Clustering analysis of high-redshift Luminous Red Galaxies in Stripe 82

Clustering analysis of high-redshift Luminous Red Galaxies in Stripe 82

N. Nikoloudakis, T. Shanks, U. Sawangwit
Physics Department, University of Durham, South Road, Durham, DH1 3LE, UK
National Astronomical Research Institute of Thailand, Chiang Mai, 50200, Thailand

We present a clustering analysis of Luminous Red Galaxies (LRGs) in Stripe 82 from the Sloan Digital Sky Survey (SDSS). We study the angular two-point autocorrelation function, , of a selected sample of over 130 000 LRG candidates via colour-cut selections in with the K band coverage coming from UKIRT Infrared Deep Sky Survey (UKIDSS) LAS. We have used the cross-correlation technique of Newman (2008) to establish the redshift distribution of the LRGs. Cross-correlating them with SDSS quasi-stellar objects (QSOs), MegaZ-LRGs and DEEP2 galaxies, implies an average redshift of the LRGs to be with space density, hMpc. For (corresponding to hMpc), the LRG significantly deviates from a conventional single power-law as noted by previous clustering studies of highly biased and luminous galaxies. A double power-law with a break at hMpc fits the data better, with best-fit scale length, hMpc and slope at small scales and hMpc and at large scales. Due to the flat slope at large scales, we find that a standard cold dark matter (CDM) linear model is accepted only at , with the best-fit bias factor, . We also fitted the halo occupation distribution (HOD) models to compare our measurements with the predictions of the dark matter clustering. The effective halo mass of Stripe 82 LRGs is estimated as hM. But at large scales, the current HOD models did not help explain the power excess in the clustering signal.

We then compare the results to the results of Sawangwit et al. (2011) from three samples of photometrically selected LRGs at lower redshifts to measure clustering evolution. We find that a long-lived model may be a poorer fit than at lower redshifts, although this assumes that the Stripe 82 LRGs are luminosity-matched to the LRGs. We find stronger evidence for evolution in the form of the LRG correlation function with the above flat 2-halo slope maintaining to hMpc. Applying the cross-correlation test of Ross et al. (2011), we find little evidence that the result is due to systematics. Otherwise it may represent evidence for primordial non-Gaussianity in the density perturbations at early times, with .

galaxies: clustering – luminous red galaxies: general – cosmology: observations – large-scale structure of Universe.
pagerange: LABEL:firstpageLABEL:lastpagepubyear: 2012

1 Introduction

The statistical study of the clustering properties of massive galaxies provides important information about their formation and evolution which represent major questions for cosmology and astrophysics. The correlation function of galaxies remains a simple yet powerful tool for implementing such statistical clustering studies. (e.g. Peebles, 1980).

A lot of interest has been concentrated specifically on measuring the clustering correlation function of luminous red galaxies (LRGs) (Eisenstein et al., 2001) (see e.g Zehavi et al., 2005b; Blake, Collister, & Lahav, 2008; Ross et al., 2008; Wake et al., 2008; Sawangwit et al., 2011). LRGs are predominantly red massive early-type galaxies, intrinsically luminous () (Eisenstein et al., 2003; Loh & Strauss, 2006; Wake et al., 2006) and thought to lie in the most massive dark matter haloes. They are also strongly biased objects (Padmanabhan et al., 2007) and this coupled with their bright luminosity makes their clustering easy to detect out to high redshifts. For linear bias, the form of the LRG correlation function will trace that of the mass but even in this case the rate of correlation function evolution will depend on the bias model (e.g. Fry, 1996), which in turn depends on the galaxy formation process.

The passive evolution of the LRG LF and slow evolution of the LRG clustering (Wake et al., 2008; Sawangwit et al., 2011) seen in SDSS, 2SLAQ and Surveys already presents a challenge for hierarchical models of galaxy formation as predicted for a cold dark matter (CDM) universe. Since the LRG clustering evolution with redshift has been controversial, a major goal is to use the angular correlation function to test if the slow clustering evolution trend continues out to .

The uniformity of the LRG Spectral Energy Distributions (SEDs) with their 4000 break, offer the ability to apply a colour-colour selection algorithm for our candidates. This technique has been successfully demonstrated primarily by Eisenstein et al. in SDSS in the analysis of LRG clustering at low redshift and then in 2SLAQ (Cannon et al., 2006) and (Ross et al., 2008) LRG surveys at higher redshifts. For our study, the available deep optical-IR ugrizJHK imaging data from the SDSS + UKIDSS LAS/DXS surveys in Stripe 82 will be used. This combination of NIR and deep optical imaging data, on a moderate sample size of area deg, results in a sample of LRG candidates at redshift .

The main tool for our clustering analysis will be the two-point angular correlation function, , which has been frequently used in the past, usually in cases where detailed redshift information was not known. Hence, selecting Stripe 82 LRGs based on colour-magnitude criteria, correspond to a rough photometric redshift (photo-z) estimation based on the 4000 break shifting through the passbands. We shall apply the cross-correlation technique which was introduced by Newman (2008) to measure the redshift distribution, , of our photometrically selected samples. One of the main advantages of is that it only needs the of the sample and then through Limber’s formula (Limber, 1953) it can be related to the spatial two-point correlation function, .

In recent clustering studies, it was noted that the behaviour of , which has previously been successfully described by a single power-law of the form , significantly deviates from such a power-law at hMpc. The break in the power-law, can be interpreted in the framework of a halo model, as arising from the transition between small scales (1-halo term) to larger than a single halo scales (2-halo term). Currently, our theoretical understanding of how galaxy clustering relates to the underlying dark matter is provided by the halo occupation distribution model (HOD, see, e.g Jing, Mo, & Boerner 1998; Ma & Fry 2000; Peacock & Smith 2000; Seljak 2000; Scoccimarro et al. 2001; Berlind & Weinberg 2002) via dark matter halo bias and halo mass function. Furthermore, the evolution of HOD can also give an insight into how certain galaxy populations evolve over cosmic time (White et al., 2007; Seo, Eisenstein, & Zehavi, 2008; Wake et al., 2008; Sawangwit et al., 2011).

The outline of this paper is as follows. In Section 2, we briefly describe the SDSS and UKIDSS data used in this paper, while in Section 3 we describe the angular function correlation function estimators and their statistical uncertainties. In Section 4, we estimate the redshift distribution through cross-correlations and then present the correlation results together with their power-law fits, CDM model and a halo model in Section 5. Section 6 is devoted to interpretation of the clustering evolution. In section 7, we explore potential systematic errors that might affect the large scale clustering signal. We then argue that, if real, an observed large-scale clustering excess may be due to the scale-dependent bias caused by primordial non-Gaussianity and compare our results to other previous works in Section 8. Finally, in Section 9 we summarize and conclude our findings.

Throughout this paper, we use a flat -dominated cosmology with , , h=0.7, and magnitudes are given in the AB system unless otherwise stated.

2 Data

2.1 LRG sample selection

We perform a -band selection of high redshift LRGs in Stripe 82 based on the combined optical and IR imaging data, ugrizJHK, from SDSS DR7 (Abazajian et al., 2009) and UKIDSS LAS surveys (Lawrence et al., 2007; Warren et al., 2007), respectively. In previous studies, gri and riz colours have been used to select low to medium redshift LRGs, such as SDSS (Eisenstein et al., 2001), 2SLAQ (Cannon et al., 2006) and AA (Ross et al., 2008) LRGs surveys up to . In this work we aim to study LRGs at , thus we use the izK colour magnitude limits for our selection in order to sample the 4000 break of the LRGs’ SED as it moves across the photometric filters (Fukugita et al. 1996; Smith et al. 2002) taking advantage of the NIR photometry coverage from UKIDSS LAS. Coupling the UKIDSS LAS to with the SDSS ugriz imaging to in Stripe 82 produces an unrivaled combination of survey area and depth. Our selection criteria are :

Figure 1: iz vs zK colour-colour plot. Priority A and B correspond to the and LRG samples, respectively. Objects with which is typical for M stars are plotted as black circles where as those with are plotted in green. Evolutionary tracks for single burst (red line) and (blue line) are overplotted from to 1.6 with symbols indicate interval of 0.2. The evolutionary track of late type galaxies (magenta line) is also shown for comparison.

The photometric selection of LRGs at requires a combination of optical and NIR photometry as the band straddles the z band. The selection of high-redshift LRGs is done on the basis of SDSS photometric data and the LAS band data (Fig. 1). LRG evolutionary models of Bruzual & Charlot (2003) are overplotted for single burst and galaxy models indicating the izk plane area where we should apply our selections in order to study the high-z LRG candidates.

Late-type star contamination is a major problem in selecting a photometric sample of LRGs. Here the colour also helps to distinguish the M stars colour locus from those of galaxies. From Fig.1, we see that most of the M stars lie at the bottom of the colour plane. We identify these M stars by assuming their typical NIR colour, . However, this means that our selection criteria must involve band data and would reduce the sky coverage due to the data availability. Therefore we choose to exclude these M stars by applying a cut in colour plane with the condition in Eq. 1.

All magnitudes and colours are given in SDSS AB system and are corrected for extinction using the Galactic dust map of Schlegel, Finkbeiner, & Davis (1998). All colours described below refer to the differences in ‘model’ magnitudes (see Lupton et al., 2001, for a review on model magnitudes) unless otherwise stated.

Applying the above selection criteria (Eq. 1) on the SDSS DR7, we have two main LRG samples with a total observed area (after masking) of . The first sample has 130819 LRGs candidates with a sky surface density of and the second one 44543 with a sky density of . The LRG sample was selected in such a way to check if the redshift distribution implied by cross-correlations is higher than the LRG sample.

3 The 2-Point Angular Correlation Function Measurements and Errors

3.1 Estimators

The probability of finding a galaxy within a solid angle on the celestial plane of the sky at a distance from a randomly chosen object is given by(e.g. Peebles, 1980)


where n is the mean number of objects per unit solid angle. The angular two-point correlation function (2PCF) in our case, actually calculates the excess probability of finding a galaxy compared to a uniform random point process.

Different estimators can be used to calculate , so to start with we use the minimum variance estimator from Landy & Szalay (1993),


where is the number of LRG-LRG pairs, and are the numbers of LRG-random and random-random pairs, respectively with angular separation summed over the entire survey area. is the total number of random points, is the total number of LRGs and is the normalisation factor. For our calculation we used two LRG samples (as explained in § 2.1) with different sky density, thus the density of the random catalogue that we use is times and times the number of the real galaxies for the first and second LRG samples, respectively. Using a high number density random catalogue helps to ensure the extra shot noise is reduced as much as possible.

We also compute by using the Hamilton (1993) estimator which does not depend on any normalisation and is given by,


The Landy-Szalay estimator when used with our samples gives negligibly different results to the Hamilton estimator. Note that the Landy-Szalay estimator is used throughout this work except in §7.1 where we used both estimators to test for any possible gradient in number density of our samples.

For the computation of the cross-correlations in §4 and §7 we use the estimator (Guo, Zehavi, & Zheng (2011)) :


where the subscript G and S stands for the contribution in the pairs of the quantities that are cross-correlated in each case.

3.2 Error Estimators

To determine statistical uncertainties in our methods, we used three different methods to estimate the errors on our measurements. Firstly, we calculated the error on by using the Poisson estimate


Secondly, we used the field-to-field error which is given by


where N is the total number of subfields, is an angular correlation function estimated from the ith subfield and is measured using the entire field. For this method we divide our main sample to 36 subfields of equal size . We also reduce the number of subfields down to 18 with sizes of as we want to test how the results could deviate by using different sets of subsamples. While Stripe 82 has only deg height, our subfields with their deg and deg widths are a reasonable size for estimating the correlation function up to scales of deg.

Our final method is jackknife resampling, which is actually a bootstrap method. This technique has been widely used in clustering analysis studies with correlation functions (see, e.g Scranton et al. 2002; Zehavi et al. 2005a; Ross et al. 2007; Norberg et al. 2009; Sawangwit et al. 2011). The jackknife errors are computed using the deviation of the measured from the combined 35 subfields out of the 36 subfields (or 17 out of 18 when 18 subfields are used). The subfields are the same as used for the estimation of the field-to-field error above. is calculated repeatedly, each time leaving out a different subfield and hence results in a total 36 (or 18) measurements. The jackknife error is then


where is a measurement using the whole sample except the ith subfield and is approximately 35/36 (or 17/18) with slight variation depending on the size of resampling field. A comparison of the error estimators can be seen in Fig. 2. Poisson errors are found to be much smaller compared to jackknife errors particularly at larger scales. Field-to-field errors give similar results as jackknife errors, except at where the FtF errors underestimate the true error due to missing cross-field pairs. Since the jackknife errors are better at a scale of order which are of prime interest here, these are the error estimators that will be used in this work unless otherwise stated.

When calculated in small survey areas, can be affected be an ‘integral constraint’, . Normally has a positive signal at small scales and if the surveyed area is sufficiently small, this will cause a negative bias in at largest scales (Groth & Peebles, 1977), i.e. . The integral constraint can be calculated from (see e.g. Roche & Eales 1999):


where for the we assume the standard model in the linear regime (§5.3). No integral constraint is initially applied to our full sample results as the expected magnitude of is smaller than the amplitudes at scales analysed in this paper. This position will be reviewed when we move on to discuss models with excess power at large scales in §7.

To provide robust and accurate results from the correlation functions, we are also interested in model fitting to the observed (see in §5.2, §5.4 and §5.3). Hence, for model fitting we will use the covariance matrix, which is calculated by:


where the is the correlation function measurement value excluding the subsample and the factor corrects from the fact that the realizations are not independent (Myers et al. 2007; Norberg et al. 2009; Ross, Percival, & Brunner 2010; Crocce et al. 2011; Sawangwit et al. 2011). The jackknife errors are the square-root of the diagonal elements of the covariance matrix, so we can now calculate the correlation coefficient, which is defined in terms of the covariance,


where (see Fig. 3). We can see that the bins are strongly correlated at large scales. The covariance matrix is more stable when we use 36 Jackknife subfields instead of 18, so we will use only the covariance matrix for the case of 36 subfields.

Figure 2: Comparison of the measured error ratios of the Jackknife, field-to-field and the Poisson errors for the measurements of the Stripe 82 LRG sample. Two different resampling sets have been used for the Jackknife and field-to-field errors, the first one based on 36 subfields and the second from 18 subfields.
Figure 3: The correlation coefficients , showing the level of correlation between each angular separation bin for the Stripe 82 LRG sample as calculated by using 36 subfields.

3.3 Angular Mask and Random Catalogue

To measure the observed angular correlation function we must compare the actual galaxy distribution with a catalogue of randomly distributed points. The random catalogue must follow the same geometry as the real galaxy catalogue, so for this reason we apply the same angular mask. The mask is constructed from ‘BEST’ DR7 imaging sky coverage111 Furthermore, regions excluded in the quality holes defined as ‘BLEEDING’,  ‘TRAIL ’, ‘BRIGHT_STAR ’ and ‘HOLE’. The majority of the holes in the angular mask is from the lack of K coverage in Stripe 82. The final mask is applied to both our data and random catalogue (see Fig. 4).

Figure 4: A fraction of the total observed area in Stripe 82. LRG candidates (red) and random objects (blue), follow the same angular selection. Empty sky patches resulting from the lack of K-band coverage in the combined optical-IR data.

For generating the randomly distributed galaxies/points, we tried two different ways in order to modulate the surface density of the random points to follow the number density and the selection function of the real data. The selection function of the random catalogue mimics only the angular selection of the real data.

For the first method, we use a uniform density for the random points across the Stripe 82 area, so the normalization factor, , to be and for the and the LRG samples, respectively. A second random catalogue was created by dividing Stripe 82 into six smaller subfields ( each) and normalizing the density of random points to the density of galaxies within each subfield. The difference between the measured angular correlation function when we use the ‘global’ or the ‘local’ random catalogue is negligible. We will use the ‘global’ random catalogue for the clustering analysis. A d-trees code (Moore et al., 2001) has been used to minimise the computation time required in the pair counting procedure.

Figure 5: Normalised redshift distributions of MEGAz-LRGs, DEEP2 galaxies and SDSS QSOs in Stripe 82 that are used in the cross-correlations with the LRG samples.

4 LRG N(z) via Cross-Correlations

Even if the redshift of individual galaxies is not available, the 3-D clustering information can yet be recovered if the sample’s redshift distribution, n(z), is known. This can be achieved using Limber’s inversion equation (Limber, 1953) which can project the spatial galaxy correlation function, , to the angular correlation function given the n(z) of the sample:


where f(x) is the galaxy redshift selection function. For our photometric selected LRG samples, only a very small fraction has a measured redshift, thus it is vital to estimate the n(z) of the Stripe 82 LRG samples.

One method for estimating the redshift distribution of the sample could be based on the various popular programs that derive photometric redshifts (photo-z’s). Photo-z estimates are based on the deep multi-band photometry coverage, and work by tracing some specific spectral features across the combination of filters which are then compared with different type of objects SED templates. Indeed, our izK selection is a rough photo-z cut as we follow the movement of the break across the selected bands. In order to use the angular correlation function and the information that is encoded we need the n(z) of our sample, hence we follow the technique of Newman (2008) for reconstructing the LRG redshift distribution from cross-correlations.

4.1 Redshift distribution reconstruction

We employ Newman’s method, which is about determining the underlying redshift distribution of a sample of objects (LRGs in our case) through cross-correlation with a sample of known redshift distribution. By cross-correlating the sample (or samples) with known redshift and the sample under consideration, if both samples lie at the same distance, this will give a strong clustering signal. If the two samples that we are cross-correlating are separated and are at different distances, no cross-correlation signal will result. Thus, through the cross-correlations we can infer our photometrically selected LRG sample z ranges.

Following Newman (2008) the probability distribution function of the redshift of the Stripe 82 LRG samples, , is:


where is the integrated cross correlation function, , of the LRG photometric samples with the samples of known spectroscopic redshift (see §4.2), where is the Gamma function, is the comoving angular distance and is the comoving distance at redshift z. The comoving distance corresponds to the maximum angle at given redshift, which must be large enough to avoid nonlinear biasing effects.

To derive via Eq. 13 we must estimate , since the angular size distance, and the comoving distance are given by the assumed cosmology. Thus we now require only knowledge of the and parameters as function of redshift. Fortunately under the assumption of linear biasing, the cross-correlation of the two samples under consideration is the result of the geometric mean of the autocorrelation functions of the samples, i.e. , hence we can use the information provided by autocorrelation measurements for each sample to break the degeneracy between correlation strength and redshift distribution.

Newman investigates the effect of systematics such as: different cosmologies, bias evolution, errors from the autocorrelation measurements and field-to-field zero points variations in the final redshift probability distribution result. These issues could be more important in the case of future photometric surveys aimed at placing constraints on the equation of dark energy.

4.2 Cross-Correlation data sets

Newman’s angular cross-correlation technique requires the use of a data sample with known spectroscopic, or sufficiently accurate photometric, redshifts. For this reason we use a variety of samples with confirmed spectroscopic and photometric redshifts for the cross-correlations with Stripe 82 LRGs. The data samples that we use are: DEEP2 DR3 galaxies (Davis et al., 2003, 2007) , MegaZ-LRGs (Collister et al., 2007), SDSS DR6 QSOs (Richards et al., 2009) and SDSS DR7 QSOs (Schneider et al., 2010). In Fig. 5 we show the normalised redshift distributions of all the samples and in Table 1 we present the number of objects in each redshift bin.

DEEP2 MegaZ-LRGs DR6 Photometric Sample DR7 Spectroscopic sample
0.4 - 0.6 - 30503 436 456
0.6 - 0.8 3152 - 695 526
0.8 - 1.0 5512 - 1199 547
1.0 - 1.2 3620 - 1630 729
1.2 - 1.4 - - 1312 820
1.4 - 1.6 - - 2646 854
1.6 - 1.8 - - 1193 803
1.8 - 2.0 - - 1990 668
Table 1: Number of objects in each separate redshift-bin used for the cross-correlations with Stripe 82 LRGs
Figure 6: (a): Cross-correlation measurements of the Stripe 82 LRG sample with spectroscopic SDSS QSOs. (b): Same as (a) but now photometric SDSS QSOs are involved in the cross-correlations. Measurement uncertainties are jackknife errors.
Figure 7: Cross-correlation measurements of the and Stripe 82 LRG samples with DEEP2 galaxies in (a) and (b), respectively. Uncertainties are jackknife errors.
Figure 8: Cross-correlation measurements of the (green diamond) and (purple star) Stripe 82 LRGs with MEGAz-LRGs, along with jackknife errors.
Figure 9: (a) The probability distribution function of the redshift, , of the and Stripe 82 LRGs as estimated through cross-correlations with MEGAz-LRGs, DEEP2 galaxies and spectroscopic SDSS QSOs. (b) Same as in (a) but now using photometric SDSS QSOs instead of spectroscopic in the cross-correlations. Error bars shown in both cases are jackknife summed up to .

By using the above data sets for cross-correlation we satisfy the principal requirements of Newman’s method, with the most important being that the sky coverage of the data sets overlap the Stripe 82 LRGs. It must be mentioned though that not all the redshift surveys have the same sky coverage as Stripe 82 LRGs, so we reconstruct two redshift distributions via the cross-correlations providing us with the opportunity to check how much the n(z) cross-correlation technique is affected by area selection. One is reconstructed by using all the data sets, the other by using only SDSS QSOs in the cross-correlations.

4.2.1 Sdss Dr6 DR7 QSOs

QSO surveys are the main samples that we used for our cross-correlation measurements and they span the redshift range . When we refer to QSO data sets, we separate them into spectroscopic and photometric samples.

For the spectroscopic QSO sample we use the fifth edition of the SDSS Quasar Catalog, which is based on the SDSS DR7 (Schneider et al., 2010). The original data set contains 105,783 spectroscopically confirmed QSOs, from which only 5,403 in Stripe 82 have been used at for cross-correlations (Table 1) with ( of QSOs at ).

The photometric QSO sample comes from the photometric imaging data of the SDSS DR6 (Richards et al., 2009). The parent catalogue contains QSOs candidates from which we use 11,101 with in Stripe 82 and in the same redshift range as the spectroscopic QSOs.

In Fig. 6 we plot the cross-correlations between the Stripe 82 LRGs and the SDSS QSOs. We show only the case for cross-correlations of the Stripe 82 LRG sample with the spectroscopic and photometric SDSS QSOs. Cross-correlation with the LRG sample does not differ much. Errors shown here and for the other cross-correlation cases are jackknife errors.

4.2.2 DEEP2 Sample

The next sample of galaxies that we use is DEEP2 DR3 galaxies (Davis et al., 2003, 2007). The survey coverage in Stripe 82 is with . Galaxies in DEEP2 are split in three redshift bins with 0.2 step in the redshift range . The redshift distribution of the DEEP2 DR3 sample is shown in Fig. 5, with 12,284 galaxies in total. In Fig. 7 we show the results of the cross-correlations of the and LRG samples with the DEEP2 galaxies in the three aforementioned redshift bins.

4.2.3 MegaZ-LRG sample

The last sample that we use are LRGs from the MegaZ-LRG photometric catalogue (Collister et al., 2007). MegaZ-LRGs are used only in the redshift range of with . This sample offers us the ability to check the clustering properties of our high-redshift LRG candidates with another sample of LRGs. The total number of MEGAz LRGs that we use for cross-correlations is 30,503. In Fig. 8 are shown the cross-correlations between the Stripe 82 LRGs and the MEGAz LRGs.

4.3 Cross-Correlation results for n(z)

Having estimated the clustering signal from the cross-correlations of the above samples, we proceed to the reconstruction of the redshift distribution of the photometrically selected Stripe 82 LRG candidates. To estimate the probability distribution function of the redshift, , for the high-z LRG candidates we use equation (13). The pair-weighted clustering signal of the cross-correlations has been integrated up to for each redshift bin.

In Fig. 9 we can see the two cases of the estimated probability distribution function of the redshift for the high-z LRG candidates. For the first case, has been estimated by using the spectroscopic SDSS QSOs whereas in the other case, is estimated using only the photometric SDSS QSOs (DEEP2 galaxies and MEGAz-LRGs are also always used). For both cases we plot the errors estimated for each point in the redshift bin from the contributed cross-correlated sample.

To estimate the redshift distribution, n(z), we use the weighted mean for the in each redshift bin, calculated through :


where k is the total number of bins at that redshift, is the measured probability distribution function of each cross-correlation data set in the ith bin and the error on that measurement.

The spectroscopic QSO in Fig. 9a compared to the photo-z case in Fig. 9b, gives increased probability at . This may be explained by the SDSS QSO spectroscopic redshifts being more precise. For this reason, in our analysis and in fitting models to our results, we will use only the spectroscopic n(z) for higher accuracy.

In Fig. 10 we plot the normalized redshift distribution of the and LRGs samples as calculated from Eq. 1314. When we selected the two LRG samples from the colour-plane, we applied a redder selection for the sample (see Eq. 1), aiming for a sample with a slightly higher redshift peak in the distribution as predicted from the evolutionary tracks in Fig. 1. This small difference may be seen between the spectroscopic n(z) of the and samples where the bluer cut has an average of where for the redder sample the average is . But since the LRG sample has higher statistical accuracy in the n(z) determination, the majority of our analysis will be focused in this sample.

Figure 10: Weighted normalised redshift distribution of the Stripe 82 LRGs candidate samples when we use the spectroscopic SDSS QSOs along with the DEEP2 and MEGAz-LRG data sets. As expected the sample (solid green line) n(z) peak is lower when compared with the sample (dashed blue line).

5 Results

5.1 Measured and comparisons

In Fig. 11 we compare the observed angular correlation function of the LRG in Stripe 82 with Sawangwit et al. (2011) results. The measurements are presented with 1 Jackknife errors.

Figure 11: The angular correlation function, , from the Stripe 82 LRGs (star), AA LRGs (square), 2SLAQ LRGs (triangle) and SDSS LRGs (diamond). At small scales all of the measurements show similar clustering behaviour, but at large scales the Stripe 82 clustering slope appears to be flatter than the lower samples.

The work of Sawangwit et al. involved three LRG data sets at :

  1. SDSS LRGs at

  2. 2SLAQ LRGs

  3. AA LRGs

From Fig. 11 we can see that at small scales, , the clustering trend for all the samples is similar but with decreasing amplitude for increasing redshift. At larger scales, we note that the of the Stripe 82 LRGs seems to have a flatter slope than the other samples, departing from the expected behaviour for the correlation function.

Further comparisons below with the LRG clustering results of Sawangwit et al. will focus on the slope and amplitude of the results, with an initial view to interpret any changes in terms of evolution. It is therefore of interest to see how the Stripe 82 sample match to the LRG samples used in previous studies in terms of luminosity and comoving space density.

A pair-weighted galaxy number density is given by (see e.g. Ross & Brunner, 2009) :


where is the observed area of the sky, is the comoving distance to redshift and is the speed of light. The observed space density for the 700deg Stripe 82 sample is found to be . The quoted error has been estimated from the difference of the number density as calculated through Eq. 15 and by converting Fig. 10 into a plot of number density as a function of z (by dividing its bin by its corresponding volume).

Within the uncertainties of our , the 700deg sample appears to have similar space density to that of the AA LRG sample (see Table 2 in §5.2). However, in this study we do not yet have redshift information for individual LRGs, not even for a subset of the sample. Hence it is more uncertain if our sample has similar luminosity as the LRG samples used by Sawangwit et al. (2011). We therefore take the fact that the samples are number-density matched to imply that they are also approximately luminosity matched which may turn out to be a reasonable assumption (see e.g. Sawangwit et al. 2011). This then should enable us to compare the clustering slopes and amplitudes of the AA and Stripe 82 and infer any evolution independently of luminosity dependence.

5.2 and power-law fits

Our first aim here is to fit power-laws to the Stripe 82 to provide a simple parameterisation of the results. Our second aim is to make comparisons of the 3-D correlation amplitudes and slopes to measure evolution. Both aims will require application of Limber’s formula to relate the 2-D and 3-D correlation functions.

We begin by noting that the simplest function fitted to correlation functions is a single power-law with amplitude and slope . In previous studies, the spatial correlation function has been frequently described by a power-law of the form:


The angular correlation function as a projection of can be written as , commonly with a slope fixed at . The amplitude of the angular correlation function, , can be related with the correlation length through Limber’s formula (Eq. 12) using the equation (Blake, Collister, & Lahav, 2008):


where is the redshift distribution, is the comoving radial coordinate at redshift z and the numerical factor .

Sample Single power-law Double power-law
AA LRGs 0.68 42.8 1.3 3.4
Stripe82 LRGs 1.0 5.89 2.38 3.65
Table 2: Best fit parameters for the single and double power-law fits to the angular correlation function.

A deviation from a single power law at has been measured in previous studies (Shanks et al., 1983; Blake, Collister, & Lahav, 2008; Ross et al., 2008; Kim et al., 2011; Sawangwit et al., 2011) and can be explained by the the 1-halo and 2-halo terms imprinted in the clustering signal under the assumption of the halo model (see §5.4). To parameterise the clustering characteristics of our sample, we fit a single-power law and a double-power law to our measured angular correlation function. The double power-law form is given as:


with to be the break point at where the power-law slope changes from being steeper at small scales (), to flatter at large scales.

The power-laws are fitted in the range using the -minimization with the full covariance matrix constructed from the jackknife resampling (see §3.2):


where is the number of angular bins, is the difference between the measured angular correlation function and the model for the th bin, and is the inverse of the covariance matrix.

Figure 12: The best-fit single power law (diamond) and double power law (triangle), for the LRGs candidates overplotted on the angular correlation function (square) with the Jackknife error. Lower panel shows the fitting residuals.
Figure 13: a) The AA LRG raw measurements (red square) with predictions from comoving evolution model (dashed red line), using the best-fit double power-law values with Limber’s formula as Sawangwit et al. (2011) calculated. We then evolve the AA best-fits utilising the estimated Stripe 82 LRG n(z) under the assumption of comoving evolution (dashed-dot black line) clustering. The observed Stripe 82 LRG ) is shown as well (blue star). b) Same raw measurements as above, but now compared to the virialised evolution clustering model. Stripe 82 LRG measurements are described more accurate with comoving evolution at small and large scales compared to virialised evolution as it can been seen from the lower panel, where are plotted the residuals of the observed Stripe 82 versus the comoving evolution (black star) and virialised evolution (green star) models, respectively.

For the single power-law, our best-fit spatial clustering length and clustering slope pair from Limber’s formula are measured to be and with associated reduced . The pairs for the double power-law are and at small scales and and at large scales with a reduced . From the intersection of the 2 power law for , we have calculated the break scale, . This is higher than the estimated from the SDSS, 2SLAQ and LRG surveys (Sawangwit et al., 2011).

In Fig. 12 we show the data points including the Jackknife errors with the best-fitting power laws where the largest scale considered in the fitting was , which corresponds to at for the LRG sample. Fig. 12 confirms that the double power-law clearly gives a better fit to the data than the single power-law. Note that in the case of the single power-law and the double power-law at small scales, our results give values consistent with outcomes from previous studies. However, at large scales the Stripe 82 slope () is significantly flatter than the AA result ().

Fig. 13 shows the double power-law fits for AA (dashed red lines) taken from Sawangwit et al. and then evolved (black and green dot-dashed lines) to the Stripe 82 depth using Eq. 17 under the assumptions of comoving and virialised clustering, respectively. We shall interpret the amplitude scaling in the discussion of evolution in §6.1 later. At this point we again note that the biggest discrepancy seems to be at large scales where the Stripe 82 slope is increasingly too flat relative to the AA result. Fitted parameters are given in Table 2, where the best-fit power-law parameters for the LRG sample (Sawangwit et al., 2011) are also presented for comparison.

We note here that Kim et al. (2011) studied the clustering of extreme red objects (EROs) at in the SA22 field and they report a similar change of the large scale slope. Gonzalez-Perez et al. (2011) tried to fit clustering predictions from semi-analytic simulations to the Kim et al. ERO but found that the model underpredicts the clustering at large scales.

5.3 CDM model fitting in the linear regime

Since the standard CDM model was found to give a good fit to the lower redshift LRG samples of Sawangwit et al. (2011), we now check to see whether the flatter large-scale slope of the Stripe 82 LRG leads to a statistically significant discrepancy with the CDM model at . We generate matter power spectra using the ‘CAMB’ software (Lewis, Challinor, & Lasenby, 2000), including the case of non-linear growth of structure correction. For this reason we use the ‘HALOFIT’ routine (Smith et al., 2003) in ‘CAMB’. Our models assume a CDM Universe with , , , , and . Then we transform the matter power spectra to obtain the matter correlation function, , using:


The relationship between the galaxy clustering and the underlying dark-matter clustering is given by the bias, :

Figure 14: The best-fit spatially flat, CDM model assuming compared to the observed of Stripe 82 LRGs in the linear regime. The standard model cannot explain the large scale power excess in the angular correlation function of the Stripe 82 LRGs. The shaded area corresponds to jackknife error. Also shown is a spatially flat CDM model with the same parameters as before except for a lower value of and an arbitrary normalisation. The model appears to give a better fit than the standard model.

As we are interested in the linear regime, we fit the projected to the Stripe 82 LRG in the range , corresponding to comoving separations Mpc. By fitting the model predictions to the measured it will result with the best linear bias factor, the only free parameter in this case. For our fitting, the -minimization with the full covariance matrix constructed from the jackknife resampling (see §3.2) has been used.

The best-fit linear bias parameter is estimated to be with . The upper limit of our fitted range in was varied, while the lower limit stayed constant to avoid any contribution from the non-linear regime. Thus, for the range the best-fit bias is with and at is with . In Fig. 14 we plot the LRG with the error and the CDM model with the best-fit bias. For low values of the upper limit of the fitting range, the measured biases are in approximate agreement with other results in the literature. But in terms of the flat slope of at large scales, the standard CDM linear model is inconsistent with the data at the level. One of the aims of the next section will be to see if a HOD model can explain the flat large-scale slope of the Stripe 82 LRGs.

5.4 Halo model analysis

We are going to use the approach of the halo model (see Cooray & Sheth, 2002, for a review) of galaxy clustering to finally fit our angular correlation function results. Under the halo-model framework we can examine the way the dark matter haloes are populated by galaxies through the Halo Occupation Distribution (HOD). Various studies have used this model to fit their results (e.g. Masjedi et al., 2006; White et al., 2007; Blake, Collister, & Lahav, 2008; Wake et al., 2008; Brown et al., 2008; Ross & Brunner, 2009; Zheng et al., 2009; Sawangwit et al., 2011; Gonzalez-Perez et al., 2011) as a way to explain the galaxy correlation function and gain insight into their evolution. Specifically, we shall investigate whether the HOD model may be able to explain the flatter slope of the correlation function observed here.

In the halo model, the clustering of galaxies is expressed by the contribution of number of pairs of galaxies within the same dark matter halo (one-halo term, ) and to pairs of galaxies in two separate haloes (two-halo term) :


The 1-halo term dominates on small scales .

The fundamental ingridient in the HOD formalism of galaxy bias is the probability distribution , for the number of galaxies N to hosted by a dark matter halo as a function of its mass M.

We use the so-called centre-satellite three-parameter HOD model (e.g. Seo, Eisenstein, & Zehavi, 2008; Wake et al., 2008; Sawangwit et al., 2011) which distinguishes between the central galaxy and the satellites in a halo. This separation has been shown in simulatations (Kravtsov et al., 2004) and has been commonly used in semi-analytic galaxy formation models in the last years (Baugh, 2006).

Different HODs are applied for the central and satellite galaxies. We assume that only haloes which host a central galaxy are able to host satellite galaxies. The fraction of haloes of mass M with centrals is modelled as:


In such haloes, the number of satellite galaxies follows a Poisson distribution (Kravtsov et al., 2004) with mean:


To describe the distribution of the satellite galaxies around the halo centre we use the NFW profile (Navarro, Frenk, & White, 1997). So, the mean number of galaxies residing in a halo of mass is:


and the predicted galaxy number density from the HOD is then:


where is the halo mass function, where in our case we use the model of Sheth & Lemson (1999).

From the HOD we can derive useful quantities which are the central fraction :


and the satellite fraction of the galaxy population:


as . We can also determine the effective mass, , of the HOD:


and the effective large-scale bias:


where is the halo bias, for which we use the ellipsoidal collapse model of *Sheth01 and the improved parameters of Tinker et al. (2005).

() () () (per cent)
AA 0.68 13.6
Stripe82 1.0 2.4
Stripe82 1.0 2.3
Stripe82 1.0 3.1
Stripe82 ) 1.0 3.6
Table 3: Best-fit HOD parameters.
Figure 15: (a) The mean number of LRGs per halo as a function of halo mass at . The total, central and satellite contributions are shown by the solid, dashed and dotted lines, respectively. (b) The measured angular correlation function for the LRG sample with the best HOD fit (black star). The 1, 2 and Jackknife errors are shown in red, blue and green, respectively.

As the galaxy correlation function is the Fourier transform of the power spectrum, the 1-halo term and the 2-halo term of the clustering functions can be written as:


Moreover the 1-halo term can be distinguished from the contribution of the central-satellite pairs, , and satellite-satellite pairs, , (see e.g. Skibba & Sheth, 2009):




where is the NFW density profile in Fourier space and we have simplified the number of satellite-satellite pairs to since the satellites are Poisson-distributed.

The 2-halo term is evaluated as:


where is a non-linear matter power spectrum. We derive the mass limit, , using the ‘-matched’ approximation of (Tinker et al., 2005), which accounts the effect of halo exclusion: different haloes cannot overlap. is the restricted galaxy number density (Eq. B13 of Tinker et al. (2005)).

For the scale-dependent halo bias, , we use the model given by Tinker et al. (2005):


where is the non-linear matter correlation function. For the 2-halo term, we need to correct the galaxy pairs from the restricted galaxy density to the entire galaxy population.

By using Limber’s formula to project the predicted spatial galaxy correlation function to the angular correlation function and we fit for a variety of the three-parameter halo model (, , ).

The best-fit model for each of our sample is then determined from the minimum value of the -statistic using the full covariance matrix. We use the full covariance matrix over the range in our fitting. Smaller scales are excluded in the fitting because any uncertainty in the model can have a strong effect on due to the projection. To determine the error on the fits, the region of parameter space from the best fits with ( for 1 degree of freedom) is considered. For , , and which depend on all the three main parameters, the considered region of the parameter space becomes .

Fig. 15a shows the resulting best-fit HOD of the mean number of LRGs per halo along with the central and satellite contributions. The best-fitting values for , and where , and , respectively. The associated values for , , and are given in Table 3.

We see that the of the LRGs flatten at unity, as expected from the assumption satellite galaxies are hosted by halos with central galaxies. The LRGs as expected populate massive dark matter haloes with the masses . With the fraction of LRGs that are satellites being less than , we therefore find that % of LRGs are central galaxies in their dark matter haloes. The best fit linear bias, , agrees with the prediction from Sawangwit et al. (2011) in the case of a long lived model for the LRGs and indicates that the LRGs are highly biased tracers of the clustering pattern. The effective mass, , confirms that LRGs are hosted by the most massive dark matter haloes. Despite the fact that we use a higher redshift LRG sample, our best-fit HOD parameters are statistically not too dissimilar to those found in previous LRG studies (eg see Table 3).

In Fig. 15b we show the best-fit model for , compared to the data. The first thing we notice is that while at small scales the best-fit HOD are in good agreement with the measurements, at large scales the model fits only at . The flatter slope at large scales is responsible for that and we still are not able to say if this can be explained by evolution in the linear regime or any kind of systematic effect. In §7 we will check systematic errors that could affect our results.

Moreover, due to the high value of the best-fit reduced , we also try to fit the HOD models at different scales by using 4 different maximum bins of the covariance matrix in our fits, which we present in Table 3. The fits at large scales did not improve and above there was not any change in the best-fit HOD measurements.

Considering the two-halo term in the HOD model, one can see that the bias in this regime is mostly scale-independent and the correction factor is in fact having the opposite effect on the slope. The scale-independent bias is simply the average of the halo bias, , weighted by the halo mass function and the mean number of galaxies hosted by the corresponding halo. One way to boost the large-scale amplitude is to increase and therefore increase the mass range of the halo where most galaxies occupy and hence linear bias and amplitude of the two-halo term. However, to compensate for the increase numbers of satellite galaxies (and consequently small-scale clustering amplitude) one must also increase , the mass at which a halo hosts one satellite galaxy on average. And in order to produce the overall flatter slope one needs to increase . However, this would still overpredict the clustering amplitude in the intermediate scales, . Note that our best-fit HOD gives , consistent with previous results for lower redshift LRGs of (Sawangwit et al., 2011) and (Wake et al., 2008). However, as noted earlier including bins at larger and larger scales does not change the best-fit parameters which means that also remains unchanged due to the reason discussed above. We therefore conclude that the HOD prescription in the framework of standard CDM cannot explain the observed large-scale slope in of the LRG sample.

6 Clustering Evolution

6.1 Intermediate scales

First, we compare the clustering of the Stripe 82 LRG sample to the lower redshift LRG sample. We recall that these LRG samples have approximately the same space density and so should be approximately comparable. We follow Sawangwit et al. (2011) and by using our best-fit and we make comparison with their data and models via the integrated correlation function in a sphere, .

LRG results are described better with the long-lived model of Fry (1996). Fry’s model assumes no merging in the clustering evolution of the galaxies while they move within the gravitational potential, hence the comoving number density is kept constant. The bias evolution in such a model is given by:


where D(z) is the linear growth factor.

However, the flat slope beyond 1hMpc causes a highly significant, %, rise in above the as we can see in Fig. 16 (see also Figs. 13a,b). If we assume that the 2 samples are matched then we would conclude that all of the models discussed by Sawangwit et al. (2011) were rejected.

Figure 16: The LRG measurements as a function of redshift and luminosity from Sawangwit et al. (2011). Lowest redshift data are early-type galaxies from 2dFGRS (Norberg et al. 2002). Stars represent the brighter samples (SDSS, 2SLAQ* and -LRG), where the lower luminosity samples, triangles, have been lowered by 0.2 for clarity. The 700 deg Stripe 82 LRGs measurement is at (square).

One possibility is that the 700deg LRG sample is closer to the SDSS and LRG space density of hMpc because the LRG fits the extrapolated models better there. If so, then this would imply that the Stripe 82 LRG width was underestimated in the cross-correlation procedure and this would then increase the deprojected amplitude of , suggesting that this explanation may not work. Similarly a larger correction for stellar contamination would also produce a higher Stripe 82 clustering amplitude. We do not believe that looking further into the evolution of the bias (Papageorgiou et al., 2012) and DMH is warranted until we understand the flat slope of the Stripe 82 at large scales.

6.2 Small scales

At smaller scales () the situation is less complicated by the flat large-scale slope. Here Sawangwit et al. found that a virialised model gave a better fit to the slightly faster evolution needed to fit the small-scale correlation function amplitudes than a comoving model. But in the present case, the scaling between the AA and Stripe 82 LRGs in Fig. 13a,b, shows that here the comoving model is preferred at small scales over the faster virialised evolution. This fits with the more general picture of the Stripe 82 LRGs presenting a higher amplitude than expected all the way down to the smallest scales. Unfortunately the remaining uncertainty in the Stripe 82 LRG luminosity class is still too large to make definitive conclusions on this evolution possible.

6.2.1 HOD Evolution

Given the uncertainty in caused by the flat slope on intermediate - large scales, we will extend further the studies at small-scales, using the HOD model to interpret the small-scale clustering signal of the LRGs. Based on the HOD fit at , we again follow Sawangwit et al. (2011), (and references therein) and test long-lived and merging models by comparing the predictions of these models to the SDSS HOD fit from Sawangwit et al.. These authors and also Wake et al. (2008) found that long-lived models were more strongly rejected at small scales ( hMpc) than at intermediate-large scales.

Again we follow the approach of Wake et al. (2008)) and Sawangwit et al. (2011) who assumed a form for the conditional halo mass function Sheth & Tormen (2002) and a sub-Poisson distribution for the number of central galaxies in low-redshift haloes of mass such that


where ,


and is the expression of Sheth & Tormen (2002) for the conditional halo mass which generalize those of Lacey & Cole (1993). The mean number of satellite galaxies in the low-redshift haloes is then given by




and the main parameter is which is the fraction of un-merged low- satellite galaxies which were high- central galaxies.

This model is called ‘central-central mergers’ in Wake et al. (2008). More massive high-z central galaxies are more likely to merge with one another or the new central galaxy rather than satellite-satellite mergers.

Figure 17: The predicted SDSS LRG ’s at for the case of passively () evolving the best-fit HOD of Stripe 82 LRGs sample from and the case where central galaxies merging is allowed from (), in green dot-dashed line and blue long-dashed line, respectively. The bottom panel shows the ratios of the evolved ’s to the SDSS best-fit, the shaded regions signify the uncertainties.

Setting means that there is no merging of initial central galaxies in subsequently merged haloes, so it is similar to the passive/long-lived model. equals to 0 means that all the central galaxies in haloes at high redshift merge to form new central and/or satellite galaxies in the low redshift haloes. In the analysis below, we use the best-fit HOD model values as estimated for scales up to (see Table 3).

The case is shown as the passive model in Fig. 17 and is clearly rejected by the data at (see lower panel). Best-fit HOD predictions of the satellite fraction in the case of the passively evolved LRGs from to is whereas Sawangwit et al. measured % for a brighter selection of LRGs at . We see that both these results, for the long-lived model, are significantly higher compared to the best-fit SDSS HOD, %. The difference in the number of the satellite galaxies is explained as the predicted clustering amplitude at small scales (1-halo term) for the passive model, is higher compared to the SDSS HOD fit as it is clearly shown in Fig. 17. Higher clustering signal at small scales indicates the presence of too many satellite galaxies in the low-redshift haloes.

The merger model is described by as presented in Fig. 17 and clearly fits the data well. For this model the satellite fraction at estimated to be % and is in a good agreement with Sawangwit et al. Moreover, the best-fit HOD model values for the evolved LRGs to for bias and galaxy number density are and , respectively. Compared to the SDSS best-fit model, with and , the number of galaxies at have been decreased by almost due to central-central merging. The evolved linear bias and galaxy number density are consistent with the best-fit HOD of Sawangwit et al. at level.

Note that the agreement at large scales in Fig. 17 is somewhat artificial given the underestimation of by the HOD model in Fig. 15b which remains unexplained in the HOD formalism. But at these smaller scales the result that the merging model fits better than the long-lived or indeed the virialised clustering model of Fig. 13b may be more robust, given the reasonable fit of the HOD model at small scales () in Fig. 15b.

7 Tests For Systematic Errors

In this section we will present an extended series of checks for systematic errors that might have affected our clustering analysis, with the major issue being the flatter slope at large scales as estimated in §5.2, §5.3 and §5.4. Tests for possible systematics that will be discussed here are:

  • data gradient artefacts,

  • estimators bias,

  • survey completeness,

  • observational parameters ; such as star density, galactic extinction, seeing etc.

7.1 Data gradients and estimator bias

A false clustering signal at large scales can arise from artificial gradients in the data, as the correlation function is very sensitive to such factors. In attempting to explain the behaviour of the observed at large scales, first we divide the LRG sample area in 6 equal subfields in RA. Then the angular correlation function of each subfield has been calculated using the Landy Szalay, Hamilton and the Peebles estimator - the standard estimator. Furthermore, we average the results of the 6 subfields as measured by each estimator and we compare them with LRG full sample results (see Fig. 18).

From these comparisons, it is clear that when we use the Landy Szalay and Hamilton estimators, we do not find any significant difference in the amplitude of the measured between the averaged subfields’ or between the full samples’ measurements. When the averaged measurements are compared with those from the full sample, only a very slightly smaller clustering signal in the averaged ’s is seen, barely visible in Fig. 18. Furthermore, this is only the amount expected from the integral constraint (see §3.2) on , if the above Landy Szalay estimate is assumed to apply in a single sub-field area. The standard estimator is known to be subject to larger statistical errors at large scales and here the signal is actually stronger when compared with the other two estimators.

Moreover, in Fig. 19 we display the results of the measurements from the 6 subfields individually against the full sample measurements as estimated with the Landy Szalay estimator in all cases. Even now we cannot see any major trend through the subfields’ correlation function measurements, except possibly for the subfield which has a steeper slope at larger scales.

Figure 18: ’s from Landy Szalay, Hamilton and standard estimator of the LRG sample. For comparison, the averaged ’s from the 6 subfields (see text for more detail), are overplotted as measured from each estimator. Landy Szalay and Hamilton estimators, give similar results for the average subfields and the full sample measurements, respectively. The standard estimator is more biased, at larger scales.
Figure 19: results of the 6 equal size subfields ( each) across Stripe 82, the total area as estimated by using the Landy Szalay estimator and the averaged clustering signal from the 6 subfields. In the bottom panel are displayed the ratios of the of each subfield compared to the total area.
17.0-17.2 4894
17.2-17.4 11096
17.4-17.6 22490
17.6-17.8 38659
17.8-18.0 53680
Table 4: K-limited sub-samples used for auto-correlations in Fig. 20.
Figure 20: Auto-correlation functions from Landy-Szalay estimator for the LRG K-limited sub-samples from Table 4. Total sample is overplotted for comparison.

7.2 Magnitude incompleteness

Another issue that we want to address is how the clustering signal can be affected by magnitude incompleteness. The colour-selection used for the LRGs, applied up to the faintest limits of the SDSS-UKIDSS LAS surveys (see §2.1). To account for this, first we divide the LRG sample in 5 magnitude bins in the range . The number of LRGs in each magnitude bin is shown in Table 4.