H(z), D_{A}(z), and f(z)\sigma_{8}(z) from SDSS DR7 LRGs

# Modeling the Anisotropic Two-Point Galaxy Correlation Function on Small Scales and Single-Probe Measurements of H(z), DA(z), and f(z)σ8(z) from the Sloan Digital Sky Survey DR7 Luminous Red Galaxies

Chia-Hsun Chuang and Yun Wang
Homer L. Dodge Department of Physics & Astronomy, Univ. of Oklahoma, 440 W Brooks St., Norman, OK 73019, U.S.A.
MultiDark Fellow; E-mail: chia-hsun.chuang@uam.es
July 7, 2019
###### Abstract

We present a simple and efficient phenomenological model for the two-dimensional two-point galaxy correlation function that works well over a wide range of scales, from large scales down to scales as small as 25Mpc. Our model incorporates nonlinear effects, a scale-dependent galaxy bias on small scales, and allows the redshift-space distortions to be scale and direction dependent. We validate our model using LasDamas mock catalogs, and apply it to the Sloan Digital Sky Survey (SDSS) DR7 Luminous Red Galaxies (LRGs). Using only the monopole and quadrupole of the correlation function measured from the SDSS DR7 LRGs, we obtain improved measurements , , and at , using the scale range of Mpc. We expect our results and model to be useful in tightening dark energy and gravity constraints from the full analysis of current and future galaxy clustering data.

###### keywords:
cosmology: observations, distance scale, large-scale structure of Universe
\@maketitle

## 1 Introduction

In our quest to solve the mystery of the observed cosmic acceleration (Riess et al., 1998; Perlmutter et al., 1999), galaxy clustering plays an increasingly important role as a probe of both dark energy and gravity, the two main classes of possible explanations for cosmic acceleration. Current data from the Sloan Digital Sky Survey (SDSS) Data Release Seven (DR7) (Abazajian et al., 2009), WiggleZ (Blake et al., 2009), and BOSS (Eisenstein et al., 2011) are allowing us to place very useful constraints on dark energy. The planned space mission Euclid will survey 60 million emission-line galaxies at over 15,000 square degrees (Cimatti et al., 2009; Wang et al., 2010; Laureijs et al., 2011), and provide potentially revolutionary bounds on the nature of cosmic acceleration.

The SDSS data have been analyzed using both the power spectrum method (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. 2012; Padmanabhan et al. 2012). Although these two methods are simple Fourier transforms of one another, the analysis processes are quite different and the results cannot be converted using Fourier transform directly because of the finite size of the survey volume.

The power of galaxy clustering as a dark energy probe lies in the fact that the Hubble parameter, , the angular diameter distance, , can in principle be extracted simultaneously from data through the measurement of the baryon acoustic oscillation (BAO) scale in the radial and transverse directions (Blake & Glazebrook, 2003; Seo & Eisenstein, 2003; Wang, 2006). The inclusion of information from full galaxy clustering goes beyond BAO only, and enables significantly enhanced constraints on and . Most importantly, it allows the measurement of the growth rate of cosmic large scale structure, (where denotes the linear redshift-space distortion (RSD) factor (Kaiser, 1987), and denotes galaxy bias), required for using galaxy clustering to test gravity (Guzzo et al., 2008; Wang, 2008).

In fact, it is possible to measure (Song & Percival, 2009) or (Wang, 2012) without facing the difficulty of measuring galaxy bias.

In Chuang & Wang (2012), we made significant improvements in modeling galaxy clustering from previous studies (Okumura et al., 2008; Cabre & Gaztanaga, 2009; Kazin et al., 2010b), and succeeded in making the first simultaneous measurements of and from data, using the full 2D correlation function of a sample of SDSS DR7 LRGs (Eisenstein et al., 2001), and without assuming a dark energy model or a flat Universe. Xu et al. (2013) measured and at from the SDSS DR7 LRGs by applying density-field reconstruction to an anisotropic analysis of the BAO peak. Anderson et al. (2013) applied the same method on SDSS III Baryon Oscillation Spectroscopic Survey (SDSS-III/BOSS) DR9 sample. Regarding the measurements of growth constraints, Samushia et al. (2012) measured from SDSS DR7 LRG sample with CMB + SNIa priors. Reid et al. (2012) measured , , and at from the monopole and quadrupole of the 2D 2PCF of the SDSS-III/BOSS DR9 sample, assuming CMB priors. Most recently, Chuang et al. (2013) applied similar analysis as this paper on SDSS-III/BOSS DR9 sample to measure , , and without CMB priors.

In Chuang & Wang (2013), we extended our method by exploring the use of the multipoles of the correlation function to measure , , and . The obvious advantage of using multipoles of the correlation function instead of the full 2D correlation function is the reduced number of data points used to obtain similar amount of information.

The proper modeling of RSD is required in order to measure or from galaxy clustering data. Recent work on improving the modeling of RSD include that of Jennings, Baugh, & Pascoli (2011) and Reid & White (2011). In this paper, we focus on the detailed phenomenological modeling of the correlation function on smaller scales to obtain improved constraints on or . We use the multipoles of the 2D correlation function for speed and efficiency.

In Section 2, we introduce the galaxy sample used in our study. In Section 3, we describe the details of our new model. In Section 4, we describe the details of our methodology. In Section 5, we present our improved measurements from SDSS DR7 LRGs. We summarize and conclude in Sec. 6.

## 2 Data

The SDSS has observed one-quarter of the entire sky and performed a redshift survey of galaxies, quasars and stars in five passbands and with a 2.5m telescope (Fukugita et al., 1996; Gunn et al., 1998, 2006). We use the public catalog, the NYU Value-Added Galaxy Catalog (VAGC) (Blanton et al., 2005), derived from the SDSS II final public data release, Data Release 7 (DR7) (Abazajian et al., 2009). We select our LRG sample from the NYU VAGC with the flag bit mask set to . K-corrections have been applied to the galaxies with a fiducial model (CDM with and ), and the selected galaxies are required to have rest-frame -band absolute magnitudes (Blanton & Roweis, 2007). The same selection criteria were used in previous papers (Zehavi et al., 2005; Eisenstein et al., 2005; Okumura et al., 2008; Kazin et al., 2010a). The sample we use is referred to as “DR7full” in Kazin et al. (2010a). Our sample includes 87000 LRGs in the redshift range 0.16-0.44.

Spectra cannot be obtained for objects closer than 55 arcsec within a single spectroscopic tile due to the finite size of the fibers. To correct for these “collisions”, the redshift of an object that failed to be measured would be assigned to be the same as the nearest successfully observed one. Both fiber collision corrections and K-corrections have been made in NYU-VAGC (Blanton et al., 2005). The collision corrections applied here are different from what has been suggested in Zehavi et al. (2005). However, the effect should be small since we are using relatively large scale which are less affected by the collision corrections.

We construct the radial selection function as a cubic spline fit to the observed number density histogram with the width . The NYU-VAGC provides the description of the geometry and completeness of the survey in terms of spherical polygons. We adopt it as the angular selection function of our sample. We drop the regions with completeness below to avoid unobserved plates (Zehavi et al., 2005). The Southern Galactic Cap region is also dropped.

## 3 Modeling 2D Correlation Function

In this section, we describe our model which encompasses the linear scale to the nonlinear scale.

### 3.1 Modeling 2D Correlation Function for Large Scales

We compute the linear matter power spectra, , by using CAMB (Lewis, Challinor, & Lasenby, 2000). The linear power spectrum can be composed to two parts:

 Plin(k)=Pnw(k)+PlinBAO(k), (1)

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

 PnlBAO(k,μk)=PlinBAO(k)⋅exp(−k22k2⋆[1+μ2k(2f+f2)]), (2)

where could be computed by (Crocce & Scoccimarro, 2006; Matsubara, 2008)

 k⋆=[13π2∫Plin(k)dk]−1/2. (3)

The dewiggled power spectrum is

 Pdw(k,μk)=Pnw(k)+PnlBAO(k,μk), (4)

is the cosine of the angle between and the line of sight (LOS). Note that Eqs.(1)-(4) are the same as Eq.(2) in Chuang & Wang (2012), except for the addition of the direction-dependent terms in the exponent of the damping factor in Eq.(2), but are somewhat more intuitive.

Next, we include the linear RSD as follows to obtain the galaxy power spectrum in redshift space at large scales (Kaiser, 1987):

 Psg(k,μk) = b2(1+βμ2k)2Pdw(k,μk), (5) = Psg,nw(k,μk)+Psg,BAO(k,μk), (6)

where is the linear galaxy bias. Note that we have defined

 Psg,nw(k,μk)=b2(1+βμ2k)2Pnw(k) Psg,BAO(k,μk)=b2(1+βμ2k)2PnlBAO(k,μk) = b2(1+βμ2k)2PlinBAO(k)⋅exp(−k22k2⋆[1+μ2k(2f+f2]). (8)

Analogous to Eq.(6), the galaxy correlation function can be decomposed into no-wiggle and wiggled parts as follows:

 ξsg,dw(σ,π)=ξsg,nw(σ,π)+ξsg,BAO(σ,π). (9)

While can be obtained by Fourier-transforming , doing so involves two-dimensional integrals, and thus is time-consuming and inefficient. Instead, we can Fourier transform each term in Eq.(6) separately, using Legendre polynomial expansions and integral convolutions that only involve one-dimensional integrals.

The no-wiggle galaxy correlation function in redshift space can be computed by Fourier transforming Eq.(3.1), which gives (Hamilton, 1992)

 ξsg,nw(σ,π)=b2(ξnw0(s)P0(μ)+ξnw2(s)P2(μ)+ξnw4(s)P4(μ)), (10)

where , is the cosine of the angle between and the LOS, and are Legendre polynomials. The multipoles of are defined as

 ξnw0(r) = (1+2β3+β25)ξnw(r), (11) ξnw2(r) = (4β3+4β27)[ξnw(r)−¯ξ(r)], (12) ξnw4(r) = 8β235[ξnw(r)+52¯ξnw(r)−72¯¯¯¯¯¯ξnw(r)], (13)

where is the linear RSD parameter and

 ¯ξnw(r) = 3r3∫r0ξnw(r′)r′2dr′, (14) ¯¯¯¯¯¯ξnw(r) = 5r5∫r0ξnw(r′)r′4dr′, (15)

where is obtained by fourier transforming .

The wiggled part of the galaxy correlation function in redshift space, , is obtained by Fourier transforming Eq.(8). Note that the -dependent damping factor in -sapce in Eq.(8) becomes a Gaussian convolution in configuration space:

 ξsg,BAO(σ,π)=∫∞−∞ξ⋆(σ,π−x)f⋆(x)dx, (16)

where is the Fourier transform of , and

 f⋆(x)=1σ⋆√πexp(−x2σ2⋆), (17)

where

 σ2⋆=4f+2f2k2⋆. (18)

can be obtained using Eq. (10)-(15), but replace (the Fourier transform of ) with the Fourier transform of .

Table. 3 shows the performance of our convolution method by comparing with the results using fast Fourier transform (FFT) directly. One can see our method is much more efficient. The two-dimensional dewiggle model has a obvious feature at the BAO scale for the normalized quadrupole, (see Samushia et al. 2012, where

 Q(s)=ξ2(s)ξ0(s)−(3/s3)∫s0ξ0(s′)s′2ds′. (19)

Fig. 1 shows that the results from the FFT method converges to the one from the convolution method. While these tests are performed on a single machine, the grid size used for FFT method is limited by the memory size. One can still see some fluctuations at small scales for the maximum grid size (=). Therefore, our method not only provide a much faster way but also use much less resources to compute the theoretical model. With a multi-cores machine (FFT can only use single core unless the machine’s memory is much larger), our method could be hundreds times faster than FFT method. It is crucial while doing Markov Chain Monte Carlo (MCMC) analysis.

### 3.2 Modeling 2D Correlation Function on Small Scales

For small scales, we need to model three effects: the nonlinear matter correlation function, the scale-dependent galaxy bias, and the RSD from the random galaxy pairwise velocities. It is well known that the small scale galaxy correlation function is well described by a powerlaw (Peebles, 1980). Since the galaxy correlation function is given by on small scales, we model the combination of nonlinear matter correlation function and the scale-dependent galaxy bias at small scales by multiplying with the following factor

 bnl(r)=rbAF(r), (20)

where is a constant. is a function which is close to 1 for small and close to 0 when is large; we choose

 F(r)=11+(rbB)bC, (21)

where we choose Mpc and ; these are motivated by the fact that the galaxy correlation function is a powerlaw at small scales (i.e. Mpc) and the scale-dependent effects (including nonlinear effects and scale-dependent galaxy bias) are negligible at larger scales, Mpc. The overall scale-dependent effects are included when computing the no-wiggle galaxy correlation function by replacing with in applying Eq. (10)-(15). The resultant correlation function is denoted as .

We now obtain the 2D correlation function that incorporate nonlinear effects, galaxy bias, and linear RSD:

 ~ξ(σ,π)=ξs,nlg,nw(σ,π)+ξsg,BAO(σ,π), (22)

where is given by Eq.(16).

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

 ξ(σ,π)=∫∞−∞~ξ(σ,π−vH(z)a(z))f(v)dv, (23)

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

 f(v)=1σv(s′,μ′2)√2exp(−√2|v|σv(s′,μ′2)), (24)

where is the pairwise peculiar velocity dispersion, and . We find that the 2D correlation functions measured from LasDamas mocks can be well fitted by

 σv(s′,μ′2)=σv,0(1+Cμμ′2)(1+cσ1e−cσ2σ2), (25)

where is the dispersion corresponding to the truly random motion and , , and (with unit of Mpc) terms describe the dependence on direction and separation. The -dependence is similar to that found by Cabre & Gaztanaga (2009). They found that the 2D correlation functions from the MICE N-body simulations are fitted well with a pairwise velocity distribution which is large when Mpc. We have added the direction-dependent term, , to model the high amplitude of at small scales (see Fig. 5).

## 4 Methodology

In this section, we present the methodology and results of testing our model described in the previous section.

### 4.1 Mock Catalogs Used

We use the 160 mock catalogs from the LasDamas simulations (McBride et al., in preparation) to test our model. LasDamas provides mock catalogs matching SDSS main galaxy and LRG samples. We use the LRG mock catalogs from the LasDamas gamma release with the same cuts as the SDSS LRG DR7full sample, and . We have diluted the mock catalogs to match the radial selection function of the observational data by randomly selecting the mock galaxies according to the number density of the data sample. We calculate the multipoles of the correlation functions of the mock catalogs and construct the covariance matrix (see Chuang & Wang (2013) for details).

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

We convert the measured redshifts of galaxies to comoving distances by assuming a fiducial model, CDM with . We use the two-point correlation function estimator given by Landy & Szalay (1993):

 ξ(σ,π)=DD(σ,π)−2DR(σ,π)+RR(σ,π)RR(σ,π), (26)

where is the separation along the line of sight (LOS), 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 in a given distance range. The LOS is defined as the direction from the observer to the center of a pair. The bin size we use here 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 number of random data. The number of random data we use is 10 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 selection function and Mpc (Eisenstein et al., 2005).

Fig 2 shows the averaged 2D correlation function measured from the mock catalogs. We use the averaged radial selection function to construct the random catalog since it is closer to the true mean density. Clearly, our model provides an excellent fit to data over a wide range of scales, from the largest scales where data are not too noisy, to the smallest scales plotted (except very near the LOS).

### 4.3 Multiples of the Correlation Function

As in Chuang & Wang (2013), the effective multipoles of the correlation function are defined by

 ^ξl(s)≡∑s−Δs2<√σ2+π2

where Mpc in this work, and

 σ=(n+12)h−1Mpc,n=0,1,2,... (28)
 π=(m+12)h−1Mpc,m=0,1,2,... (29)
 μ≡π√σ2+π2. (30)

Note that both the measurements and the theoretical predictions for the effective multipoles are computed using Eq.(27). We do not use the conventional definitions of multipoles to extract parameter constraints as they use continuous integrals. Bias could be introduced if the definitions of multipoles are different between measurements from data and the theoretical model.

Fig. 3, 4, and 5 show the effective monopole (), quadrupole (), and hexadecapole () measured from the LasDamas mock catalogs comparing to our full model and a simpler model (linear model + 1D dewiggle damping + constant velocity dispersion). In Fig. 3, one can see how our model completely corrects the scale-dependent effects in the measured monopole. Fig.4 shows that our model provides a reasonable fit to the measured quadrupole. In Fig. 5, we find that angle-dependent term, , significantly improves the fitting of hexadecapole at small scales (Mpc). However, at larger scales(Mpc), the LasDamas mocks show some oscillatory features while the theoretical models are flat. It is likely due to the dewiggle damping not being adequate enough to model and one might need higher order term (i.e. ).  Therefore, we do not include to measure parameters in this study.

### 4.4 Covariance Matrix

We construct the covariance matrix as

 Cij=1N−1N∑k=1(¯Xi−Xki)(¯Xj−Xkj), (31)

where is the number of the mock catalogs, is the mean of the element of the vector from the mock catalog multipoles, and is the value in the elements of the vector from the mock catalog multipoles. The data vector is defined by

 X={^ξ(1)0,^ξ(2)0,...,^ξ(N)0;^ξ(1)2,^ξ(2)2,...,^ξ(N)2;...}, (32)

where is the number of data points in each measured multipole; while using the scale range, Mpc. The length of the data vector depends on how many multipoles are used.

### 4.5 Likelihood

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

 χ2≡NX∑i,j=1[Xth,i−Xobs,i]C−1ij[Xth,j−Xobs,j] (33)

where is the length of the vector used, is the vector from the theoretical model, and is the vector from the observational data (we use the mock catalogs as the observational data to test the model in this section).

As explained in Chuang & Wang (2012), instead of recalculating the observed correlation function for different theoretical models, we rescale the theoretical correlation function to avoid rendering values arbitrary. The rescaled theoretical correlation function is computed by

 T−1(ξth(σ,π))=ξth⎛⎝DA(z)DfidA(z)σ,Hfid(z)H(z)π⎞⎠, (34)

where is given by eq. (23). Hence can be rewritten as

 χ2 ≡ NX∑i,j=1{T−1Xth,i−Xfidobs,i}C−1fid,ij⋅ (35) ⋅{T−1Xth,j−Xfidobs,j},

where is a vector given by eq. (34) with replaced by its effective multipoles (defined by eq. (27)), and is the corresponding vector from observational data measured assuming the fiducial model in converting redshifts to distances. See Chuang & Wang (2012) for a more detailed description of our rescaling method.

### 4.6 Markov Chain Monte-Carlo Likelihood Analysis

We use CosmoMC in a Markov Chain Monte-Carlo likelihood analysis (Lewis & Bridle, 2002). The parameter space that we explore spans the parameter set of , , , , , , , , f(0.35), , , , , . Only , , , , are well constrained by the mock data.

We marginalize over the other parameters, , , , , , , , , , with flat priors over the ranges of , , Mpc, , , skm, , , Mpc, where the flat priors of and are centered on the measurements from WMAP7 and has width of (with from Komatsu et al. (2010)). These priors are wide enough to ensure that CMB constraints are not double counted when our results are combined with CMB data (Chuang, Wang, & Hemantha, 2012).

### 4.7 Validation of the Model Using Mock catalogs

We apply our method on the averaged correlation function from LasDamas SDSS LRG mock catalogs to validate our methodology. Table 2 shows the measurements of , , , , , , from the averaged correlation function from LasDamas SDSS LRG mock catalogs using and the scale range, Mpc, comparing with the input values of the simulations. We find the input values of the simulations are well recovered by our methodology.

## 5 Measurements from SDSS DR7 LRG

Table 3 lists the mean and rms variance of the parameters, , , , , , , , derived in an MCMC likelihood analysis from the measured of the correlation function of the SDSS LRG sample with the scale range, Mpc and Mpc. Table 4 and 5 gives the corresponding normalized covariance matrices.

While we are modeling the correlation function on small scales, the uncertainties would still become larger when smaller scales are included. Although one could obtain tighter constraints by using very small scales, to be conservative, we only fit the measurements using scales larger than 25 Mpc and check the consistency with the measurements using the scales larger than 40 Mpc.

Our measurements are consistent between two scale ranges considered which shows no hint of systematics. We choose the results using the scale range, Mpc, as our fiducial results. As expected, the constraints become tighter when including smaller scales. Notice that the correlations between and , also increase. It is due to the fact that our measurements gain more constraining power from the overall shape beyond the BAO peak region.

## 6 Conclusion and Discussion

We have presented and validated a simple and efficient phenomenological model for the two-dimensional two-point galaxy correlation function that works well over a wide range of scales, from large scales down to small scales not used in our previous work (where we restricted ourselves to scales larger than 40Mpc). Applying this model to the SDSS LRGs over the scale range of Mpc, We obtain the measurements: , , and at , which summarize the cosmological constraints extracted from the SDSS DR7 LRG sample. We also provide the covariance matrix needed to use these measurements (see Table 4).

Our model incorporates the overall nonlinear effects via the use of the “dewiggled” galaxy power spectrum, as in Chuang & Wang (2012), but we now include the enhanced damping along the line of sight (see Eqs.(1)-(4)). We also introduce a much efficient way to compute this model which is crucial for MCMC analysis. On small scales, the nonlinear effect and scale-dependent galaxy bias are degenerate, and we model these as an overall scale-dependent correction. Most significantly, we allow the RSD to be scale and direction dependent in our model. Our model provides excellent fit to mock data (see Fig.2).

We expect our methodology and results to be useful in tightening dark energy and gravity constraints from the full analysis of current and future galaxy clustering data.

## Acknowledgements

We are grateful to the LasDamas project for making their mock catalogs publicly available. The computing for this project was performed at the OU Supercomputing Center for Education and Research (OSCER) at the University of Oklahoma (OU). This work used the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation grant number OCI-1053575. This work was supported by DOE grant DE-FG02-04ER41305. C.C. was also supported by the Spanish MICINNâs Consolider-Ingenio 2010 Programme under grant MultiDark CSD2009-00064 and grant AYA2010-21231, the Comunidad de Madrid under grant HEPHACOS S2009/ESP-1473, and the Spanish MINECOâs âCentro de Excelencia Severo Ochoaâ Programme under grant SEV-2012-0249.

Funding for the Sloan Digital Sky Survey (SDSS) has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Aeronautics and Space Administration, the National Science Foundation, the U.S. Department of Energy, the Japanese Monbukagakusho, and the Max Planck Society. The SDSS Web site is http://www.sdss.org/.

The SDSS is managed by the Astrophysical Research Consortium (ARC) for the Participating Institutions. The Participating Institutions are The University of Chicago, Fermilab, the Institute for Advanced Study, the Japan Participation Group, The Johns Hopkins University, Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, University of Pittsburgh, Princeton University, the United States Naval Observatory, and the University of Washington.

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