On the clustering of compact galaxy pairs in dark matter haloes

Wang, Y.; Brunner, R. J.

doi:10.1093/mnras/stu1562

Abstract

We analyse the clustering of photometrically selected galaxy pairs by using the halo-occupation distribution (HOD) model. We measure the angular two-point auto-correlation function, ω(θ), for galaxies and galaxy pairs in three volume-limited samples and develop an HOD to model their clustering. Our results are successfully fit by these HOD models, and we see the separation of ‘one-halo’ and ‘two-halo’ clustering terms for both single galaxies and galaxy pairs. Our clustering measurements and HOD model fits for the single galaxy samples are consistent with previous results. We find that the galaxy pairs generally have larger clustering amplitudes than single galaxies, and the quantities computed during the HOD fitting, e.g. effective halo mass, M_eff, and linear bias, b_g, are also larger for galaxy pairs. We find that the central fractions for galaxy pairs are significantly higher than single galaxies, which confirms that galaxy pairs are formed at the centre of more massive dark matter haloes. We also model the clustering dependence of the galaxy pair correlation function on redshift, galaxy type, and luminosity. We find early–early pairs (bright galaxy pairs) cluster more strongly than late–late pairs (dim galaxy pairs), and that the clustering does not depend on the luminosity contrast between the two galaxies in the compact group.

galaxies: formation, galaxies: interactions, cosmology: observations

INTRODUCTION

Studying the clustering of galaxies provides insights into both galaxy formation and evolution, especially when analysed as a function of the intrinsic properties of a galaxy such as mass, size, morphology, gas content, star-formation rate, nuclear activities, characteristic velocity and metallicity (Kauffmann et al. 2004; Yang et al. 2012; Watson et al. 2014). Traditional clustering analyses can connect the visible galaxies to the unseen dark matter haloes, thereby providing insight into the assembly history of galaxies. The standard picture in an ΛCDM universe is that galaxies are formed from gas that cools and condenses at the centres of dark matter haloes (White & Rees 1978). As smaller haloes fall into more massive ones, their central galaxies become satellites of the new, larger hosts, sometimes merging to form new central galaxies in the core of the resultant dark matter halo. In this model, each halo will contain a central, dominant galaxy at the bottom of the gravitational potential well, surrounded by other, smaller satellite galaxies that orbit around the dominant galaxy.

While successful in predicting the clustering behaviour of single galaxies (Zehavi et al. 2004), galaxy clustering studies typically have not analysed groups of galaxies, yet we know that many galaxies live in dense environments. While Berlind et al. (2006) did study the multiplicity functions of groups that are selected from Sloan Digital Sky Survey (SDSS), in this paper, however, we extend our previous SDSS clustering results (Wang, Brunner & Dolence 2013) to the study of the clustering of galaxy groups. As a result, we first construct a catalogue of photometrically selected compact galaxy groups, and, secondly, use this new catalogue to measure and analyse the clustering behaviour of small galaxy groups in the same framework traditionally applied to isolated galaxies. Compact groups contain a small number of galaxies with compact angular configurations (Hickson 1982) where galaxy–galaxy interactions are the dominant processes. On the other hand, rich groups or galaxy clusters evolve under both galaxy–galaxy interactions (Toomre & Toomre 1972; Farouki & Shapiro 1980) and galaxy–environment interactions (Gunn & Gott 1972; Nulsen 1982).

As a result, the abundance of compact groups, their spatial distribution, and their intrinsic properties are entwined with their halo merger history, which is closely related to the underlying cosmology. For example, the evolution of merger rates is sensitive to the cosmic matter density, while the mass distribution of merging objects depends on the linear power spectrum of the initial density of fluctuations (Lacey & Cole 1993). Thus, the properties of compact groups can be used to constrain cosmological parameters. On the other hand, the physical processes driving galaxy evolution have a strong effect on satellite galaxies. For example, galaxy colours are affected by the stripping of gas within the galaxies during interaction events, which can suppress star formation; while the galaxy structures and morphologies can be modified by the gravitational and hydrodynamical interactions between galaxies.

Modelling galaxy clustering is complicated by the fact that the gravitational framework is supplied by the invisible dark matter, thus we must have a means to connect the visible tracers of the gravitational field to the invisible dark matter structure. Currently, there are two methods that have been widely used to provide this connection. First is the use of a semi-analytic model (SAM) that employs a simplified representation of the relevant astrophysics to follow galaxy growth within the evolving dark matter halo population. The SAM can be used to predict the detailed properties of both central and satellite galaxies (White & Frenk 1991; Springel et al. 2001; Kang et al. 2005) for comparison with an observed galaxy population. The second approach is the semi-empirical technique that provides an insightful description of the relations between the galaxies and their host dark matter haloes via the observed galaxy properties. There are several different semi-empirical techniques in use, including abundance matching (Conroy, Wechsler & Kravtsov 2006; Hearin et al. 2013) and age matching (Hearin & Watson 2013); in this paper, however, we use the halo occupation distribution (hereafter HOD; Jing, Mo & Boerner 1998; Peacock & Smith 2000; Zehavi et al. 2004; Zheng et al. 2005). An HOD model quantifies the central and satellite galaxy populations of dark matter haloes as a function of the host halo mass by optimizing the model fit to the measured galaxy correlation function from large surveys.

Since galaxies live in a variety of different environments, the measurement of their clustering properties as a function of scale has displayed a variance since they were initially measured (Peebles 1980). Initially, this difference in clustering was described by fitting a simple power-law function to the two-point galaxy correlation functions on small scales (Peebles 1980). However, more recent large area surveys have measured the clustering pattern accurately enough to detect deviations from the traditional power-law model (e.g. Zehavi et al. 2004; Watson, Berlind & Zentner 2011). This deviation provides an important insight into the growth of large-scale structure; and, the deviations have been more accurately interpreted in terms of dark matter haloes, which link the small-scale clustering to the underlying dark matter haloes. A well-defined HOD, along with accurate correlation function measurements, can be used to statistically describe how galaxies populate dark matter haloes as a function of halo mass.

With the enormous data being provided by large surveys, such as the recently completed SDSS (York et al. 2000), this type of analysis can also now be extended to the study of galaxy groups. To date, there have been two primary approaches used to extract galaxy groups from the SDSS. First, Yang et al. (2007) selected galaxy groups from the SDSS spectroscopic data, which has a limiting apparent magnitude of r ∼ 17.7, finding significant disagreement with a mock SAM catalogue previously created by Croton et al. (2006). A second approach was developed by Wang & White (2012); see also Tal et al. (2012b) and Tal, Wake & van Dokkum (2012a), who combined the spectroscopic and photometric SDSS data to identify a spectroscopic central galaxy with photometric satellites, which allowed them to study the luminosity and mass functions of the satellite galaxies.

In this paper, we extend the previous galaxy clustering analyses to galaxy groups. We start with the clean galaxy catalogue we generated (Wang et al. 2013) from the photometric data released in the seventh data release (DR7) of the SDSS (Abazajian et al. 2009). We first select compact galaxy groups, following Hickson (1982) to obtain a flux-limited compact galaxy group catalogue. Since this is a photometric only data set, we do not have spectroscopic redshifts to remove contamination from line-of-sight galaxy interlopers. While we would prefer to remove these galaxies, the standard techniques of bootstrap background correction (Lorrimer et al. 1994) or quantifying the galaxy cluster probabilistic membership via photometric redshifts (Brunner & Lubin 2000) do not provide a reliable method to identify and remove interloping galaxies at the low redshifts of the galaxies in our photometric sample as shown in Fig. 1. However, the correlation functions we will measure for the galaxy groups are much stronger than the correlation functions we measure for single galaxies. In addition, the group correlation functions still show deviations from a power-law model, thus we will still be able to connect the distribution of compact galaxy groups with their parent dark matter haloes by developing a compact group HOD.

Figure 1.

The normalized galaxy redshift distribution for the SDSS DR7 galaxies in the Wang et al. (2013) clean galaxy catalogue restricted to the range 18 < r < 21. We assume a Gaussian photometric redshift PDF for each galaxy with the Gaussian μ and σ given by the photometric redshift and its associated error. The individual photometric redshift PDFs are stacked to construct this smooth figure. The mode of the redshift distribution lies at z = 0.176 and the median redshift is < z > =0.24 with a dispersion of 0.16.

Open in new tab Download slide

We first describe the clean galaxy sample and both the luminosity-limited and volume-limited sub-samples in Section 2. Next, in Section 3 we introduce the selection criteria for the angular-compact galaxy groups and discuss the basic properties of our group catalogues. In Section 4, we measure the correlation functions of the galaxy groups and quantify their dependence on group richness and galaxy type. We present our new HOD model in Section 5, where we first apply a basic HOD model as shown in Ross & Brunner (2009) to the isolated galaxies selected from the SDSS DR7, and, secondly, we apply our new HOD model to our galaxy pair catalogue, and to our galaxy pair catalogue split by galaxy type. Finally, we present the modelling result in Section 6, and conclude with a discussion of our results in Section 7.

DATA

The first phase of the SDSS was a photometric and spectroscopic survey designed to map one-fourth of the entire sky to produce a large data set to analyse large-scale structure and to study the underlying cosmic evolution. The survey was conducted by the Astrophysical Research Consortium at the Apache Point Observatory in New Mexico, and provided photometric observations in five bands: u, g, r, i, and z (Fukugita et al. 1996). The last data release from the first phase was DR7, which was released in 2008 November, and included approximately 10⁸ photometric galaxies to a 5σ detection limit of r ∼ 23.1 covering approximately 10⁴ deg² of the sky (Abazajian et al. 2009).

As a starting point, we use our previously constructed clean galaxy sample from the SDSS DR7 (Wang et al. 2013). To construct this catalogue, we analysed a number of different SDSS flags to optimally minimize the impact of galaxy image deblending and we tested the best magnitude range to optimally select a complete galaxy catalogue by comparing source detections from the single-image and stacked image Stripe 82 data. We also identified how to optimally mitigate the systematic effects of atmosphere seeing variations and Galactic extinction. A complete, detailed description of how the clean galaxy catalogue was generated is outlined in appendix A of Wang et al. (2013).

In this work, we follow our previous guidelines and impose a seeing cut of <1.3 arcsec and a reddening cut of <0.15 mag. In the end, the initial catalogue we use contains approximately 21 million galaxies with extinction-corrected r-band model magnitudes in the range 17 < r ≤ 21. The normalized galaxy redshift distribution for the entire galaxy catalogue is presented in Fig. 1. For this figure, we assume each galaxy is represented in redshift space by a Gaussian photometric redshift probability density function (PDF) with μ and σ given by the photometric redshift value and error.

Volume-limited sub-samples

To better understand the evolution of compact galaxy groups and their luminosity dependencies, we generate three volume- and luminosity-limited sub-samples according to the following prescriptions:

I: 0.0 < z ≤ 0.3, M_r < −19.0,
II: 0.0 < z ≤ 0.3, M_r < −19.5, and
III: 0.3 < z ≤ 0.4, M_r < −19.5.

We denote these three sub-samples as ZI, ZII, and ZIII, and they contain approximately 4 million, 2.2 million, and 1.8 million galaxies, respectively. These explicit redshift ranges were selected to result in a ratio of 1: 1.2 for the cosmic volumes of the 0.0 < z ≤ 0.3 and 0.3 < z ≤ 0.4 samples. These samples were k-corrected before construction, and the final galaxy redshift distributions are shown in Fig. 2, where the photometric redshift PDFs were used in the same manner as for Fig. 1.

Figure 2.

The normalized galaxy redshift distribution for the three volume-limited samples: ZI (dashed), ZII (dash–dotted) and ZIII (dotted). The individual distributions are computed in the same manner as Fig. 1.

Open in new tab Download slide

COMPACT GALAXY GROUPS

The primary focus of this paper is to analyse the clustering of compact galaxy groups. In this section, we present the method used to select compact galaxy groups from the SDSS DR7 clean galaxy catalogue, which was discussed in Section 2, and provide a brief analysis of this compact galaxy group catalogue.

Selection criteria

Based on the criteria presented by Hickson (1982), we select isolated, compact groups of galaxies that satisfy the following four tests:

2 < θ_s ≤ 15 arcsec,
N = 2, 3, 4, or ≥5,
θ_n ≥ 3θ_g, and
μ_g < 26.0 mag arcsec⁻².

This process, which is graphically demonstrated in Fig. 3, first selects close galaxy pairs, where θ_s is the separation between the two galaxies in the pair, which should be greater than 2 arcsec for the SDSS DR7 since the SDSS deblending algorithm rarely separates galaxies reliably at separations closer than this value (Infante et al. 2002). N is the number of galaxies within 3 mag of the brightest galaxy, and we create the final compact group catalogue by compressing the selected galaxy pairs that have common members. θ_g is the angular radius of the isolated group, and θ_n is the maximum angular distance from the group centre to the nearest neighbour galaxy. μ_g is the averaged surface brightness of these N galaxies within the group angular radius.

Figure 3.

A visual demonstration of the criteria we use to select isolated groups of galaxies. Left: the green star shapes are the galaxies in our ZI sample. Middle: the selected galaxies that are separated by between two and 15 arcsec are circled in red. Right: the compact, isolated groups are circled in blue with both R_g and R_n indicated by arrows for one group. This sample group is located at 115.331,+31.793.

Open in new tab Download slide

Compact group catalogues

By applying the above selection criteria to the three volume-limited sub-samples, which were described in Section 2.1, we construct catalogues consisting of isolated field galaxies (N = 1), galaxy pairs (N = 2), galaxy triplets (N = 3), and galaxy quads (N = 4). In Table 1 we present the number of compact galaxy groups as a function of group richness. The different catalogues all display a similar decrease with increasing group size. They also show a decrease in the total number of groups of a given size as the total number of galaxies in the input sample decreases (i.e. going from ZI to ZII/ZIII). As the final two sub-samples have similar number of galaxies (within 10 per cent), it is reassuring to see they have nearly identical group populations.

Table 1.

Open in new tab

The number of compact galaxy groups in the SDSS DR7 for the three volume-limited samples presented in this paper.

Catalogue	N = 1	N = 2	N = 3	N = 4
ZI	3665 035	134 155	8137	669
ZII	2045 095	52 792	2568	198
ZIII	1699 581	42 943	2506	222

Table 1.

Open in new tab

The number of compact galaxy groups in the SDSS DR7 for the three volume-limited samples presented in this paper.

Catalogue	N = 1	N = 2	N = 3	N = 4
ZI	3665 035	134 155	8137	669
ZII	2045 095	52 792	2568	198
ZIII	1699 581	42 943	2506	222

Estimating group redshifts

Since we are identifying compact galaxy groups directly from photometric data, we do not have spectroscopic redshifts. While we could use photometric redshifts, their relatively large errors bars, especially at the low redshifts probed by the SDSS, make it difficult to quantify a group redshift. We, therefore, explore four different techniques for estimating the redshift of a compact galaxy group. Several of these techniques treat a photometric redshift in a similar manner as a spectroscopic redshift, while others treat each photometric redshift estimate as a Gaussian PDF, where the Gaussian mean and width are given by the estimated photometric redshift and error. We note that, if available, these approaches would likely be improved by using more accurate photometric redshift PDFs (e.g. Carrasco Kind & Brunner 2013, 2014).

The first technique is to compute the average photometric redshift of all galaxies in the compact group. The second technique sums the individual galaxy photometric redshift PDFs, while the third technique multiplies the photometric redshift PDFs to generate the group redshift estimate. The fourth technique uses the estimated photometric redshifts to identify the brightest, in absolute magnitude as computed by using the photometric redshifts, compact group galaxy. The photometric redshift of this brightest galaxy is used as a proxy for the group redshift.

We compare these four techniques for the galaxy pair sample over the full redshift range of the main volume-limited sample (0.0 < z ≤ 0.4) to the main galaxy redshift distribution in Fig. 4. We find that only one method, averaging the photometric redshifts, performs poorly, when compared to the main galaxy redshift distribution. Two techniques, the summation of the different PDFs and simply selecting the brightest group member's photometric redshift, seem to perform the best. As a result, we choose the simplest technique, and use the photometric redshift of the brightest group member as a proxy for the group's photometric redshift (highlighted in blue). Given our chosen definition for group redshift, it is not surprising that the normalized redshift distributions for the pairs are nearly identical to the galaxy redshift distributions in each volume limited sub-sample, which are shown in Fig. 2.

Figure 4.

A comparison, using only close galaxy pairs, of the four methods we developed to compute the photometric redshift of a compact galaxy group to the redshift distribution of the main galaxy sample. The method we use is highlighted in blue.

Open in new tab Download slide

CLUSTERING PROPERTIES

To understand the clustering of compact galaxy groups, especially in comparison with normal galaxies, we will use the two-point correlation function. In this section, we present how we compute the correlation functions for different samples and briefly discuss the results.

Methodology

The two-point angular correlation function (TPACF) is a simple, yet effective technique for quantifying the clustering of a spatial point process. The TPACF measures the excess probability over random that one object will be located at a certain distance from another object. In our case, the object can be a galaxy, a galaxy pair, a galaxy triple, or a galaxy quad. Since the compact groups have already been preselected to be clustered (i.e. they are already overdense), we expect their clustering signals to be enhanced relative to the full galaxy sample, which corresponds to them being more strongly correlated.

To speed up the bin counting process when estimating the correlation functions, we use our publicly available, fast two-point correlation code¹ that implements a two-dimensional quad-tree structure to vastly reduce the computational time. More details about this code can be found in Dolence & Brunner (2008) and Wang et al. (2013).

For the three samples described in Section 2.1, the correlation function is calculated between 0| $_{.}^{\circ}$|01 and 10° with a logarithmic binning of 25 bins in total angle separation. It is processed by cutting into 32 sub-samples, which can be used to calculate jackknife errors. We find 32 sub-samples are sufficient enough to create a stable covariance matrix. To maintain a sufficient signal-to-noise ratio in the correlation measurements, we restrict the angular ranges for the correlation functions and the covariance matrices to be between 0| $_{.}^{\circ}$|01 and 1| $_{.}^{\circ}$|5, which provide 19 bins in total.

The correlation function estimators

Given the computed pair-counts from our fast correlation function estimator code, we use the (Landy & Szalay 1993) estimator:

\begin{equation} \omega (\theta )=\frac{N_{\rm dd}-2N_{\rm dr}+N_{\rm rr}}{N_{\rm rr}} \end{equation}

(1)

where N_dd is the normalized number of data–data pairs counted within a certain angular separation., θ ± δθ, over all SDSS DR7 fields. N_dr and N_rr stand for the number of data–random pairs and random–random pairs, respectively. The data in the estimator can either stand for a galaxy or a galaxy pair. In all cases, we use approximately 10 times as many randoms as data.

Errors and covariance matrices

We calculate error bars by using the delete one jackknife method, and we use a jackknife defined covariance matrix to fit the measured correlation functions. We determine the covariance matrix for N jackknife samples by using

\begin{equation} C_{jk}(x_i, x_j) = \frac{N - 1}{N} \sum _{k=1}^{N} (x_i^k - \bar{x_i})(x_j^k - \bar{x_j}) \end{equation}

(2)

where x_i is the ith measure of the statistic of N total samples, and the mean expectation of x is given by

\begin{equation} \bar{x_i} = \sum _{k=1}^{N} \frac{x_i^k}{N}. \end{equation}

(3)

To determine the best-fitting model for each measured correlation function, we perform a chi-squared minimization:

\begin{equation} \chi ^2 = \frac{1}{N_{\rm dof}} \sum _{i,j} [\omega (\theta _i) - \omega _{\rm m}(\theta _i)] C_{i,j}^{-1} [\omega (\theta _j) - \omega _{\rm m}(\theta _j)], \end{equation}

(4)

where ω(θ) is the measured two-point galaxy angular correlation function, ω_m(θ) is the model two-point galaxy angular correlation function, and C_{i, j} are the elements of the covariance matrix calculated by using equation (2).

The angular correlation functions of galaxies and isolated galaxy pairs

We have previously (Wang et al. 2013) demonstrated that the angular correlation function for galaxies drawn from the SDSS DR7 sample can be approximated with a power law. In the three panels of Fig. 5, we now present the angular correlation functions measured for the individual galaxies and isolated galaxy pairs that are selected from the ZI, ZII, and ZIII samples (as defined in Section 2.1). As expected from hierarchical clustering models, the galaxy pairs are more strongly clustered than individual galaxies. In addition, we clearly see structure in the data that are traditionally interpreted as the transition from a one-halo to the two-halo term in the correlation function, but in this case for a pair of galaxies.

Figure 5.

From left to right: the TPACF for all individual galaxies and galaxy pairs selected from the samples ZI, ZII, and ZIII. The best-fitting power-law models are shown by the dashed lines.

Open in new tab Download slide

Before proceeding with a more detailed analysis of the halo model interpretation of the galaxy pair angular correlation function, we first perform power-law fits to all correlation functions by minimizing the χ²-statistic that is described in equation (4). The fit parameters, amplitude (A_ω), and slope (γ), are shown in Table 2. In Section 6, we will compare these fit parameters to the results generated by using our new HOD model.

Table 2.

Open in new tab

The power-law fit parameters to the angular correlation function of galaxies and galaxy pairs selected from the ZI, ZII, and ZIII sub-samples.

		log₁₀(A_ω)	γ	\|$\chi _{pl}^2 / \text{dof}$\|
ZI	Galaxies	−1.883±0.019	1.797	15.03
	Pairs	−1.556±0.016	1.968	8.33
dZII	Galaxies	−1.830±0.017	1.835	7.29
	Pairs	−1.453±0.015	2.014	5.42
ZII	Galaxies	−1.857±0.018	1.8827	19.4
	Pairs	−1.536±0.015	2.1146	10.29

		log₁₀(A_ω)	γ	\|$\chi _{pl}^2 / \text{dof}$\|
ZI	Galaxies	−1.883±0.019	1.797	15.03
	Pairs	−1.556±0.016	1.968	8.33
dZII	Galaxies	−1.830±0.017	1.835	7.29
	Pairs	−1.453±0.015	2.014	5.42
ZII	Galaxies	−1.857±0.018	1.8827	19.4
	Pairs	−1.536±0.015	2.1146	10.29

Table 2.

Open in new tab

The power-law fit parameters to the angular correlation function of galaxies and galaxy pairs selected from the ZI, ZII, and ZIII sub-samples.

		log₁₀(A_ω)	γ	\|$\chi _{pl}^2 / \text{dof}$\|
ZI	Galaxies	−1.883±0.019	1.797	15.03
	Pairs	−1.556±0.016	1.968	8.33
dZII	Galaxies	−1.830±0.017	1.835	7.29
	Pairs	−1.453±0.015	2.014	5.42
ZII	Galaxies	−1.857±0.018	1.8827	19.4
	Pairs	−1.536±0.015	2.1146	10.29

		log₁₀(A_ω)	γ	\|$\chi _{pl}^2 / \text{dof}$\|
ZI	Galaxies	−1.883±0.019	1.797	15.03
	Pairs	−1.556±0.016	1.968	8.33
dZII	Galaxies	−1.830±0.017	1.835	7.29
	Pairs	−1.453±0.015	2.014	5.42
ZII	Galaxies	−1.857±0.018	1.8827	19.4
	Pairs	−1.536±0.015	2.1146	10.29

HOD MODEL FRAMEWORK

A HOD model quantifies the probability distribution for the number of galaxies, N, hosted by a dark matter halo as a function of the halo mass. The galaxy HOD model separates the two-point clustering signal of galaxies into two components: the distribution of galaxies within individual haloes, which dominates small-scale clustering, and the clustering of galaxies between two different haloes, which dominates large-scale clustering. By combining these two components, an HOD model can accurately model the observed scale-dependent clustering signal. Throughout this paper, we assume a flat cosmology with Ω_m = 0.27, h = 0.7, σ₈ = 0.8, and Γ = 0.15.

More formally, we use the halo model of galaxy clustering to generate a model spatial correlation function, ξ(r), and subsequently project this three-dimensional spatial function to the two-dimensional angular sky coordinates. By comparing this projected angular clustering signal to the measured ω(θ) from large area, photometric surveys, we can place constraints on the model parameters. The relationship between the angular and spatial correlation functions is provided via Limber's equation (assuming a flat Universe):

\begin{equation} \omega (\theta ) = \frac{2}{c} \int _{0}^{\infty } H(z)(\mathrm{d}n/\mathrm{d}z)^2 \mathrm{d}z\int _0^{\infty } \xi (r = \sqrt{u^2 + x^2(z) \theta ^2}) \mathrm{d}u. \!\!\!\!\!\!\!\!\! \end{equation}

(5)

The theoretical galaxy distributions, however, are more easily generated in Fourier space. Thus we need to convert theoretical power spectra into real-space correlations via the Fourier transform of the power spectrum:

\begin{equation} \xi (r) = \frac{1}{2\pi ^2}\int _{0}^{\infty }P(k,r)k^2 \frac{sin\ kr}{kr} \mathrm{d}k. \end{equation}

(6)

In keeping with our HOD model, we split the theoretical galaxy power spectrum into two components: the contribution due to galaxies that reside within a single halo (the one-halo term), and the contribution due to galaxy pairs located in separate haloes (the two-halo term):

\begin{equation} P(k,r) = P_{\rm 1h}(k,r) + P_{\rm 2h}(k,r). \end{equation}

(7)

In the rest of this section, we derive these theoretical one-halo and two-halo terms, along with other important, associated quantities that are used in our HOD model. In the following equations, we will use the notation N(M) ≡ 〈N|M〉, N_c(M) ≡ 〈N_c|M〉 and N_s(M) ≡ 〈N_s|M〉, where 〈N|M〉 is the probability distribution for the number of galaxies N hosted by a dark matter halo with mass M, and likewise for the associate probabilities for central, 〈N_c|M〉, and satellite, 〈N_s|M〉, galaxies.

The one-halo term

To compute the power spectrum for the one-halo term, we recognize that we have two distinct components: the central–satellite galaxy component, P_cs(k, r), and the satellite–satellite component, P_ss(k):

\begin{equation} P_{\rm 1h}(k,r) = P_{\rm cs}(k,r) + P_{\rm ss}(k). \end{equation}

(8)

Where r is the comoving separation between the galaxies. These individual theoretical power spectra terms, P_cs and P_ss, are computed by using the following relations:

\begin{equation} P_{\rm cs}(k,r) = \int _{M_{\rm vir}(r)}^{\infty } n(M)N_{\rm c}(M)\frac{Ns(M)\mu (k|M)}{n_{\rm g}^2/2} \mathrm{d}M \end{equation}

(9)

\begin{equation} P_{\rm ss}(k) = \int _{M_{\rm vir}(r)}^{\infty } n(M)N_{\rm c}(M)\frac{(Ns(M)\mu (k|M))^2}{n_{\rm g}^2} \mathrm{d}M. \end{equation}

(10)

M_vir(r) is the minimum virial mass that can hold the corresponding central–satellite or satellite–satellite galaxy pairs at a separation r for a given halo in equations (9) and (10):

\begin{equation} M_{\rm vir}(r) = \frac{4}{3}\pi r^3 \bar{\rho } \Delta . \end{equation}

(11)

In the previous equation, the comoving background density of the Universe is defined as |$\bar{\rho }$| = 2.78 × 10¹¹Ω_m h² M_⊙ Mpc⁻³. We follow Blake, Collister & Lahav (2008) to define Δ = 200 as the critical over-density for virialization; thus we can express the virial radius in terms of the halo mass, |$r_{\rm vir} = {3M}/({\Delta \times 4\pi \bar{\rho }})$|⁠.

Other terms in equations (9) and (10) include n(M), the halo mass function, which will be discussed in more detail in Section 5.3; N_c(M), the mean occupation for central galaxies at halo mass M; and N_s(M), the mean number of satellite galaxies within a single halo. From these terms, we can compute the model galaxy number density:

\begin{equation} n_{\rm g,mod} = \int _{0}^{\infty } n(M)\ N(M)\,\mathrm{d}M. \end{equation}

(12)

We can directly compare our model, n_{g, mod}, and observed galaxy number densities to place constraints on our specific HOD model parameters (see Section 5.4 for more details). We can thus match the n_{g, mod} with n_{g, obs} to determine one HOD parameter, M_cut, for a given (M₀, α, σ_cut) set.

Finally, μ(k|M) is the Fourier transform of the NFW (Navarro, Frenk & White 1997) halo density profile, ρ(r|M):

\begin{equation} \rho (r|M) = \frac{\rho _{\rm s}}{(r/r_{\rm s})(1 + r/r_{\rm s})^2} \end{equation}

(13)

where r_s is the scale radius of the halo. ρ_s is the dark matter density at the scale radius:

\begin{equation} \rho _{\rm s} = \frac{M}{4\pi r_{\rm s}^3}[ln(1+c)-\frac{c}{(1+c)}]^{-1} \end{equation}

(14)

where c is the concentration parameter defined as c = r_vir/r_s. Bullock et al. (2001) and Zehavi et al. (2004) have shown that this term can be expressed as:

\begin{equation} c(M,z) = \frac{11}{(1+z)}(\frac{M}{M_{\rm c}})^{-0.13} \end{equation}

(15)

where M_c is a parametrized cutoff mass in units of h⁻¹ M_⊙. For our assumed cosmology, it is quantified by log₁₀(M_c) = 12.56.

The two-halo term

The two-halo power spectrum can be computed as:

\begin{eqnarray} P_{\rm 2h}(k,r) &=& P_{\rm m}(k)\nonumber \\ &&\times\,\bigg [\int _{0}^{M_{\rm lim}(r)} n(M)\ b(M,r)\ \frac{N(M)}{n_{\rm g}^\prime } \mu (k|M) \mathrm{d}M\bigg ]^2 \end{eqnarray}

(16)

where P_m(k) is the non-linear matter power spectrum at the survey redshift and all other terms, other than M_lim(r) and b(M, r), are as defined before for the one-halo term. The mass limit, M_lim(r), is calculated by employing the ‘|$n^\prime _{\rm g}$|-matched’ approximation (Tinker et al. 2005), which considers halo exclusion effects and matches the restricted galaxy number density, |$n^\prime _{\rm g}(r)$|⁠, as a function of physical separation, r. We can compute |$n^\prime _{\rm g}(r)$| for our HOD model:

\begin{equation} n_{\rm g}^\prime (r) = \int _{0}^{M_{\rm lim}(r)} n(M)\ N(M)\,\mathrm{d}M. \end{equation}

(17)

On the other hand, we can also compute |$n_{\rm g}^\prime (r)$| by employing the halo exclusion:

\begin{eqnarray} n_{\rm g}^{\prime \ 2}(r) &=& \int _{0}^{\infty } n(M_1) N(M_1)\,\mathrm{d}M_1 \nonumber \\ &&\times\,\int _0^{\infty } n(M_2) N(M_2) P(r, M_1, M_2)\,\mathrm{d}M_2 \end{eqnarray}

(18)

where P(r, M₁, M₂) measures the probability of non-overlapping haloes of masses M₁ and M₂ at separation r.

By analysing simulations, Tinker et al. (2005) obtained P(y) = 3y² − 2y³ for 0 < y < 1, P(y) = 0 for y < 0, and P(y) = 1 for y > 1, where y is connected to the virial radii:

\begin{equation} y(r) = \frac{x(r) - 0.8}{0.29},\ x(r) = \frac{r}{R_1 + R_2} \end{equation}

(19)

where R₁ and R₂ are the virial radii corresponding to masses M₁ and M₂. Given a galaxy density |$n^\prime _{\rm g}(r)$| computed from equation (18), we can increase M_lim(r) in equation (17) until we obtain the same galaxy density for our HOD model. Having determined this mass limit, we can subsequently use M_lim(r) in equation (16) to calculate the two-halo power spectrum term.

The last remaining undefined term in equation (16), b(M, r), is the scale-dependent bias, which can be written in the following form (Tinker et al. 2005):

\begin{equation} b(M,r)^2 = b^2(M)\frac{[1 + 1.17\xi _{\rm m}(r)]^{1.49}}{[1+0.69\xi _{\rm m}(r)]^{2.09}}, \end{equation}

(20)

where ξ_m(r) is the non-linear, matter correlation function, and b(M) is the halo bias function that quantifies the relative bias of a halo of mass M with respect to the overall dark matter distribution (Sheth, Mo & Tormen 2001; Tinker et al. 2005):

\begin{eqnarray} b(\nu ) &=& 1 + \frac{1}{\delta _{\rm sc}} \nonumber \\ &&\times\, \left[ q\nu + s(q\nu )^{1-t} - \frac{q^{-1/2}}{1 + s(1-t)(1-\frac{t}{2})(q\nu )^{-t}}\right] \end{eqnarray}

(21)

where the three parameters are assigned the following values: q = 0.707, s = 0.35, and t = 0.8.

The model spatial correlation function can be calculated from the two-halo term by using the Fourier transform of the power spectra:

\begin{equation} \xi ^\prime _{\rm 2h}(r) = \frac{1}{2\pi ^2}\int _{0}^{\infty }P_{\rm 2h}(k,r)k^2 \frac{\sin \ kr}{kr} \mathrm{d}k. \end{equation}

(22)

As shown by Tinker et al. (2005), we can correct |$\xi ^\prime _{\rm 2h}(r)$| to the true spatial correlation function, ξ_2h(r), via:

\begin{equation} 1 + \xi _{\rm 2h}(r) = \left[\frac{n_{\rm g}^\prime (r)}{n_{\rm g}}\right]^2 [1 + \xi _{\rm 2h}^\prime (r)]. \end{equation}

(23)

We note that this correction only modifies the ξ_2h(r) measurement at small spatial separations, where the ξ_1h(r) signal is dominated by the one-halo term. Therefore, this correction is a negligible correction (<1 per cent) to our final HOD model correlation functions.

Halo mass function

The halo mass function, n(M), quantifies the number density of haloes as a function of mass M, originally described by Press & Schechter (1974):

\begin{equation} n(M)\mathrm{d}M = \frac{\bar{\rho }}{M} f(\nu )\,\mathrm{d}\nu . \end{equation}

(24)

The mass function, f(ν), is typically modelled in the following form:

\begin{equation} f(M) = \frac{1}{2 \nu }a_1 \text{exp} (- |\ln \sigma ^{-1} + a_2|^{a_3}) \end{equation}

(25)

where a₁ = 0.315, a₂ = 0.61, and a₃ = 3.8 as determined from simulations by Tinker et al. (2005). In this form, σ is defined by |$\sigma (M,z)\equiv \delta _{\rm sc} / \sqrt{\nu }$|⁠, which is the variance of the linear power spectrum within a spherical top hat that contains average mass M:

\begin{equation} \sigma ^2(M,z) = \frac{D^2(z)}{2 \pi ^2}\int _0^{\infty } k^2 P_{\rm lin}(k) W^2(kR)\,\mathrm{d}k. \end{equation}

(26)

The functional terms in equation (26) include W(x) = (3/x³)[sin x − x cos x]; |$R = (3\ M/(4\ \pi \ \bar{\rho }))^{1/3}$|⁠; D(z), which is the linear growth factor at redshift z; and P_lin(k), which is the linear power spectrum at redshift zero.

Halo occupation distribution model

The number of galaxies that populate a halo of mass M can be described as

\begin{equation} N(M) = N_{\rm c}(M) \times (1+N_{\rm s}(M)) \end{equation}

(27)

where N_c(M) is the mean occupations for central galaxies, and N_s(M) is the mean occupation for satellite galaxies, for which we assume a Poisson distribution (Kravtsov et al. 2004). The form of equation (27) implies that a halo can only host a satellite galaxy if it already hosts a central galaxy. We follow Ross & Brunner (2009) to model the HOD with a softened transition for both the central and satellite galaxies:

\begin{equation} N_{\rm c}(M) = 0.5\ \left[1 + \text{erf}\left(\frac{\log _{10} (M/M_{\rm cut})}{\sigma _{\rm cut}}\right)\right] \end{equation}

(28)

\begin{equation} N_{\rm s}(M) = 0.5\ \left[1 + \text{erf}\left(\frac{\log _{10} (M/M_{\rm cut})}{\sigma _{\rm cut}}\right)\right] \times \left(\frac{M}{M_0}\right)^\alpha \end{equation}

(29)

where M_cut is the halo mass that can host a single, central galaxy, and M₀ is the halo mass where the halo starts to host satellite galaxies in addition to the central galaxy. This model has four free parameters, one of which, M_cut, we can remove by matching the observed galaxy density, n_g, to the model-derived number density of galaxies by using equation (12).

To compare with previous results, we also derive two additional quantities from our HOD: the effective mass, M_eff:

\begin{equation} M_{\rm eff} = \int M n(M) \frac{N(M)}{n_{\rm g}}\mathrm{d}M \end{equation}

(30)

and the effective large-scale bias, b_g:

\begin{equation} b_{\rm g} = \int n(M) b(M) \frac{N(M)}{n_{\rm g}}\mathrm{d}M. \end{equation}

(31)

We also use the halo mass function to weight the HOD to determine the average fraction of central galaxies in the sample:

\begin{equation} f_{\rm c} = \frac{\int n(M) N_{\rm c}(M)\,\mathrm{d}M}{\int n(M) N(M)\,\mathrm{d}M}. \end{equation}

(32)

The fraction for satellite galaxies is given by f_s = 1 − f_c.

HOD FOR INDIVIDUAL GALAXIES AND GALAXY PAIRS

We fit the three-parameter halo model (α, M₀, σ_cut) to the observed angular correlation functions, fixing the remaining parameter M_cut by matching the model galaxy number density to the observed galaxy number density. By comparing the change in these parameters between different galaxy or galaxy pair samples, we can quantify the dependence of the clustering of galaxies on these parameters. We determine all HOD parameters by using a Monte Carlo Markov-Chain (MCMC) sampler to locate the minimum value of the χ² fitting statistic by using the information from the full covariance matrix. Specifically, we use the emcee code (Foreman-Mackey et al. 2013) for which we set nwalkers to 288, and we use at least 100 steps within each walk, which we have found to be sufficient for convergence.

We use the peak probability as the location of the optimal fitting. Nominally this is given by the value where the χ² measured by the fitting process is minimized. However, the MCMC approach can often give similar χ² values at slightly different parameter combinations as we sample near the peak probability. We have verified that these different parameter configurations do not significantly change the results presented in the rest of this paper. We determine the errors on the fit parameters by finding the one sigma range around the peak probability. The best-fitting HOD values and the standard deviations of each set of model parameters are listed in Table 3. The number of galaxy triplets and quads is too small within our samples to obtain correlation functions with sufficiently small error bars to make model constraints. Therefore, we only perform HOD fittings to the single galaxy and galaxy pair samples in the rest of this paper.

Table 3.

Open in new tab

The best-fitting HOD model parameters for the galaxies and galaxy pairs selected from the three volume limited samples. All masses are in units of M_⊙ h⁻¹.

		α	log₁₀(M₀)	σ_cut	log₁₀(M_cut)	\|$\chi ^2 / \text{dof}$\|	log₁₀(M_eff)	b_g	f_c
dZI	Galaxies	1.16\|$\pm ^{0.03}_{0.02}$\|	12.99\|$\pm ^{0.029}_{0.025}$\|	0.19\|$\pm ^{0.22}_{0.18}$\|	11.63	22.41	13.40	1.13	0.76
	Pairs	1.54\|$\pm ^{0.04}_{0.03}$\|	14.20±0.03	0.28\|$\pm ^{0.13}_{0.10}$\|	13.18	9.31	13.78	1.90	0.84
ZII	Galaxies	1.18\|$\pm ^{0.025}_{0.021}$\|	13.19\|$\pm ^{0.02}_{0.018}$\|	0.10\|$\pm ^{0.12}_{0.09}$\|	11.87	14.26	13.45	1.21	0.76
	Pairs	1.73±0.03	14.41\|$\pm ^{0.028}_{0.024}$\|	0.08\|$\pm ^{0.13}_{0.07}$\|	13.36	13.02	13.92	2.11	0.89
ZIII	Galaxies	1.46\|$\pm ^{0.04}_{0.03}$\|	13.59\|$\pm ^{0.024}_{0.02}$\|	0.16\|$\pm ^{0.13}_{0.11}$\|	12.37	26.19	13.51	1.45	0.79
	Pairs	1.03\|$\pm ^{0.028}_{0.031}$\|	14.86\|$\pm ^{0.014}_{0.012}$\|	0.07\|$\pm ^{0.12}_{0.06}$\|	13.87	7.64	13.94	2.47	0.89

		α	log₁₀(M₀)	σ_cut	log₁₀(M_cut)	\|$\chi ^2 / \text{dof}$\|	log₁₀(M_eff)	b_g	f_c
dZI	Galaxies	1.16\|$\pm ^{0.03}_{0.02}$\|	12.99\|$\pm ^{0.029}_{0.025}$\|	0.19\|$\pm ^{0.22}_{0.18}$\|	11.63	22.41	13.40	1.13	0.76
	Pairs	1.54\|$\pm ^{0.04}_{0.03}$\|	14.20±0.03	0.28\|$\pm ^{0.13}_{0.10}$\|	13.18	9.31	13.78	1.90	0.84
ZII	Galaxies	1.18\|$\pm ^{0.025}_{0.021}$\|	13.19\|$\pm ^{0.02}_{0.018}$\|	0.10\|$\pm ^{0.12}_{0.09}$\|	11.87	14.26	13.45	1.21	0.76
	Pairs	1.73±0.03	14.41\|$\pm ^{0.028}_{0.024}$\|	0.08\|$\pm ^{0.13}_{0.07}$\|	13.36	13.02	13.92	2.11	0.89
ZIII	Galaxies	1.46\|$\pm ^{0.04}_{0.03}$\|	13.59\|$\pm ^{0.024}_{0.02}$\|	0.16\|$\pm ^{0.13}_{0.11}$\|	12.37	26.19	13.51	1.45	0.79
	Pairs	1.03\|$\pm ^{0.028}_{0.031}$\|	14.86\|$\pm ^{0.014}_{0.012}$\|	0.07\|$\pm ^{0.12}_{0.06}$\|	13.87	7.64	13.94	2.47	0.89

Table 3.

Open in new tab

The best-fitting HOD model parameters for the galaxies and galaxy pairs selected from the three volume limited samples. All masses are in units of M_⊙ h⁻¹.

		α	log₁₀(M₀)	σ_cut	log₁₀(M_cut)	\|$\chi ^2 / \text{dof}$\|	log₁₀(M_eff)	b_g	f_c
dZI	Galaxies	1.16\|$\pm ^{0.03}_{0.02}$\|	12.99\|$\pm ^{0.029}_{0.025}$\|	0.19\|$\pm ^{0.22}_{0.18}$\|	11.63	22.41	13.40	1.13	0.76
	Pairs	1.54\|$\pm ^{0.04}_{0.03}$\|	14.20±0.03	0.28\|$\pm ^{0.13}_{0.10}$\|	13.18	9.31	13.78	1.90	0.84
ZII	Galaxies	1.18\|$\pm ^{0.025}_{0.021}$\|	13.19\|$\pm ^{0.02}_{0.018}$\|	0.10\|$\pm ^{0.12}_{0.09}$\|	11.87	14.26	13.45	1.21	0.76
	Pairs	1.73±0.03	14.41\|$\pm ^{0.028}_{0.024}$\|	0.08\|$\pm ^{0.13}_{0.07}$\|	13.36	13.02	13.92	2.11	0.89
ZIII	Galaxies	1.46\|$\pm ^{0.04}_{0.03}$\|	13.59\|$\pm ^{0.024}_{0.02}$\|	0.16\|$\pm ^{0.13}_{0.11}$\|	12.37	26.19	13.51	1.45	0.79
	Pairs	1.03\|$\pm ^{0.028}_{0.031}$\|	14.86\|$\pm ^{0.014}_{0.012}$\|	0.07\|$\pm ^{0.12}_{0.06}$\|	13.87	7.64	13.94	2.47	0.89

		α	log₁₀(M₀)	σ_cut	log₁₀(M_cut)	\|$\chi ^2 / \text{dof}$\|	log₁₀(M_eff)	b_g	f_c
dZI	Galaxies	1.16\|$\pm ^{0.03}_{0.02}$\|	12.99\|$\pm ^{0.029}_{0.025}$\|	0.19\|$\pm ^{0.22}_{0.18}$\|	11.63	22.41	13.40	1.13	0.76
	Pairs	1.54\|$\pm ^{0.04}_{0.03}$\|	14.20±0.03	0.28\|$\pm ^{0.13}_{0.10}$\|	13.18	9.31	13.78	1.90	0.84
ZII	Galaxies	1.18\|$\pm ^{0.025}_{0.021}$\|	13.19\|$\pm ^{0.02}_{0.018}$\|	0.10\|$\pm ^{0.12}_{0.09}$\|	11.87	14.26	13.45	1.21	0.76
	Pairs	1.73±0.03	14.41\|$\pm ^{0.028}_{0.024}$\|	0.08\|$\pm ^{0.13}_{0.07}$\|	13.36	13.02	13.92	2.11	0.89
ZIII	Galaxies	1.46\|$\pm ^{0.04}_{0.03}$\|	13.59\|$\pm ^{0.024}_{0.02}$\|	0.16\|$\pm ^{0.13}_{0.11}$\|	12.37	26.19	13.51	1.45	0.79
	Pairs	1.03\|$\pm ^{0.028}_{0.031}$\|	14.86\|$\pm ^{0.014}_{0.012}$\|	0.07\|$\pm ^{0.12}_{0.06}$\|	13.87	7.64	13.94	2.47	0.89

Our photometric sample could potentially be affected by projection effects as the volumes probed by the angular cones increase. Furthermore, when working with galaxy pairs, we might expect a second projection effect to result when two distinct galaxy pairs appear too close to reside in the same physical halo. For example, we might consider imposing a minimum angular separation of at least 0| $_{.}^{\circ}$|028 or 0| $_{.}^{\circ}$|014 at z = 0.05 or 0.1 to force galaxy pairs into two, distinct massive haloes. To test the effects of these projection concerns, we measure the correlation functions for the galaxy pairs between 0| $_{.}^{\circ}$|01 and 1°, and we also measure these same correlation functions between 0| $_{.}^{\circ}$|02 and 0| $_{.}^{\circ}$|8, which equates to removing the three smallest and two largest angular bins from the full angular range probed by the first set of correlation functions. We compared the HOD model fits to these two sets of correlation functions and found minimal variation between the measured HOD model parameters; in all cases the changes were less than one sigma, and were generally much less than this. Thus, for simplicity we simply used the full angular range for all correlation measurements of both the single galaxies and the isolated galaxy pairs.

Model fits for the three samples

We fit the halo model to galaxy and galaxy pair samples selected in three volume-limited samples: ZI (0.0 < z ≤ 0.3, M_r < −19.0), ZII (0.0 < z ≤ 0.3, M_r < −19.5), and ZIII (0.3 < z ≤ 0.4, M_r < −19.5). The full set of HOD fit parameters for these six samples is presented in Table 3, and the data and the best-fitting HOD models are shown in Fig. 6.

Figure 6.

The measured angular correlation function (top row) and HOD models for single galaxies (orange circles) and galaxy pairs (magenta triangles) selected from the ZI (left column), ZII (middle column) and ZIII (right column) samples. The dashed lines indicate the best-fitting HOD models. The middle row displays the ratio of the best-fitting HOD models (dashed line) and measured correlation function (points) to the best-fitting power-law models. The bottom row displays the HOD of the best-fitting HOD models for the three single galaxy (orange) and galaxy pair (magenta) samples.

Open in new tab Download slide

We display in the top panel of Fig. 6 the measured ω(θ) for galaxies and galaxy pairs that lie within the ZI, ZII and ZIII samples. The HOD model fits to all single galaxy correlation functions are computed between 0| $_{.}^{\circ}$|01 and 1° (i.e. 16 degrees of freedom), and yield |$\chi ^2/\text{dof}$| values ranging from 14.26 to 26.19. The model fits to the galaxy pairs are generally more accurate than for the single galaxies for all three samples; however, since these correlation functions tend to have larger error bars, this result is not surprising. With 16 degrees of freedom, we have |$\chi ^2/\text{dof}$| values ranging from 7.64 to 13.02; and no systematic discrepancies are found with any of the model fits, implying that there are no major biases present in the model fits.

We also note that the |$\chi ^2/\text{dof}$| values for the HOD model fits are not necessarily smaller than the |$\chi ^2_{pl}/\text{dof}$| values for the best power-law fits, which are tabulated in Table 3. The error bars on the correlation functions are extremely small for most angular bins, which will result in large χ² fitting values even when there are only small deviations between the model and the data.

To highlight the differences between our best-fitting HOD and the power-law models to the measured correlation functions, we display the ratio of the best-fitting HOD model (dashed line) and measured correlation function (points) to the best-fitting power-law model in the middle panels of Fig. 6. The deviations from unity within these panels clearly indicate that the HOD models successfully reproduce the observed deviations from the power-law model and that the shape of the one-halo term (small scales) and the two-halo term (large scales) are closer or almost identical to the observed ω(θ). Furthermore, we see that the HOD models capture the variation shown in the transition regions between the one-halo and two-halo terms for both single galaxies and galaxy pairs, which is missed by the power-law model.

In the bottom panels of Fig. 6, we display the best-fitting HODs for individual galaxies and galaxy pairs in the three volume-limited samples, assuming the galaxy populations defined by equations (27). The HOD for individual galaxies shows inflection points around M_cut = 10^11.63 h⁻¹ M_⊙ for the ZI sample, M_cut = 10^11.87 h⁻¹ M_⊙ for the ZII sample and around M_cut = 10^12.37 h⁻¹ M_⊙ for the ZIII sample; this parameter defines the mass scale at which haloes host central galaxies. Likewise, these model fits define M₀ = 10^12.99 h⁻¹ M_⊙, M₀ = 10^13.19 h⁻¹ M_⊙, and M₀ = 10^13.59 h⁻¹ M_⊙ for the ZI, ZII and ZIII samples, respectively, which is the mass scale at which haloes start to host satellite galaxies in addition to a central galaxy. Similarly, we find that the best-fitting galaxy pair parameters are M_cut = 10^13.18 h⁻¹ M_⊙ and M₀ = 10^14.20 h⁻¹ M_⊙ for the ZI sample, M_cut = 10^13.36 h⁻¹ M_⊙ and M₀ = 10^14.41 h⁻¹ M_⊙ for the ZII sample and M_cut = 10^13.87 h⁻¹ M_⊙ and M₀ = 10^14.86 h⁻¹ M_⊙ for the ZIII sample.

From the figures and the tabulated fit parameters, we see that the M_cut parameter for galaxy pairs is close to the value at which a halo hosts two galaxies as measured from the single galaxy correlation function. Naively, one might think the mass scale at which haloes host one central galaxy pair should be the same as the mass scale at which haloes host one central and one satellite galaxy. However, since we define our galaxy pair catalogue as isolated galaxy pairs (i.e. no nearby galaxies within a distance that is three times the group radius), we have parent haloes that are unlikely to reside in an environment that contains both small and large haloes. Therefore, our catalogue preferentially selects more massive dark matter haloes that have higher probabilities to form isolated pairs. Thus the haloes for our pair catalogue that contain one central galaxy pair [two galaxies, i.e. where the N(M) = 1, usually at the point ≳M_cut] must be larger than the M₀ for the single galaxies. The average number of single galaxies hosted by the halo mass M = 10¹³ h⁻¹M_⊙ is 2.0, 1.5, and 1.2 in the ZI, ZII and ZIII samples, respectively, and the average number of galaxy pairs hosted by the halo mass M=10^14.5 h⁻¹M_⊙ is likewise 4.9, 2.4 and 1.5.

We compare the weighted fraction of central (given by equation (32)) and satellite galaxies and galaxy pairs as a function of the dark matter halo mass for all three galaxy samples in Fig. 7. In all samples, we see that the transition from central to satellite occurs at higher masses for galaxy pairs. This observation agrees with the standard picture that galaxy pairs are formed in the cores of more massive dark matter haloes than single galaxies. We also note that since our group selection criteria select isolated galaxy pairs, we expect that there are a large fraction of galaxy pairs that are the only galaxies that populate a parent dark matter halo. Furthermore, if these galaxy pairs are real physical pairs at close separation, we will only see central galaxy pairs accompanied by satellite galaxy pairs in the most massive haloes. At these mass scales, the haloes are large enough to host two galaxy pairs, while still allowing the two pairs to remain separated at a large physical range within the halo. Thus, the central and satellite galaxy pairs reside in subhaloes of the parent halo.

$The weighted fraction of central (solid lines) and satellite (dashed lines) components for the HOD models for single galaxies (orange) and galaxy pairs (magenta).$

Figure 7.

The weighted fraction of central (solid lines) and satellite (dashed lines) components for the HOD models for single galaxies (orange) and galaxy pairs (magenta).

Open in new tab Download slide

In Table 3 we list the derived values of the effective halo mass, M_eff, and the galaxy bias factor, b_g, that are calculated by using equations (30) and (31). The effective mass and the linear bias are M_eff = (10^13.40, 10^13.45, 10^13.51) h⁻¹M_⊙ and b_g = (1.13, 1.21, 1.45) for the ZI, ZII, and ZIII samples, respectively. These results imply that the linear bias increases with luminosity and redshift, which agrees with fits to the large-scale power spectrum of the main SDSS DR7 galaxy samples (Hayes & Brunner 2013). We compare our results in more detail with previous work in Section 7.

The effective halo mass and the linear bias for the galaxy pairs are M_eff = (10^13.78, 10^13.92, 10^13.94) h⁻¹M_⊙ and b_g = (1.90, 2.11, 2.47), which, in comparison with our values for single galaxies, confirms that galaxy pairs are hosted by massive dark matter haloes and are highly biased tracers of the underlying dark matter distribution. The difference of M_eff between the single galaxies and isolated galaxy pairs is consistently around 10^∼0.4, or 2.5. This ratio implies that the haloes hosting isolated galaxy pairs are, on average, slightly greater than twice the mass of the haloes that host single galaxies. We note that the systematic increase in galaxy bias between the ZI and the ZII and ZIII samples indicates that the bias is more dependent on sample redshift than on luminosity, at least within the redshift range probed by our volume limited samples. In addition, the similarity in M_eff and the change in b_g between the ZII and ZIII samples suggest that galaxy pairs in the higher redshift sample are more strongly clustered than the galaxy pairs in the lower redshift sample and that the parent dark matter haloes for the ZIII pairs likely reside in more overdense environments.

Finally, we note that the fitting parameter, α, is systematically larger for galaxy pairs than for the single galaxies in the two lower redshift samples: ZI and ZII. α is the slope of the HOD at increasing mass, which controls the rate of increase of satellite galaxies in a halo. This trend implies that isolated galaxy pairs at low redshift are more likely to be strongly clustered, and are probably embedded within even more massive dark matter haloes than single galaxies. On the other hand, the highest redshift sample, ZIII, has a shallower slope (or smaller value of α), than the single galaxy sample. This implies that the largest haloes in this sample are more likely to be composed of single galaxies than isolated galaxy pairs. Thus, over time we see an increase in the clustering of galaxy pairs in the same mass haloes, which is consistent with the expected assembly history of galaxies and haloes.

Redshift dependence

By comparing model fits to the correlation functions in the ZII (0.0 < z ≤ 0.3, M_r < −19.5) and ZIII (0.3 < z ≤ 0.4, M_r < −19.5) samples, we can study the redshift evolution of our HOD model (a complete analysis of the three samples is in Section 6.1). We display in the top panels of Fig. 8 the measured ω(θ) for galaxies and pairs from the ZII and ZIII samples, and we display the ratio of the ω(θ) from two samples in the bottom panels. The ratios for the single galaxies and pairs show a similar trend, although the ratio for pairs shows more fluctuations that are due to the larger error bars in the pair correlations. Since the relative bias between two populations is defined by: |$b^2_{1,2} = \omega _1/\omega _2$|⁠, we see that the deviations from unity indicate bias evolution between the ZII and ZIII samples. We also see scale dependence in the relative bias, as b(θ) changes with scale, becoming largest within the one- and two-halo regions, while being similar within the transition region between these two terms.

Figure 8.

The measured angular correlation function (top row) and HOD models for single galaxies (left column) and galaxy pairs (right column) that are selected from ZII sample (orange circles) and ZIII sample (magenta triangles). The dashed lines indicate the best-fitting HOD models. The bottom row displays the ratio of the best-fitting HOD models (dashed line) and measured correlation function (points) of the ZIII sample to the ZII sample.

Open in new tab Download slide

Type dependence

We now focus our investigation on the dependence of the clustering of galaxy pairs on galaxy type. In order to ensure sufficient statistics, we only use the ZI sample, which we subdivide into the following two subsamples:

EE: a galaxy pair in which both galaxies are identified as early-type galaxies, and
LL: a galaxy pair in which both galaxies are identified as late-type galaxies.

The galaxy type is determined by using the m_u − m_r colour cut, where m_u and m_r are dereddened model magnitudes. Following Strateva et al. (2001), if this colour is >2.2, the galaxy is classified as an early-type galaxy, while if it is <2.2 it is classified as a late-type galaxy.

These two sub-samples contain 68 316 and 17 291 isolated galaxy pairs, respectively. In Fig. 9 we present the normalized number density distributions for the two sub-samples as a function of redshift measured by using the technique discussed in Section 3.3. To reduce the magnitude of the error bars on our correlation function measurements, we only use 10 angular bins for all correlation function and associated covariance matrix measurements for these two sub-samples, and for the two sub-samples, L0.2 and L0.8, that will be discussed in Section 6.4. However, to generate stable covariance matrices, we still use 32 jackknife samples.

Figure 9.

The normalized number density distribution for early–early galaxy pairs (EE, red), and late–late galaxy pairs (LL, blue).

Open in new tab Download slide

We present the measured correlation function for the galaxies and galaxy pairs selected from the ZI sub-sample in the top panel of Fig. 10. Also shown in these plots are the best-fitting HOD models. The early–early galaxy pairs show stronger clustering over all scales than the late–late galaxy pairs, albeit with a steeper slope, in agreement with galaxy morphology-density analyses (see e.g. Dressler 1980).

Figure 10.

The measured angular correlation function (top panel) and HOD models for galaxy pairs selected from ZI as early–early pairs (red up triangles) and late–late pairs (blue down triangles). The dashed lines indicate the best-fitting HOD models. The middle panel displays the ratio of the best-fitting HOD models (dashed line) and measured correlation function (points) to the parent sample. The bottom panel displays the HOD of the best-fitting HOD models for the two samples.

Open in new tab Download slide

These fit parameters are also listed in Table 4. To quantify the dependence of these HOD model results for all galaxy pairs as a function of galaxy type, we divide both the measured correlation functions and the best-fitting HOD models for our EE and LL subsamples by the best-fitting HOD model for all galaxy pairs. The result is shown in the middle panel of Fig. 10, which, following the discussion on relative bias in Section 6.2, provides an indication of the square of the relative bias between these samples. The early–early type galaxy pairs show much stronger bias with respect to all galaxy pairs, especially at the halo centre, while the late–late type galaxy pairs show less strong bias over all scales.

Table 4.

Open in new tab

The best-fitting HOD model parameters for galaxy pairs selected from the ZI sample as early–early pairs (EE) or late–late pairs (LL). All masses are in units M_⊙ h⁻¹.

	α	log₁₀(M₀)	σ_cut	log₁₀(M_cut)	\|$\chi ^2 / \text{dof}$\|	log₁₀(M_eff)	b_g	f_c	log₁₀(A_ω)	γ	\|$\chi _{pl}^2 / \text{dof}4$\|
EE	1.40\|$\pm ^{0.05}_{0.04}$\|	14.32\|$\pm ^{0.04}_{0.03}$\|	0.30 ± 0.17	13.42	6.82	13.97	2.09	0.83	−1.66 ± 0.01	2.09 ± 0.02	2.88
LL	0.77\|$\pm ^{0.028}_{0.03}$\|	15.48\|$\pm ^{0.04}_{0.05}$\|	1.17\|$\pm ^{0.17}_{0.13}$\|	14.76	1.27	13.79	1.77	0.99	−1.69 ± 0.02	1.67 ± 0.01	1.73

	α	log₁₀(M₀)	σ_cut	log₁₀(M_cut)	\|$\chi ^2 / \text{dof}$\|	log₁₀(M_eff)	b_g	f_c	log₁₀(A_ω)	γ	\|$\chi _{pl}^2 / \text{dof}4$\|
EE	1.40\|$\pm ^{0.05}_{0.04}$\|	14.32\|$\pm ^{0.04}_{0.03}$\|	0.30 ± 0.17	13.42	6.82	13.97	2.09	0.83	−1.66 ± 0.01	2.09 ± 0.02	2.88
LL	0.77\|$\pm ^{0.028}_{0.03}$\|	15.48\|$\pm ^{0.04}_{0.05}$\|	1.17\|$\pm ^{0.17}_{0.13}$\|	14.76	1.27	13.79	1.77	0.99	−1.69 ± 0.02	1.67 ± 0.01	1.73

Table 4.

Open in new tab

The best-fitting HOD model parameters for galaxy pairs selected from the ZI sample as early–early pairs (EE) or late–late pairs (LL). All masses are in units M_⊙ h⁻¹.

	α	log₁₀(M₀)	σ_cut	log₁₀(M_cut)	\|$\chi ^2 / \text{dof}$\|	log₁₀(M_eff)	b_g	f_c	log₁₀(A_ω)	γ	\|$\chi _{pl}^2 / \text{dof}4$\|
EE	1.40\|$\pm ^{0.05}_{0.04}$\|	14.32\|$\pm ^{0.04}_{0.03}$\|	0.30 ± 0.17	13.42	6.82	13.97	2.09	0.83	−1.66 ± 0.01	2.09 ± 0.02	2.88
LL	0.77\|$\pm ^{0.028}_{0.03}$\|	15.48\|$\pm ^{0.04}_{0.05}$\|	1.17\|$\pm ^{0.17}_{0.13}$\|	14.76	1.27	13.79	1.77	0.99	−1.69 ± 0.02	1.67 ± 0.01	1.73

	α	log₁₀(M₀)	σ_cut	log₁₀(M_cut)	\|$\chi ^2 / \text{dof}$\|	log₁₀(M_eff)	b_g	f_c	log₁₀(A_ω)	γ	\|$\chi _{pl}^2 / \text{dof}4$\|
EE	1.40\|$\pm ^{0.05}_{0.04}$\|	14.32\|$\pm ^{0.04}_{0.03}$\|	0.30 ± 0.17	13.42	6.82	13.97	2.09	0.83	−1.66 ± 0.01	2.09 ± 0.02	2.88
LL	0.77\|$\pm ^{0.028}_{0.03}$\|	15.48\|$\pm ^{0.04}_{0.05}$\|	1.17\|$\pm ^{0.17}_{0.13}$\|	14.76	1.27	13.79	1.77	0.99	−1.69 ± 0.02	1.67 ± 0.01	1.73

The bottom panel of Fig. 10 displays the best-fitting HODs for the two sub-samples. We find the late–late type galaxy pairs have larger M₀ and M_cut values than the early–early type pairs, which is consistent with the results found for individual galaxies (Ross & Brunner 2009). We note that the slope of the late–late galaxy pair HOD is smaller than the early–early galaxy pair HOD. Thus, the fraction of late–late galaxy pairs is largest in small mass haloes and smallest in high-mass haloes.

Luminosity dependence

We complete our analysis of the clustering of galaxies and galaxy pairs by examining the clustering dependence of galaxy pairs on galaxy luminosity. As presented in Section 6.3, to maintain sufficient statistics, we only use the ZI sample, which we subdivide into the following six subsamples:

B singles: bright single galaxies with M_r < −20.0,
D singles: dim single galaxies with −20.0 < M_r < −19.0,
B pairs: bright galaxy pairs where both galaxies satisfy M_r < −20.0,
D pairs: dim galaxy pairs where both galaxies satisfy −20.0 < M_r < −19.0,
L0.2: galaxy pairs where the luminosity ratio for the two galaxies is less than 20 per cent, and
L0.8: galaxy pairs where the luminosity ratio for the two galaxies is greater than 80 per cent.

These six sub-samples contain 992 275, 2983 258, 16 340, 67 153, 14 263, and 25 435 single galaxies or isolated galaxy pairs, respectively. In Fig. 11 we present the normalized number density distributions for the six sub-samples as a function of redshift measured by using the technique discussed in Section 3.3. As mentioned in Section 6.3, we only use 10 angular bins for the sub-samples L0.2 and L0.8 to reduce the magnitude of the error bars. For the other four sub-samples, we use 19 angular bins; and in all cases, we still use 32 jackknife samples to generate the covariance matrices.

Figure 11.

The normalized number density distribution for bright single galaxies (B singles), dim single galaxies (D singles), bright galaxy pairs (B pairs), dim galaxy pairs (D pairs), large luminosity contrast pairs (L0.2), and small luminosity contrast pairs (L0.8).

Open in new tab Download slide

In the top-panel of Fig. 12, we present the measured correlation function for the six sub-samples as well as the best-fitting HOD models. We first see that the bright samples for both single galaxies and galaxy pairs (B singles and B pairs) show stronger clustering than their corresponding dim samples (D singles and D pairs). On the other hand, the large and small luminosity contrast pairs demonstrate similar clustering, within the 1σ error bars, over all scales. The small luminosity contrast sample, however, does have a local minimum near 0| $_{.}^{\circ}$|1, which we interpret as a result of the transition between the one-halo and two-halo terms.

Figure 12.

The measured angular correlation function (top row) and HOD models for (left column) bright single galaxies (B singles, orange down triangles) and dim single galaxies (D singles, magenta up triangles) selected from the ZI sample, for (middle column) bright galaxy pairs selected from ZI (B pairs, orange down triangles) and dim galaxy pairs (D pairs, magenta up triangles), and for (right column) galaxy pairs selected from ZI as large luminosity contrast pairs (L0.2, orange down triangles) and small luminosity contrast pairs (L0.8, magenta up triangles). In all cases, the dashed lines indicate the best-fitting HOD models. The middle row displays the ratio of the best-fitting HOD models (dashed line) and measured correlation function (points) to the parent samples. The bottom row displays the HOD of the best-fitting HOD models for the two samples displayed in the first row.

Open in new tab Download slide

The HOD model fits for the six luminosity sub-samples are all well constrained with minimum χ² ranging from 0.46 to 11.25 (16 degrees of freedom for the first four samples and seven degrees of freedom for the L0.2 and L0.8 samples). Together with these HOD model parameters, we list the effective halo mass and the galaxy bias factor for all sub-samples in Table 5.

Table 5.

Open in new tab

The best-fitting HOD model parameters for bright single galaxies (B singles), dim single galaxies (D singles), bright galaxy pairs (B pairs), dim galaxy pairs (D pairs), large luminosity contrast pairs (L0.2), and small luminosity contrast pairs (L0.8). All masses are in units M_⊙ h⁻¹.

	α	log₁₀(M₀)	σ_cut	log₁₀(M_cut)	\|$\chi ^2 / \text{dof}$\|	log₁₀(M_eff)	b_g	f_c
B singles	1.3\|$\pm ^{0.031}_{0.028}$\|	13.43±0.03	0.21\|$\pm ^{0.17}_{0.14}$\|	12.23	4.04	13.42	1.34	0.77
D singles	1.54\|$\pm ^{0.039}_{0.042}$\|	13.46\|$\pm ^{0.043}_{0.041}$\|	0.18\|$\pm ^{0.17}_{0.15}$\|	11.81	11.25	13.41	1.10	0.87
B pairs	0.96\|$\pm ^{0.031}_{0.035}$\|	14.89\|$\pm ^{0.038}_{0.042}$\|	0.63\|$\pm ^{0.25}_{0.26}$\|	14.07	1.85	14.06	2.17	0.93
D pairs	1.41\|$\pm ^{0.053}_{0.051}$\|	14.58\|$\pm ^{0.052}_{0.047}$\|	0.34 ± 0.16	13.46	3.04	13.80	1.91	0.92
L0.2	0.83\|$\pm ^{0.044}_{0.49}$\|	15.40\|$\pm ^{0.037}_{0.034}$\|	0.78\|$\pm ^{0.35}_{0.26}$\|	14.15	0.46	13.94	1.88	0.98
L0.8	1.83\|$\pm ^{0.057}_{0.055}$\|	14.85\|$\pm ^{0.049}_{0.53}$\|	0.001\|$\pm ^{0.140}_{0.001}$\|	13.71	2.10	14.11	2.38	0.95

	α	log₁₀(M₀)	σ_cut	log₁₀(M_cut)	\|$\chi ^2 / \text{dof}$\|	log₁₀(M_eff)	b_g	f_c
B singles	1.3\|$\pm ^{0.031}_{0.028}$\|	13.43±0.03	0.21\|$\pm ^{0.17}_{0.14}$\|	12.23	4.04	13.42	1.34	0.77
D singles	1.54\|$\pm ^{0.039}_{0.042}$\|	13.46\|$\pm ^{0.043}_{0.041}$\|	0.18\|$\pm ^{0.17}_{0.15}$\|	11.81	11.25	13.41	1.10	0.87
B pairs	0.96\|$\pm ^{0.031}_{0.035}$\|	14.89\|$\pm ^{0.038}_{0.042}$\|	0.63\|$\pm ^{0.25}_{0.26}$\|	14.07	1.85	14.06	2.17	0.93
D pairs	1.41\|$\pm ^{0.053}_{0.051}$\|	14.58\|$\pm ^{0.052}_{0.047}$\|	0.34 ± 0.16	13.46	3.04	13.80	1.91	0.92
L0.2	0.83\|$\pm ^{0.044}_{0.49}$\|	15.40\|$\pm ^{0.037}_{0.034}$\|	0.78\|$\pm ^{0.35}_{0.26}$\|	14.15	0.46	13.94	1.88	0.98
L0.8	1.83\|$\pm ^{0.057}_{0.055}$\|	14.85\|$\pm ^{0.049}_{0.53}$\|	0.001\|$\pm ^{0.140}_{0.001}$\|	13.71	2.10	14.11	2.38	0.95

Table 5.

Open in new tab

The best-fitting HOD model parameters for bright single galaxies (B singles), dim single galaxies (D singles), bright galaxy pairs (B pairs), dim galaxy pairs (D pairs), large luminosity contrast pairs (L0.2), and small luminosity contrast pairs (L0.8). All masses are in units M_⊙ h⁻¹.

	α	log₁₀(M₀)	σ_cut	log₁₀(M_cut)	\|$\chi ^2 / \text{dof}$\|	log₁₀(M_eff)	b_g	f_c
B singles	1.3\|$\pm ^{0.031}_{0.028}$\|	13.43±0.03	0.21\|$\pm ^{0.17}_{0.14}$\|	12.23	4.04	13.42	1.34	0.77
D singles	1.54\|$\pm ^{0.039}_{0.042}$\|	13.46\|$\pm ^{0.043}_{0.041}$\|	0.18\|$\pm ^{0.17}_{0.15}$\|	11.81	11.25	13.41	1.10	0.87
B pairs	0.96\|$\pm ^{0.031}_{0.035}$\|	14.89\|$\pm ^{0.038}_{0.042}$\|	0.63\|$\pm ^{0.25}_{0.26}$\|	14.07	1.85	14.06	2.17	0.93
D pairs	1.41\|$\pm ^{0.053}_{0.051}$\|	14.58\|$\pm ^{0.052}_{0.047}$\|	0.34 ± 0.16	13.46	3.04	13.80	1.91	0.92
L0.2	0.83\|$\pm ^{0.044}_{0.49}$\|	15.40\|$\pm ^{0.037}_{0.034}$\|	0.78\|$\pm ^{0.35}_{0.26}$\|	14.15	0.46	13.94	1.88	0.98
L0.8	1.83\|$\pm ^{0.057}_{0.055}$\|	14.85\|$\pm ^{0.049}_{0.53}$\|	0.001\|$\pm ^{0.140}_{0.001}$\|	13.71	2.10	14.11	2.38	0.95

	α	log₁₀(M₀)	σ_cut	log₁₀(M_cut)	\|$\chi ^2 / \text{dof}$\|	log₁₀(M_eff)	b_g	f_c
B singles	1.3\|$\pm ^{0.031}_{0.028}$\|	13.43±0.03	0.21\|$\pm ^{0.17}_{0.14}$\|	12.23	4.04	13.42	1.34	0.77
D singles	1.54\|$\pm ^{0.039}_{0.042}$\|	13.46\|$\pm ^{0.043}_{0.041}$\|	0.18\|$\pm ^{0.17}_{0.15}$\|	11.81	11.25	13.41	1.10	0.87
B pairs	0.96\|$\pm ^{0.031}_{0.035}$\|	14.89\|$\pm ^{0.038}_{0.042}$\|	0.63\|$\pm ^{0.25}_{0.26}$\|	14.07	1.85	14.06	2.17	0.93
D pairs	1.41\|$\pm ^{0.053}_{0.051}$\|	14.58\|$\pm ^{0.052}_{0.047}$\|	0.34 ± 0.16	13.46	3.04	13.80	1.91	0.92
L0.2	0.83\|$\pm ^{0.044}_{0.49}$\|	15.40\|$\pm ^{0.037}_{0.034}$\|	0.78\|$\pm ^{0.35}_{0.26}$\|	14.15	0.46	13.94	1.88	0.98
L0.8	1.83\|$\pm ^{0.057}_{0.055}$\|	14.85\|$\pm ^{0.049}_{0.53}$\|	0.001\|$\pm ^{0.140}_{0.001}$\|	13.71	2.10	14.11	2.38	0.95

In the middle row of Fig. 12, we display the ratio of the measured correlation functions (points) and the best-fitting HOD models (curves) to the same measurements from the parent ZI sample. As discussed previously, these ratios are the square of the relative bias between the two subsamples. Interestingly, in the left panel we see that the bright single galaxies have a larger bias value, which increases towards the halo centre, while the dim single galaxies show a lower bias at all scales, which decreases towards the halo centre. This is not surprising if we recall that the parent sample contains galaxies of all magnitudes, so that we are seeing luminosity-dependent bias.

In the middle panel, we see the pairs follow a similar trend to their corresponding single galaxy sample; however we now see a stronger deviation in the data for the two-halo term. Despite larger error bars that can introduce uncertainties in the HOD model fitting, this trend is still seen within the 1σ deviations. We interpret this feature as an indication that bright galaxy pairs are more strongly clustered at scales larger than the halo size, thus they display a stronger bias on two-halo scales. In the right-hand panel of the middle row, we see that the relative bias of the galaxy pairs with large luminosity contrast (orange) to the parent sample is nearly uniform, while the galaxy pairs with similar luminosities (magenta) deviates both at the halo centre and in the transition region.

The bottom panel of Fig. 12 displays the best-fitting HODs for the six sub-samples. We find the bright samples have larger M₀ and M_cut values than the dim samples. We note that the slope of all of the galaxy pairs’ HODs are smaller than that of the single galaxies, thus these pairs are less likely to form larger haloes than single galaxies. Therefore, the fraction of galaxy pairs at the centre of a halo is larger than fraction of single galaxies. The weighted fractions of the central galaxies and galaxy pairs from the six sub-samples (B singles, D singles, B pairs, D pairs, L0.2, and L0.8) are 0.77, 0.87, 0.93, 0.92, 0.98, and 0.95, respectively.

In Table 5, we present the HOD model fit parameters and the derived values of the effective halo mass, galaxy bias factor, and the fraction of central galaxies or galaxy pairs. Interestingly, we find that the L0.8 sub-sample has a much larger bias than the L0.2 sub-sample, although they have similar measured angular correlation functions. In fact, the b_g for the L0.2 sub-sample is similar to that of the ZI pair sample, which implies that large luminosity contrast pairs occupy similar dark matter haloes as normal pairs, while the large b_g for the L0.8 sub-sample implies that pairs with similar luminosities occupy much larger dark matter haloes than the average ZI galaxy pair.

CONCLUSIONS

We have measured the angular, two-point correlation function for galaxies drawn from three volume-limited samples of SDSS DR7: (ZI) 0.0 < z < 0.3, M_r < −19.0, (ZII) 0.0 < z < 0.3, M_r < −19.5, and (ZIII) 0.3 < z < 0.4, M_r < −19.5. We also have selected isolated galaxy pairs from these three volume-limited galaxy samples, and have measured the angular two-point correlation function for these galaxy pairs. By modelling the angular two-point correlation function, we have computed, for the first time, the best-fitting halo model for photometrically selected isolated galaxy pairs. These models quantify that the galaxy pairs have larger effective mass and higher bias values than single galaxies. The central fraction for galaxy pairs is also higher than for single galaxies, which implies that galaxy pairs preferentially reside in dark matter haloes as central rather than satellite galaxies.

Furthermore, we have explored the dependence of the correlation function and best-fitting HOD models on redshift, galaxy type, and luminosity. We find that (1) early–early pair galaxy pairs have stronger clustering than late–late type galaxy pairs, (2) bright galaxy pairs have stronger clustering than dim galaxy pairs, and that (3) the clustering of large luminosity contrast pairs is similar (within the 1σ error bars) to that of small luminosity contrast pairs.

We can directly compare our measured galaxy angular two-point correlation functions and associated best-fitting HOD models to the results from Ross & Brunner (2009), who performed a similar best-fitting HOD model analysis for galaxies selected from the SDSS DR5. The galaxy samples they use have similar luminosity and redshift cuts that approximate ours (e.g. their Z3 sample corresponds to our ZII sample). We also note that their sample shares a similar median redshift with our catalogue (ZII: |$\bar{z}\ \sim 0.21$|⁠, Z3: |$\bar{z}\ \sim 0.25$|⁠), strengthening the comparison.

First, we note that the effective mass for single galaxies in their Z3 sample is M_eff = 10^13.11 h⁻¹ M_⊙, where we find a larger value of M_eff = 10^13.45 h⁻¹ M_⊙. The discrepancy is likely a result of the differences in the number densities of the two samples; our number density for single galaxies in ZII is smaller (n_g ∼ 0.0053 h³ Mpc^{− 3}) than the number density for their Z3 galaxies (n_g ∼ 0.0102 h³ Mpc^{− 3}). Thus, our catalogue contains fewer low-mass haloes. The smaller n_g in our catalogue is a result of the stronger requirements we developed to create our clean galaxy catalogue that has minimal systematic effects (Wang et al. 2013). Secondly, the satellite galaxy fraction that we find for ZII, f_s = 1 − f_c = 0.24, is slightly higher than their results (f_{s, Z3} = 0.15), but this value is in agreement with other results (see e.g., fig. 5 Zheng, Coil & Zehavi 2007). We have luminosity thresholds of L/L_★ = 0.27 for M_r < −19.0 and L/L_★ = 0.42 for M_r < −19.5, which correspond to f_s =(0.29, 0.24) for single galaxies in our ZI and ZII samples.

On the other hand, we do not see the local minimum for the late-type single galaxies in our best-fitting HOD model as seen previously (Ross & Brunner 2009; Zehavi et al. 2011). There are two reasons for this difference. First, we use different HOD models for the late-type galaxies. For example, Zehavi et al. (2011) separate the central and satellite galaxies and measure the fractions of late-type galaxies seen in each case, while we simply apply the HOD model to the late–late galaxy pairs directly. Our HOD models for the late–late galaxy pairs are excellent fits, which implies that our model is sufficient enough to describe the distributions of late–late type galaxy pairs within dark matter haloes. Secondly, we only compute HOD fits to galaxy pairs within our catalogue. The local minimum for the late-type single galaxies comes from low-mass haloes, where the fraction of the single central late-type galaxy decreases as the mass increases. However, the halo mass is insufficient to host satellite galaxies as required to increase the HOD. On the other hand, for the late–late type galaxy pairs, the parent haloes are massive enough to host additional galaxies, and they thus produce similar HOD models as the other sub-samples, which do not include an obvious inflection point.

Our clustering measurements of the early–early galaxy pairs show that they cluster more strongly than the late–late galaxy pairs. This is in agreement with the results from Hearin & Watson (2013) and Watson et al. (2014) who studied the co-evolution of galaxies and haloes. In their approach, they introduce a parameter, z_starve, that quantifies the epoch in the halo's mass assembly history at which the star formation in the resident galaxies of the halo becomes inefficient. By implementing an age distribution match, they find z_starve, on general, is larger for redder (i.e. our early-type) galaxies than bluer (i.e. our late-type) galaxies, which implies that redder galaxies tend to reside in older haloes. As a result, we can expect that early–early galaxy pairs also preferentially reside in more massive haloes where the halo clustering strength is larger, which is what we observe.

We note this work only considered angularly selected isolated galaxies, which can suffer from line-of-sight contamination. Any interloper galaxies will systematically decrease the correlation functions that we use for the HOD modelling, and will, therefore, lower the final HOD statistics. One technique to overcome this limitation would be to adopt accurate photometric redshift probability distribution functions (PDFs; Carrasco Kind & Brunner 2013, 2014) to place stronger spatial constraints on galaxy pairs within our sample. The use of photometric redshift PDFs will allow for a more reliable determination of the evolution of galaxy pairs within dark matter haloes.

Finally, we note that we have cut our galaxy catalogue into several subsamples: individual galaxies, galaxy pairs, and galaxies by type and luminosity. We have not, however, measured the cross-correlations between these subsamples. Such an investigation would be of direct physical interest as they will likely yield important insights into the clustering dependence of galaxy groups on richness, redshift, type, and luminosity. We intend to conduct a detailed study of these relationships once we have also more thoroughly addressed the projection effect issues.

The authors acknowledge support from the National Science Foundation Grant No. AST-1313415. RJB has been supported in part by the Institute for Advanced Computing Applications and Technologies faculty fellowship at the University of Illinois. This work also used resources from the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation grant number OCI-1053575. We also thank Ashley Ross and Chris Blake for helpful discussions and comments that helped this work.

Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS Web Site is http://www.sdss.org/.

The SDSS is a joint project of the American Museum of Natural History, Astrophysical Institute Potsdam, University of Basel, University of Cambridge, Case Western Reserve University, University of Chicago, Drexel University, Fermilab, the Institute for Advanced Study, the Japan Participation Group, Johns Hopkins University, the Joint Institute for Nuclear Astrophysics, the Kavli Institute for Particle Astrophysics and Cosmology, the Korean Scientist Group, the Chinese Academy of Sciences (LAMOST), Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, Ohio State University, University of Pittsburgh, University of Portsmouth, Princeton University, the United States Naval Observatory, the University of Washington.

1

http://lcdm.astro.illinois.edu/code

REFERENCES

Abazajian

K. N.

et al. ,

ApJS

,

2009

, vol.

182

pg.

543

Crossref

Search ADS

Berlind

A. A.

et al. ,

ApJS

,

2006

, vol.

167

pg.

1

Crossref

Search ADS

Blake

C.

Collister

A.

Lahav

O.

,

MNRAS

,

2008

, vol.

385

pg.

1257

Crossref

Search ADS

Brunner

R. J.

Lubin

L. M.

,

AJ

,

2000

, vol.

120

pg.

2851

Crossref

Search ADS

Bullock

J. S.

Kolatt

T. S.

Sigad

Y.

Somerville

R. S.

Kravtsov

A. V.

Klypin

A. A.

Primack

J. R.

Dekel

A.

,

MNRAS

,

2001

, vol.

321

pg.

559

Crossref

Search ADS

Carrasco

Kind M.

Brunner

R. J.

,

MNRAS

,

2013

, vol.

432

pg.

1483

Crossref

Search ADS

Carrasco

Kind M.

Brunner

R. J.

,

MNRAS

,

2014

, vol.

438

pg.

3409

Crossref

Search ADS

Conroy

C.

Wechsler

R. H.

Kravtsov

A. V.

,

ApJ

,

2006

, vol.

647

pg.

201

Crossref

Search ADS

Croton

D. J.

et al. ,

MNRAS

,

2006

, vol.

365

pg.

11

Crossref

Search ADS

Dolence

J.

Brunner

R.

,

9th LCI International Conference on High-Performance Clustered

,

2008

Computing

Dressler

A.

,

ApJ

,

1980

, vol.

236

pg.

351

Crossref

Search ADS

Farouki

R.

Shapiro

S. L.

,

ApJ

,

1980

, vol.

241

pg.

928

Crossref

Search ADS

Foreman-Mackey

D.

Hogg

D. W.

Lang

D.

Goodman

J.

,

PASP

,

2013

, vol.

125

pg.

306

Crossref

Search ADS

Fukugita

M.

Ichikawa

T.

Gunn

J. E.

Doi

M.

Shimasaku

K.

Schneider

D. P.

,

AJ

,

1996

, vol.

111

pg.

1748

Crossref

Search ADS

Gunn

J. E.

Gott

J. R.

III

,

ApJ

,

1972

, vol.

176

pg.

1

Crossref

Search ADS

Hayes

B.

Brunner

R.

,

MNRAS

,

2013

, vol.

428

pg.

3487

Crossref

Search ADS

Hearin

A. P.

Watson

D. F.

,

MNRAS

,

2013

, vol.

435

pg.

1313

Crossref

Search ADS

Hearin

A. P.

Zentner

A. R.

Berlind

A. A.

Newman

J. A.

,

MNRAS

,

2013

, vol.

433

pg.

659

Crossref

Search ADS

Hickson

P.

,

ApJ

,

1982

, vol.

255

pg.

382

Crossref

Search ADS

Infante

L.

et al. ,

ApJ

,

2002

, vol.

567

pg.

155

Crossref

Search ADS

Jing

Y. P.

Mo

H. J.

Boerner

G.

,

ApJ

,

1998

, vol.

494

pg.

1

Crossref

Search ADS

Kang

X.

Jing

Y. P.

Mo

H. J.

Börner

G.

,

ApJ

,

2005

, vol.

631

pg.

21

Crossref

Search ADS

Kauffmann

G.

White

S. D. M.

Heckman

T. M.

Ménard

B.

Brinchmann

J.

Charlot

S.

Tremonti

C.

Brinkmann

J.

,

MNRAS

,

2004

, vol.

353

pg.

713

Crossref

Search ADS

Kravtsov

A. V.

Berlind

A. A.

Wechsler

R. H.

Klypin

A. A.

Gottlöber

S.

Allgood

B.

Primack

J. R.

,

ApJ

,

2004

, vol.

609

pg.

35

Crossref

Search ADS

Lacey

C.

Cole

S.

,

MNRAS

,

1993

, vol.

262

pg.

627

Crossref

Search ADS

Landy

S. D.

Szalay

A. S.

,

ApJ

,

1993

, vol.

412

pg.

64

Crossref

Search ADS

Lorrimer

S. J.

Frenk

C. S.

Smith

R. M.

White

S. D. M.

Zaritsky

D.

,

MNRAS

,

1994

, vol.

269

pg.

696

Crossref

Search ADS

Navarro

J. F.

Frenk

C. S.

White

S. D. M.

,

ApJ

,

1997

, vol.

490

pg.

493

Crossref

Search ADS

Nulsen

P. E. J.

,

MNRAS

,

1982

, vol.

198

pg.

1007

Crossref

Search ADS

Peacock

J. A.

Smith

R. E.

,

MNRAS

,

2000

, vol.

318

pg.

1144

Crossref

Search ADS

Peebles

P. J. E.

,

The Large-Scale Structure of the Universe

,

1980

Princeton, NJ

Princeton Univ. Press

pg.

435

Google Scholar

Google Preview

OpenURL Placeholder Text

WorldCat

Press

W. H.

Schechter

P.

,

ApJ

,

1974

, vol.

187

pg.

425

Crossref

Search ADS

Ross

A. J.

Brunner

R. J.

,

MNRAS

,

2009

, vol.

399

pg.

878

Crossref

Search ADS

Sheth

R. K.

Mo

H. J.

Tormen

G.

,

MNRAS

,

2001

, vol.

323

pg.

1

Crossref

Search ADS

Springel

V.

White

S. D. M.

Tormen

G.

Kauffmann

G.

,

MNRAS

,

2001

, vol.

328

pg.

726

Crossref

Search ADS

Strateva

I.

et al. ,

AJ

,

2001

, vol.

122

pg.

1861

Crossref

Search ADS

Tal

T.

Wake

D. A.

van Dokkum

P. G.

,

ApJ

,

2012a

, vol.

751

pg.

L5

Crossref

Search ADS

Tal

T.

Wake

D. A.

van Dokkum

P. G.

van den Bosch

F. C.

Schneider

D. P.

Brinkmann

J.

Weaver

B. A.

,

ApJ

,

2012b

, vol.

746

pg.

138

Crossref

Search ADS

Tinker

J. L.

Weinberg

D. H.

Zheng

Z.

Zehavi

I.

,

ApJ

,

2005

, vol.

631

pg.

41

Crossref

Search ADS

Toomre

A.

Toomre

J.

,

ApJ

,

1972

, vol.

178

pg.

623

Crossref

Search ADS

Wang

W.

White

S. D. M.

,

MNRAS

,

2012

, vol.

424

pg.

2574

Crossref

Search ADS

Wang

Y.

Brunner

R. J.

Dolence

J. C.

,

MNRAS

,

2013

, vol.

432

pg.

1961

Crossref

Search ADS

Watson

D. F.

Berlind

A. A.

Zentner

A. R.

,

ApJ

,

2011

, vol.

738

pg.

22

Crossref

Search ADS

Watson

D. F.

Hearin

A. P.

Berlind

A. A.

Becker

M. R.

Behroozi

P. S.

Skibba

R. A.

Reyes

R.

Zentner

A. R.

,

2014

preprint (arXiv e-prints)

White

S. D. M.

Frenk

C. S.

,

ApJ

,

1991

, vol.

379

pg.

52

Crossref

Search ADS

White

S. D. M.

Rees

M. J.

,

MNRAS

,

1978

, vol.

183

pg.

341

Crossref

Search ADS

Yang

X.

Mo

H. J.

van

den Bosch F. C.

Pasquali

A.

Li

C.

Barden

M.

,

ApJ

,

2007

, vol.

671

pg.

153

Crossref

Search ADS

Yang

X.

Mo

H. J.

van den Bosch

F. C.

Zhang

Y.

Han

J.

,

ApJ

,

2012

, vol.

752

pg.

41

Crossref

Search ADS

York

D. G.

et al. ,

AJ

,

2000

, vol.

120

pg.

1579

Crossref

Search ADS

Zehavi

I.

et al. ,

ApJ

,

2004

, vol.

608

pg.

16

Crossref

Search ADS

Zehavi

I.

et al. ,

ApJ

,

2011

, vol.

736

pg.

59

Crossref

Search ADS

Zheng

Z.

et al. ,

ApJ

,

2005

, vol.

633

pg.

791

Crossref

Search ADS

Zheng

Z.

Coil

A. L.

Zehavi

I.

,

ApJ

,

2007

, vol.

667

pg.

760

Crossref

Search ADS

Download all slides

Month:	Total Views:
November 2016	1
July 2017	1
August 2017	1
November 2017	1
December 2017	10
January 2018	1
February 2018	8
March 2018	8
April 2018	7
May 2018	6
June 2018	6
July 2018	10
August 2018	9
September 2018	1
October 2018	3
November 2018	2
December 2018	2
January 2019	3
February 2019	4
March 2019	5
April 2019	6
May 2019	6
June 2019	2
July 2019	7
August 2019	4
September 2019	4
October 2019	3
November 2019	7
December 2019	5
January 2020	3
February 2020	6
March 2020	3
April 2020	4
June 2020	5
July 2020	3
August 2020	5
September 2020	4
October 2020	3
November 2020	1
December 2020	2
February 2021	2
March 2021	4
April 2021	3
May 2021	1
June 2021	5
July 2021	5
August 2021	2
September 2021	3
October 2021	4
November 2021	4
January 2022	1
February 2022	6
March 2022	1
April 2022	4
June 2022	1
July 2022	11
August 2022	9
September 2022	6
October 2022	7
November 2022	4
December 2022	1
January 2023	2
February 2023	1
March 2023	3
April 2023	1
May 2023	2
June 2023	1
July 2023	4
August 2023	7
November 2023	2
December 2023	7
January 2024	9
February 2024	13
March 2024	10
April 2024	7
May 2024	3
July 2024	2
August 2024	2
October 2024	9
November 2024	8
December 2024	1
January 2025	3
February 2025	6
March 2025	20

Article Contents

On the clustering of compact galaxy pairs in dark matter haloes

Abstract

INTRODUCTION

DATA

Volume-limited sub-samples

COMPACT GALAXY GROUPS

Selection criteria

Compact group catalogues

Estimating group redshifts

CLUSTERING PROPERTIES

Methodology

The correlation function estimators

Errors and covariance matrices

The angular correlation functions of galaxies and isolated galaxy pairs

HOD MODEL FRAMEWORK

The one-halo term

The two-halo term

Halo mass function

Halo occupation distribution model

HOD FOR INDIVIDUAL GALAXIES AND GALAXY PAIRS

Model fits for the three samples

Redshift dependence

Type dependence

Luminosity dependence

CONCLUSIONS

REFERENCES

Citations

Views

Altmetric

Email alerts

Astrophysics Data System

Citing articles via

Latest

Most Read

Most Cited

Article Contents

On the clustering of compact galaxy pairs in dark matter haloes

Abstract

INTRODUCTION

DATA

Volume-limited sub-samples

COMPACT GALAXY GROUPS

Selection criteria

Compact group catalogues

Estimating group redshifts

CLUSTERING PROPERTIES

Methodology

The correlation function estimators

Errors and covariance matrices

The angular correlation functions of galaxies and isolated galaxy pairs

HOD MODEL FRAMEWORK

The one-halo term

The two-halo term

Halo mass function

Halo occupation distribution model

HOD FOR INDIVIDUAL GALAXIES AND GALAXY PAIRS

Model fits for the three samples

Redshift dependence

Type dependence

Luminosity dependence

CONCLUSIONS

REFERENCES

Citations

Views

Altmetric

Email alerts

Astrophysics Data System

Citing articles via

Latest

Most Read

Most Cited

This Feature Is Available To Subscribers Only