sections 4.1-4.4we computed the logarithmic growth f (z) required for our model of the full shape of the clustering wedges in the context of GR.Here we relax this assumption and parametrize its redshift evolution as f (z) = Ωm(z) γ , with the exponent γ treated as a free parameter.As described in Linder & Cahn (2007), GR predicts a value of γ ≃ 0.55, with small corrections depending on the value of wDE.Then, the constraints on this parameter from the full shape of the LOWZ and CMASS clustering wedges can be used to set limits on potential deviations from the predictions of GR (Guzzo et al. 2008).This analysis is only possible using anisotropic clustering measurements such as the clustering wedges, as the effect of f (z) and the bias parameter are degenerate in angle-averaged quantities.
The blue short-dashed line in Fig. 13 corresponds to the one-dimensional marginalized constraints on γ obtained from the combination of the Planck CMB measurements with the DR11 CMASS ξ∆(s).In this case we obtain γ = 0.77±0.20.Although a wide range of values of this parameter c 0000 RAS, MNRAS 000, 000-000 are allowed by the data, these results are consistent within one σ with the predictions from GR, indicated by the dotted line.As shown by the red solid line, adding the information from the clustering wedges of the LOWZ sample improves the constraints to γ = 0.69 ± 0.15, which illustrates the importance of including the low-redshift measurements.As shown in Table 5, these constraints are not modified when the Planck data are extended with the high-ℓ CMB measurements, or when the BAO and SN data sets are included in the analysis.The grey long-dashed line in Fig. 13 shows the constraints obtained by means of the WMAP9+BOSS ξ∆(s) combination, which leads to a constraint of γ = 0.64 ± 0.15, in good agreement with the result obtained using the data from the Planck satellite.
Our constraints are in good agreement with those inferred in our companion papers.Samushia et al. (2013b) use the full shape of the CMASS monopole-quadrupole pair in combination with recent CMB measurements to find γ = 0.69 ± 0.11, while Beutler et al. (2013) derive a constraint of γ = 0.772 +0.124  −0.097 from the combination of Planck data with the multipoles of the CMASS power spectrum.
We also tested the effect of allowing for simultaneous variations of wDE (assumed time independent) and γ.Fig. 14 presents the two-dimensional marginalized constraints in the wDE-γ plane obtained in this case by means of the ePlanck + BOSS ξ∆µ(s) combination (solid lines), and when these data are combined with the BAO and SN data sets (dot-dashed lines).Including γ as a free parameter leads to a degeneracy between this quantity and the dark energy equation of state, degrading the constraints on these parameters.In this case we find wDE = −1.15± 0.11 and γ = 0.88 ± 0.22.As dis- cussed in Sánchez et al. (2013), assuming that f (zm) follows the predictions of GR implies that the relative amplitude of ξ ⊥ (s) and ξ (s) provides information on Ωm that improves the obtained constraints.However, when this assumption is relaxed including γ as a free parameter, this extra constraining powers is lost, leading to weaker limits.Including the SN and additional BAO measurements reduces the degeneracy present in the CMB + BOSS ξ∆µ(s) constraints, leading to wDE = −1.055± 0.057 and γ = 0.75 ± 0.17, in agreement with the results obtained when these parameters are varied separately.

CONCLUSIONS
We have analysed the cosmological implications of the angle-averaged correlation functions, ξ(s), and the clustering wedges, ξ ⊥ (s) and ξ (s), of the LOWZ and CMASS samples corresponding to SDSS-DR10 and DR11.We use a simple parametrization, based on renormalized perturbation theory, as a tool to extract cosmological information from the full shape of these measurements for s 40 h −1 Mpc.We combine this information with CMB, SN, and additional BAO measurements to derive constraints on the parameters of the standard ΛCDM model and a number of potential extensions, including curvature, alternative dark energy models, massive neutrinos, additional relativistic species and deviations from the predictions of general relativity.As shown by Sánchez et al. (2013), we find that the extra information provided by the clustering wedges is most useful when the dark energy equation of state is treated as a free parameter.
The constraints on H(z) and DA(z) from the clustering wedges are consistent with the predictions of the c 0000 RAS, MNRAS 000, 000-000 best-fitting ΛCDM model to the Planck CMB measurements.Assuming that the growth of structures follows the predictions of GR, the full shape of the LOWZ clustering wedges imply DA(zm) = (965 ± 37) r d /r fid d Mpc and H(zm) = (82.5 ± 3.5) r fid d /r d km s −1 Mpc −1 at the mean redshift zm = 0.32, while the CMASS results give DA(zm) = (1387 ± 22) r d /r fid d Mpc and H(zm) = (94.3± 2.4) r fid d /r d km s −1 Mpc −1 at zm = 0.57.Relaxing the assumption of GR, we find the marginalized constraints of f σ8(z = 0.32) = 0.48 ± 0.10 and f σ8(z = 0.57) = 0.417 ± 0.045.These values are in good agreement with those reported in our companion papers (Anderson et al. 2013b;Chuang et al. 2013b;Samushia et al. 2013b;Beutler et al. 2013), indicating the robustness of our results with respect to details in the methodology implemented in the analysis.
As can be seen in Table 5, our results show no significant evidence for a deviation from the standard ΛCDM model, which provides a good description of the full shape of all our clustering measurements.In particular, the ePlanck+BOSS ξ∆µ(s) combination alone is sufficient to constrain the curvature of the Universe to Ω k = 0.0010 ± 0.0029, the total neutrino mass to mν < 0.24 eV (95 per cent CL), the effective number of relativistic species to N eff = 3.31 ± 0.27, and the dark energy equation of state to wDE = −1.051± 0.076.Adding the information from our BAO and SN data sets further improves these constraints.
The assumption that the dark energy component can be characterized by a constant equation of state specified by wDE(z) = −1 has strong implications on the constraints obtained in several parameter spaces.Allowing this parameter to vary weakens the constraints obtained from the combination of CMB information with our BOSS ξ(s) measurements.These cases illustrate the extra constraining power of the clustering wedges, which lead in general to similar constraints than the ones derived under the assumption that dark energy behaves as a cosmological constant.
The information from the full shape of the clustering wedges can be used to constrain potential deviations from the predictions of GR.Assuming that f (z) = Ωm(z) γ , the combination of the ePlanck CMB measurements with the DR11 LOWZ and CMASS clustering wedges gives a constraint of γ = 0.69 ± 0.15, consistent with no deviation from the GR prediction of γ = 0.55 within one σ.The assumption that f (z) follows the predictions of GR implies that the relative amplitude of the clustering wedges contains information on Ωm.When this assumption is relaxed, this additional constraining power is lost, affecting the constraints on other parameters.For example, if γ and wDE are varied simultaneously, the ePlanck+BOSS ξ∆µ(s) combination implies that γ = 0.88 ± 0.22 and wDE = −1.15± 0.11.When the additional BAO and Union2.1 SN measurements are included in the analysis we find wDE = −1.055± 0.057 and γ = 0.75±0.17, in agreement with the results obtained when these parameters are varied separately.
Combining our clustering measurements with the WMAP9 or Planck CMB measurements leads to results that, although consistent, differ at the one σ level.This highlights the importance of a detailed analysis of the origin of the differences between WMAP9 and Planck as these could indicate the presence of systematic errors.Similar differences are observed when the Union2.1 SN compilation is replaced by the SNLS sample, which leads to changes in the obtained constraints of the same order or larger than the associated statistical errors.This situation might change in the near future, as results based on new calibrations of these SN samples (Betoule et al. 2013) become available.
As SDSS-III approaches the end of observations in June 2014, the galaxy and quasar samples from BOSS are close to being completed.Besides the improvement in the statistical power of the final data sets, these samples will allow us to perform an improved analysis of the potential systematic errors affecting our measurements.The final galaxy samples from BOSS will deliver the most accurate views of the LSS of the Universe at z < 0.7, thus providing invaluable cosmological information.

Figure 2 .
Figure 2. The same as Figure 1, but for the LOWZ and CMASS DR11 galaxy samples.

Figure 3 .
Figure 3. Two-dimensional marginalized constraints in the D A (zm) r fid d /r d -H(zm) r d /r fid d shows the two-dimensional marginalized constraints on DA(zm) r fid d /r d and H(zm) r d /r fid d at zm = 0.32 (left panel) and zm = 0.57 (right panel), derived from the DR11 LOWZ and CMASS samples, respectively.The long-dashed lines correspond to the constraints derived from the angle-averaged correlation function, which correspond to a degeneracy of constant DV(zm)/r d .For each sample, the extra information contained in the clustering wedges breaks this degeneracy, leading to separate constraints on DA and H shown by the solid lines.

Figure 4 .
Figure 4. Two-dimensional marginalized constraints in the Ωmh plane.The blue dashed lines correspond to the constraints from the Planck (upper panel) and WMAP9 (lower panel) CMB measurements, which follow a degeneracy of constant Ωmh 3 , as indicated by the dotted lines.The red solid lines show the results obtained when these measurements are combined with the information from the LOWZ and CMASS DR11 clustering wedges.

Figure 5 .
Figure 5. Left panel: marginalized 68 and 95 per cent CL in the Ωm-w DE plane for the ΛCDM parameter set extended by including the redshift-independent value of w DE as an additional parameter.The contours correspond to the results obtained using the WMAP9-only (blue long-dashed lines), the WMAP9+BOSS ξ(s) combination (grey short-dashed lines) and the WMAP9+BOSS ξ ∆µ (s) case (red solid lines).The right panel shows the results obtained when the WMAP9 measurements are replaced by the Planck CMB data set.The dotted line in both panels corresponds to the ΛCDM model value of w DE = −1.

Figure 7 .
Figure 7. Marginalized constraints in the Ωm-Ω k plane when the ΛCDM model is extended to allow for non-flat models.The contours show the 68 and 95 per cent CL obtained using the ePlanck CMB data alone (blue dashed lines) and the ePlanck+BOSS ξ ∆µ (s) combination (red solid lines).The dotted line corresponds to flat universes, with Ω k = 0.

Figure 8 .
Figure8.Marginalized constraints in the w DE -Ω k plane for the ΛCDM parameter set extended by allowing for simultaneous variations on both of these parameters.The contours correspond to the 68 and 95 per cent CL derived from the combination of ePlanck data alone (blue dashed lines), ePlanck plus the clustering wedges of the LOWZ ans CMASS DR11 samples (red solid lines), and when the BAO and SN data sets are added to the later combination (green dot-dashed lines).The dotted lines correspond to the values of these parameters in the ΛCDM model.

Figure 10 .
Figure10.One-dimensional marginalized constraints on mν obtained by means of the ePlanck CMB measurements (blue short-dashed line) and the combination of these data with the BOSS ξ ∆µ (s) (red solid line).The grey long-dashed and green dot-dashed lines correspond to the results obtained when the ePlanck CMB measurements are replaced by WMAP9.

Figure 11 .
Figure11.Marginalized constraints in the w DE -Σmν plane obtained when ΛCDM parameter set is extended by treating these quantities as free parameters.The contours correspond to the 68 and 95 per cent CL derived by the ePlanck CMB data alone (blue short-dashed lines) and the ePlanck+BOSS ξ(s) (grey longdashed lines), ePlanck+BOSS ξ ∆µ (s) (red solid lines), and Full data combinations (green dot-dashed lines).

Figure 12 .
Figure 12.Marginalized constraints in the Ωm-N eff plane obtained when ΛCDM parameter set is extended by treating the effective number of relativistic species as a free parameter.The dashed and solid lines correspond to the 68 and 95 per cent CL derived by the ePlanck CMB data and its combination with the full shape of the DR11 LOWZ and CMASS clustering wedges.The dotted line indicates the standard value of N eff = 3.046

Figure 13 .
Figure 13.One-dimensional marginalized constraints on the power-law index of the structure growth rate parameter, assuming f (z) = Ω γ m .The blue dashed lines correspond to the results obtained by combining the ePlanck CMB data with the DR11 CMASS clustering wedges.The red solid line shows the improvement obtained by including the information of the clustering wedges of the LOWZ sample in the analysis.Replacing the ePlanck CMB measurements by the WMAP9 data leads to similar constraints (grey dashed lines).

Figure 14 .
Figure 14.Marginalized constraints in the w DE -γ plane obtained when ΛCDM parameter set is extended by treating these quantities as free parameters.The red solid lines correspond to the 68 and 95 per cent CL obtained by combining the ePlanck CMB data with the DR11 LOWZ and CMASS clustering wedges.The green dot-dashed lines show the result of including the additional BAO and SN data sets in the analysis.

Table 1 .
Correction factors of equation (4) to account for the bias in the estimation of the inverse covariance matrix.

Table 2 .
Cosmological parameters considered in our analysis.The upper part lists the parameters of the standard ΛCDM model while the middle section lists a number of possible extensions of this parameter space.The lower part lists a number of important quantities whose values can be derived from the first two sets.Present-day dark energy equation of state, w DE wa Time-dependence of w DE (assuming w DE (a) = w 0 + wa(1 − a)) mν Total sum of the neutrino masses N eff Effective number of relativistic species γ Power-law index of the structure growth-rate parameter, assuming f (z) = Ω γ 8 Linear-theory rms mass fluctuations in spheres of radius 8 h −1 Mpc fν Dark matter fraction in massive neutrinos f (zm) Structure growth-rate parameter, f (z) = d ln D d ln a a Quoted at the pivot wavenumber of k 0

Table 4 .
Marginalized 68 per cent constraints on geometrical quantities and the growth of structure derived from the full shape of our clustering wedges, when the assumption that f (zm) follows the predictions of GR is relaxed.We have rescaled our results by the sound horizon at the drag redshift for our fiducial cosmology, r fid

Table 5 .
Marginalized 68% constraints on the most relevant cosmological parameters of the parameter spaces analysed in Sections 4.1 to 4.5, obtained using different combinations of the data sets described in Section 2. A complete list of the constrains obtained in each case can be found in Appendix A.

Table A1 .
Marginalized 68% constraints on the cosmological parameters of the standard ΛCDM model, obtained using different combinations of the data sets described in Section 2.

Table A2 .
Marginalized 68% constraints on the cosmological parameters of the ΛCDM model extended by including w DE (assumed constant) as a free parameter, obtained using different combinations of the datasets described in Section 2.

Table A3 .
The marginalized 68% constraints on the cosmological parameters of the ΛCDM model extended by allowing for variations on w DE (a) (parametrized according to equation 17), obtained using different combinations of the datasets described in Section 2.

Table A5 .
The marginalized 68% constraints on the cosmological parameters of the ΛCDM model extended by allowing for simultaneous variations on w DE and Ω k , obtained using different combinations of the datasets described in Section 2.

Table A7 .
The marginalized 68% constraints on the cosmological parameters of the ΛCDM model extended by allowing for simultaneous variations on mν and w DE , obtained using different combinations of the datasets described in Section 2. ePlanck+BOSS ξ(s) ePlanck+BOSS ξ ∆µ (s) ePlanck + BOSS ξ ∆µ (s) +BAO+SN

Table A8 .
Marginalized 68% constraints on the cosmological parameters of the ΛCDM model extended by including N eff as a free parameter, obtained using different combinations of the datasets described in Section 2. ePlanck+BOSS ξ(s) ePlanck+BOSS ξ ∆µ (s) ePlanck + BOSS ξ ∆µ (s) +BAO+SN