# The clustering of galaxies in the SDSS-III Baryon Oscillation Spectroscopic Survey: single-probe measurements and the strong power of on constraining dark energy

###### Abstract

We present measurements of the anisotropic galaxy clustering from the Data Release 9 (DR9) CMASS sample of the SDSS-III Baryon Oscillation Spectroscopic Survey (BOSS). We analyze the broad-range shape of the monopole and quadrupole correlation functions to obtain constraints, at the effective redshift z=0.57 of the sample, on the Hubble expansion rate , the angular-diameter distance , the normalized growth rate , the physical matter density , and the biased amplitude of matter fluctuation . We obtain , , , , = kmsMpc, Mpc, , , and their covariance matrix as well. The parameters which are not well constrained by our galaxy clustering analysis are marginalized over with wide flat priors. Since no priors from other data sets (i.e., CMB) are adopted and no dark energy models are assumed, our results from BOSS CMASS galaxy clustering alone may be combined with other data sets, i.e. CMB, SNe, lensing or other galaxy clustering data to constrain the parameters of a given cosmological model. We show that the major power on constraining dark energy from the anisotropic galaxy clustering signal, as compared to the angular-averaged one (monopole), arises from including the normalized growth rate . In the case of the cosmological model assuming a constant dark energy equation of state and a flat universe (CDM), our single-probe CMASS constraints, combined with CMB (WMAP9+SPT), yield a value for the dark energy equation of state parameter of . Therefore, it is important to include while investigating the nature of dark energy with current and upcoming large-scale galaxy surveys.

###### keywords:

cosmology: observations - distance scale - large-scale structure of Universe - cosmological parameters## 1 Introduction

The cosmic large-scale structure from galaxy redshift surveys provides a powerful probe of dark energy and the cosmological model that is highly complementary to the cosmic microwave background (CMB) (e.g., Hinshaw et al. 2012), supernovae (SNe) (Riess et al., 1998; Perlmutter et al., 1999), and weak lensing (e.g. see Van Waerbeke & Mellier 2003 for a review).

The scope of galaxy redshift
surveys has dramatically increased in the last decade. The 2dF Galaxy Redshift Survey (2dFGRS)
obtained 221,414 galaxy redshifts at (Colless et al., 2001, 2003),
and the Sloan Digital Sky Survey (SDSS, York et al. 2000) has collected
930,000 galaxy spectra in the Seventh Data Release (DR7) at (Abazajian et al., 2009).
WiggleZ has collected spectra of 240,000 emission-line galaxies at over
1000 square degrees (Drinkwater et al., 2010; Parkinson et al., 2012), and the Baryon Oscillation Spectroscopic Survey (BOSS, Dawson et al. 2013) of the SDSS-III (Eisenstein et al., 2011) is surveying 1.5 million
luminous red galaxies (LRGs) at over 10,000 square degrees.
The first BOSS data set has been made publicly available recently in SDSS data release 9 (DR9, Ahn et al. 2012).
The planned space mission Euclid^{1}^{1}1http://sci.esa.int/euclid will survey over 60 million emission-line galaxies at over 15,000 deg
(e.g. Laureijs et al. 2011) and the upcoming ground-based experiment BigBOSS^{2}^{2}2http://bigboss.lbl.gov/ will survey 20 million galaxy redshifts up to and
600,000 quasars () over 14,000 deg (Schlegel et al., 2011).
The WFIRST satellite would map 17 million galaxies in the redshift
range over 3400 deg, with a larger area
possible with an extended mission (Green et al., 2012).

Large-scale structure data from galaxy redshift surveys can be analyzed using either the power spectrum or the two-point correlation function. Although these two methods are Fourier transforms of one another, the analysis processes, the statistical uncertainties, and the systematics are quite different and the results cannot be converted using Fourier transform directly because of the finite size of the survey volume. The SDSS-II LRG data have been analyzed, and the cosmological results delivered, using both the power spectrum (see, e.g., Tegmark et al. 2004; Hutsi 2005; Padmanabhan et al. 2007; Blake et al. 2007; Percival et al. 2007, 2010; Reid et al. 2010; Montesano et al. 2011), and the correlation function method (see, e.g., Eisenstein et al. 2005; Okumura et al. 2008; Cabre & Gaztanaga 2009; Martinez et al. 2009; Sanchez et al. 2009; Kazin et al. 2010a; Chuang, Wang, & Hemantha 2012; Samushia et al. 2011; Padmanabhan et al. 2012; Xu et al. 2013). Similar analysis have been also applied on the SDSS-III BOSS CMASS sample and obtained the most precise measurements to date (Anderson et al., 2012; Manera et al., 2013; Nuza et al., 2012; Reid et al., 2012; Samushia et al., 2012; Tojeiro et al., 2012).

Galaxy clustering allows us to differentiate smooth dark energy and modified gravity as the cause for cosmic acceleration through the simultaneous measurements of the cosmic expansion history , and the growth rate of cosmic large scale structure, (Guzzo et al., 2008; Wang, 2008; Blake et al., 2012). However, to measure , one must measure the galaxy bias , which requires measuring higher-order statistics of the galaxy clustering (see Verde et al. 2002). Song & Percival (2009) proposed using the normalized growth rate, , which would avoid the uncertainties from the galaxy bias. Percival & White (2009) developed a method to measure and applied it on simulations. Wang (2012) estimated expected statistical constraints on dark energy and modified gravity, including redshift-space distortions and other constraints from galaxy clustering, using a Fisher matrix formalism.

In principle, the Hubble expansion rate , the angular-diameter distance , the normalized growth rate , and the physical matter density can be well constrained by analyzing the galaxy clustering data alone. Eisenstein et al. (2005) demonstrated the feasibility of measuring and an effective distance, , from the SDSS DR3 LRGs, where corresponds to a combination of and . Chuang & Wang (2012a) measured and simultaneously using the galaxy clustering data from the two dimensional two-point correlation function of SDSS DR7 LRGs. Chuang & Wang (2012b, c) improved the method and modeling to measure , , , and from the same data.

Samushia et al. (2011) measured from the SDSS DR7 LRGs. Blake et al. (2012) measured , , and from the WiggleZ Dark Energy Survey galaxy sample. Reid et al. (2012) measured , , and from the SDSS BOSS DR9 CMASS and Samushia et al. (2012) derived the cosmological implications from these measurements to test deviations from the concordance CDM model and general relativity (see also Nesseris et al. (2011) for using to constrain modified gravity theories).

In this study, we apply the similar method and model as Chuang & Wang (2012b, c) to measure , , , and which extracts a summary of the cosmological information from the large-scale structure of the SDSS BOSS DR9 CMASS alone. One can combine our single-probe measurements with other data sets (i.e. CMB, SNe, etc.) to constrain the cosmological parameters of a given dark energy model. We also explore the strong power of adding to the two dimensional galaxy clustering analysis on constraining dark energy.

This study is part of a series of papers performing anisotropic clustering analysis on the BOSS DR9 CMASS galaxy sample. Instead of using multipoles, Sanchez et al. (2013) present a different method taken from the ’clustering wedges’ measurements by Kazin et al. (2013) and combine the results with CMB, SNe, and Baryon Acoustic Oscillations (BAO) to obtain constraints on the cosmological parameters. Anderson et al. (2013) present anisotropic analysis using two approaches, multipoles and wedges, to obtain robust measurement of the BAO signal.

This paper is organized as follows. In Section 2, we introduce the SDSS-III/BOSS DR9 galaxy sample and mock catalogues used in our study. In Section 3, we describe the details of the methodology that constrains cosmological parameters from our galaxy clustering analysis. In Section 4, we present our single-probe cosmological measurements and demonstrate how to use our results assuming different cosmological models or combining other data sets. In Section 5, we compare our results with previous or parallel works. In Section 6, we discuss the requirements to provide single-probe measurements. In Section 7, we apply some systematic tests to our measurements. We summarize and conclude in Sec. 8.

## 2 Data Set

### 2.1 The CMASS Galaxy Sample

The Sloan Digital Sky Survey (SDSS; Fukugita et al. 1996; Gunn et al. 1998; York et al. 2000; Smee et al. 2012) mapped over one quarter
of the sky using the dedicated 2.5m Sloan Telescope (Ahn et al., 2012).
The Baryon Oscillation Sky Survey (BOSS, Eisenstein et al. 2011; Bolton et al. 2012; Dawson et al. 2013) is part of the SDSS-III survey.
It is collecting the spectra and redshifts for 1.5 million galaxies, 160,000 quasars and
100,000 ancillary targets. The Data Release 9 has been made publicly available^{3}^{3}3http://www.sdss3.org/.
We use galaxies from the SDSS-III BOSS DR9 CMASS catalogue in the redshift range .
’CMASS’ samples are selected with an approximately constant stellar mass threshold (Eisenstein et al., 2011).
The sample we are using includes a total of 264,283 galaxies with 207,246 in the north and 57,037 in the
south Galactic hemispheres. The median redshift of the sample is .
The details of generating this sample are described in Dawson et al. (2013).

### 2.2 The Mock Catalogues

Manera et al. (2013) created 600 mock catalogues for DR9 CMASS sample. They created 2nd-order Lagrangian perturbation theory matter fields from which they populate haloes with mock galaxies using a halo occupation distribution prescription which has been calibrated to reproduce the clustering measurements on scales between 30 and 80 Mpc (White et al., 2011). We use these mock catalogues to construct the covariance matrix in our analysis.

## 3 Methodology

In this section, we describe the measurement of the multipoles of the correlation function from the observational data, construction of the theoretical prediction, and the likelihood analysis that leads to constraining cosmological parameters and dark energy.

### 3.1 Measuring the Two-Dimensional Two-Point Correlation Function

We convert the measured redshifts of the BOSS CMASS galaxies to comoving distances by assuming a fiducial model, i.e., flat CDM with and which is the same model for constructing the mock catalogues (see Manera et al. 2013). We use the two-point correlation function estimator given by Landy & Szalay (1993):

(1) |

where is the separation along the light of sight (LOS) and is the separation in the plane of the sky. DD, DR, and RR represent the normalized data-data, data-random, and random-random pair counts, respectively, for a given distance range. The LOS is defined as the direction from the observer to the center of a galaxy pair. The bin size we use is MpcMpc. The Landy and Szalay estimator has minimal variance for a Poisson process. Random data are generated with the same radial and angular selection functions as the real data. One can reduce the shot noise due to random data by increasing the amount of random data. The number of random data we use is more than 15 times that of the real data. While calculating the pair counts, we assign to each data point a radial weight of , where is the radial number density and Mpc (see Feldman et al. 1994).

### 3.2 Theoretical Two-Dimensional Two-Point Correlation Function

First, we adopt the cold dark matter model and the simplest inflation model (adiabatic initial condition). Thus, we can compute the linear matter power spectra, , by using CAMB (Code for Anisotropies in the Microwave Background, Lewis, Challinor, & Lasenby 2000). The linear power spectrum can be decomposed into two parts:

(2) |

where is the “no-wiggle” or pure CDM power spectrum calculated using Eq.(29) from Eisenstein & Hu (1998). is the wiggled part defined by the equation itself. The nonlinear damping effect of the “wiggled” part, in redshift space, can be well approximated following Eisenstein, Seo, & White (2007) by

(3) |

where is the cosine of the angle between and the LOS, is the growth rate, and is computed following Crocce & Scoccimarro (2006); Matsubara (2008) by

(4) |

The dewiggled power spectrum is

(5) |

Next, we include the linear redshift distortion as follows in order to obtain the galaxy power spectrum in redshift space at large scales (Kaiser, 1987), i.e.,

(6) |

where is the linear galaxy bias and is the linear redshift distortion parameter.

We compute the theoretical two-point correlation function, , by Fourier transforming the non-linear power spectrum . This task is efficiently performed by using Legendre polynomial expansions and one-dimensional integral convolutions as introduced in Chuang & Wang (2012c).

We convolve the 2D correlation function with the distribution function of random pairwise velocities, , to obtain the final model following Peebles (1980) by

(7) |

where the random motions (fingers of god) are represented by an exponential form (e.g., Ratcliffe et al. 1998; Landy 2002)

(8) |

where is the pairwise peculiar velocity dispersion.

The cosmological parameter set that we use to compute the theoretical correlation function is , , , , , , , , where and are the matter and baryon density fractions, is the power-law index of the primordial matter power spectrum, is the dimensionless Hubble constant ( km sMpc), and is the normalization of the power spectrum. The linear redshift distortion parameter can be expressed as . Thus, one can derive from the measured and . On the scales we use for comparison with the BOSS CMASS data, the theoretical correlation function only depends on cosmic curvature and dark energy through the parameters , , , and assuming that dark energy perturbations are unimportant (valid in the simplest dark energy models). Thus we are able to extract constraints from clustering data that are independent of a dark energy model and cosmic curvature.

### 3.3 Effective Multipoles of the Correlation Function

The traditional multipoles of the two-point correlation function, in redshift space, are defined by

(9) | |||||

where

(10) | |||||

(11) |

and is the Legendre Polynomial (0 and 2 here). We integrate over a spherical shell with radius , while actual measurements of are done in discrete bins. To compare the measured and our theoretical model, the last integral in Eq.(9) should be converted into a sum. This leads to the definition for the effective multipoles of the correlation function (Chuang & Wang, 2012b):

(12) |

where Mpc in this work, and

(13) |

(14) |

Both the measurement and the theoretical prediction for the effective multipoles are computed using Eq.(12), with given by the measured correlation function (see Eq.1) for the measured effective multipoles, and Eq.(7) for the theoretical predictions. We do not use the conventional definitions of multipoles to extract parameter constraints as they use continuous integrals (see Eq. 9). Bias of the result could be introduced if the definitions of multipoles differ between measurements from data and the theoretical model.

### 3.4 Covariance Matrix

We use the 600 mock catalogues created by Manera et al. (2013) for the BOSS CMASS DR9 to estimate the covariance matrix of the observed correlation function. We calculate the multipoles of the correlation functions of the mock catalogues and construct the covariance matrix as

(15) |

where is the number of the mock catalogues, is the mean of the element of the vector from the mock catalogue multipoles, and is the value in the elements of the vector from the mock catalogue multipoles. The data vector is defined by Eq.(19).

### 3.5 Likelihood

The likelihood is taken to be proportional to (Press et al., 1992), with given by

(16) |

where is the length of the vector used, is the vector from the theoretical model, and is the vector from the observed data.

As explained in Chuang & Wang (2012a), instead of recalculating the observed correlation function while computing for different models, we rescale the theoretical correlation function to avoid rendering the values arbitrary. It can be considered as an application of Alcock-Paczynski effect (Alcock & Paczynski, 1979). The rescaled theoretical correlation function is computed by

(17) |

where is computed by eq. (7), and can be rewritten as

(18) | |||||

where is the vector computed by eq. (12) from the rescaled theoretical correlation function, eq. (17). is the vector from observed data measured with the fiducial model (see Chuang & Wang 2012a for more details regarding the rescaling method).

### 3.6 Markov Chain Monte-Carlo Likelihood Analysis

We perform Markov Chain Monte-Carlo likelihood analyses using CosmoMC (Lewis & Bridle, 2002). The parameter space that we explore spans the parameter set of , , , , , , , , . Only , , , , are well constrained using the BOSS CMASS alone in the scale range of interest. We marginalize over the other parameters, , , , , with the flat priors , , km/s, , where the flat priors of and are centered on the WMAP7 measurements with a width of ( is taken from Komatsu et al. 2010). These priors are sufficiently wide to ensure that CMB constraints are not double counted when our results are combined with CMB data (Chuang, Wang, & Hemantha, 2012).

## 4 Results

### 4.1 Measurement of multipoles

Fig.1 and 1 show the effective monopole () and quadrupole () measured from the BOSS CMASS galaxy sample compared with the theoretical model given the parameters measured. We are using the scale range, Mpc, and the bin size is 5 Mpc. The data points from the multipoles in the scale range considered are combined to form a vector, , i.e.,

(19) |

where is the number of data points in each measured multipole; here . The length of the data vector depends on the number of multipoles used.

### 4.2 Measurement of Cosmological Parameters from BOSS CMASS only

We now present the dark energy model independent measurements of the parameters , , , , and b, obtained by using the method described in previous sections. We also present the derived parameters including , , , , , , and

(20) |

(21) |

(22) |

and

(23) |

where is the comoving sound horizon at the drag epoch calculated using eq. (6) in Eisenstein & Hu (1998). is the effective distance which can be measured from the spherical averaged correlation function or power spectrum (e.g. see Eisenstein et al. 2005). is a robust measurement while including small scales (e.g. see Blake et al. 2012). and are the dilation and wrapping parameters between the true and fiducial cosmology models (e.g. see Xu et al. 2013).

Table 1 lists the mean, rms variance, and 68% confidence level limits for , , , , b, , , , , , and derived in an MCMC likelihood analysis from the measured of the DR9 CMASS correlation function.

Table 2 gives the normalized covariance matrix for this parameter set measured using . It is clear that the correlation between and , , or are close to zero, since the dependency on is removed by dividing or multiplying .

For this measurement, we use 48 bins (), 9 fitting parameters (see Sec. 3.6), and scale range Mpc Mpc. The per degree of freedom (d.o.f.) is 0.51. This low value might indicate not only a good modeling but also possible over-estimation for the covariance matrix constructed with the mock catalogues.

Measured | mean | lower | upper | |
---|---|---|---|---|

87.6 | 7.2 | 80.8 | 94.3 | |

1396 | 74 | 1324 | 1470 | |

0.126 | 0.019 | 0.116 | 0.134 | |

0.367 | 0.084 | 0.287 | 0.446 | |

1.19 | 0.14 | 1.05 | 1.33 | |

Derived | ||||

0.0454 | 0.0031 | 0.0426 | 0.0482 | |

8.95 | 0.27 | 8.69 | 9.22 | |

13.54 | 0.29 | 13.26 | 13.82 | |

0.428 | 0.069 | 0.362 | 0.494 | |

0.436 | 0.017 | 0.419 | 0.454 | |

1.024 | 0.022 | 1.002 | 1.045 | |

0.015 | 0.029 | -0.012 | 0.042 |

1.0000 | -0.2203 | 0.5224 | 0.1701 | 0.1534 | 0.8502 | |

-0.2203 | 1.0000 | -0.7799 | 0.4611 | -0.2938 | 0.2530 | |

0.5224 | -0.7799 | 1.0000 | -0.4100 | 0.5746 | 0.0237 | |

0.1701 | 0.4611 | -0.4100 | 1.0000 | -0.7220 | 0.4219 | |

0.1534 | -0.2938 | 0.5746 | -0.7220 | 1.0000 | -0.0952 | |

0.8502 | 0.2530 | 0.0237 | 0.4219 | -0.0952 | 1.0000 | |

0.4038 | 0.5739 | -0.0033 | 0.3153 | 0.1183 | 0.4874 | |

-0.5020 | 0.2651 | -0.0200 | -0.1470 | 0.2124 | -0.5828 | |

0.3565 | 0.4387 | -0.1612 | 0.8495 | -0.2768 | 0.5300 | |

0.0802 | -0.3945 | 0.7669 | -0.4649 | 0.6870 | -0.3090 | |

-0.5020 | 0.2651 | -0.0200 | -0.1470 | 0.2124 | -0.5828 | |

-0.7989 | -0.4034 | -0.0102 | -0.4397 | 0.0348 | -0.9476 |

0.4038 | -0.5020 | 0.3565 | 0.0802 | -0.5020 | -0.7989 | |

0.5739 | 0.2651 | 0.4387 | -0.3945 | 0.2651 | -0.4034 | |

-0.0033 | -0.0200 | -0.1612 | 0.7669 | -0.0200 | -0.0102 | |

0.3153 | -0.1470 | 0.8495 | -0.4649 | -0.1470 | -0.4397 | |

0.1183 | 0.2124 | -0.2768 | 0.6870 | 0.2124 | 0.0348 | |

0.4874 | -0.5828 | 0.5300 | -0.3090 | -0.5828 | -0.9476 | |

1.0000 | 0.4220 | 0.5389 | 0.2301 | 0.4220 | -0.7316 | |

0.4220 | 1.0000 | -0.0515 | 0.5431 | 1.0000 | 0.3081 | |

0.5389 | -0.0515 | 1.0000 | -0.1512 | -0.0515 | -0.6026 | |

0.2301 | 0.5431 | -0.1512 | 1.0000 | 0.5431 | 0.1678 | |

0.4220 | 1.0000 | -0.0515 | 0.5431 | 1.0000 | 0.3081 | |

-0.7316 | 0.3081 | -0.6026 | 0.1678 | 0.3081 | 1.0000 |

### 4.3 Using Our Results from CMASS only

In this section, we describe the steps to combine our results with other data sets assuming some dark energy models. For a given model and cosmological parameters, including the linear galaxy bias , one can compute , , , , and . From Table 1 and 2, one can derive the covariance matrix, , of these five parameters. Then, can be computed by

(24) |

where

(25) |

and

(26) |

One can use a subset of these parameters (measured and derived) and their covariance matrix to derive the cosmological parameters. For example, if one is only interested in the cosmological parameters but not in the galaxy bias, , one can use only four parameters, , , , and to compute to constrain the parameters of a given model. In Sec. 4.5, we use , , and to explore the power on constraining dark energy from .

In addition, we use , , and instead of , to be more general. For example, while combining the supernovae data, which do not have as a parameter of the cosmological model, it is simpler to use than use .

We also provide the code for using CosmoMC that includes BOSS CMASS clustering alone^{4}^{4}4http://members.ift.uam-csic.es/chuang/BOSSDR9singleprobe.

### 4.4 Assuming Dark Energy Models

In this section, we present examples of combining our CMASS-only clustering results with CMB data sets assuming specific dark energy models.

Table 3, 4, 5, and 6 show the cosmological constraints assuming
CDM, oCDM (non-flat CDM), CDM (constant equation of state of dark energy), and oCDM (non-flat universe with a constant equation of state of dark energy) models.
In this study, we only list the parameters which can be well constrained by galaxy clustering.
We also present the results of the combination of CMASS and CMB data.
The CMB data we use includes
WMAP7 and WMAP9, which are the previous and the newest data release from the Wilkinson Microwave Anisotropy Probe collaboration;
(Komatsu et al., 2010; Bennett et al., 2012; Hinshaw et al., 2012).
We are also using the newest data release from the South Pole Telescope (SPT) collaboration (Story et al., 2012; Hou et al., 2012).
For WMAP7 only and WMAP9 only data, we download the Markov chains from the WMAP website^{5}^{5}5WMAP7:http://lambda.gsfc.nasa.gov/product/map/dr4/parameters.cfm
^{6}^{6}6WMAP9:http://lambda.gsfc.nasa.gov/product/map/dr5/parameters.cfm.
While using WMAP9+SPT, we obtain the Markov chains by using CosmoMC (Lewis & Bridle, 2002) with the data and likelihood code provided
by WMAP (Bennett et al., 2012; Hinshaw et al., 2012) and SPT (Story et al., 2012; Hou et al., 2012; Keisler et al., 2011) collaborations.

One can see that the measurements from BOSS CMASS-only dataset are consistent with those from CMB, and adding CMASS to CMB produces significantly tighter constraints than using CMB data alone. While adding SPT to WMAP9 in a CDM model, is decreased as found in Story et al. (2012) (although they used WMAP7). It is interesting that the mean values from WMAP9+SPT in a CDM model are much closer to those from WMAP7 than from WMAP9.

Figure 2 shows how CMASS clustering breaks the degeneracy between and constrained by CMB in the oCDM model, resulting in a much tighter constraint. Figure 3 demonstrates how CMASS clustering also breaks the degeneracy between and constrained by CMB in the CDM model, resulting in a much better constraint in which is consistent (within 1 ) with (cosmological constant model). This statement is true independent of which CMB data set (see Table 5). Figure 4 shows that adding the CMASS and the CMB data improves the constraints on and significantly in the oCDM model, and the results are consistent with and (i.e. a flat CDM model). This statement holds regardless of which CMB data set is used (see Table 6).

CDM | CMASS only | WMAP7 only | WMAP7+CMASS | WMAP9 only | WMAP9+CMASS | WMAP9+SPT | WMAP9+SPT+CMASS |
---|---|---|---|---|---|---|---|

oCDM | CMASS only | WMAP7 only | WMAP7+CMASS | WMAP9 only | WMAP9+CMASS | WMAP9+SPT | WMAP9+SPT+CMASS |
---|---|---|---|---|---|---|---|

CDM | CMASS only | WMAP7 only | WMAP7+CMASS | WMAP9 only | WMAP9+CMASS | WMAP9+SPT | WMAP9+SPT+CMASS |
---|---|---|---|---|---|---|---|

oCDM | CMASS only | WMAP7 only | WMAP7+CMASS | WMAP9 only | WMAP9+CMASS | WMAP9+SPT | WMAP9+SPT+CMASS |
---|---|---|---|---|---|---|---|