The Occurrence of Compact Groups of Galaxies through Cosmic Time

As it is not possible to observe a sample of CGs spanning all redshifts, the only available path forward is to utilize cosmic simulations of galaxy evolution. The Millennium Simulation (Springel et al. 2005; Lemson & Virgo Consortium 2006) provides a tool to study the history of CGs in the universe. It used 2160³ particles with mass M = 8.6 × 10⁸ h⁻¹ M_⊙ to trace the evolution of a comoving cube with side length 500 h⁻¹ Mpc, where h is the Hubble constant parameterization such that H₀ = 100 h km s⁻¹ Mpc⁻¹. The Millennium Simulation used a ΛCDM model with cosmological parameters Ω_M = 0.25, Ω_b =0.045, Ω_Λ = 0.75, and h = 0.73, consistent with observational results from the Carnegie Hubble Program (Freedman et al. 2012) and the WMAP (Hinshaw et al. 2013) mission.¹⁰

The Millennium Simulation used only dark matter particles to which associated properties were later assigned. Dark matter halos were identified using a friends-of-friends algorithm as described in Springel et al. (2005). Galaxies were added to the simulation post facto using semianalytic techniques (e.g., De Lucia & Blaizot 2007; Guo et al. 2010, 2011, 2013). Galaxies are initially associated with individual dark matter halos, which may become subhalos of larger structures over time. The techniques of De Lucia & Blaizot (2007), for example, expanded on the methods of Croton et al. (2006) and incorporated the effects of rapid star formation and gas loss in galaxy mergers, changes in galaxy properties with varying stellar mass functions and stellar populations, and attenuation due to dust.

The simulation results were saved in a collection of 64 redshift "snapshots." The identified dark matter halos were then assigned galaxy properties using the semianalytic techniques discussed in the references above. The snapshots, combined with the high resolution of the simulation and the sophisticated semianalytic galaxy formation and evolution models, provide an excellent tool for studying the history of CGs over cosmic time. Throughout this paper, we adopt the De Lucia & Blaizot (2007) catalog of galaxies created through this semianalytic approach.

A number of previous studies have utilized the Millennium Simulation galaxy catalogs to investigate CGs. These studies have been limited to relatively low redshifts to facilitate comparison to observations, such as the well-known Hickson catalog (Hickson 1982; Hickson et al. 1992). By using a light cone in the Millennium Simulation galaxy catalogs (Henriques et al. 2012), it has also been possible to create "mock" catalogs of CGs, which can be compared to observations and are subject to analogous limitations (e.g., interloping foreground or background galaxies; McConnachie et al. 2008; Díaz-Giménez & Mamon 2010; Díaz-Giménez & Zandivarez 2015; Farhang et al. 2017).

By comparing mock CGs in projection in the Millennium Simulation to observed Hickson compact groups (HCGs), McConnachie et al. (2008) found that ∼29% of mock CGs identified from "images" alone are physically dense systems of three or more galaxies, and that the remaining projected CGs are the result of chance alignments. Díaz-Giménez & Mamon (2010) found that the fractions of 3D dense groups among 2D CGs depend on the semianalytical model (SAM) used, the consideration of galaxy blending, and the criterion used to define a dense group. Their Table 5 indicates that, for particle-based group definitions and their optimal definition of a dense group combining line-of-sight size and elongation, the fractions of CGs selected in projection that are physically dense range from 20% for the De Lucia & Blaizot (2007) SAM to 43% for the Bower et al. (2006) SAM. These fractions rise to over half and up to 3/4, respectively, for CGs that survive the velocity filter. The differences with the results from the McConnachie team arise from different definitions of what constitutes a physically dense group.

Finally, by comparing CGs observed in the Two Micron All Sky Survey (2MASS) catalog to those in a mock light cone from a Millennium Simulation galaxy catalog, Díaz-Giménez & Zandivarez (2015) found that only about a third of CGs are embedded in larger structures, i.e., the majority are truly isolated systems, yet Díaz-Giménez & Mamon (2010) find that only 11% of their velocity-filtered CGs are constituted by the Friends-of-Friends group in 3D and hence are isolated.

We developed an open-source, publicly available code¹¹ to identify and characterize CGs in Millennium Simulation galaxy catalogs. This process is subdivided into several simple algorithms with tunable parameters defining the properties of galaxy groups considered "compact," where we use "compact" to refer to the 3D spatial separation. The general outline of our methodology (see also Figure 1) is as follows:

• 1.  
  Download galaxy data from a chunk (snapshot at a given redshift) of the Millennium Simulation, utilizing the De Lucia & Blaizot (2007) catalog.
• 2.  
  Use a tunable clustering algorithm to identify clusters and groups of galaxies in the simulation chunk.
• 3.  
  Filter out identified group members that are dwarf or "fly-by" galaxies.
• 4.  
  Measure statistics of the discovered galaxy groups.
• 5.  
  Use tunable parameters to filter the identified galaxy groups to select only "CGs."
• 6.  
  Save the statistics of identified CGs and their galaxy members.

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These algorithms contain several tunable parameters that can be used to adjust how CGs are selected from the Millennium Simulation galaxy data. These are briefly outlined here and described in greater detail in the following sections:

• 1.  
  Neighborhood, nh—the neighborhood parameter of the DBScan algorithm (similar to the "linking length" parameter of friends-of-friends algorithms).
• 2.  
  Minimum number of members, ${N}_{\min }$—the minimum number of nondwarf galaxy members an identified candidate group must have in order to be considered a CG.
• 3.  
  Maximum shell density ratio, sr—the maximum allowable ratio of the virial mass¹² density of galaxy subhalos in a shell surrounding a CG to the virial mass density of galaxy subhalos within that CG.
• 4.  
  Galaxy mass ratio, mr—the minimum allowable virial mass ratio of the secondary+tertiary dark matter subhalos to the primary galaxy dark matter subhalos in a group.
• 5.  
  Critical velocity difference, $| {\rm{\Delta }}v|$—the velocity difference between a galaxy and the median velocity of a CG for the galaxy to be considered a "fly-by" and not a member of the CG.
• 6.  
  Dwarf limit—the stellar mass limit for a galaxy to be considered a dwarf galaxy and thus excluded from consideration.

To identify clusters and groups of galaxies, we used DBscan, which employs an intuitive, number-density-based clustering approach (Ester et al. 1996). DBscan is very adept at handling arbitrary shapes, as it does not depend on any density-smoothing processes. In clustering terminology, each galaxy constitutes a "node." The algorithm works by searching the defined radial neighborhood of each node and counting the number of neighbors that are within this radius. If the neighborhood contains more than, or equal to, the minimum number of members, then it is flagged. This is repeated for each node/member until all nodes within the neighborhood have been searched, thus classifying the resulting flagged collection of nodes as a "cluster" (Birant & Kut 2006), which in our case is an identified CG. The group's center is defined as the median of the positions of the identified galaxy members. The group radius is then the greatest distance from this center to any of the member galaxies.

Thus, DBscan requires two initial parameters, neighborhood and minimum number of members. Neighborhood is the radius starting on each identified node and used to search for adjacent nodes. Minimum number of members specifies the required minimum number (density) of galaxies that are in the neighborhood of an identified node (a "central" galaxy), for the system to be considered a CG.

One caveat about the DBscan algorithm is that it is only deterministic under certain conditions. The minimum number of members must be less than or equal to three. If this condition is met, which it is in this study, then the algorithm is fully deterministic.

We selected our CGs in 3D space from the SAM outputs in a similar manner to how Hickson selected his CGs in redshift space. The selection criteria Hickson used were based on compactness, relative luminosity, minimum number of members, and isolation. The particular parameters were chosen to identify dense systems of multiple galaxies while excluding substructure within galaxy clusters.

Using imaging alone, Hickson (1982) defined his CGs with a membership of at least four galaxies (although subsequent studies relaxed this restriction to three members; Barton et al. 1996). The requirement for compactness was that the surface brightness averaged over the smallest circle enclosing the group galaxy centers be brighter than 26 mag arcsec⁻². The isolation criterion was that there should be no galaxies brighter than the faintest galaxy within 3 times the angular radius of the group. This aimed to exclude CGs that are associated with galaxy clusters or other regions whose "external" galaxies may strongly influence group galaxies. As these restrictions are subject to projection effects, the presence of foreground or background galaxies can influence the initial identification of CGs (by falsely including or excluding them; Mamon 1986; McConnachie et al. 2008; Brasseur et al. 2009; Díaz-Giménez & Mamon 2010).

In addition, the spectroscopic study of Hickson et al. (1992) introduced the further criterion that an "accordant" member galaxy should be within 1000 km s⁻¹ of the median group velocity. This criterion was meant to remove distant foreground and/or background galaxies that appear in projection along the line of sight.

Our selection criteria were motivated by those used for Hickson's original catalog, modified to take full advantage of the 3D information in the Millennium Simulation galaxy catalogs produced by the SAM of De Lucia & Blaizot (2007).

We describe subsequently in this section our choices for the input parameter values to DBScan based on the known properties of HCGs that have been determined empirically since their identification. The nh parameter is the search radius around a galaxy, effectively determining the degree of group compactness, and is the initial parameter used by the DBscan algorithm to identify clustered galaxies. sr is a group isolation criterion, while mr is a criterion characterizing the dominance of the most massive galaxy in a group among the top-ranked group galaxies.

2.4.1. Neighborhood Parameter: nh

2.4.2. Minimum Number of Members: ${N}_{{\min }}$

2.4.3. Limit on Mass in Surrounding Shell: sr

2.4.4. Mass Ratio of Group Members: mr

2.4.5. Relative Velocity Restriction

2.4.6. Exclusion of Dwarf Galaxies

In our analysis we impose a galaxy mass threshold to exclude galaxies with stellar masses that may be too low. Specifically, only galaxies that have masses greater than 5 × 10⁸ h⁻¹ M_⊙ are included. This corresponds to a median dark matter halo mass of 3.4 × 10¹⁰ M_⊙ or ∼40 dark matter particles. Any "galaxies" with zero stellar mass are also removed from the analysis, as they are artifacts of the De Lucia & Blaizot (2007) SAM. As shown in Figure 2, even for redshifts of z ≲ 0.5, this results in the exclusion of ∼60% of the mass systems identified in the Millennium Simulation galaxy catalog. At earlier cosmic times, the relative fraction of such "dwarf" galaxies that are excluded owing to this mass cutoff becomes increasingly important, reaching more than 80% at z ≳ 5.

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2.4.7. Significance of Selection Criteria

In order to both mitigate the effect of a set of rigid selection criteria and also investigate how CG demographics may depend on these criteria, we vary nh, sr, and mr over a range of values about a "default" set (see Table 1). We first test each nondefault value by keeping all other criteria at their default values and then also use combinations of the most "lenient" and most "restrictive" parameter values, in order to constrain the most extreme CG populations. In the case of nh, lower values reduce the size of the search area for neighboring galaxies, so that more CGs will be selected, and conversely for higher values. For very large values, galaxy clusters will start to be included. Our default value is 50 h⁻¹ kpc (see Section 2.4.1), and we also use the values 25 and 75 h⁻¹ kpc.

Table 1.  Parameter Grid and Results

Parameter	nh	log(sr)	mr	Median Galaxy	Median Group	Median σv,3D	Max.	z	Ngroups	ngroups
Set				Separation	Radius	(3D)	Pop.	of Peak	at z = 0	at z = 0
	(${h}^{-1}$ kpc)			(${h}^{-1}$ kpc)	(${h}^{-1}$ kpc)	(km s−1)				(10−5 h3 Mpc−3)
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)	(11)
A	25	−4	0.10	19	17	99	0.18%	1.0	3854	3.1
B	75	−4	0.10	47	45	95	1.03%	1.9	16708	13.5
C	50	−3	0.10	37	34	102	0.83%	1.9	13940	11.3
D	50	−5	0.10	29	26	97	0.29%	1.5	5973	4.8
E	50	−4	0.05	37	34	118	1.27%	1.5	24134	19.5
F	50	−4	0.20	35	31	83	0.21%	2.4	3355	2.7
Default	50	−4	0.10	37	33	99	0.66%	1.9	11222	9.1
Restrictive	25	−5	0.20	18	17	82	0.05%	1.4	1032	0.8
Lenient	75	−3	0.05	57	53	119	3.13%	1.5	53288	43.1
HCGs	...	...	...	39	35	331	...	...	...	9.5

Note. Column (1): label of the observational or computational parameter set, characterized by the parameter values that were varied, as specified in Columns (2), (3). and (4). Columns (2)–(4): DBscan algorithm neighborhood radius, maximum mass–density ratio of galaxies in a shell surrounding a candidate group to the candidate group's galaxies, and minimum mass ratio of the second and third galaxies combined to the most massive galaxy in a candidate group, respectively. The remaining columns show results for HCGs or CGs identified in the De Lucia & Blaizot (2007) catalogs produced from the Millennium Simulation, following the adoption of the selection criteria in Columns (2)–(4). Column (5): median galaxy–galaxy separation in a group. Column (6): median group radius. The group radius is defined relative to the group center, taken to be the median of the positions of the identified galaxy members. The group radius is simply the greatest distance from this center to any of the member galaxies. Column (7): median 3D galaxy velocity dispersion, ${\sigma }_{v,3{\rm{D}}}\equiv \sqrt{{\sigma }_{v,x}^{2}+{\sigma }_{v,y}^{2}+{\sigma }_{v,z}^{2}}$. Column (8): maximum fraction of galaxies that were in a CG at the redshift given in column (9). Column (9): redshift at which the number of CGs was at its maximum. Column (10): number of CGs at the present time. Column (11): volume number density of CG galaxies at the present time.

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Our default value for log(sr) is −4, and we also use −3 and −5. For mr the default value is 0.10, and we also use 0.05 and 0.20.

To be able to analyze the full simulation in a time-efficient manner, we developed optimization strategies. The resulting time requirements are extreme: the full simulation catalog has over 26 million galaxies in its most populous snapshot. In order to make the algorithm more computationally efficient, we divide a given snapshot into roughly equivalent overlapping boxes. Each box has a side length of 102 ${h}^{-1}$ Mpc, corresponding to about 1/125 of the total simulation volume. Only the central 100 ${h}^{-1}$ Mpc were sampled in each box, since, when conducting the density analysis, a buffer zone of 1 ${h}^{-1}$ Mpc is needed at the edge. Without this buffer, the groups on the edge of the box would contain less volume in the spherical density analysis than required to determine a group's isolation. This process of division allowed us to run the algorithm in an extensively parallelized fashion on as many cores as desired.

To track the galaxies in any snapshot that are, or have ever been, in CGs, each galaxy had to be monitored throughout the simulation. The Millennium Simulation employs a standard naming convention for galaxies and also provides the descendant ID. Once the groups are identified in the main algorithm, they can then be traced through the simulation forward or backward in time to determine the fraction of galaxies living in CGs at any given time step.

Starting from the beginning of the simulation (cosmic time = 0), galaxies and their descendants are placed into a list, which is then searched for repeated galaxies (e.g., when a galaxy merges with another one). This process is repeated for every galaxy in a given treeID. The latter is a grouping assigned by De Lucia & Blaizot (2007) to categorize related groups of galaxies, so that all galaxies in a given CG will have the same treeID. By dividing the data set according to treeID, further extensive parallelization was achieved.

4.1.1. Mass Ratio of Group

The restriction on the mass ratio of secondary+tertiary to primary galaxies (mr) has a strong impact on the fraction of galaxies considered to be in CGs at z ≲ 5. A mass ratio of mr = 0.05 is the most lenient value and thus results in the largest fraction of galaxies in CGs. Conversely, mr = 0.20 is the most restrictive and reduces the relative number of galaxies considered to be in a CG by nearly an order of magnitude for z < 5 (see Figure 3). All three values of mr result in the fraction of galaxies in CGs peaking between z ∼ 1.5 and 2.4, with the more restrictive values of mr peaking at higher redshifts. The most liberal value of mr = 0.05 (with other parameters set to their default values, set E) results in a maximum fraction of galaxy membership in CGs of ∼1.3%.

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4.1.2. Shell Density Ratio

Varying the shell density ratio (sr) selection criterion resulted in similar patterns to those seen with respect to changing mr. Despite varying sr by two orders of magnitude, the resulting impact on the fraction of galaxies in CGs is not as strong as seen from only changing mr by a factor of four. However, for z < 5 different values of sr do strongly influence CG selection (Figure 4). For the most restrictive (i.e., most isolated) value of log(sr) = −5 (model D), the relative number of galaxies in CGs never reached values greater than ∼0.3%. The most liberal value of log(sr) = − 3, with other parameters at their default values (Model C), results in a peak fraction of galaxies in CGs of ∼0.83% at z = 1.9.

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4.1.3. Neighborhood

4.1.4. Most Lenient and Restrictive Parameter Sets

In addition to the relative number of galaxies that are in CGs at any given redshift, we can also determine the relative number of galaxies that are currently in or have ever been members of a CG over cosmological time.

Here we track the galaxies that are in a CG at any given instant and continue to count them as part of the total number of galaxies as the groups evolve, even if they should no longer be considered to belong to a CG, due to mergers of constituent galaxies causing the group to have fewer than three members. By tracking the CGs' descendants, we can determine the number and properties of galaxies in the present day that once were a part of a CG in their evolutionary history.

As shown in the right panel of Figure 5, the maximal fraction of galaxies in the present-day universe that have ever been part of a CG exceeds 10%. Unlike the rest of the parameter sets, the most lenient and most restrictive ones lead to a particularly large variation in the final percentage. The final percentages range from 0.4% to 16.1% across the full parameter range.

As can be seen in the right panels of Figures 3 and 6, both of the mr and nh criteria have a moderate effect on the final fraction. On the other hand, the right panel of Figure 4 shows that the sr criteria have little effect on the final fraction of galaxies that have ever been in CGs.

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Finally, we trace the evolution of z = 2 CGs down to z = 0. We find that the overwhelming majority (16,071 out of 16,797 CGs, or 96%) have merged into a single galaxy by z = 0. The distribution of these galaxies at z = 0 in color–magnitude space is shown in Figure 7. Taken at face value, this color–magnitude distribution is notably different from the general Sloan Digital Sky Survey (SDSS) sample presented by Blanton et al. (2005); the CG descendants show a clump of red and luminous galaxies and two distinct plumes that are not seen in the general SDSS population. However, we caution that there is no guarantee that the semianalytic model used will in general reproduce the colors of the SDSS population. Perhaps not surprisingly, this distribution is more similar to that observed for galaxies in CGs (Walker et al. 2013), but there remain clear differences—including the plume of CG descendants across a range in g − r color at an I-band magnitude of ∼−23. However, the caveat that applies to SDSS colors still applies here. Thus, these results tentatively and qualitatively suggest that the products of CG evolution may have properties statistically distinct from the general galaxy population, which warrants an in-depth follow-up study.

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Interestingly, we also identify a small minority of CGs (49 out of the 726 that have not merged by z = 0) that have at least one galaxy separated by 250 kpc, or more, from other group members. Such member galaxies are reminiscent of NGC 7320C, located in the northeast quadrant of Stephan's Quintet (HCG 92) and likely to have passed through the group a few times ∼10⁸ yr ago. It has been suggested that NGC 7320C is responsible for the tidal tail of NGC 7319 (Moles et al. 1997). Such galaxies may thus be transient group members that nevertheless could have a significant effect on CG galaxy evolution.

We compare the properties of the CGs that we found in the De Lucia & Blaizot SAM output to those of the CGs found via the observational work of Hickson and collaborators. In particular, we compare the space densities, the median radii, and the median 3D galaxy velocity dispersions of the groups, ${\sigma }_{v,3{\rm{D}}}\equiv \sqrt{{\sigma }_{v,x}^{2}+{\sigma }_{v,y}^{2}+{\sigma }_{v,z}^{2}}$. The results are summarized in Table 1.

The number of CGs identified in Hickson's catalog can be compared to the number of CGs identified in the Millennium Simulation galaxy catalogs for specific parameter sets by making a few assumptions. While Hickson et al. (1992) identified a total of 92 groups¹³ with at least three accordant members within z ≤ 0.14 over 67% of the sky, the median redshift of groups in the catalog is z = 0.03, which suggests that the catalog becomes increasingly incomplete for larger values of z. Further, Díaz-Giménez & Mamon (2010) found that the velocity-filtered HCG catalog is only 8% complete even at z = 0.03. To this redshift, and over 67% of the sky, there are 47 detected HCGs with accordant velocities in Hickson et al. (1992). Therefore, the expected number of CGs over 67% of the sky out to z = 0.03 is N_HCG = 47/0.08 = 587.5, if we assume that the Hickson catalog is 8% complete out to this distance. On the other hand, the comoving volume to z = 0.03 is $V\,={({\rm{\Delta }}{\rm{\Omega }}/3)({D}_{L}/(1+z))}^{3}$, where D_L = 92.1 h⁻¹ Mpc is the luminosity distance to this redshift and ΔΩ = 8π/3 is the solid angle for a 67% sky coverage. This gives a completeness-corrected expected number density of HCGs to z = 0.03 of n_HCG = 2.9 × 10⁻⁴ h³ Mpc⁻³.

The volume of a snapshot of the Millennium Simulation is V_snap = (500 h⁻¹ Mpc)³ = 1.25 × 10⁸ h⁻³ Mpc³, and there is one snapshot at z = 0. Hence, if the simulation has the same volume density of groups as HCGs, we expect the total number of identified groups in the simulation to be N_sim = 36,250 at z = 0. This number is between the number produced by set E and the "lenient" parameter set (see Table 1, Column (10)). Thus, if one assumes that the space density of HCGs up to z = 0.03 is representative, a more lenient set of parameters is preferred.

The median projected group separation in the Hickson et al. (1992) catalog is R_HCG = 39 h⁻¹ kpc. Thus, if the 3D separation is 2π/5 larger than the projected 2D separation, then the expected 3D HCG galaxy separation is ∼49 h⁻¹ kpc. Several of the parameter sets produce median group radii that are within 20% of this value, including the "default," C, D, E, and F parameter sets. All of these parameter sets share a common neighborhood (nh) of 50 h⁻¹ kpc, unlike all the rest of the parameter sets.

While there are parameter sets that reasonably reproduce the observed space density and sizes of HCGs, the 3D velocity dispersion presents an issue. Hickson et al. (1992) observed a median 1D velocity dispersion of 200 km s⁻¹ and, after considering the uncertainties in the individual velocities, inferred a 3D dispersion of σ_v,3D,HCG = 331 km s⁻¹. The parameter sets tested here, however, all produce significantly smaller 3D velocity dispersions that are all <120 km s⁻¹, and typically ≲100 km s⁻¹. We note that for groups with only three members, on average, the 3D velocity dispersion we obtain is 81 km s⁻¹, whereas for groups with four members or more, on average, the 3D velocity dispersion is 159 km s⁻¹. This shows that, even if we had set ${N}_{\min }$ = 4, we would still not be able to reconcile our results with the HCG value of 331 km s⁻¹.

We have performed a series of further tests to assess whether any specific modifications in our approach might result in substantially different velocity dispersions, but we were only able to obtain negligible changes. Specifically, we compared results from DBscan both including and excluding dwarf galaxies. We also computed 3D velocity dispersions using the biased dispersion, unbiased dispersion, and gapper techniques, resulting in median velocity dispersions at z = 0 of ∼100, ∼125, and ∼150 km s⁻¹, respectively, which fall well short of the value observed for HCGs.

We postulate that this disagreement may be due to the inherent differences in selecting observed groups based on 2D spatial projections as opposed to actual 3D information available in the simulations. One hypothesis is that this key difference has its origin in the more restrictive selection criteria in 3D space, resulting in more tightly bound groups than the ones in the Hickson et al. (1992) catalog.