Ultimate age-dating method for galaxy groups; clues from the Millennium Simulations

Abstract

1 INTRODUCTION

The age determination for galaxy systems in the hierarchical structure formation is not trivial because, in this paradigm, more massive galaxy systems such as galaxy clusters are formed through the mergers of smaller galaxy systems and are thus generally young galaxy systems. A galaxy group, however, can be recently formed or forming while some could be relatively old if they have not being subject to a substantial merging with other galaxy systems. In fact, the answer to the question of age determination should come from the cosmological simulations where evolutionary history of galaxy systems can be studied. The cosmological dark matter simulations and the implemented semi-analytic galaxy models offer tools, needed to develop an insight into this subject.

Before these robust simulations became available in cosmological scales, an observational method for identification of old galaxy groups was developed in a pioneering study by Ponman et al. (1994). This was motivated by earlier numerical simulations suggesting that in most cases, members of compact galaxy groups could merge to form a single elliptical galaxy in a few billion years (Barnes 1989). An elliptical galaxy formed by the merger of such a group retains its X-ray emitting halo of hot gas, which is unaffected by merging (Ponman et al. 1994). Such groups were called fossil groups in which the essential observational tracer has been identified as the large luminosity gap between the two brightest group members.

According to the convention introduced by Jones et al. (2003), fossil groups have a magnitude gap of at least 2 mag within 0.5 virial radius and L_{X, bol} ≈ 10⁴² |$h_{50}^{-2}$| erg s⁻¹. Since then there have been many studies focused on the detailed characterization and properties of fossil groups (Khosroshahi, Jones & Ponman 2004; Sun et al. 2004; Ulmer et al. 2005; Khosroshahi, Ponman & Jones 2006b), based on X-ray and optical observations. However, due to limited number of adequate X-ray surveys and their suitability, most studies selected the magnitude gap as the fossil identifier in theoretical and observational studies (Yoshioka et al. 2004; Milosavljevic et al. 2006; Van den Bosch et al. 2007), hydrodynamic simulations (D'Onghia et al. 2005; Cui et al. 2011), dark matter simulations (Von Benda-Beckmann et al. 2008) and combined with semi-analytical models (SAMs) for galaxies (Sales et al. 2007; Díaz-Giménez, Muriel & Mendes de Oliveira 2008; Dariush et al. 2010). Khosroshahi, Ponman & Jones (2007) show that for a given optical luminosity, fossil groups are not only more X-ray luminous than the general population of galaxy groups, but also they have a more concentrated halo as well as hotter intergalactic medium (IGM) for a given halo mass. The study of scaling laws of fossil groups also show that they mostly follow the trend of galaxy clusters which is likely to be driven by their dynamical relaxed state, although Voevodkin et al. (2010) have shown that in the cluster regime there are no noticeable difference between the X-ray luminosity of the fossils and non-fossils for a given optical luminosity. In addition, some other studies support the same conclusion employing other cluster samples (Aguerri et al. 2011; Harrison et al. 2012; Girardi et al. 2014).

One of the largest cosmological simulations, the Millennium Simulation (Springel et al. 2005) joined with SAMs for galaxy formation, provide a useful tool to address open questions, regarding the age determination of the galaxy systems. Dariush et al. (2007) show that the luminosity gap is a fairly good indicator of the halo mass assembly, such that in galaxy groups with a large luminosity gap (Δm₁₂ ≥ 2), the haloes accumulate 50 per cent of their mass at current epoch by z = 1 and thus are relatively older than their counterparts with small luminosity gap. Dariush et al. (2010) introduce an alternative optical criterion Δm₁₄ ≥ 2.5, i.e. the luminosity gap between the first and fourth brightest galaxies within 0.5R₂₀₀, which is found to be more efficient in identifying early-formed groups than the conventional criterion, Δm₁₂ ≥ 2.0.

Following the discovery of the fossils groups with different masses, from surveys or serendipitous observations, there have been attempts to study and compare various halo, IGM and galaxy properties of fossil and non-fossil groups (Khosroshahi et al. 2007; Dupke et al. 2010; Miller et al. 2012). The velocity dispersion of the member galaxies in groups (Herbst et al. 2012; Madrid & Donzelli 2013), the globular cluster distribution and colour (Alamo-Martínez et al. 2012), the galaxy luminosity function (Cypriano, Mendes de Oliveira & Sodré 2006; Lieder et al. 2013), the AGN activities of the groups (Hess, Wilcots & Hartwick 2012; Miraghaee et al. 2014), merger history (Díaz-Giménez et al. 2008; Eigenthaler & Zeilinger 2013) and halo concentration (Khosroshahi et al. 2007; Deason et al. 2013) are amongst these properties. The basic argument upon which these studies are based, is the early formation of fossils groups. The older age of fossil group has been argued in the earlier studies (Ponman et al. 1994; Jones et al. 2003) and demonstrated in the cosmological simulations (D'Onghia et al. 2005; Dariush et al. 2007). However, there is still room to develop age-dating algorithms with a high efficiency using measurable parameters accessible from routine optical observations and other convenient methods.

In this paper, we first show the extent at which the large luminosity gap can discriminate old and the young galaxy groups. We then introduce some parameters in which the old and young galaxy groups can be identified more distinctively. We finally define a parameter space which consists of optically measurable parameters of group galaxies, that allow age dating with a very high efficiency.

Section 2 describes Millennium Simulation and SAM. In Section 3, we describe our multiparameter analysis to search for old and young groups using the optical measurements. Finally, in Section 4 we present summary of our result and give our conclusion.

2 THE SIMULATION

In this study, we use the public release of the Millennium Simulation (hereafter MS; Springel et al. 2005). The cosmological model adopted in simulation is a Λ cold dark matter (CDM) with the following parameters: Ω_m = 0.25, Ω_b = 0.045, Ω_Λ = 0.75, h=0.73, n=1,σ₈ = 0.9 [note that the value of σ₈ is assumed to be greater than its present value of 0.82 given by Wilkinson Microwave Anisotropy Probe 9 data (Hinshaw et al. 2013) that is not strongly affects in this study]. The simulation box (500 h^{− 1}Mpc)³ contains 2160³ particles and presents the mass resolution of 8.6× 10⁸h⁻¹ M_⊙.

The dark matter merger trees within each simulation snapshot (64 snapshots) are expanded approximately logarithmically in time between z = 127 and z = 0 and extracted from the simulation using a combination of friends-of-friends (Davis 1985) and SUBFIND (Springel et al. 2001) halo finders algorithm. The gas and stellar components of galaxies in dark matter haloes are constructed semi-analytically, based on difference in the phenomenological recipes.

For galaxy properties, we use Guo et al. (2011) SAM, in which the treatments of many of the physical processes have been improved in comparison to an earlier model by De Lucia & Blaizot (2007). This model provides good fits to the observed luminosity and stellar mass functions of galaxies, from the Sloan Digitized Sky Survey (SDSS; York et al. 2000) data and recent determinations of the abundance of satellite galaxies around the Milky Way and the clustering properties of galaxies as a function of stellar mass, as well. Data available in the Millennium data base (Lemson & Springel 2006), contains ∼51 000 haloes with masses above 10¹³h⁻¹ M_⊙ and ∼5 million galaxies from which we only select galaxies brighter than −14 in r-band absolute magnitude for completeness.

3 ANALYSIS AND RESULTS

3.1 Age dating based on luminosity gap

3.1.1 Δm₁₂

As mentioned earlier, the magnitude gap between the brightest and second brightest galaxies in a galaxy group is often used as an indicator of the dynamical age of groups of galaxy. Dariush et al. (2010) investigated the assembly of groups and clusters of galaxies using the Millennium dark matter simulation and semi-analytic catalogues of galaxies. The study aimed at verifying the argument that galaxy groups with a large magnitude gap are formed earlier than the galaxy groups with a small magnitude gap. They used mass accumulation of the group halo as a proxy for early/late halo formation. They selected galaxy groups and clusters at the present time with dark matter halo mass M(R₂₀₀) > 10¹³ h⁻¹ M_⊙, and trace their properties to z ≃ 1. In addition, they applied an X-ray luminosity criterion to keep those galaxy systems for which the X-ray luminosity L_{X, bol} > 0.25 × 10⁴² h^{− 2} erg s^{− 1} at redshift z = 0. They argued that while it is true that a large magnitude gap between the two brightest galaxies of a particular group often indicates that a large fraction of its mass was assembled at an early epoch, it is not a necessary condition. More than 90 per cent of fossil groups defined on the basis of their magnitude gaps (at any epoch between 0 < z < 1) cease to be fossils within 4 Gyr, mostly because other massive galaxies are assembled within their cores, even though most of the mass in their haloes might have been assembled at early times.

As a definition here (and Dariush et al. 2007), a galaxy group which formed more than 50 per cent of its total mass by z ≈ 1 is labelled as old. A group is labelled young if less than 30 per cent of its final mass is formed by z ≈ 1. In Fig. 1 top panel, we present the luminosity gap Δm₁₂ (within 0.5R₂₀₀) as a function of the brightest group galaxy (BGG) magnitude in r band, M_r(BGG), given for all 39 132 groups (i.e. groups with M(R₂₀₀) ≥ 10¹³ h⁻¹ M_⊙ and exist within both z = 0 and z = 1) using Guo et al. (2011) SAM at the present epoch (z = 0). The groups are colour-coded according to their α_{0, 1} parameters(defined as ‘age’ of systems), where α_{0, 1} = M_{z ≈ 1}/M_z = 0. The horizontal line subdivides groups into magnitude gap bins. Those with Δm₁₂ ≥ 2 are conventional fossil groups. Groups with Δm₁₂ ≤ 0.5 are labelled as control groups which are known to be young galaxy groups (Dariush et al. 2007, 2010). Vertical lines bin the groups according to the luminosity of their brightest galaxy (BGG) in 4 mag bins from −20.5 to −24.5. We note that the systems with large magnitude gap and faint BGGs are modest galaxies with some dwarf satellites, similar to the Milky Way. As these systems do not present galaxy groups, satisfying fossil groups condition, we limit our analysis to BGGs which are at least as bright as M_r < −21.5, i.e giant galaxies. This diagram shows that the galaxy luminosity gap combined with the luminosity of the BGG for all halo mass over 10¹³ M_⊙h⁻¹ is success to identify old groups with a probability of 61 per cent and young galaxy group with a probability of 92 per cent.

Figure 1.

Top: distribution of the galaxy groups in the plane of luminosity gap Δm₁₂ within 0.5R₂₀₀ and the r-band magnitude of the Brightest Group Galaxy, M_r(BGG), in the Millennium Simulations with Guo et al. (2011) semi-analytic model. Data point are colour-coded according to the ratio of the group halo mass at redshift z ≈ 1 to its mass at z = 0 (α_{0, 1}). The red box defines fossil groups region, e.g. groups dominated by a giant galaxy and Δm₁₂ ≥ 2, while the blue box defines control groups with Δm₁₂ ≤ 0.5). The plane has been subdivided into blocks within which the probability that the halo is old or young, is given. In this diagram, panels (5), (9) and (10) contain mostly old systems while panels (3), (4) and(8) are mostly occupied by young systems. By our definition, a group is old if its halo has over 50 per cent of its final mass at z = 1 and its young if this fraction is less than 30 per cent. Bottom: same as the top panel, for the exception of Δm₁₄ within 0.5R₂₀₀, is used (see the text), as an age indicator (Dariush et al. 2010). In this diagram, panels (5), (9) and (10) contain mostly old systems while panels (3), (4) and (8) are mostly occupied by young systems.

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3.1.2 Δm₁₄

The assembly time of a dark matter halo defined as the look-back time when its main progenitor reaches a mass that is more than 50 per cent of the present halo mass. Dariush et al. (2010) showed that the alternative optical criterion Δm₁₄ ≥ 2.5 (in the r band, within 0.5R₂₀₀ of the centre of the group) is more efficient in identifying early-formed groups than the conventional criterion Δm₁₂ ≥ 2.0 and it is more success to identify at least 50 per cent more early-formed groups. However, the conventional criterion (Δm₁₂ ≥ 2.0) performs marginally better at finding early-formed groups at the high-mass end of groups. Fig. 1 Bottom panel shows distribution of luminosity gap Δm₁₄ within 0.5R₂₀₀ combined with the BGG magnitude in r band. In this diagram, panels (5), (9) and (10) contain mostly old systems while the panels (3), (4) and (8) are mostly occupied by young systems. The mentioned panels can identify old groups with a probability of at most 74 per cent and young galaxy group with a probability, up to 88 per cent.

3.2 Other age indicators

3.2.1 Halo concentration

Considering the relation between halo mass and concentration, lower mass haloes universally form at earlier epochs relative to higher mass haloes (Gao et al. 2008; Prada et al. 2012). Therefore, the lower mass haloes must be more concentrated (Wechsler et al. 2002). Due to this close connection between halo concentration and assembly history; group haloes that formed most of their halo mass at early times are also more concentrated in comparison to the haloes with later formation epoch (Deason et al. 2013). Observationally, for a given optical luminosity, fossil groups have more concentrated haloes as well as hotter IGM for a given halo mass (Khosroshahi et al. 2007).

Using the MSs, Ludlow et al. (2012) demonstrated that the virial-to-half-mass radius, R₂₀₀/r_h in which r_h is the half-mass radius, is a reliable tracer for the halo concentration. We thus estimate the halo concentration as the ratio between the R₂₀₀ and half-mass radius for all candidates for old and young systems. As expected, in Fig. 2 , the haloes of early-formed groups show higher concentration compared to the same in late-formed groups. While the bimodal distribution of the halo concentration between the old and the young groups (Fig. 2) appears to be very attractive for discriminating these two type of groups, there are significant observational limitations.

Figure 2.

A comparison between the halo concentration (C) of old (red line) and young (blue dashed line) groups in MS using Guo et al. (2011) SAM for different halo masses. We estimate the halo concentration as the ratio between the R₂₀₀ and half-mass radius for all candidates for old and young systems. The old systems show higher halo concentration than the young galaxy systems.

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Observationally, the halo concentration can be obtained from any observations that can provide halo profile measurement, such as the X-ray observations of the hot gas, strong/weak gravitational lensing and galaxy distribution within the group/cluster halo.

The X-ray method of measuring the halo mass/density profile is built upon the hydrostatic equilibrium assumption where the gravitational contraction is balanced by the hot gas pressure. This requires measurements of the gas density and the temperature profile which can be obtained from X-ray observations. In practice, additional assumptions are made, such as the spherical symmetry (Khosroshahi et al. 2004; Humphrey et al. 2006) to obtain the halo mass/density profile (Fabricant, Rybicki & Gorenstein 1984; Pratt & Arnaud 2005; Khosroshahi et al. 2006a). Usually an NFW profile (equation 1) is fitted to the total gravitational mass density which results in the halo concentration parameter according to equation (2).

The gravitational lensing method of mass profile reconstruction is based on the observations of strong arc shaped features of a distant background source (Fort & Mellier 1994) or the observation of lens-induced shear on the shape of the source galaxies by a lens galaxy cluster (Miralda-Escude 1991; Kaiser & Squires 1993; Wilson, Kaiser & Luppino 2001). These methods are observationally demanding, requiring high-quality photometric and spectroscopic measurements, and can only provide a projected mass density for the lensing cluster (see Leauthaud et al. 2007, 2010). In particular, the case of galaxy groups is challenging due to a low lensing signal and stacking of groups may be necessary (Hoekstra et al. 2001). The lensing method of halo mass measurement is only efficient in the intermediate-redshift range (Smith et al. 2005; Oguri et al. 2010), while there are many sources of systematic errors in weak lensing (Mandelbaum et al. 2005; Johnston et al. 2007).

There are a limited number of observations which can provide halo concentration of galaxy groups and clusters with a reasonable uncertainty. One of the largest samples of galaxy clusters, studied as part of the LoCuSS project (Smith et al. 2010), points at limited statistics on galaxy clusters for which the halo concentration is measured and also a typically 20 per cent uncertainty in X-ray measurements of the halo concentration, C₅₀₀ (Sanderson, Edge & Smith 2009). Similarly, Khosroshahi et al. (2006a) in the study of fossil galaxy cluster reported a large uncertainty in the concentration, as high as ∼40 per cent. Thus, halo concentration measurements obtained from gravitational lensing and/or X-ray observations may be subject to large uncertainties (Mandelbaum et al. 2005, 2006a,b; Johnston et al. 2007; Oguri et al. 2010; Shan, Qin & Zhao 2010b; Shan et al. 2010a).

3.2.2 Luminosity decentring

Typically, galaxy formation models assume that the BGG is located at the centre of group haloes; however, Skibba et al. (2011) demonstrated that the BGG may not always be the central galaxy. In fact, merging systems are dynamically unrelaxed which can result in a significant separation between the brightest galaxy and the halo centre. This quantity has been used as a tracer of the dynamical age of the galaxy groups. For instance, Rasmussen et al. (2012) in a study of ongoing star formation of groups, use the separation of the BGG from the luminosity centroid of member galaxies in the group as an age indicator for the galaxy groups. The X-ray emission peak, or the centroid, and the mass centroid from the gravitational lensing observations, have also been used as indicators of the halo centre (Oguri et al. 2010; Dietrich et al. 2012).

One of the interesting properties of the fossil groups is revealed by their X-ray morphology. The X-ray peak coincides very well with the centre of the dominant giant elliptical galaxy (Khosroshahi et al. 2007). Relaxed morphology of the clusters or absence of substructures are good indicators of the dynamical state of the galaxy systems (Smith et al. 2005). Smith et al. (2010) show that clusters with dominant galaxy at the centre are the one with least substructure and are thus dynamically old.

In the absence of X-ray or lensing mass maps, one can resort to optical luminosity distribution. We thus define the luminosity weight of each groups as X_L = ∑X_iL_i/∑L_i, where L_i is the luminosity of galaxy in group in r band and X_i is the projected coordinate of each galaxy. The decentring parameter is thus defined as the projected physical separation between the luminosity centroid of the group and the location of the BGG. As shown in Fig. 3, younger systems show larger separation between BGG and the luminosity centroid of the galaxies in comparison to the old groups. This parameter can also be replaced with the separation between the BGG and other halo centroid indicators such as the one obtained from gravitational lensing and/or X-ray observations; however, our choice has the advantage of being optically measurable.

Figure 3.

The distribution of luminosity decentring the distance between the location of the BGG and the luminosity centroid (luminosity decentring) in various mass bins. Galaxy systems with a large luminosity decentring are dynamically unrelaxed and thus younger.

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3.3 Age dating based on optical observations: a four-dimensional parameter space

Age dating based on halo concentration and other halo parameters method will rarely be possible to apply in practice, thus we look for a method using only parameters which are commonly available.

A four-dimensional parameter space can be constructed based upon galaxy properties and using optical observations, only. It has been shown that the total luminosity of a galaxy group is a fair indicator of the halo mass (Tavasoli et al. 2012). Fig. 4 shows the correlation between groups total luminosity (within R₂₀₀) and halo mass. Using a linear fit to our data, we find that the mass ranges of 10¹³, 10^13.5 and 10¹⁴ h⁻¹ M_⊙ generally correspond to total luminosity ranges of 1.9 × 10¹⁰, 20.27 × 10¹⁰, 57.2 × 10¹⁰ h⁻² L_⊙, respectively. While the halo concentration is found to be a good age indicator, in this four-dimensional approach, we assume that only photometric observations of galaxy groups is available. These observations should be such that the two brightest groups members can be identified, for instance through spectroscopic observations. The total luminosity can be estimated through red sequence fitting or similar methods which allow statistical memberships identifications.

Figure 4.

The relation between total optical luminosity within R₂₀₀ for each semi-analytic group and its halo mass. The red dashed line, fitted to the data, implies that masses of 10¹³, 10^13.5 h⁻¹ M_⊙ correspond to total luminosities of 1.9 × 10¹⁰, 20.27 × 10¹⁰, 57.2 × 10¹⁰ h⁻² L_⊙, respectively.

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Fig. 5 shows the probability for a system to be considered as old (top) and young (bottom) based on the observable four-dimensional parameter space. The parameter space consists of the total group optical luminosity, luminosity of the BGG, luminosity gap (Δm₁₂) and luminosity decentring. In this figure, size of the symbols (balls) indicate the oldness probability for given subregion in the parameter space. The luminosity decentring and the radius at which the magnitude gap, Δm₁₂, has been estimated are all projected. We note that the Δm₁₄ improves the statistical probability of old/young groups (Fig. 1); however, we prefer to use Δm₁₂ due to relative ease of its measurement in spectroscopic surveys. For instance, the groups for which the Δm₁₄ is measured in SDSS (DR10; Ahn et al. 2014) accounts for 10 per cent of the groups for which the Δm₁₂ can be obtained. In addition, Δm₁₂ has been used widely in the majority of the studies to date.

Figure 5.

The probability for a system to be considered as old (top) and young (bottom) based on the observable four-dimensions parameter space. The parameter space consists of the total group optical luminosity, luminosity of the BGG, luminosity gap and luminosity decentring. Size of the symbols (balls) indicate the oldness probability for a given subregion in the parameter space. For example within the total luminosity range 1.9 × 10¹⁰ h⁻² L_⊙ through 20.27 × 10¹⁰ h⁻² L_⊙, galaxy groups with log(separation)( − 2 ≥ log(separation) ≥ −2.5) and M_r(BGG)≈−22 mag, and Δm₁₂ ≈ 4.5 are 100 per cent old (see Table 1), according to our definition of age, based on mass accumulation. As seen low total luminosity systems show the highest population of old galaxy groups. In addition for a galaxy group to be old, the luminosity gap should be large while the luminosity segregation should be small. Same as top panel for young galaxy groups. Largest symbol indicates highest probability of galaxy groups been young. As shown youngest systems are found in high total luminosity galaxy groups where the luminosity gap is small and the luminosity segregation is large.

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As an example, galaxy groups with a total luminosity of 1.9 × 10¹⁰ h⁻² L_⊙ through 20.27 × 10¹⁰ h⁻² L_⊙, and − 2 ≥ log(separation) ≥ −2.5 and M_r(BGG)≈−22 mag, with a Δm₁₂ ≈ 4.5 are 100 per cent old, according to our definition of age, which is based on mass accumulation history of the halo. As Fig. 5 reveals, systems with low total luminosity show the highest population of old galaxy groups. Moreover, for a galaxy group to be considered as old, the luminosity gap should be large while the luminosity segregation should be small.

Bottom panel in Fig. 5 shows the same probability for young galaxy groups. Largest symbol indicates highest probability of galaxy groups been young. As shown youngest systems are found in high total luminosity galaxy groups where the luminosity gap is small and the luminosity segregation is large.

In Table 1, we report few of the highest probability regions to find old and young galaxy groups; and the average mass-to-light ratio(M₂₀₀/L_r) for a given region. Statistically, the old group haloes have higher mass-to-light ratio relative to the young group haloes.

4 CONCLUSIONS

Assigning age to galaxy systems in the ΛCDM hierarchical structure formation paradigm is not trivial, but essential, especially when one aims to study the connection between the evolution of the haloes to the evolution of galaxies and the properties of the IGM. We quantify the probability of finding galaxy groups with a given halo mass accumulation history, namely old and young galaxy groups, in the parameter space of the luminosity gap and the BGG luminosity for a given halo mass. We show that there is a limited success in identifying galaxy groups based on the luminosity gap as the statistical probability of finding old groups using this parameter is only 60  percent, in the MSs (based on Guo et al. 2011 semi-analytic model for the galaxies).

We define a four-dimensional parameter space consisting, the optical luminosity of the BGG, the optical luminosity difference between the two brightest galaxies in the groups (the luminosity gap), the total luminosity of the group (as a proxy to the group mass) and the physical separation between the luminosity centroid of the galaxy group and its BGG. Using this multidimensional parameter space, we are able to achieve a very high probability in statistical identification of the evolved, e.g. old, and evolving, e.g. young, galaxy groups using purely optical observations. These parameters can be obtained from the optical observations of galaxies alone which are now available from various imaging/spectroscopic surveys.

We note that while the conventional definition of the fossil groups, based on luminosity gap, can result in a contaminated sample of old groups, other probes such as the relaxed X-ray morphology and the cocentring of the X-ray emission and the dominant galaxy in fossil increase the probability of these galaxy groups being old. Observationally, most of the well-studied fossil groups (Khosroshahi et al. 2004; Sun et al. 2004; Khosroshahi et al. 2006b, 2007) meet these criteria.

We show that old systems have more concentrated haloes than young ones in various mass bins, in agreement with Khosroshahi et al. (2007) which showed that for a given optical luminosity, fossil groups have a more concentrated halo, compared to non-fossil groups. However, measuring the halo concentration is still subject to large observational uncertainties, which prevents us to use this parameter for a reliable halo age dating in combination with other observables, routinely available from optical observations of galaxies.

We note that there may some differences in the galaxy properties assigned to sub-haloes in different semi-analytic models (Bower et al. 2006; De Lucia & Blaizot 2007; Guo et al. 2011) but this can only have a minor impact on the values of the boundaries between the subregions in the parameter space introduced here.

The MS data base used in this paper and the web application providing online access to them were constructed as part of the activities of the German Astrophysical Virtual Observatory. We thankfully acknowledge Gerard Lemson for facilitating the access to the data.

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