THE BARYONIC ACOUSTIC FEATURE AND LARGE-SCALE CLUSTERING IN THE SLOAN DIGITAL SKY SURVEY LUMINOUS RED GALAXY SAMPLE

To measure the two-point correlation function, we use the Landy & Szalay (1993) estimator:

$$\begin{equation} \xi (s)=\frac{DD(s)-2DR(s)+RR(s)}{RR(s)}, \end{equation} \tag{ 1 }$$

which compares the normalized number of data pair counts to randomly distributed points. The quantity s refers to the mean redshift-space separation for each bin. If we define r to be the ratio of the number of random points to data and N_DD(s) to be the total number of galaxy pairs separated by values (s − ds/2, s + ds/2] (where ds is the width of the bin), then the normalized number of pairs are DD = N_DDr², DR = N_DRr, and RR = N_RR. Here, DR and RR stand for data–random and random–random pairs, respectively. The random points account for the effects introduced by survey boundary, holes within the data set, and sector incompleteness.

Additionally, in our counting of pairs, we apply weights to each galaxy. Appendices A and B.2 discuss the details of distributing the random points and the pair-count weighting. In Appendix B.3, we also compare this estimator to other known methods, showing excellent agreement with the method proposed by Hamilton (1993) on large scales, and substantial differences with those proposed by Davis & Peebles (1983) and Peebles & Hauser (1974).

We use mock galaxy catalogs produced from the LasDamas simulations (C. K. McBride et al. 2010, in preparation) to measure the uncertainty covariance matrix, as well as to investigate systematic errors in the two-point correlation estimators. These mock catalogs provide 160 light-cone redshift-space realizations of an SDSS-sized volume, with appropriate number densities and clustering properties for comparison to the observed data.

The LasDamas simulations are designed to model the clustering of SDSS DR7 in a wide luminosity range. In this suite of simulations, galaxies are artificially placed in dark matter halos specifically using the formalism of the halo occupation distribution (HOD; Berlind & Weinberg 2002) with parameters chosen to match an observational SDSS sample. For complete details, see C. K. McBride et al. (2010, in preparation); for distribution visit, the World Wide Web.¹²

We use the "gamma" release of mock LRG catalogs produced from simulations. The Oriana simulation consists of 40 N-body dark matter simulations; each realization contains 1280³ particles of mass 45.73 × 10¹⁰ h⁻¹ M_☉ in a box of length 2400 h⁻¹ Mpc. The softening parameter is 53 h⁻¹ kpc. The simulations assume a flat ΛCDM cosmology with total matter density Ω_M0 = 0.25, baryon density Ω_b0 = 0.04, Hubble expansion rate H₀ = 70 km s⁻¹ Mpc⁻¹, σ₈ = 0.8, and spectral index n_s = 1. The HOD parameters are adjusted to reproduce the observed number density as well as the projected two-point correlation function w_p(r_p) at projected separations 0.3 h⁻¹ Mpc < r_p < 30 h⁻¹ Mpc, well below the scales we consider here. The choice of HOD affects the resulting ξ(s) shape on small scales, but on large scales the HOD choice primarily affects the amplitude as expected from local galaxy bias arguments (Fry & Gaztanaga 1993; Coles 1993; Scherrer & Weinberg 1998; Narayanan et al. 2000).

The LasDamas team has divided the sky into two alternative footprints: one can use either the SDSS Northern Galactic and Southern Galactic Caps together, or only the Northern. We use the latter option because in that case each simulation produces four samples (as opposed to two in the former) resulting in twice as many realizations. In summary, we analyze 40 × 4 = 160 LasDamas mock catalogs for each of the luminosity subsamples DR7-Bright and DR7-Dim. We note that the angular distribution of galaxies in these mocks is similar to the angular distribution of the data, aside from the observational incompleteness. In Appendix A.1, we explain how we account for the incompleteness within the observational data.

Our only manipulation of the catalogs provided by C. K. McBride et al. (2010; in preparation) is in the radial direction, where we randomly subsample to match the SDSS comoving volume density n(z), to better represent the Poisson noise. This subsampling reduces the number of LRGs in DR7-Dim by 15% and in DR7-Bright by 7%. In Appendix A.2, we show that this subsampling does not noticeably affect either ξ or its uncertainties. For more details regarding the radial selection function, please refer to that appendix.

In our analysis, we use the standard χ² fitting algorithm based on a covariance matrix, which is constructed from the LasDamas mock realizations.

Because the uncertainties in the ξ values for the bins are interdependent, we build a covariance matrix C_ij to measure the dependence of the ith bin on the jth. We construct C_ij from the individual mock realizations as follows:

$$\begin{equation} C_{ij} = \frac{1}{N_{\mathrm{mocks}}-1}\cdot \sum _{k=1}^{N_{\mathrm{mocks}}}\left(\overline{\xi }_i-\xi _i^k\right) \left(\overline{\xi }_j-\xi _j^k \right), \end{equation} \tag{ 2 }$$

where $\overline{\xi }_m$ is the correlation value for the m^th bin of the mock mean and ξ^k_m is the same for the k^th mock realization. In all calculations here, N_mocks = 160.

We can then estimate χ² for our observational result ξ^obs relative to models ξ^model using:

$$\begin{equation} \chi ^2(\vec{\theta })= \sum _{i,j=1}^{N_s} \big(\xi ^{\mathrm{obs}}_{i}-\xi ^{\mathrm{model}}_{i}(\vec{\theta })\big) C_{ij}^{-1} \big(\xi ^{\mathrm{obs}}_{j}-\xi ^{\mathrm{model}}_{j}(\vec{\theta })\big), \end{equation} \tag{ 3 }$$

where $\vec{\theta }$ are parameters of the model, and N_s is the number of separation bins used.

To demonstrate consistency with previous studies, we first compare our analysis of an earlier data release, DR3, with that of Eisenstein et al. (2005). For this analysis, we create a DR3 LRG sample with the same criteria used by Eisenstein et al. (2005), and a corresponding random sample. In this subsection, we assume Ω_M0 = 0.3 in calculations.

In Figure 3, we reproduce the redshift space ξ of the SDSS LRGs first measured by Eisenstein et al. (2005). There is good agreement between our results (green diamonds) and theirs (red crosses). To investigate the minute remaining differences, we also test using their random catalogs with our data set and vice versa. We conclude that the remaining subtle variations are due to Poisson noise in the random catalogs. As previously presented in Eisenstein et al. (2005), we obtain a narrow peak at an apparent separation s > 100 h⁻¹ Mpc (see Section 4.5 for our analysis of peak position value), and a steep slope that flattens at ∼135 h⁻¹ Mpc.

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The dashed lines in Figure 3 display our result for the final data release DR7 (sample DR7-Full), which runs through the same galaxy selection algorithm and data analysis as DR3. The thick blue dashed line uses the same cosmology as in DR3; for the thin black dashed line, we alter that cosmology to Ω_M0 = 0.25, Ω_Λ0 = 0.75. The binning in all cases is the same chosen by Eisenstein et al. (2005), with the exception that for DR7 we extend the signal a bit further. In both DR7 cases, the resulting position of the baryonic acoustic feature is in fair agreement (Section 4.5). However, we clearly see stronger power on large scales, yielding a wider peak. In the next section, we examine the significance of this strong large-scale signal relative to our ΛCDM model predictions.

We test the DR7 LRG clustering against a ΛCDM model by comparing the observed ξ to those yielded by the LasDamas mock LRG catalogs (Section 3.2). We run the same analysis as before on each subsample of DR7: DR7-Dim and DR7-Bright (see Table 1 and Figure 2). We also analyze DR7-Full, which has the disadvantage of being flux limited at z > 0.36 but can probe larger scale modes. We find that DR7-Dim is in clear agreement with the model used in the simulations. DR7-Bright also has a strong signal at s > 150 h⁻¹ Mpc.

We first investigate the quasi-volume-limited subsample DR7-Dim showing the SDSS results compared to the 160 LasDamas mock realizations. This subsample has a similar ξ(s) to DR7-Full at most radii s < 145 h⁻¹ Mpc (see Figure 4 for direct comparison). Our results are presented in Figure 5. The mock mean is indicated by the green solid line and the SDSS results are the diamonds. The light gray shaded region represents the area containing 68.2% of the realizations lying closest to the mean (that is, the 1σ uncertainties). The dark gray area is the same for 95.4% (2σ uncertainties). The dotted blue lines indicate the strongest and weakest signal of all mocks at each separation (not one realization in particular).

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The observed signal is clearly within the 1σ–2σ significance level at all scales. To quantify the significance of the strong signal at large scales (s > 130 h⁻¹ Mpc), we reproduce this last result to s < 400 h⁻¹ Mpc (top inset; larger separation binning). Although the expected signal at the largest scales is very small relative to the noise, in the top inset we include data as far out as possible without significantly reducing pair counts due to edge effects. For a discussion regarding edge effects, please refer to Appendix B.3.

We also show in Figure 5 a histogram of the mean correlation on large scales ${\langle \xi \rangle }_{s=[130,400]\,h^{-1}\;{\rm Mpc}}$ (bottom inset; red dashed line is SDSS, solid line is the mock mean), where the brackets denote an average over all separation bins in the indicated range. To further quantify the significance of the large-scale signal, we measure the χ² difference between the observed ξ and the mock mean using Equations (2) and (3). In the last equation, we limit our bins to those between 130 h⁻¹ Mpc < s < 400 h⁻¹ Mpc. Using the 10 bins in the top inset (dof = 10), we measure a normalized value of χ²/dof = 0.721. We have tried several definitions for the significance of large-scale power and all agree that there is a 1.5σ consistency with respect to the ΛCDM plus HOD model used here.

Figure 6 shows several hand-picked mock realizations to demonstrate the effects of variance. We caution the reader to avoid interpreting these particular realizations as "typical." Instead, we have chosen them to demonstrate the variety of signals that could be reasonably expected from a survey the size of SDSS DR7-Dim. Whereas some realizations (e.g., green-dotted and red-solid lines) have a similar clustering signal to that observed (symbols), others do not show evidence of the baryonic acoustic feature (e.g., the orange dashed and cyan dot-dashed lines).

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Visually examining all 160 DR7-Dim realizations, we find that at least 17 (∼10%) of the mocks have no peak at the expected s ∼ 100 h⁻¹ Mpc. We examine ξ results and define a peak-less realization as one with no sign of a peak within 95 h⁻¹ Mpc < s < 120 h⁻¹ Mpc. We took a liberal approach, so this result should be considered a lower bound; i.e., realizations with subtle peaks were counted as having a peak. If we ask how many of the mocks have very clear signs of a peak, we find about 75 out of the 160 do. This designation is subjective, of course. It is also difficult to compare these numbers with theoretical expectations, e.g., using random Gaussian field statistics is not enough as weak peaks in the Gaussian case (corresponding to the linear density field) are washed out by non-linear motions.

To verify the fraction of featureless realizations, we also checked the mock realizations in another independent mock LRG catalog. In the Horizon-Run mock catalog (Kim et al. 2009), LRG positions are determined by identifying physically self-bound dark matter sub-halos that are not tidally disrupted by larger structures. After adjusting their mock galaxy catalogs to fit the SDSS radial function n(z), we ran the same analysis as for LasDamas and found that, of their 32 realizations, five showed no sign of the baryonic acoustic feature, comparable to the result obtained with LasDamas mock catalogs.

Figure 4 also shows that the correlation function of DR7-Full (dashed line) is stronger than DR7-Dim (green diamonds) at scales of 150 h⁻¹ Mpc < s < 200 h⁻¹ Mpc. To help understand the difference between these samples, we examine another subsample, DR7-Bright.

The correlation function ξ of the brighter volume-limited sample, DR7-Bright (blue crosses), is stronger on all scales than DR7-Full and DR7-Dim, which is expected, as bias is known to be a function of luminosity (see, e.g., Zehavi et al. 2005). For example, Figure 7 shows the relative redshift-space bias ratios, defined as

$$\begin{equation} b \equiv \sqrt{\xi _{\rm{DR7-Bright}}/\xi _{\rm{DR7-Dim}}}, \end{equation} \tag{ 4 }$$

showing that DR7-Bright is biased by a factor of 1.14 on most scales relative to DR7-Dim, in agreement with the mock catalogs (by design, of course, at scales r_p < 30 h⁻¹ Mpc). The gray bands show one and two σ distributions of the mock results when relating each of the DR7-Bright samples to the same realization DR7-Dim mock sample. The dotted lines are the outermost values for each bin. According to these uncertainties, the relative bias of DR7-Bright is significantly stronger (by at least 2σ) than DR7-Dim up to s ∼ 40 h⁻¹ Mpc.

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As discussed in Section 4.5, Figure 4 shows good agreement in the baryonic acoustic peak position among the DR7-Full and DR7-Dim. However, the relative strengths of the large-scale signal is worth investigating. Figure 8 is similar to Figure 5 for DR7-Bright. The figure and its insets show that its signal is significantly stronger than the mock values on scales s > 130 h⁻¹ Mpc.

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The bottom inset of Figure 8 is the histogram of $\langle \xi \rangle _{s=[130,500]\,h^{-1}\;{\rm Mpc}}$ for the mocks, compared to the SDSS value (red vertical dashed line). We perform the same χ² comparison as in DR7-Dim, but this time for scales up to 500 h⁻¹ Mpc (see Appendix B.3 for a justification of the larger scale). This test yields χ²/dof = 2.5 for 11 degrees of freedom (separation bins of top inset), showing an unlikely fit with the model used here.

We note that this result does not rule out all possible ΛCDM and HOD models, because we have compared to only one choice of parameters designed to fit statistics on smaller scales (see Section 3.2 for details).

We also split DR7-Bright into two subsamples (as indicated in Table 1): DR7-Bright-Near (0.16 < z < 0.36; ∼16, 500 LRGs) and DR7-Bright-Far (0.36 < z < 0.44; ∼13, 800 LRGs). Each produce noisy results for ξ, so we cannot draw concrete conclusions regarding large-scale clustering and the baryonic acoustic feature. For example, we find, as shown in Cabré & Gaztañaga (2009), that the distant subsample DR7-Bright-Far has no clear sign of a baryonic acoustic feature.

To examine the effect of the luminosity cuts on the strong signal in DR7-Dim in the range 120 h⁻¹ Mpc < s < 155 h⁻¹ Mpc, we exclude the DR7-Bright-Near subsample from DR7-Dim by creating in the same volume (0.16 < z < 0.36) a subsample including only galaxies dimmer than M_g = −21.8. This dimmer subsample (45,426 LRGs, −21.8 < M_g < −21.2) and DR7-Bright-Near yield low S/N results, so we cannot conclude significant differences.

In summary, we find that the apparent strong signal is consistent with sample variance in all cases. We conclude that, given the current measurement uncertainties, effects of potential systematics, such as different sample redshifts, volumes, or intrinsic brightness of the galaxy tracers, are sub-dominant and not statistically significant.

Given the large scales and small signals we are probing here, we need to test whether our results are sensitive to systematic errors in the calibrations as a function of angle. Eisenstein et al. (2001) cautions that the number count of LRG targets is sensitive to small variations in color cuts, especially in the g and i bands. Since the SDSS targeting catalog is known to have calibration errors at the 1% level, the true color cuts applied vary across the survey (Padmanabhan et al. 2008). Such a variation might introduce an artificial clustering signal in our analysis.

To test the possible effect of these errors on ξ, we subsample the LasDamas DR7-Bright sample in a spatially varying fashion, and analyze the correlation function of the resulting sample (using the original, un-subsampled random catalogs). In detail, the subsampling factor varies from 95% to 100% in sinusoidal waves along the declination direction with a wavelength of 10°. This choice is motivated by the fact that the targeting catalog is separately calibrated in each stripe, which spans 25 and are (very roughly) parallel to lines of constant declination.

Comparing the resulting ξ with the full mock catalogs, we find insignificant effects for individual mock realizations and no difference when averaging over all 160. We tested this both on the full DR7-Bright and on a subsample DR7-Bright-Far (0.36 < z < 0.44; see Table 1). Effects are significant only when enhancing the subsampling factor amplitude from 5% to 15%, well above the expected systematic uncertainties due to calibration. Even at this unrealistic uncertainty, the baryonic acoustic feature is noticeable, on average, though with a weaker amplitude.

Notably, we cannot obtain the strong large-scale signal as observed in Figure 8 using any realistic level of calibration uncertainty.

Here, we measure the baryonic acoustic peak position s_p. We first construct a model:

$$\begin{equation} \xi ^{\rm model}(s)=\beta \cdot \overline{\xi }(\alpha \cdot s), \end{equation} \tag{ 5 }$$

where the template $\overline{\xi }$ is based on the mock mean, β represents a bias term, and α represents a change in length scale. We minimize χ² over α and β (see Equations (2) and (3)), using the observed ξ at scales near the peak. After β and α are determined, we use a spline interpolation on the (very smooth) best-fit model to pinpoint s_p. This procedure is run on the observed ξ and, as explained below, for the individual mock realizations.

The advantage of using this approach over linear models is that clustering non-linearities, clustering bias, and redshift distortions are already taken into account in the simulations, as well as the angular mask and radial selection. A disadvantage is that we are not self-consistently altering the cosmology for each model. However, in Section 4.6, when using s_p to determine cosmological distances, we expect the uncertainties in assumptions we make to be small relative to the data uncertainties.

Figure 9 displays our result for DR7-Dim using $\overline{\xi }=\overline{\xi }_{160}$, the mock mean over all 160 realizations. The upper left panel shows the distribution of the best-fit peak position for the mocks and the observations. The vertical red line is the best fit for the observations, and the vertical green line is for the mean mock catalog. The upper right panel exhibits the distribution of χ² per degree of freedom. The vertical red line marks the results from the observations and the histogram is that of 75 clear-peaked mocks. The bottom left panel is the normalized covariance matrix $C_{ij}/\sqrt{C_{ii}C_{jj}}$. The bottom right panel compares the best-fit model ξ^model (blue solid line) to the observations (symbols), and the red arrow points to the fit peak value.

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The mean peak value for the mocks (green line on the top left panel) is s^mock_peak = 106.31 ± 3.19 h⁻¹ Mpc, where the 1σ uncertainties (68.2% CL) are calculated from mock realizations only with clear signs of a peak (about 75 out of the 160). The individual mock s_p results are shown as the histogram on the upper left panel and their normalized χ² result in the upper right plot. We obtain the uncertainty on the observations by scaling these uncertainties for this mock catalog distribution by  s_p/s^mock_p.

Using the observed DR7-Dim sample between 60 h⁻¹ Mpc < s < 150 h⁻¹ Mpc, we find the peak to be at s_p = 101.74 ± 3.05 h⁻¹ Mpc (3%) where these 1σ uncertainties are scaled from our mock catalog results. We caution that these uncertainties rely on the chosen fiducial cosmology of our mocks and can potentially vary—although we expect such variations to be small within observationally motivated ΛCDM models (see Appendix B.3 for details).

For this fit, we find χ²/dof = 1.26, where we use dof = 19 (the number of data points used minus two fitting parameters β, α). Thus, the fit is within expectations. The bottom right panel of Figure 9 shows a best fit that undershoots the data at all points. This appears to be a poor fit only because ones eye does not account for the strong covariances among the bins (see C_ij in lower left panel of Figure 9).

If we limit ourselves to the region 60 h⁻¹ Mpc < s < 135 h⁻¹ Mpc, we find  s_p = 105.0 ± 3.1 h⁻¹ Mpc. Although both results are consistent, the dependence of s_p on fitting range indicates that this sample is still limited in its power to constrain s_p. We also changed the lower limit to various values between [55, 75] h⁻¹ Mpc, constraining the upper bound to 150 h⁻¹ Mpc, and find s_p does not vary more than 1.35 h⁻¹ Mpc. If the lower limit is increased to 80 h⁻¹ Mpc, we obtain s_p = 100.7 ± 3.3 h⁻¹ Mpc, and increasing to 85 h⁻¹ Mpc our mechanism for detecting of the peak position breaks down, due to not using the full dip feature around 80 h⁻¹ Mpc.

We perform the same analysis on 24 jackknife subsamples of the observed sample. They are obtained by dividing the R.A.–decl. map into N_JK = 24 regions with equal number of LRGs and excluding one region, each in turn. For each jackknife subsample, we calculate ξ and run it through the peak finder algorithm to obtain $s_{p_i}$. The jackknife uncertainty σ_JK is obtained by

$$\begin{equation} \sigma _{\rm{JK}}^2=\frac{{N_{\rm{JK}}}-1 }{ {N_{\rm{JK}}}}\sum _{i=1}^{{N_{\rm{JK}}}} \left(s_{p_i}-\overline{s}_{p}^{\rm{JK}} \right)^2, \end{equation} \tag{ 6 }$$

where $\overline{{s}}_{p}^{\rm{JK}}$ is the jackknife mean. We measure σ_JK = 3.3 h⁻¹ Mpc. As a test, we jackknife a random (but with an apparent strong baryonic acoustic feature) mock realization, and obtain similar results. A more in-depth analysis of jackknife uncertainties would include testing for different values N_JK and geometries for multiple realizations. Our simple test shows jackknife uncertainties of s_p similar to that obtained by sample variance of the mocks. We trust the sample variance of s_p estimated from mock catalogs, as this is obtained by measuring a suite of realistic mocks of the full DR7-Dim volume.

Another template we test is $\overline{\xi }=\overline{\xi }_{75}$, the mock mean of 75 hand-picked realizations with a clear peak. $\overline{\xi }_{160}$, used before, yields the expected peak given a DR7-Dim size sample, where $\overline{\xi }_{75}$ produces the same given a realization with a clear peak (as in the case of the observations). In Figure 10, we show the correlation functions of $\overline{\xi }_{75}$ (dashed bright green) and $\overline{\xi }_{160}$ (solid black). As expected, both have roughly the same clustering at all scales, except that the latter has a weaker peak. The position of the baryonic acoustic feature is nearly the same (107.01 h⁻¹ Mpc for $\overline{\xi }_{75}$ versus 106.31 h⁻¹ Mpc for $\overline{\xi }_{160}$), and they have roughly the same width. The bottom panel shows the residual $\overline{\xi }_{75}-\overline{\xi }_{160}$. When fitting each template to the data we use the same covariance matrix constructed from all 160 realizations.

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When using $\overline{\xi }_{75}$ for the data between 60 h⁻¹ Mpc < s < 150 h⁻¹ Mpc we obtain s_p =1 − 2.03 ± 2.77 h⁻¹ Mpc, very similar to that obtained when using $\overline{\xi }_{160}$. The χ²/dof is 1.21. This shows that the choice of template $\overline{\xi }_{160}$ or $\overline{\xi }_{75}$ does not effect the end result in a DR7-Dim size sample.

Besides sample variance, another source of uncertainty in s_p is due to random shot noise. In Appendix B.1, we show that choices of different random catalogs (ratio of r ∼ 15.6 random points per data) yield slightly different s_p. To reduce this effect, we choose to use r ∼ 50 for our final results.

We also test the effect of dilution on s_p, i.e., choosing a different cosmology when converting redshifts to comoving distances (Padmanabhan & White 2008). In all above results, we apply a flat cosmology with Ω_M0 = 0.25. When using a different value Ω_M0 = 0.30 (but maintaining mocks as before), we obtain s_p = 100.40 h⁻¹ Mpc at χ²/dof = 1.05. As this result is well within the 1σ variance and systematics explained above, we conclude that the choice of cosmology does not change our results, but may need to be considered when variance is reduced in future surveys.

We find that the $\overline{\xi }$ for the DR7-Bright subsample, in the same binning used for DR7-Dim, is quite noisy and not useful to measure s_p. Although DR7-Bright covers a larger volume than DR7-Dim, its density is over four times lower yielding a sample with less than half the number of galaxies, severely increasing noise in our measurement.

We applied the same algorithm (same DR7-Dim covariance matrix) on DR7-Full and DR3, and find consistency with these results. A more in-depth analysis would involve building a new covariance matrix for DR7-Full, as it has a larger volume than DR7-Dim. We do not perform this analysis, because the LasDamas mocks do not extend to that volume. DR3 has fewer LRGs than DR7-Dim resulting in a noisier signal, so we do not find reason to do further analysis.

As Bashinsky & Seljak (2004) describe, the peak position s_p is closely related to r_s, the physical comoving sound-wave radius at the end of the baryon drag epoch. r_s is determined to high precision in physical units from CMB measurements (e.g., Komatsu et al. 2009), whereas we determine s_p in redshift space. Therefore, we can compare the two to measure the relationship between redshift and distance, as proposed by Eisenstein et al. (1999).

There are three important effects we must consider in such a determination. First, the relationship between the redshift-space peak position s_p and r_s is sensitive to non-linear clustering, redshift distortions and galaxy bias, the dominant effect being a broadening of the peak (Eisenstein et al. 2007a, 2007b; Crocce & Scoccimarro 2008; Smith et al. 2008; Angulo et al. 2008; Seo et al. 2008; Kim et al. 2009) and a dominant shift toward small scales that is below current statistical uncertainties.

Second, the measured s_p is necessarily associated with a fiducial cosmology used to interpret the redshifts and angular positions in terms of comoving distances. The fiducial cosmology we assume is the same concordance flat cosmology as the mocks used ([Ω^fid_M0, Ω^fid_b0, h^fid] = [0.25, 0.04, 0.7]). Technically, this means that we use cosmological assumptions in two steps of our algorithm: selection of LRGs by magnitude cuts, and converting observed redshifts to comoving distances. For more details, please refer to Appendix B.3

Third, we need to specify what "distance" we seek to measure, since the definition of cosmological distance within the context of general relativity is not unique. The relevant distance measures in this context are the angular diameter distance D_A(z) and the Hubble constant H(z) at redshift z (Hogg 1999). The former would be ideally constrained by measuring s_p in a thin shell at radius z, and H(z) by measuring the line-of-sight clustering (Matsubara 2004). These measurements individually are hard to perform with the current data set (for attempts in DR6, please refer to Okumura et al. 2008; Gaztanaga et al. 2009). Instead, we will constrain a combination of the two following the standard approach in the literature (see Equation (8) below).

To account for the above effects, the standard interpretation of the acoustic peak in the correlation function is as follows (with an analogous argument made for the power spectrum; Percival et al. 2007). First, we assume the proportionality:

$$\begin{equation} r_s = \gamma s_p. \end{equation} \tag{ 7 }$$

Mock catalogs can be used to determine γ, as long as the assumed cosmology is not far from the true one. We do so here for the LasDamas mock catalogs, whose cosmological parameters are well motivated from the CMB and other constraints. We calculate r^fid_s(z_d) = 159.75 Mpc for the mocks using Equation (1) from Blake & Glazebrook (2003), and calculate the sound speed c_s and the end of the drag epoch z_d = 1012.13 using Equations (4) and (5) from Eisenstein & Hu (1998). We insert Θ_2.7 = 2.725/2.7 as the temperature ratio of the CMB in their Equation (5). Using s^mock_p from Section 4.5 and h = 0.7, we obtain γ = 1.052.

Second, we construct the "distance" quantity:

$$\begin{equation} D_V\equiv \left[(1+z)^2D_A^2cz/H(z)\right]^{\frac{1}{3}}. \end{equation} \tag{ 8 }$$

This quantity is designed in such a way that the ratio D_V(〈z〉)/s_p(〈z〉) is approximately independent of the choice of the fiducial cosmological model (Eisenstein et al. 2005; Percival et al. 2007; Padmanabhan & White 2008).

Third, we can rearrange these relationships to obtain:

$$\begin{equation} \frac{D_V(\langle {z}\rangle)}{r_s} = \frac{D_V(\langle {z}\rangle,\mathrm{fid})}{\gamma s_p(\langle {z}\rangle)}, \end{equation} \tag{ 9 }$$

where s_p(〈z〉) is understood to be the inferred s_p from Section 4.5 given the fiducial cosmology. Measuring s_p thus yields the ratio of the distance at redshift 〈z〉 to the acoustic scale. Given the acoustic scale r_s in Mpc from the CMB, we then can determine the distance D_V(〈z〉) to that redshift. As Equation (8) shows, this distance measurement constrains a combination of the angular diameter distance D_A and the Hubble expansion rate H(〈z〉).

Given the results in Section 4.5 for DR7-Dim, we find r_s/D_V = 0.1389 ± 0.0043 at 〈z〉 = 0.278, with 1σ uncertainties of around 3%. This value is in excellent agreement with the analysis of the power spectrum of a similar sample by Percival et al. (2009). Combining our result with the sound horizon r_s = 153.3 Mpc obtained from WMAP5 CMB (Komatsu et al. 2009), we find D_V(0.278) = 1103 ± 43 Mpc.

Figure 11 presents our result compared with those obtained by other studies, as well as predictions of flat ΛCDM cosmologies. Our data point is the black diamond where we choose to use the mean sample redshift 〈z〉 = 0.278. The red and orange crosses are the values published in Percival et al. (2009) and Percival et al. (2007): r_s/D_V(z = 0.2) = 0.1981 ± 0.0071, r_s/D_V(0.275) = 0.1390 ± 0.0037, r_s/D_V(0.35) = 0.1094 ± 0.0040 where the small red crosses results are indicated in Table 3 of Percival et al. (2009). The cyan cross is r_s/D_V(0.35) = 0.1097 ± 0.0039 obtained by Reid et al. (2009) when analyzing the P(k) of the reconstructed halo density field of DR7 LRGs. The blue square is the result obtained by Eisenstein et al. (2005; D_V(0.35) = 1370 ± 64 Mpc, if we use the CMB r_s value as before). We also add the result obtained by Sánchez et al. 2009 (purple triangle), who analyze the DR7 LRG ξ. Using only LRG clustering, they obtain D_V(0.35) = 1230 ±  220 Mpc. They show that when combining ξ with CMB measurements they obtain a tighter constraint D_V(0.35) = 1300 ± 31 Mpc. We plot the latter where we use r_s from CMB for plotting purposes.

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Our result is in perfect agreement with that obtained by Percival et al. (2009) at z = 0.275. Please keep in mind that both results, as well as those obtained by other above studies that analyze the SDSS LRGs, are not independent, as we use roughly the same data set. We are encouraged, nevertheless, that our consistency check yields a result in agreement with Percival et al. (2009).

We comment that if we use the median redshift z_med = 0.287 rather than the mean 〈z〉, we obtain r_s/D_V(z = 0.287) = 0.1349 ± 0.0043 and D_V(z_med) = 1136 ± 35 Mpc.

When comparing these results to cosmological predictions, we assume a flat ΛCDM, and fix ω_M ≡ Ω_M0h² = 0.1358. This constraint is motivated by its low (2.7%) uncertainty in the WMAP5 CMB temperature measurements (Komatsu et al. 2009). D_V depends both on Ω_M0 and h independently, thus its values vary. Meanwhile, r_s depends on ω_M (and Ω_bh²) so it is kept constant. The 1σ results of r_s/D_V indicate that, constraining w_M from CMB, the preferred region of the matter density lies in the range Ω_M0 = [0.25, 0.34], and h = [0.63, 0.73] in agreement with CMB and others. This is consistent for both choices of redshift (mean or median). If we would plot median redshift result (z = 0.287) in Figure 11, it would appear along the Ω_M0 = 0.28 line. We defer a full analysis of the cosmological implications.

Our ξ results for DR7 LRGs as well as the covariance matrix may be be obtained on the World Wide Web.¹³

The SDSS LRGs cover roughly 20% of the sky, mostly in the Northern Galactic Cap, and partially in the Southern (Figure 1). After an image of the sky is obtained in the u, g, r, i, z bands, objects are targeted for a spectroscopic observation to measure their redshifts. Here we refer to targeted quasars, Main galaxies, and LRGs as targeted objects. Zehavi et al. (2005), in their appendix, discuss reasons that some targeted objects fail to receive spectra, and their method (which is also used in Eisenstein et al. 2005) of correcting for this small incompleteness. Please refer to their study for definitions of "sector," "sector completeness," "fiber-collisions," and the polygon method with which the complicated angular geometry is expressed in DR3 and later releases.

We choose a slightly different definition for sector completeness than Zehavi et al. (2005). We define it as the number of targeted objects (Quasars+LRG+Main) that have obtained spectra (after fiber-collision corrections) divided by the total number of targeted objects. By "fiber-collision corrections" in the number of targeted objects which obtained spectra, we mean that before calculating sector completeness, we up-weight objects with spectra which are within 55'' of objects that did not. For example, if we find a group of four objects in 55'' proximity and three of them get spectra, all three are fiber-collision weighted by 4/3. This up-weighting plays two roles in our algorithm: defining sector completeness and while counting galaxy pairs.

We mentioned above our definition for sector completeness. If, for example, the only objects which did not get spectra within a sector are due to fiber-collisions, the sector is considered fully complete. A rare peculiarity occurs due to 55'' neighbors straddling the border between sectors, in which one (or more) obtain spectra and another does not. In these cases, some sectors might have a completeness larger than unity, meaning they obtain a partial completeness fraction from the neighboring sector. In summary, the average completeness of all sectors is 98% for DR7. If we define Main+LRG as "targeted objects" (excluding quasars) the completeness yields the same, and if we define targeted objects as LRGs only we obtain 96%. We find that these different definitions for completeness result in subtle mismatches of the DR3 LRGs compared sample to Eisenstein et al. (2005), all of which less than 0.7% of the ∼47, 000 galaxies, and negligible effects on ξ compared to shot noise of random points used. As in Eisenstein et al. (2005), we limit ourselves to sectors with >60% completeness. This results in 29 sectors (that have a non-zero completeness) with a total area of 13 deg² (0.16% of targeted sky) and a total of 364 targeted LRGs. As in Eisenstein et al. (2005), we used the completeness as a probability to exclude random points from each sector. They also up-weight both data and random points in each sector by the reciprocal of the completeness value. With the high completeness of the survey, we do not expect differences in the resulting ξ between the methods.

To account for fiber-collisions while pair counting, we use a slightly different population than before. LRGs get up-weighted if they are 55'' neighbors of a targeted LRG (not quasar or Main galaxy) which did not get spectra. This effectively increases our sample LRG number by ∼2.2% for DR7-Full and ∼1.8% for DR3, which is important to take into account when calculating normalized number of pairs DD, DR, and RR.

The LasDamas mock catalogs match the survey geometry as described by the polygon description in the NYU-VAGC. Since the "gamma" release mocks do not model fiber collisions nor missing sectors, completeness is defined to be 100% within the SDSS area. For more details, please refer to C. K. McBride et al. (2010; in preparation).

The LRG sample used here is the largest quasi-volume-limited spectroscopic sample of its kind today. That said, the radial selection function, n(z), is not constant (meaning not volume limited), as one would expect in a homogeneous universe (up to Poisson noise and radial clustering). Instead, it is quasi-volume limited up to z ∼ 0.36 and flux limited thereafter. In what follows we show that the features in the n(z) of DR7-Dim do not affect our results. On a more technical note, we discuss distributing random points in the radial direction.

We test the LasDamas mock catalogs for the features in the observed radial selection function. The original mocks do not trace the n(z) of the corresponding SDSS redshift and luminosity cuts. The top left inset in Figure 13 shows this difference in shape for the DR7-Dim sample, as LasDamas provides an n(z) with a slight negative slope (cyan; average over 160 realizations). This slope in the LasDamas mocks is a consequence of applying fixed HOD parameters to a light-cone halo catalog, as it neglects the evolution of the dark matter halo mass (i.e., lower halo masses as higher redshift result in fewer artificial galaxies and hence the negative slope). The survey result (thick green histogram) is the same as in Figure 2. Two features noticeable are the negative slope between 0.16 < z < 0.28, and the positive slope to the peak at z ∼ 0.34. We show that these features have a negligible effect on ξ and the rms σ_ξ compared to the distribution given by the original mocks, which is closer to volume limited. To validate this claim, we exclude mock LRGs such that they match the SDSS radial selection function. For the DR7-Dim, this means excluding 15% of the mock galaxies (n(z) average over 160 realization shown as black histogram) and for DR7-Bright it means excluding 7%, yielding, on average, a similar number count and volume density as in Table 1. In the main plot of Figure 13, we compare results before and after exclusion of the mock galaxies, and find that the mean ξ of 160 realizations agrees on all scales. The right inset shows that the diagonal terms of the covariance matrix $\sigma _\xi \equiv \sqrt{C_{ii}}$ is slightly higher for the galaxy-excluded sample as expected from slightly larger Poisson noise. We conclude that the shape of observed n(z) for the DR7-Dim sample does not appear to affect the results obtained in this study.

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We would also like to address the issue of distributing redshifts to random catalogs, a non-trivial step in using random point based ξ estimators, when dealing with non-volume-limited data. Two popular methods for distributing redshifts (or comoving distances) to random points are redistributing the actual data redshifts randomly, and assigning random distances, so that the overall n(z) shape matches that of the data (i.e., using the data n(z) as a probability function). We test both random point distribution with and without radial weighting.

We show results of both methods of distributing random points in Figure 14. To clarify differences, in the top panel we plot s · ξ and the ratio between all cases to that chosen in this study (weighted Data-n(z); ξ_ref) in the bottom panel. For convenience, we define the case of distributing data redshifts to random points as "Data-Redshift" (diamonds), and the case of distributing random points to match the data n(z) as "Data-n(z)" (crosses). The data used in this analysis are the original DR7-Dim mocks averaged over eight realizations. We compare results without radial weighting (black) as well as with (bright green, shifted by 2 h⁻¹ Mpc for visual clarity, shift not calculated within s · ξ). The radial weighting when pair counting is explained bellow. The figure shows that "Data-Redshift" is noticeably weaker than "Data-n(z)" until s ≲ 110 h⁻¹ Mpc. This is a clear display of the diminishing of the radial clustering modes. Although this effect is very small relative to our current 1σ measurements, it could be important when these are reduced in future surveys. On the larger scales, the differences are diminished.

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In conclusion, we choose to radially distribute random points to match the same n(z) shape of the data, using the same weight, and not distributing the data redshifts to the random points, as to preserve the radial modes.

After deciding to use the survey n(z) to analyze the mock catalogs, we have two options of use of random points: (1) compare all data points to one random set, which has the mean n(z) of the data; or (2) imitate the observation for each realization by comparing each to a random catalog with a tailor-made random catalog. By "tailor-made," we refer to the adjustment of the redshifts z, so that each random catalog yields a similar n(z) to that of the mock data. We applied this difference when analyzing DR7-Dim and DR7-Bright and find no difference in the ξ mean results. As for the rms σ_ξ, we find very small differences in both. On scales up to s < 200 h⁻¹ Mpc, the DR7-Dim shows fluctuations smaller than 4% where DR7-Bright shows fluctuations below 8%, with no particular preference for either method. This might be a result of the fact that the transverse modes dominate the radial ones. For completeness, we note that in our mock results we use the former method, i.e., we apply one random catalog for all the mock data, where the ratio of random to data is r ∼ 50.

To weight the data approximately according to volume, we need to account for the radial selection function n(z) (see Appendix A.2 and Figure 2). To do so, we apply the standard weighting technique (Feldman et al. 1994).

For the SDSS LRGs, we calculate n(z) in bins of Δz = 0.015 and use a spline interpolation to calculate the radial selection value of each LRG and random point. Each point is then assigned a radial weight of 1/(1 + n(z) · P_w) where we use P_w = 4 · 10⁴ h⁻³ Mpc³, as in Eisenstein et al. (2005). The weights are applied to both the data and the random catalogs. The choice for Δz does not affect the measured ξ (as also shown in Cabré & Gaztañaga 2009). We also examined for differences between choosing the observed n(z) for weighting and the model used in Zehavi et al. (2005) and found no difference.

When comparing between different ξ results obtained by various systematics, it is important to differentiate biases from variances obtained by random shot noise.

Applying five different random catalogs to the observed DR7-Dim, at a ratio of ∼15.6 random points for every data point, and assigning radial weighting (similar to that performed in the analysis) we show in Figure 15 that the random shot noise is minimal on most scales of concern. The top panel shows ξ for five different random catalogs against the same observed sample (black-dotted, green-dashed, red-dot-short-dashed, orange-long-dashed, blue-triple-dotted-dashed). The bottom panel shows the ξ ratios with respect to the first random catalog. For the chosen binning, the baryonic acoustic region seems to yield less than a 10% difference. We reran our s_p algorithm on all random catalogs and find the peak positions [101.6, 101.8, 103.4] h⁻¹ Mpc. This shows that, although the overall shape seems very consistent when using the different random catalogs, the random shot noise has a small effect on pinpointing peak position. This is currently smaller than the survey 1σ, but should be considered when statistical uncertainties improve. The fitting normalized χ² range for all five catalogs is χ²/dof = (1.24–1.58). To reduce this effect on our measurement in Section 4.5, we used a ratio of number of random to data of ∼50. The peak position for this marked by the top arrow at s_p =  101.7 h⁻¹ Mpc. In the region between 50 < s < 90 h⁻¹ Mpc, we see up to 10% differences, which is expected in a sharp logarithmic slope. On smaller scales s < 50 h⁻¹ Mpc, differences yield up to 3% difference in amplitude.

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In order to optimally measure the correlation function and to account for the fiber collision effects, we apply weighting algorithms. All differences due to choices about how we weight turn out to be much smaller than current 1σ variances. Angular weighting schemes (fiber-collision correction and sector completeness) are explained in Appendix A.1. Here, we explain our algorithm for the radial weighting and show results for both.

Figure 16 shows the effects of various angular and radial weighting schemes on the large-scale ξ of DR7-Full (for clarity, we present s · ξ in the top panel). The weighting schemes compared are as follows:

• 1.  
  No weighting at all (black).
• 2.  
  Radial weighting only (cyan; shifted by 1.75 h⁻¹ Mpc for clarity).
• 3.  
  Radial weighting + fiber-collision (red; shifted by 3.5 h⁻¹ Mpc).

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As mentioned in Appendix A.1, sector completeness is taken into account in the distribution of the random points. The bottom panel shows the ratio of each of the above options ξ_i over that chosen in this study (Fiber-Col+Rad; ξ_ref). To avoid possible random shot noise (see Appendix B.1), we use the same set of random points in each case. The weight is modified for each option for the data and random as indicated in the plot. The fiber-collision weight for the random set is always w = 1.

We clearly see from the above three options the effect of radial weighting on scales s ≲ 170 h⁻¹ Mpc. We can grossly divide this range into a smaller one of s ≲ 115 h⁻¹ Mpc in which the radial weight adds some power to the signal, and a larger scale region in which the signal is reduced. The fiber-collision correction weights ("Fiber-Coll") are important when dealing with scales s ≲ 40 h⁻¹ Mpc at the few percent level, but not at the scales discussed in this study. Beyond 170 h⁻¹ Mpc, the effects of the weighting schemes used here are minimal. Most importantly, as shown in Cabré & Gaztañaga (2009), the apparent strong ξ signal at large scales, and the baryonic acoustic feature position are consistent among the various weighting methods.

These weighting effects are not substantial in Figure 14, because the n(z) used there (full DR7-Dim LasDamas catalog without dilution to match SDSS n(z)) does not have the same complex features seen in DR7-Full (the mocks used in that figure are instead close to volume-limited).

We have checked a few other possible systematics collectively described here.

The SDSS DR7 sample is mostly contiguous, with only 9.8% of the surveyed area not contained in the main part of the Northern Galactic Cap sample (Figure 1). We examined three choices for which footprint to use: the full survey (100% of LRG sample), the Northern Galactic Cap only (90.9%), and the Northern Galactic Cap without the small "island" (90.2%). We found no significant difference in the resulting DR7 ξ for these samples.

In order to calculate the separation in comoving space between galaxy pairs, we must use the observed redshifts and assume a certain fiducial cosmology. For most of this study, we assumed a flat ΛCDM model with Ω_M0 = 0.25. When analyzing differences between DR3 and DR7-Full, we also tested the cosmology assumed by Eisenstein et al. (2005) Ω_M0 = 0.3. We also examine each cosmology on each subsample and found that the cosmology does not affect the resulting ξ or the measured s_p significantly relative to our other uncertainties. For a direct comparison of both cosmologies in DR7 see Figure 3.

We also tested how the choice of cosmology affects the selection of LRGs, due to the fact that M_g is sensitive to cosmology. Within DR7-Dim (0.16 < z < 0.36, $-23.2<\mathrm{\it {M}_g}<-21.2$), we count 61,899 LRGs when using Ω_M0 = 0.25 and 61,102 when Ω_M0 = 0.30, a 1.3% difference. When probing the full sample, DR7-Full, we find an agreement in number of LRG selected in the two cosmologies to better than 1%. This implies that for large range of redshift an approximate cosmology should not change the number count very much.

As mentioned in Section 3, the ξ calculation requires the choice of a particular estimator. We tested the Landy & Szalay (1993) estimator against those proposed by Hamilton (1993); Davis & Peebles (1983); Peebles & Hauser (1974). Figures 17 and 18 show our ξ and σ_ξ results, when using DR7-Dim. In Figure 17, the mock mean (over 160 realizations) are the lines, while the SDSS result are the symbols. The different colors indicate the different estimators used, where (Landy & Szalay 1993; black solid, LS93 hereon) and (Hamilton 1993; green triple-dot-dashed, HAM93) are not distinguishable by eye for the most part. The inset shows the noise-to-signal ratio. Bear in mind that the spikes around 140 h⁻¹ Mpc simply result from the signal crossing zero around that scale. In Figure 18, the notation is the same when plotting σ_ξ(s) for the different estimators. The inset in this plot shows the ratio of each estimator relative to that of LS93.

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We find the following:

• 1.  
  Mean value $\overline{\xi }$: averaging over all 160 mocks, we see that LS93 and HAM93 agree on all scales. The inset of Figure 18 shows that (Peebles & Hauser 1974; PH74) deviates from the latter by 10% at scales of ∼75 h⁻¹ Mpc and (Davis & Peebles 1983; DP83) does the same at 85 h⁻¹ Mpc. We also note that LS93 and HAM93 asymptote faster to zero than the others.
• 2.  
  rms σ_ξ: the rms of the signal varies among the estimators on various scales. PH74 yields the same σ_ξ as LS93 up to scales of ∼40 h⁻¹ Mpc before it strongly starts branching off. This is clearly seen in the main plot of Figure 18, and indicated in the other plots by the strong variation of the observed result. Both HAM93 and DP83 have a larger variance than LS93 at scales smaller than 10 h⁻¹ Mpc. HAM93 later matches the LS93 on all larger scales very well, where DP83 breaks off at ∼60 h⁻¹ Mpc.

We conclude that the Landy & Szalay (1993) and Hamilton (1993) estimators agree in signal for each realization on large scales, and perform much better than the other two as their variance is smaller, and converge much more quickly to zero. In particular, the other two estimators yield very large uncertainties on large scales for individual realizations. We also find that Hamilton (1993) does not perform as well as Landy & Szalay (1993) on smaller scales s < 10 h⁻¹ Mpc. Our analysis of DR7-Dim agrees with the other samples DR7-Bright, DR7-Bright-Near, and DR7-Bright-Far. Here, we use ratio of random points to data of r ∼ 15.6 for the SDSS and r ∼ 10 for mocks. Kerscher et al. (2000) shows Landy & Szalay (1993) to be the estimator with the best performance relative to the true ξ in periodic box measurements, and Labini et al. (2009) shows similar comparison results to ours.

To test the importance of binning, we use the Landy & Szalay (1993) on all 160 mock DR7-Dim realizations and find a strong dependence of the variance on the bin size, as expected. We test various bin widths Δs between Δs = [0.55, 10] h⁻¹ Mpc noting that variance changes strongly, where the lowest Δs yields the noisiest σ_ξ, due to the higher shot noise in each bin. For most of our analysis, including the baryonic acoustic feature, we used Δs = 4.44 h⁻¹ Mpc. We find that peak position s_p is not altered by more than 1σ for any choice of bin size.

We also check for boundary limits in our survey, to get an idea of at what point our ξ estimates are encountering the edges of the survey. We show in Figure 18 a negative slope for σ_ξ all the way to s < 400 h⁻¹ Mpc. A region we definitely want to avoid in analysis is one in which σ_ξ has a positive slope, which indicates an upper limit to the effective scaling given the survey volume. For this reason in the top panel of Figure 19, we continue this plot for DR7-Dim (black solid) and DR7-Bright-Near (green triple-dotted-dashed) to 800 h⁻¹ Mpc. These subsamples (both limited to the range 0.16 < z < 0.36) have a declining σ_ξ to 500 h⁻¹ Mpc, and a slightly positive slope thereafter. In the second plot from the top, we show the normalized data–data pairs DD of DR7-Dim and DR7-Bright-Near (same notation), as well as the random–random pairs RR count of DR7-Dim (blue dot-dashed). The dip feature at ∼500 h⁻¹ Mpc mentioned before appears here as the peak at that scale. In the third plot from top, we differentiate the previous results by s to better see where the number of pairs stop growing with radius. We see a clear crossover between 500and600 h⁻¹ Mpc.

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The bottom two plots of Figure 19 can be considered "sanity checks," as we verify basic statistics. Given a periodic box, we expect the number of random points N around a given point at radius s in a spherical shell of width ds to be: $N=4\pi s^2 ds \overline{n}$ where $\overline{n}$ is the mean density. RR(s), is expected to go as ∼N(s) · N_R/2, where N_R is the number of random points, and 2 takes into account double counting. Hence, dln(RR)/dln(s) should yield 2. Deviations from this result are due to boundary effects. In the fourth from top plot, we present dln(x)/dln(s), where x is RR (blue dot-dashed) and DD (solid black). The important result from this plot is the deviation of dln(RR)/dln(s) from 2 at large scales. A 5% difference is noticed at 50 h⁻¹ Mpc. At the baryonic acoustic feature scale, the deviation from 2 is ∼10%, at 200 h⁻¹ Mpc by ∼23% and at 400 h⁻¹ Mpc by 60%. Although this panel indicates our volume is far from ideal for analysis of 400 h⁻¹ Mpc, the LS93 estimator appears to be valid as it corrects for these boundary effects by comparing number of data pairs to number of expected random points given boundary conditions. We performed the same test for random points within a volume of DR7-Bright (red dot-dashed) where we extend the measurement to 500 h⁻¹ Mpc.

In the bottom plot, we determine how many cubes of length s would fit into the volume contained within DR7-Dim (black solid) and DR7-Bright (red three-dot-dashed). Our choice of s = 400 h⁻¹ Mpc yields 10.31 for the former and 18.65 for the latter. For DR7-Bright up to s = 500 h⁻¹ Mpc, we count 9.55 cubes. As for the peak position at s = 101 h⁻¹ Mpc, we count 641 cubes for the DR7-Dim volume and 1159 for a DR7-Bright volume.

All the plots in Figure 19 indicate that our choice of s < 400 h⁻¹ Mpc is a fair one within the given sample, and does not depend too much on edge effects.

We also test for uncertainties σ_ξ of the observed DR7-Dim and DR7-Bright against those predicted by Gaussianity plus shot noise (Bernstein 1994) at large scales,

$$\begin{equation} \sigma ^2_\xi (s)={2\over (2\pi)^3V} \int d^3k\ (P(k)+\bar{n}^{-1})^2\, j_0(ks)^2, \end{equation} \tag{ B1 }$$

where V is the volume of the survey (see Table 1), P(k) the galaxy power spectrum, and j₀ the spherical Bessel function of zeroth order. We first test this formula against the variance derived from the mock catalogs for both Dim and Bright samples, and find that at 50 h⁻¹ Mpc < s < 100 h⁻¹ Mpc the variance is consistent with Equation (B1). At smaller scales, non-Gaussian contributions to the uncertainties quickly make Equation (B1) an underestimate of the uncertainties, whereas at scales s > 100 h⁻¹ Mpc contributions to Equation (B1) from edge effects (not included there) become important, e.g., making the total uncertainty about 20% larger than given by Equation (B1) at 300 h⁻¹ Mpc. Since edge effects depend only on the geometry and density of the sample, we can extract their value from this comparison and apply it to the data, which has the same characteristics (but different two-point function).

From the measured two-point function in the data we use Equation (B1) including the edge effects variance to compute Gaussian uncertainties for a model that matches the observed ξ. We find uncertainties that are very close to those in the mock catalogs at s > 150 h⁻¹ Mpc, the main reason being that discreteness contributions are significant. This suggests that had we changed cosmology and HOD to give a better fit to ξ at largest scales, we would have gotten very similar uncertainties, and thus similar ∼2σ deviations away from zero-crossing for the Bright sample.