Abstract

We investigate formation processes and physical properties of globular cluster systems (GCSs) in galaxies based on high-resolution cosmological simulations with globular clusters. We focus on metal-poor clusters (MPCs) and correlations with their host galaxies by assuming that MPC formation is truncated at a high redshift (ztrun≥ 6). We find that the correlation between mean metallicities (Zgc) of MPCs and their host galaxy luminosities (L) flattens from z=ztrun to 0. We also find that the observed relation (ZgcL0.15) in MPCs can be reproduced well in the models with ZgcL0.5 at z=ztrun when ztrun∼ 10, if mass-to-light ratios are assumed to be constant at z=ztrun. A flatter LZgc at z=ztrun is found to be required to explain the observed relation for constant mass-to-light ratio models with lower z=ztrun. However, better agreement with the observed relation is found for models with different mass-to-light ratios between z=ztrun and 0. It is also found that the observed colour–magnitude relation of luminous MPCs (i.e. ‘blue tilts’) may only have a small contribution from the stripped stellar nuclei of dwarf galaxies, which have nuclei masses that correlate with their total mass at z=ztrun. The simulated blue tilts are found to be seen more clearly in more massive galaxies, which reflects the fact that more massive galaxies at z= 0 are formed from a larger number of dwarfs with stellar nuclei formed at z > ztrun. The half-number radii (Re) of GCSs, velocity dispersions of GCSs (σ) and their host galaxy masses (Mh) are found to be correlated with one another such that ReMh0.57 and σ∝Mh0.32. Based on these results, we discuss the link between hierarchical merging histories of galaxies and the physical properties of MPCs, the origin of the LZgc relation and non-homology of GCSs.

1 INTRODUCTION

A growing number of observational studies of globular cluster systems (GCSs) have revealed interesting correlations between physical properties of GCSs and those of their host galaxies (see Brodie & Strader 2006, for a recent review). For example, Strader, Brodie & Forbes (2004) found that mean colours of metal-poor globular clusters (GCs) correlate with luminosities of their host galaxies and accordingly suggested that mean GC metallicities (Zgc) depend on luminosities (L) of their host galaxies such that ZgcL0.15±0.03. Similar results (ZgcL0.16±0.04) were found using the Advanced Camera for Surveys (ACS) on the Hubble Space Telescope (HST) by Peng et al. (2006; P06). Brodie & Strader (2006) have suggested that the relation was steeper in the past. They argue that more enriched metal-poor GCs (MPCs) form first in low-mass objects (from early collapsing peaks at high redshift). These low-mass building blocks merge, ultimately forming the massive galaxies of high-density regions today. Thus the slope, particularly at the high-mass end, flattens over time.

Although such observed correlations between the physical properties of GCSs and their host galaxies have been suggested to contain fossil information of galaxy formation and evolution (e.g. Harris 1991; Forbes & Forte 2001; West et al. 2004; Brodie & Strader 2006), the details remain largely unclear owing to the lack of theoretical and numerical studies of GCSs. Based on dissipationless numerical simulations of major galaxy mergers with GCs, Bekki & Forbes (2006) first demonstrated that the GCS radial density profiles dependent on galaxy luminosity can be understood in terms of the number of major merger events experienced. The dependence of GC destruction on host galaxy mass has also been suggested to be important for understanding the inner radial density profiles (Baumgardt 1988; Vesperini et al. 2003; Capuzzo-Dolcetta 2006).

The formation of GCs in low-mass dark matter (DM) haloes at high redshifts has been investigated by numerical and theoretical studies (e.g. Broom & Clarke 2002; Mashchenko & Sills 2005). Furthermore, the physical properties of GCSs have recently been discussed by several authors based on hierarchical galaxy formation scenarios where GC formation occurs at high redshifts (e.g. Beasley et al. 2002; Santos 2003; Bekki 2005; Kravtsov & Gnedin 2005; Rhode, Zepf & Santos 2005; Yahagi & Bekki 2005; Bekki & Yahagi 2006; Bekki, Yahagi & Forbes 2006; Moore et al. 2006). For example, Beasley et al. (2002) first demonstrated that the observed bimodal colour distributions of GCSs in elliptical galaxies can be reproduced by a semi-analytic galaxy formation model. They also presented GC colour versus galaxy magnitude relations; however, they did not fully reproduce the observed trend. Although these previous works discussed some physical properties of GCSs in galaxies, they did not explore in detail the correlations between GCSs and those of their host galaxy. Thus it remains unclear how the physical properties of GCSs evolve in a hierarchical merging cosmology.

The properties of MPCs ([Fe/H]∼−1.5), such as their very old ages, low metallicities and extended spatial distribution, all suggest formation at early times, when the low-mass building blocks of galaxies formed. The purpose of this current paper is to investigate the physical properties of MPCs and their scaling relations with their host galaxies based on high-resolution cosmological simulations that follow realistic merging and accretion histories of galaxies. We investigate the structural, kinematical and chemical properties of MPCs. We also present several predictions for the expected correlations between properties of MPC systems (such as half-number radii, effective surface number densities and velocity dispersions). Since the present model is based on dissipationless simulations of MPC formation within haloes at high redshifts, the origin of metal-rich GCs (MRCs) ([Fe/H]∼−0.5) formed somewhat later during dissipative merging will not be discussed. We plan to investigate MRCs in a future paper by combining the present high-resolution cosmological simulation with semi-analytic models similar to those used by Beasley et al. (2002).

The plan of this paper is as follows. In the next section, we describe our numerical models of MPC formation and their assumed initial properties. In Section 3, we present our numerical results on (i) LZgc relations, (ii) scaling relations between properties of MPCs and (iii) metallicity–magnitude relations for luminous MPCs in massive galaxies (the so-called blue tilt). In Section 4, we discuss our results. We summarize our conclusions in Section 5. Appendices explore scaling relation dependency on MS/L and ztrun. Throughout this paper, GCSs in our simulations are composed only of MPCs.

2 THE MODEL

2.1 Identification of MPCs

We simulate the large-scale structure of GCs in a Lambda cold dark matter (ΛCDM) universe with Ω= 0.3, Λ= 0.7, H0= 70 km s−1 Mpc−1, and σ8= 0.9 by using the Adaptive Mesh Refinement N-body code developed by Yahagi (2005) and Yahagi, Nagashima & Yoshii (2004), which is a vectorized and parallelized version of the code described in Yahagi & Yoshii (2001). We use 5123 collisionless DM particles in a simulation with the box size of 70 h−1 Mpc and the total mass of 4.08 × 1016 M. We start simulations at z= 41 and follow it until z= 0 in order to investigate physical properties of old GCs outside and inside virialized DM haloes. We used cosmics (Cosmological Initial Conditions and Microwave Anisotropy Codes), which is a package of fortran programs for generating Gaussian random initial conditions for non-linear structure formation simulations (Bertschinger 1995, 2001).

Our method of investigating GC properties is described as follows. First, we select virialized DM subhaloes at a truncation redshift z=ztrun by using the friends-of-friends (FoF) algorithm (Davis et al. 1985) with a fixed linking length of 0.2 times the mean DM particle separation. The minimum particle number Nmin for haloes is set to be 10. For each individual virialized subhalo with a half-mass radius of Rh, particles within Rh/3 are labelled as ‘GC’ particles and are considered to be old MPCs. This procedure for defining GC particles is based on the assumption that energy dissipation via radiative cooling allows baryons to fall into the deepest potential well of DM haloes and finally to be converted into GCs. The value of the truncation radius (Rtr,gc=Rh/3) is chosen, because the size of the old GCs in the Galactic GC system (i.e. the radius within which most Galactic old GCs are located) is similar to Rh/3 of the DM halo in a dynamical model of the Galaxy (Bekki et al. 2005). We assume that old MPC formation is truncated completely after z=ztrun and investigate the dependences of the results on ztrun. Physical motivation for the truncation of GC formation is described later.

Secondly, we follow GC particles formed before z=ztrun until z= 0 and thereby derive locations (x, y, z) and velocities (vx, vy, vz) of GCs at z= 0. We then identify virialized haloes at z= 0 with the FoF algorithm and investigate whether each GC is within a halo. If GCs are found to be within a halo, the mass of the host halo (Mh) and physical properties of the GCS are investigated. If a GC is not in any halo, it is regarded as an intergalactic GC. The number fraction of these depends on ztrun but is typically less than 1 per cent (e.g. ∼0.3 per cent for ztrun= 10). We do not consider intergalactic GCs further in this current paper.

Thus, the present simulations enable us to investigate the physical properties only for old MPCs due to the adopted assumption of collisionless simulations. The physical properties of MRCs which may form later during dissipative merger events (e.g. Ashman & Zepf 1992) are not investigated.

2.2 Initial properties of MPCs in galaxy-scale haloes

2.2.1 Radial density profiles

We assume that the initial radial profiles of GCSs [ρ(r)] in subhaloes at z=ztrun are the same as those described by the universal ‘NFW’ profiles (Navarro, Frenk & White 1996) with ρ(r) ∝r−3 in their outer parts. The mean mass of subhaloes at z=ztrun in the present simulations is roughly similar to the total mass of dwarf galaxies today. Minniti, Meylan & Kissler-Patig (1996) found that the projected (R) density profiles of GCSs in dwarfs is approximated as ρ(R) ∝R−2, which translates roughly to ρ(r) ∝r−3 using a canonical conversion formula from ρ(r) into ρ(r) (Binney & Tremaine 1987). Therefore, the above r−3 dependency can be regarded as reasonable. Below we mainly show the fiducial model with ρ(r) similar to the NFW profiles and Rtr,gc=Rh/3.

Although we base our GC models on observational results of GCSs at z= 0, we cannot confirm whether the above ρ(r) and Rtr,gc values are reasonable for GCSs in low-mass subhaloes at z≥ 6 owing to the lack of observational data for GCSs at high redshifts. Although our present results on the LZgc relation and the blue tilt at z= 0 do not depend strongly on the initial distributions of MPCs, structural properties of GCSs do depend on the spatial distribution. For example, if we choose smaller Rtr,gc at z=ztrun, the final projected number distributions of GCSs become more compact.

2.2.2 TheLZgcrelation

In order to investigate the mean metallicities of MPCs in galaxy-scale haloes at z= 0, we need to allocate initial metallicities to all GCs formed before z=ztrun. We assume that GCs in a halo at ztrun have identical metallicities of Zgc and that the metallicities are a function of the total stellar mass (or luminosity) of the halo. Lotz, Miller & Ferguson (2004) have found that forumla, where αgc= 0.2 for B-band luminosities in dEs and dE, Ns. This relation is similar to ZgcL0.15 discovered by Strader et al. (2004). Dekel & Silk (1986) demonstrated that the stellar metallicities (Z) of dwarf galaxies embedded in massive DM haloes correlate with luminosities (L) of the dwarfs such that LZ2.7. This translates to ZgcL0.37 if GCs follow field stars in dwarfs. Prompted by these theoretical and observational studies, we adopt the following relation at high redshift between [Fe/H] (or log Zgc) of MPCs and L:  

1
formula
where αgc and βgc are set to be 0.5 and −6.0 in the present study. We show the models with these values, because they are more consistent with the observed LZgc relation.

In order to derive L for haloes with masses Mh at z= 0, we assume a mass-to-light ratio (Mh/L) dependent on Mh. Recent observational studies based on galaxy luminosity functions for luminous galaxies suggest that Mh/L depends on Mh as Mh/LMh0.33 (Marinoni & Hudson 2002). Dekel & Silk (1986) proposed that Mh/LL−0.37 for low-luminosity dwarf galaxies, which can be interpreted as Mh/LMh−0.55. These studies suggest that the Mh dependences of Mh/L are different between low- and high-luminosity galaxies (see also Zaritsky, Gonzalez & Zabludoff 2006). We thus adopt two different Mh-dependent Mh/L ratios for galaxy-scale haloes in the present study. For haloes above a threshold halo mass of Mh,th, we adopt the following:  

2
formula
where CML is a constant and the value of Mh/L at Mh=Mh,th. For haloes below a threshold halo mass of Mh,th, we adopt the following:  
3
formula
We assume that CML= 10 and Mh,th= 1011 M are reasonable values. Since the threshold mass Mh,th has not yet been so precisely determined by observational studies (e.g. Zaritsky et al. 2006), Mh,th can be a free parameter in our simulations. We, however, confirm that this Mh,th for a reasonable range is not so important as other parameters (e.g. Mh/L dependences). We thus mainly show the results of the models with Mh,th= 1011 M.

In calculating Zgc from L at z=ztrun, we need to derive L from Mh at z=ztrun. It is, however, observationally unclear what a reasonable Mh-dependent Mh/L is for haloes at z=ztrun. We accordingly investigate two extreme cases. One is that Mh/L=CML for all haloes at z=ztrun, and the other is that the Mh dependence of Mh/L is the same between z= 0 (equations 2 and 3) and z=ztrun. The total stellar mass (Ms) of a halo is estimated from Mh by using the adopted Mh-dependent Mh/L and the following stellar-mass-to-light ratio (Ms/L).

It is highly likely that Mh dependences of Mh/L are different between different redshifts, because of (i) evolution (e.g. aging) of stellar populations with different ages and metallicities and (ii) evolution of baryonic mass fraction. Although observational studies on the redshift evolution of the ‘Fundamental Plane (FP)’ of early-type galaxies (e.g. van de Ven, van Dokkum & Franx 2003) have provided some clues to stellar population evolution, it remains observationally unclear whether evolution of baryonic mass fraction is seen in galaxies: kinematical data sets for precisely estimating total masses of galaxies are currently unavailable for very high-z galaxies. The present models, which do not allow us to make explicit predictions on redshift evolution of Mh/L, have difficulties to determine which of the above two is responsible for the possible redshift evolution in Mh dependences of Mh/L. We therefore do not intend to discuss the origin of the redshift evolution in Mh dependences of Mh/L in the present paper.

The stellar population synthesis models by Vazdekis et al. (1996) for a Salpeter initial mass function show that Ms/L in the V band for stellar populations with [Fe/H]=−1.7 (0.0) is 0.22 (0.39) for 0.5 Gyr and 2.03 (4.84) for 12.6 Gyr. Although Ms/L depends strongly on age and metallicity of the stellar population, we use a constant Ms/L of 1 for all haloes because of the lack of information on stellar populations in the simulated galaxy-scale haloes. A discussion on the dependences on Ms/L is given in Appendix B.

Given the fact that most of the virialized haloes at ztrun are of low mass (<1010 M), then the low metallicities used to derive Ms/L should be reasonable. We adopt Ms/L= 1, which is the average value between 0.5 and 12.6 Gyr in metal-poor stellar populations both for ztrun and z= 0 for haloes with different masses. Although this adoption is more idealized, it helps us to derive more clearly the evolution of the MsZgc relation between ztrun and z= 0 without having complicated dependences of Ms/L on ages and metallicities of stars in haloes (that cannot be modelled directly in the present study). If we adopt a higher Ms/L (e.g. ∼4), the simulated MsZgc relation at z= 0 is shifted to lower metallicities for a given stellar mass but with the same relation slope.

2.2.3 GC luminosity function

The luminosity function of GCs has been observationally suggested to be universal and have the following form (e.g. Harris 1991):  

4
formula
where C0 is a constant, M0V=−7.23 ± 0.23 and σm= 1.25 mag (Harris 1991). We adopt this luminosity function and allocate luminosities to GC particles by generating random numbers for −11 ≤MV≤−4. We note that our results are not very sensitive to the choice of M0V or σm.

2.3 Initial properties of galactic nuclei

Previous studies have suggested that stellar galactic nuclei (GN) in nucleated low-mass galaxies might be observed as very massive GCs after the host galaxy of the GN is destroyed during merging with, and accretion on to, more massive galaxies (e.g. Zinnecker et al. 1988; Freeman 1993; Bassino, Muzzio & Rabolli 1994; Bekki & Freeman 2003; Bekki & Chiba 2004). The physical properties of very massive GCs in the Galaxy and M31 (e.g. ω Cen and G1) are observed to be different from those of ‘normal’ GCs (e.g. Freeman 1993). We identify stellar GN particles (referred to as GN particles) at z=ztrun and follow their evolution until z= 0.

The ACS Virgo Cluster Survey by Côte et al. (2006) recently found that the mass fraction of GN (stellar GN) to their host galaxies is typically 0.3 per cent in early-type galaxies. We thus adopt the following relation between the total masses of GN (Mgn) and the total stellar masses of their hosts Ms:  

5
formula
where fgn is considered to be a free parameter in the present study. Recent observations of stellar populations of SGN in nearby galaxies suggest that GN have a significant contribution from young stellar populations and thus suggest that GN are growing with time (e.g. Walcher et al. 2006). Therefore it is reasonable to adopt a fgn value significantly smaller than today's 0.3 per cent. We generally adopt fgn= 0.001 but also discuss the dependence of our present results on fgn.

Lotz et al. (2004) and Côte et al. (2006) found a correlation between Mgn (or GN luminosities) and colours of GN, which suggests a mass–metallicity relation for GN (i.e. more massive GN are more metal rich). This correlation and the above MgnMs relation implies that more luminous galaxies have more metal-rich GN. Lotz et al. (2004) found that the metallicities of field stars of dE, Ns (ZS) correlate with galaxy luminosities (L) such that ZSL0.4. Kravtsov & Gnedin (2005) showed that galaxies in their cosmological simulations exhibit a strong correlation between the stellar mass (MS) and the average metallicity (Z) of stars (described as ZSMS0.5). Guided by these studies, we assume the following relation between GN metallicities (log Zgn) and luminosities of their hosts (L):  

6
formula
where αgn and βgn are set to be the same as those adopted for MPCs denoted as αgc and βgc (i.e. 0.5 and −6.0, respectively) in the present study for consistency. We discuss whether the adopted models can provide a physical explanation for the origin of mass–metallicity relation of luminous GCs around massive galaxies in Section 4.2.

2.4 Truncation of MPC formation

Semi-analytic galaxy formation models based on a hierarchical clustering scenario have shown that models with a truncation of GC formation at z∼ 5 can better reproduce the observed colour bimodality of GCs in early-type galaxies (Beasley et al. 2002). Recent cosmological simulations have demonstrated that truncation of GC formation by some physical mechanism (e.g. reionization) is necessary to explain the very high specific frequency (SN) in cluster ellipticals, structural properties of the Galactic old stars and GCs, and the mass-dependent SN trend (Santos 2003; Bekki 2005; Bekki & Chiba 2005; Rhode et al. 2005; Bekki et al. 2006; Moore et al. 2006). Observational studies, however, have not yet found any strong evidence for the truncation of GC formation in GCSs of nearby galaxies.

If ztrun is closely associated with the completion of cosmic reionization, then ztrun may well range from 6 (Fan et al. 2003) to 20 (Kogut et al. 2003). Although we investigate models with different ztrun (i.e. 6, 10 and 15), here we present the results of the model with ztrun= 10. This is because the latest observations by Wilkinson Microwave Anisotropy Probe (WMAP) suggested that the epoch of reionization is z= 10.9+2.7−2.3 (Page et al. 2006), and because numerical simulations suggest that models with higher ztrun (≥10) are more consistent with observations of MPCs (e.g. Bekki 2005; Moore et al. 2006).

2.5 Galaxy-scale haloes

We simulate the structural, kinematical and chemical properties of 114 596 GCs formed in 25 096 low-mass haloes before ztrun= 10. The physical properties of GCSs in 2830 haloes with masses 4.0 × 109 M < Mh < 6.5 × 1014 M at z= 0 are analysed and their correlations with host properties are investigated. Since we are interested in the GCS properties in galaxy-scale haloes, we only show the results for GCSs in haloes with masses 1011 MMh≤ 1013 M (and with 1 ≤Ngc≤ 246).

We investigate the relations between three properties of GCSs with Ngc≥ 4, i.e. the half-number radii of projected GC distribution (Re), the surface number densities at Re(Ie) and the line-of-sight velocity dispersion for all MPCs in each halo (σ). We derive these three properties, because they are observationally feasible to measure. Since Ms/L is fixed at a constant value of 1 in the present study, the simulated MsZgc relation is virtually the same as the LZgc relation that is often discussed. The dependences of the present results on ztrun is important, we show them in Appendices B and C.

3 RESULT

3.1 The LZgc relation

Fig. 1 shows the large-scale (∼50 Mpc) distribution of MPCs at z= 0 and MPCs within galaxy-scale haloes with similar halo masses (Mh∼ 8 × 1012 M) in the model with ztrun= 10. It is clear from Fig. 1 that the spatial distribution of MPCs in galaxy-scale haloes is quite diverse. For example, the GCS distribution in the halo with Mh= 6.1 × 1012 M appears to be more flattened than that in the halo with Mh= 9.6 × 1012 M. The mean spatial density of MPCs within the central 50 kpc of the halo with Mh= 8.1 × 1012 M is a factor of ∼8 higher than that of the halo with Mh= 9.3 × 1012 M. These diverse GCS structures are due to the differences of merging histories of haloes (e.g. the number of low-mass subhaloes that were virialized before ztrun which merged to formed a halo). The mean metallicities and metallicity distribution functions (MDFs) of GCSs in galaxies are also diverse depending on their merging histories.

Figure 1

Spatial distributions of all MPCs projected on to the xy plane in the large-scale cosmological simulation (left) and spatial distributions of MPCs within the selected four galaxy-scale haloes with similar masses (right) for the model with ztrun= 10. The halo masses Mh are 9.3 × 1012 M, 9.6 × 1012 M, 6.1 × 1012 M and 8.1 × 1012 M, for the upper left (a), the upper right (b), the lower left (c) and lower right (d), respectively. The four circles shown by solid lines represent the locations of the four haloes in the left-hand panel. The horizontal bar is 20 kpc. The total number of MPCs range from 31 to 249 in the four haloes, which reflects the different merging histories with MPCs formed before ztrun= 10.

Figure 1

Spatial distributions of all MPCs projected on to the xy plane in the large-scale cosmological simulation (left) and spatial distributions of MPCs within the selected four galaxy-scale haloes with similar masses (right) for the model with ztrun= 10. The halo masses Mh are 9.3 × 1012 M, 9.6 × 1012 M, 6.1 × 1012 M and 8.1 × 1012 M, for the upper left (a), the upper right (b), the lower left (c) and lower right (d), respectively. The four circles shown by solid lines represent the locations of the four haloes in the left-hand panel. The horizontal bar is 20 kpc. The total number of MPCs range from 31 to 249 in the four haloes, which reflects the different merging histories with MPCs formed before ztrun= 10.

Fig. 2 shows how the initial MhZgc relation evolves between z= 10 and 0 for the model with ztrun= 10. Although the dispersion in [Fe/H] at z= 0 is significantly large for a given metallicity bin, the MhZgc relation at z= 10 becomes flatter at z= 0. This result clearly indicates that the mean metallicities of MPCs in galaxies correlate with the total masses of their host galaxies and thus with those of DM haloes. This also demonstrates that the simulated LZgc (discussed later) results from the MhZgc relation rather than the adopted assumptions on Mh/L. The dependences of this correlation on ztrun are given in Appendix A.

Figure 2

Mean metallicities ([Fe/H]) of metal-poor GCSs in simulated galaxies as a function of the total halo masses of the galaxies (log10 M h) for the model with ztrun= 10. Each small dot represents a galaxy with a GCS at z= 0. The thin blue line represents the initial relation (z= 10) between Zgc and Mh, where Zgc is the mean metallicity of their MPCs. The thin green line represents the mean value of the simulated relation today (z= 0).

Figure 2

Mean metallicities ([Fe/H]) of metal-poor GCSs in simulated galaxies as a function of the total halo masses of the galaxies (log10 M h) for the model with ztrun= 10. Each small dot represents a galaxy with a GCS at z= 0. The thin blue line represents the initial relation (z= 10) between Zgc and Mh, where Zgc is the mean metallicity of their MPCs. The thin green line represents the mean value of the simulated relation today (z= 0).

Fig. 3 shows that the initially steep LZgc relation at z= 10 becomes significantly flatter at z= 0 due to hierarchical merging of low-mass galaxies formed at high z. Fig. 3 also shows that the dispersion in [Fe/H] for a given mass range of Ms is quite large. Fig. 3 demonstrates that the observed LZgc relation (i.e. ZgcL0.16) at z= 0 can be reasonably well reproduced by the model with an initial LZgc relation of ZgcL0.5 at z= 10. We find that such a steep initial LZgc relation at ztrun is indispensable for reproducing the very flat yet significant correlation between L and Zgc at z= 0. The flattening of the initial LZgc relation can be seen also in models with different ztrun (see Appendix C).

Figure 3

Mean metallicities ([Fe/H]) of metal-poor GCSs in simulated galaxies as a function of the total stellar masses of the galaxies (log10 M s) for the model with ztrun= 10. Each small dot represents a galaxy with a GCS at z= 0. Here we use a variable M/L at z= 0 and a constant M/L at z= 10. The thin blue line represents the initial ZgcL0.5 relation, where L is the total luminosity of galaxies and Zgc is the mean metallicity of their MPCs. The thin green line represents the mean value of the simulated relation today (z= 0). For comparison, the observed LZgc relation of ZgcL0.16 (P06) is shown by a thick red line.

Figure 3

Mean metallicities ([Fe/H]) of metal-poor GCSs in simulated galaxies as a function of the total stellar masses of the galaxies (log10 M s) for the model with ztrun= 10. Each small dot represents a galaxy with a GCS at z= 0. Here we use a variable M/L at z= 0 and a constant M/L at z= 10. The thin blue line represents the initial ZgcL0.5 relation, where L is the total luminosity of galaxies and Zgc is the mean metallicity of their MPCs. The thin green line represents the mean value of the simulated relation today (z= 0). For comparison, the observed LZgc relation of ZgcL0.16 (P06) is shown by a thick red line.

Figs 4 and 5 show why the initial steep LZgc relation becomes flatter during the hierarchical growth of galaxies. Fig. 4 shows that the final (z= 0) mean metallicity of the MPCs in a galaxy with Mh= 5.0 × 1012 M is lower (Zgc=−1.39) than the initial mean metallicity (Zgc=−1.20) of its progenitor low-mass halo with Mh= 4.0 × 1010 M, which is the most massive halo among those forming this galaxy at z= 10. This galaxy grows via merging and accretion of 74 low-mass haloes, all with GCSs of different metallicities between z= 10 and 0. Since the MPCs in the accreted ‘building blocks’ are metal poor, the metallicity of the GCS in the final galaxy is lower than that of the GCS of its progenitor halo. There is thus a decrease in the mean metallicities of GCSs during galaxy evolution between ztrun and z= 0 which is driven by merging and accretion of low-mass haloes and their lower metallicity GCSs.

Figure 4

The MDF of GCSs from progenitor low-mass haloes at z= 10 that forms a galaxy-scale halo of Mh= 5.0 × 1012 M at z= 0. This halo with a mean Zgc=−1.39 (dashed line) at z= 0 originates from a low-mass halo of Mh= 4.0 × 1010 M at z= 10 with Zgc=−1.20. The GCS at z= 0 forms via hierarchical merging and accretion, between z= 10 and 0, of 74 low-mass haloes each with a GCSs containing a range of metallicities.

Figure 4

The MDF of GCSs from progenitor low-mass haloes at z= 10 that forms a galaxy-scale halo of Mh= 5.0 × 1012 M at z= 0. This halo with a mean Zgc=−1.39 (dashed line) at z= 0 originates from a low-mass halo of Mh= 4.0 × 1010 M at z= 10 with Zgc=−1.20. The GCS at z= 0 forms via hierarchical merging and accretion, between z= 10 and 0, of 74 low-mass haloes each with a GCSs containing a range of metallicities.

Figure 5

The metallicity difference between GCSs in galaxy-scale haloes at z= 0 and those in their progenitor haloes at z= 10[δZgc=Zgc(z= 10) −Zgc(z= 0)] with halo masses (Mh) at z= 0. Small dots represent GCSs in galaxy-scale haloes and big filled circles are the mean values of δZgc.

Figure 5

The metallicity difference between GCSs in galaxy-scale haloes at z= 0 and those in their progenitor haloes at z= 10[δZgc=Zgc(z= 10) −Zgc(z= 0)] with halo masses (Mh) at z= 0. Small dots represent GCSs in galaxy-scale haloes and big filled circles are the mean values of δZgc.

Fig. 5 shows the differences in mean metallicities (Zgc) between GCSs in galaxies at z= 0 and those in their progenitor low-mass haloes at z= 10[i.e. δZgc=Zgc(z= 10) −Zgc(z= 0)]. Although the dispersion in δZgc is quite large in more massive galaxies, the absolute values of δZgc are significantly larger in more massive galaxies. This means that more massive galaxies at z= 0 have experienced a larger number of merging and accretion events of lower-mass haloes with lower Zgc, so that the final Zgc at z= 0 is lower. Low-mass galaxies at z= 0, on the other hand, have not experienced so many mergers and accretion events that act to increase their masses and decrease their Zgc. Such galaxies are of low mass because they have only grown slightly through merging/accretion since their virialization at z= 10. As a result of this and forming relatively many GCs in the progenitor halo, their Zgc does not change so much from their original values at z= 10. Thus the mass dependence of δZgc is the main reason for the flattening of the LZgc relation between ztrun and z= 0.

Fig. 6 shows that if the M/L ratios in galaxies are variable at z= 10 and 0, then the simulated ZgcL0.3 relation is not a good fit to the observed one. These results, combined with those in Fig. 3, demonstrate that the model with a constant M/L at z= 10 can better reproduce the observed flat LZgc relation for a given initial LZgc relation. Given the fact that both the Mh dependence of M/L and LZgc relation at z= 10 are observationally unclear, there are two possible interpretations for the derived steeper LZgc relation at z= 0 in the variable M/L models. One possibility is that the variable M/L is more reasonable, and thus the initial LZgc relation at z= 10 should be much flatter than the adopted one of ZgcL0.5. The other possibility is that a constant M/L is more reasonable and the observed Mh dependence of M/L at z= 0 is achieved during galaxy evolution between z= 10 and 0.

Figure 6

The same as Fig. 3 but for the model with ztrun= 10 and variable M/L both at z= 0 and at z= 10.

Figure 6

The same as Fig. 3 but for the model with ztrun= 10 and variable M/L both at z= 0 and at z= 10.

An initial LZgc relation at z= 10 is required to have ZgcL0.25 to explain the observed relation in a variable M/L model, because merging flattens the slope by a factor of 1.7 in this model. This LZgc relation at z= 10 is flatter than the luminosity–metallicity relation of low-mass galaxies (ZL0.37) predicted by Dekel & Silk (1980). Given the fact that there are no observational constraints on an initial LZgc relation at z= 10, we cannot conclude whether constant M/L models with initially steep LZgc relations are better than variable M/L models with less steep LZgc relations at this stage.

The present simulations thus show two extreme cases of LZgc evolution depending on the assumed M/L at z=ztrun. The present simulations, however, confirm that the initial LZgc relation of GCSs at z=ztrun becomes significantly flatter by z= 0, irrespective of different (yet reasonable) initial Mh dependences of M/L and different ztrun. These results thus imply that the LZgc relation evolves with z in such a way that the LZgc relation becomes flatter at lower z.

3.2 Scaling relations

Fig. 7 shows the scaling relations between three projected properties of GCSs, Re (half-number radii), Ie (surface number densities at Re) and σ (line-of-sight velocity dispersion). Fig. 7 demonstrates that σ and Re strongly correlate with Mh in such a way that σ and Re are both larger for larger Mh. Ie, however, correlates weakly with Mh such that Ie is smaller for larger Mh. The least-square fits to the simulation data show that ReMh0.57, IeMh−0.65 and σ∝Mh0.32. These scaling relations appear to be different from those derived for DM haloes (Kormendy & Freeman 2004), which suggests that GCSs in our models do not simply trace the distributions of DM haloes. These scaling relations reflect the fact that MPCs in galaxy-scale haloes at z= 0 originate from the central regions of low-mass haloes at z= 10 and thus have different structures from those of DM haloes at z= 0.

Figure 7

Velocity dispersions (σ, top), effective surface number densities (Ie, middle) and half-number radii (Re, bottom) of MPCs versus the total halo masses of galaxies at z= 0 for the model with ztrun= 10. The solid line in each panel represents the least-square fit to the simulation data.

Figure 7

Velocity dispersions (σ, top), effective surface number densities (Ie, middle) and half-number radii (Re, bottom) of MPCs versus the total halo masses of galaxies at z= 0 for the model with ztrun= 10. The solid line in each panel represents the least-square fit to the simulation data.

Fig. 8 shows how the properties of GCSs (Re, Ieσ) correlate with total stellar masses (Ms) in haloes and confirm that these correlations are similar to those derived in Fig. 7. The least-square fits to the simulation data show that ReMs0.85, IeMs−0.97 and σ∝Ms0.48. The derived Ms dependences are steeper than the Mh dependences due to the adopted Mh variation with M/L in galaxies at z= 0. These results suggest that future observational studies of GCSs for a statistically significant number of GCSs can assess the validity of the present MPC formation model. The dependency of the GCS scaling relations on ztrun is given and their physical meaning is discussed in Appendix D.

Figure 8

The same as Fig. 7 but for the dependences on total stellar masses of galaxies.

Figure 8

The same as Fig. 7 but for the dependences on total stellar masses of galaxies.

Fig. 9 shows that the properties of GCSs (i.e. Re, Ie and σ) in our models do not follow a relation for dynamical systems in virial equilibrium (i.e. Re=CVPIe−1σ2). Given the fact that the simulated GCSs at z= 0 are in dynamically equilibrium, this apparent deviation from the virial relation suggests structural and kinematical non-homology in GCSs, as observationally and theoretically suggested for elliptical galaxies (e.g. Djorgovski & Davis 1987; Capelato, de Carvalho & Carlberg 1995). Re and σ of GCSs are significantly different from the half-mass radii and central velocity dispersions of DM haloes that host the GCSs. In the present simulations, MPCs within galaxy-scale haloes at z= 0 trace the particles initially within the central region of subhaloes virialized before z=ztrun. On the other hand, DM halo particles at z= 0 trace all the particle from subhaloes virialized at every z. Therefore, Re and σ of GCSs are quite different from the half-mass radii and central dispersions of DM haloes that follow the virial relation.

Figure 9

Half-number radii for projected MPCs as a function of the combination of Ie (surface-number density at Re) and σ (line-of-sight velocity dispersion for MPCs) for the model with ztrun= 10. The solid line represents the ‘virial plane’ defined as Re=CVPIe−1σ2. Note that the distribution of GCSs deviates significantly from the virial plane. Physical interpretations of this result is given in the main text.

Figure 9

Half-number radii for projected MPCs as a function of the combination of Ie (surface-number density at Re) and σ (line-of-sight velocity dispersion for MPCs) for the model with ztrun= 10. The solid line represents the ‘virial plane’ defined as Re=CVPIe−1σ2. Note that the distribution of GCSs deviates significantly from the virial plane. Physical interpretations of this result is given in the main text.

Fig. 10 shows the best fit to the three properties of Re, Ie and σ for GCSs (i.e. Re=CFPIe−0.46σ0.60). By assuming that forumla, the best values of αFP and βFP are chosen such that the dispersion along the assumed line (or plane) with αFP and βFP become the smallest. The derived relation of Re=CFPIe−0.46σ0.60 is quite different from the virial relation of Re=CVPIe−1σ2, which suggests structural and kinematical non-homology in the simulated GCSs. The more significant deviation of βFP from the virial relation suggests that the estimated line-of-sight velocity dispersions of GCSs can be quite different from (or significant smaller than) the central velocity dispersions of DM haloes required for virial equilibrium. Since these three properties of GCSs are feasible to derive observationally, it is an interesting future study to compare these simulated scaling relations with the corresponding observational ones.

Figure 10

The same as Fig. 9 but for the different abscissa. The solid line represents the ‘FP’ defined as Re=CFPIe−0.46σ0.60. The dispersion of the simulation data along this line is small suggesting that GCSs are located on a FP. Physical interpretations of this result is given in the main text.

Figure 10

The same as Fig. 9 but for the different abscissa. The solid line represents the ‘FP’ defined as Re=CFPIe−0.46σ0.60. The dispersion of the simulation data along this line is small suggesting that GCSs are located on a FP. Physical interpretations of this result is given in the main text.

3.3 Blue tilts

Next we describe the addition of accreted stellar GN to our simulated GCSs. Fig. 11 shows that the GCS in a massive halo of Mh= 3.0 × 1013 M for the model in which the fraction of the stellar galaxy mass in nuclei is fgn= 0.001. The figure shows a clear [Fe/H]–MV relation in the more luminous MPCs (MV < −9 mag). The essential reason for the simulated blue tilt is that this halo was formed from a large number of subhaloes (Nh= 62) that were virialized before ztrun and thus had GN that followed a mass–metallicity relation. Whether or not the blue tilt in a halo at z= 0 can be clearly seen is determined by how many subhaloes with GN particles merged to form the halo in the present simulations. The least-square fits to the simulation data show that MV=−10.25 − 1.24 ×[Fe/H] (i.e. ZL2.01), which means that the simulated blue tilt is significantly steeper than the observed one (e.g. ZL0.55; Harris et al. 2006).

Figure 11

Distribution of MPCs in the MV–[Fe/H] plane for the halo with Mh= 3.0 × 1013 M and Ngn= 62 in the model with fgn= 0.001. Big dots represent the mean MV for each of the five metallicity bins. This halo experienced merging/accretion of nucleated galaxies 62 times and thus now has 62 stripped nuclei in its GCS. Note that a MV–[Fe/H] relation can be clearly seen for MPCs with −12 ≤MV≤−9 (mag). This relation originates from the LZgn relation of low-mass ‘building blocks’ that formed this halo. The simulated MV–[Fe/H] relation of MPCs can be compared with the observed ‘blue tilt’. The least-square fit to the five simulation data shows ZL−2.01.

Figure 11

Distribution of MPCs in the MV–[Fe/H] plane for the halo with Mh= 3.0 × 1013 M and Ngn= 62 in the model with fgn= 0.001. Big dots represent the mean MV for each of the five metallicity bins. This halo experienced merging/accretion of nucleated galaxies 62 times and thus now has 62 stripped nuclei in its GCS. Note that a MV–[Fe/H] relation can be clearly seen for MPCs with −12 ≤MV≤−9 (mag). This relation originates from the LZgn relation of low-mass ‘building blocks’ that formed this halo. The simulated MV–[Fe/H] relation of MPCs can be compared with the observed ‘blue tilt’. The least-square fit to the five simulation data shows ZL−2.01.

Fig. 12 shows that if fgn= 0.0001, which is an order of magnitude smaller than the model in Fig. 11, the blue tilt is much less clearly seen, because low-luminosity GN particles cannot be distinguished from GC particles in the GCS. The results in Figs 11 and 12 imply that for a blue tilt to be clearly seen at z= 0, GN at z= 10 should be as massive as ∼0.1 per cent of their stellar components in their low-mass hosts. We confirm that if fgn= 0.005, a blue tilt can be clearly seen but there are too many very luminous (MV < −12 mag) MPCs in galaxy-scale haloes. The least-square fits to the simulation data show that MV=−8.22 − 0.93 ×[Fe/H] (i.e. ZL2.69), which is steeper than the blue tilt in the model with fn= 0.001 (and thus less consistent with observations). These results suggest that the presence of blue tilts gives some constraints on the destruction of low-mass nucleated galaxies around galaxies between ztrun and z= 0. The dependences of the slopes of the simulated blue tilts on the initial GC luminosity function are given in Appendix E.

Figure 12

The same as Fig. 11 but for fgn= 0.0001. The least-square fit to the five simulation data shows ZL−2.69.

Figure 12

The same as Fig. 11 but for fgn= 0.0001. The least-square fit to the five simulation data shows ZL−2.69.

Fig. 13 shows that more massive haloes will have a larger number of GN particles within their GCSs and thus massive galaxies at z= 0 are more likely to have a contribution from disrupted nuclei to their observed blue tilts. The significantly smaller dispersion in Ngn for more massive haloes with Mh > 1013 M in Fig. 13 suggests that this effect is most pronounced for GCSs in the most luminous galaxies, i.e. those at the centres of groups and clusters.

Figure 13

The number of GN versus halo mass Mh for haloes with ztrun= 10.

Figure 13

The number of GN versus halo mass Mh for haloes with ztrun= 10.

The overall distribution of MPCs in the MV–[Fe/H] plane does not reproduce well the observations for GCSs with blue tilts (Harris et al. 2006; Spitler et al. 2006; Strader et al. 2006). The stripped nuclei, in this prescription, are therefore unlikely to be the sole origin of the observed blue tilts (although they may still make a small contribution).

4 DISCUSSION

4.1 The origin of the LZgc relation

The present numerical simulations are the first to demonstrate that an initial steep LZgc relation at high z can become significantly flatter due to hierarchical merging of galaxies with GCSs. The slope of the final LZgc relation GCSs at z= 0 is similar to the observed one (Strader et al. 2004; P06) in models with ztrun= 10 and constant M/L at ztrun. Côte, Marzke & West (1998) investigated correlations between metal-poor GCSs and their host galaxy luminosities (L) in their models of GCS formation via merging/accretion of MPCs from dwarfs with a power-law galaxy luminosity function of slope α. They showed that there is a weaker correlation between the MPCs and L in the models with a steep luminosity function (α∼−1.8 in their Fig. 7). Recent high-resolution cosmological simulations on the mass function of low-mass haloes have demonstrated that the slope of the mass function is significantly steeper at higher redshifts (e.g. Yahagi et al. 2004). We suggest that the steeper mass function of haloes at high redshifts can also be an important factor for the origin of the observed slope of the LZgc relation.

The present study assumed that the initial LZgc relation at z=ztrun is similar to the LZS relation observed in Local Group dSphs today (e.g. Dekel & Silk 1986; Lotz et al. 2004). Such a steep LZgc relation at z=ztrun is important for the present model, because the initial LZgc relation becomes significantly flatter between z=ztrun and z= 0. It is, however, observationally and theoretically unclear how low-mass galaxies at high z achieve such a steep relation and what physical mechanisms are responsible. Kravtsov & Gnedin (2005) found that galaxies in their cosmological simulations exhibit a strong correlation between the stellar mass (MS) and the average metallicity (ZS) of stars (described as ZSMS0.5) similar to the observed one in dwarf galaxies (Dekel & Woo 2003). Their results imply that GCSs in low-mass galaxies at high z could also have a steep ZgcMS relation similar to that of the field stars.

It should be stressed here that the models with an initially steep LZgc relation at z=ztrun better reproduce observations only for constant M/L at z=ztrun: models with an initially flatter LZgc relation can also better explain observations for variable M/L at z=ztrun, though the required LZgc relation was not predicted by previous theoretical studies. Owing to lack of observational data sets for the LZgc relation at z=ztrun, it is currently difficult to make a robust conclusion as to whether the constant M/L models with a steep LZgc relation are better.

4.2 The origin of the blue tilt

Using the ACS on-board HST, Strader et al. (2006) and Harris et al. (2006) found that luminous blue GCs (i.e. MPCs) reveal a trend of having redder colours in giant ellipticals. This ‘blue tilt’ was interpreted as MPCs having a metallicity–luminosity relation of ZL0.55. A possible exception was NGC 4472. Spitler et al. (2006) subsequently showed that this trend is also true in the Sombrero spiral galaxy and may extend to lower GC luminosities with a somewhat shallower slope than derived by Harris et al. (2006) and Strader et al. (2006). In each of these ACS studies, the MRCs showed no corresponding trend. These observations require that any theoretical models of GC formation should explain the origin of the metallicity–luminosity relation for individual GCs, and also the apparent absence of a similar relation for MRCs (e.g. Mieske et al. 2006).

Here we have demonstrated that luminous MPCs in more massive galaxies show metallicity–luminosity relations in that brighter GCs have redder colours – a trend qualitatively similar to the observed blue tilts. In our simulations, the reason for the ‘simulated blue tilts’ is that luminous MPCs originate from stellar GN of the more massive nucleated galaxies with a luminosity–metallicity relation. The present study thus suggests that galaxies which experienced a larger number of accretion/merging events of nucleated low-mass galaxies are more likely to show a blue tilt. Since the frequency of accretion/merging events is generally higher in more massive galaxies, the blue tilts are more likely to be present in high luminosity than low luminosity galaxies. Thus not all galaxies are expected to have blue tilts. Since these simulated GCSs are composed both of stripped normal GCs (with no initial mass–metallicity relation) and of stripped GN with a mass–metallicity relation, their detection in observational data may depend on their relative numbers.

Although we find clear blue tilts in our simulations, it is not clear what role stripped GN play in the observed blue tilts to date. For example, Spitler et al. (2006) find the blue tilt is not simply a feature of luminous MPCs but seems to extend down to luminosity function as far as is measurable. Furthermore the MPC luminosity function is well fit by a standard Gaussian, or t5 function, even at the luminous end. Thus the tilt does not appear to be due to an additional population but rather a ‘tilting’ of the existing MPC system. More deep data on a variety of systems are required to confirm these trends. A direct comparison between simulations and observational data needs to account for many factors, such as the number of GN versus bona fide GCs, contamination rates, the presence of dust and measurement errors (which tend to cause spreads in both magnitude and colour).

The observed colours and metallicities of GN in dwarfs ((Lotz et al. 2004; Mieske et al. 2006) appear to suggest most GN cannot be metal-rich clusters as they have lower metallicities. The apparent absence of a MRC colour–magnitude relation may therefore reflect the fact that most MRCs do not originate from GN. Our simulated blue tilt, due to the stripped nuclei of (defunct) low-mass galaxies, is one of many possible scenarios for the origin of the blue tilt. We plan to discuss the origin of the blue tilt in a wider context of galaxy and GC formation in a future paper.

4.3 The FP of metal-poor GCSs

The scaling relations between properties of elliptical galaxies, such as ‘the FP’, have long been suggested provide valuable information on formation and evolution histories of elliptical galaxies (e.g. Djorgovski & Davis 1987). Theoretical studies based mostly on numerical simulations suggested that structural and kinematic non-homology inferred from the FP are closely associated with dynamics of galaxy merging, star formation histories dependent on galaxy masses and collapse dynamics in elliptical galaxy formation (Capelato et al. 1995; Bekki 1998; Dantas et al. 2003). The present study has shown that the scaling relation (Re=CFPIe−0.46σ0.60) among Re (projected half-number radii), Ie (surface number density at Re) and σ (velocity dispersion) of MPCs in our models is significantly different from that expected from the virial theorem (Re=CVPIe−1σ2). Although this is due partly to the way we estimate σ for a GCS in our models, the significant difference between the exponents of the Ie terms between the virial and the simulated relations strongly suggests that the deviation has a physical meaning.

MPCs are assumed to form in the central regions of low-mass haloes that are virialized before z=ztrun. The formation of MPCs is biased towards high-density peaks of the primordial matter distribution in the adopted CDM model. As a natural result of this, MPCs in galaxies at z= 0 show structural and kinematical properties different from those of the background DM haloes, the scaling relation of which should be similar to the virial relation. Recent numerical simulations with truncation of GC formation have also demonstrated that the dynamical properties of MPCs can be significantly different from those of DM haloes (Santos 2003; Bekki 2005; Moore et al. 2006). Although physical explanations for the origin of the slope of the FP in MPCs are yet to be provided, we suggest that the presence of the FP for MPCs can be one characteristic of biased formation of MPCs at high z.

A growing number of spectroscopic observations have revealed kinematic properties of GCSs, such as radial profiles of velocity dispersions and rotational velocities, maximum V/σ and kinematical differences between MPCs and MRCs, in early-type galaxies (e.g. Kissler-Patig & Gebhardt 1998; Cohen 2000; Zepf et al. 2000; Côte et al. 2001, 2002; Beasley et al. 2004; Peng, Ford & Freeman 2004; Richtler et al. 2004; Pierce et al. 2006; Romanowsky 2006). Recent photometric studies of GCSs by the HST and ground-based telescopes have also revealed structural properties of GCSs in galaxies with different Hubble types (e.g. Rhode & Zepf 2004; P06; Spitler et al. 2006). Accordingly it is worthwhile for future observational studies to investigate the FP of MPC systems. It is also an interesting observational study to investigate correlations between Re (or Ie) of MPCs and the total luminosity of their host galaxy, because such correlations are more feasible to derive observationally and thus can be compared more readily with the predicted ones.

5 CONCLUSIONS

We have investigated the physical properties of metal-poor GCSs in galaxies using high-resolution cosmological simulations based on a ΛCDM model. We particularly investigated the LZgc relation, the MV–[Fe/H] relation and MPC scaling relations in galaxies with different masses. We assumed that (i) MPC formation is truncated at z=ztrun, (ii) galaxies at z=ztrun contain MPCs and GN with masses proportional to their host galaxies, (iii) galaxies initially have a steep relation at z=ztrun, (iv) galaxies have a certain M/L at z=ztrun and (v) MPCs have a luminosity function the same as that observed today. We also assumed that both MPCs and GNs can be identified as MPCs in galaxies at z= 0, if they are within virial radii of the DM haloes. We summarize our principle results of the models as follows.

  • The original LZgc at ztrun becomes significantly flatter by z= 0 due to hierarchical merging of lower-mass galaxies. The original ZgcL0.5 at z= 10 in the model with a constant M/L at ztrun becomes ZgcL0.15 at z= 0, which is consistent with the latest observational results. The origin of the flattening of the LZgc relation reflects the fact that Zgc of GCSs in more massive galaxies at z= 0 is lower than the progenitor haloes at ztrun due to a large number of mergers and accretion of low-mass haloes of GCSs with lower Zgc. A flatter LZgc relation at z= 10 (ZgcL0.25) is required to explain the observed relation at z= 0 in the models with a variable M/L.

  • The final LZgc relation at z= 0 depends on ztrun such that it is steeper for lower ztrun. Models with constant M/L ratios (i.e. Mh/L= 10) at z=ztrun can better explain the observed LZgc relation at z= 0 for a given z=ztrun and the adopted initial LZgc relation of ZgcL0.5 at z=ztrun, though most models show a flattening of the original LZgc relation. The essential reason for the positive correlation between L and Zgc at z= 0 is that the more massive galaxies at z= 0 are formed from hierarchical merging of a larger number of more massive building blocks containing GCs with higher metallicities. Thus merging does not completely erase the original LZgc relation.

  • MPC systems in these models clearly show scaling relations between galaxy mass Ms(or halo mass Mh), Re, Ie and σ. We find ReMs0.85, Ie,gcMs−0.97,σ∝Ms0.48 in the models with ztrun= 10. Thus we find a ‘FP’ for MPCs, which can be described as ReIe−0.5σ0.6 in the models with ztrun= 10. We note that such scalings are not consistent with the virial relation of ReIe−1σ2. The ReMs relation depends on ztrun, which suggests that the scaling relation can give some constraints on ztrun. We also find that ReMh0.57 and σ∝Mh0.32.

  • Luminous MPCs show a correlation between MV and [Fe/H] if these MPCs originate from nuclei of low-mass galaxies at high z. The correlation, which is referred to as a ‘blue tilt’, can be more clearly seen in more massive galaxies with Mh∼ 1013 M. This is mainly because more massive galaxies are formed from a larger number of nucleated galaxies virialized by z=ztrun. Observations of blue tilts only loosely resemble our simulated ones suggesting stripped nuclei make a small contribution at most.

We are grateful to the anonymous referee for valuable comments, which contribute to improve the present paper. We thank J. Strader and L. Spitler for useful discussions. KB and DAF acknowledge the financial support of the Australian Research Council throughout the course of this work. HY acknowledges the support of the research fellowships of the Japan Society for the Promotion of Science for Young Scientists (17-10511). The numerical simulations reported here were carried out on Fujitsu-made vector parallel processors VPP5000 kindly made available by the Astronomical Data Analysis Center (ADAC) at National Astronomical Observatory of Japan (NAOJ) for our research project why36b.

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Appendices

APPENDIX A: DEPENDENCES OF MHZGC RELATIONS ON ZTRUN

Fig. A1 shows that the initially steep MhZgc relation at z= 6 becomes flatter in the model with ztrun= 6, though the degree of the flattening is less significant in this model than in the model with ztrun= 10. This reflects the fact that galaxies experienced a smaller number of galaxy merging (which can flatten the MhZgc relation) between z=ztrun and 0 in this model. Fig. A2 shows that the MhZgc relation is almost flat in the model with ztrun= 15, which means that hierarchical merging of low-mass galaxies between ztrun and 0 almost completely erases out the original steep MhZgc relation: this simulated MhZgc at z= 0 is inconsistent with the observed one. These results shown in Figs A1 and A2 thus suggest that the degree of the flattening in the MhZgc relation depends on ztrun in the sense that models with earlier truncation of GC formation show the larger degree of the flattening. The absence of low-mass galaxies with GCs for log 10 M h > 11.4 (M) in the model with ztrun= 15 is inconsistent with observations.

Figure A1

The same as Fig. 2 but for the model with ztrun= 6.

Figure A1

The same as Fig. 2 but for the model with ztrun= 6.

Figure A2

The same as Fig. 2 but for the model with ztrun= 15.

Figure A2

The same as Fig. 2 but for the model with ztrun= 15.

APPENDIX B: DEPENDENCES OF MSZGC RELATIONS ON MS/L

Fig. B1 shows that the original LZgc relation becomes significantly flatter between ztrun and z= 0 in the model with Ms/L= 5. The locations of galaxies on the MsZgc plane are shifted to the righter directions owing to the larger stellar masses of galaxies in this model with higher Ms/L. As a result of this, the simulated Zgc for a given Ms is slightly smaller than the observed one. The change in Ms/L, however, does not change the final slope in the LZgc relation in comparison with the model with Ms/L= 1. The flattening of the LZgc relation between ztrun and z= 0 does not depend on Ms/L, as long as we adopt assumptions of (i) a constant Ms/L at z=ztrun and (ii) an initial LZgc relation of ZgcL0.5. This means that if we adopt a slightly higher value of βgc for the higher Ms/L model, the observed relation can be better reproduced under the above assumptions. Thus high values of βgc are required in the models with higher Ms/L and with the above assumptions for the observation to be well reproduced.

Figure B1

The same as Fig. 3 but for the model with Ms/L= 5.

Figure B1

The same as Fig. 3 but for the model with Ms/L= 5.

APPENDIX C: DEPENDENCES OF MSZGC RELATIONS ON ZTRUN

Figs C1 and C2 show the simulated MsZgc relations (thus LZgc relations for the fixed Ms/L ratios) for the models with ztrun= 6 and 15, respectively. It is clear from these figures and Fig. 5 that the model with ztrun= 10 can better reproduce the observed flat LZgc relation (or MsZgc one). The model with ztrun= 15 shows the LZgc relation flatter than the observed one, whereas the model with ztrun= 6 shows the steeper one. In the model with ztrun= 15 (ztrun= 6), a larger (smaller) number of merger events for a longer (shorter) time-scale between z=ztrun and 0 can flatten the original LZgc relation at z=ztrun to a larger (smaller) extent. This is the essential reason why the simulated LZgc relation depends on ztrun.

Figure C1

The same as Fig. 3 but for the model with ztrun= 6.

Figure C1

The same as Fig. 3 but for the model with ztrun= 6.

Figure C2

The same as Fig. 3 but for the model with ztrun= 15.

Figure C2

The same as Fig. 3 but for the model with ztrun= 15.

Thus the growth of galaxies via hierarchical merging/accretion of low-mass haloes with and without MPCs between z=ztrun and 0 can flatten the original LZgc at z=ztrun owing to merging/accretion of MPCs with different metallicities. If ztrun corresponds to the epoch of reionization (zreion) and it is determined from future observations, then the results of numerical simulations shown in Figs 6, C1 and C2 imply that we can infer the original LZgc at z=zreion by comparing the simulations with the observed LZgc at z= 0.

APPENDIX D: DEPENDENCES OF SCALING RELATIONS ON ZTRUN

Fig. D1 shows that the dynamical correlations are different between different ztrun, in particular, for Ms dependences of Ie. The Ms dependence of Re is steeper for higher ztrun, whereas the Ms dependence of σ is not so different between different ztrun. The slope in the Ms dependence of Ie is negative for ztrun= 10 and 15 and positive for ztrun= 6. The derived differences in the Ms dependences of dynamical properties of GCSs between different ztrun suggest that future observational studies on the Ms dependences can provide some constraints on ztrun.

Figure D1

Dependences of line-of-sight velocity dispersions (σ, top), effective surface number densities (Ie, middle) and half-number radii (Re, bottom) of MPCs on Ms at z= 0 for the model with ztrun= 6 (solid), ztrun= 10 (dotted) and ztrun= 15 (dashed). Each of these lines represent the least-square fit to the simulation data for each of the three Ms dependences in each model.

Figure D1

Dependences of line-of-sight velocity dispersions (σ, top), effective surface number densities (Ie, middle) and half-number radii (Re, bottom) of MPCs on Ms at z= 0 for the model with ztrun= 6 (solid), ztrun= 10 (dotted) and ztrun= 15 (dashed). Each of these lines represent the least-square fit to the simulation data for each of the three Ms dependences in each model.

Structural properties of GCSs have been suggested to provide constraints on ztrun (Santos 2003; Bekki 2005; Moore et al. 2006). The present study suggested that if (i) the adopted LZgc relation at z=ztrun(ZgcL0.5) is reasonable and realistic and (ii) M/L does not depend on galaxy masses at z=ztrun, the observed LZgc relation at z= 0 could also provide constraints on ztrun. It is therefore an important observational test whether ztrun derived from the observed Ms dependences is consistent with ztrun derived from the structural and chemical properties of GCSs.

APPENDIX E: THE INFLUENCES OF MAXIMUM GC MASSES ON THE BLUE TILT

We adopted an assumption of a universal GC mass (luminosity) function with the same lower (mlow) and upper (or luminosity) mass cut off (mupp) for all haloes in the present study. Since recent observations have suggested a possible maximum mass mmax of star clusters in galaxies (e.g. Gieles et al. 2006) and correlations between mmax and physical properties of their host galaxies, such as star formation rates (e.g. Larsen & Richtler 2001), the above assumption on the fixed mupp could be less realistic. We thus investigated the models with mmaxMh relations (i.e. muppMh) suggested by numerical simulations of Kravtsov & Gnedin (2005).

Fig. E1 shows the result of the model with Mh= 3.0 × 1013 M and Ngn= 62 for mmax= 2.9 × 106 M[Mh/(1011 M)]1.29. Owing to the introduction of the mmaxMh relation, the overall distribution of MPCs on the MV–[Fe/H] in this model appears to be only slightly more similar to the observed blue tilt than the model shown in Fig. 11. The least-square fits to the simulation data show that MV=−10.74 − 1.62 ×[Fe/H] (i.e. ZL1.55). This ZL1.55 relation is, however, still significantly steeper than the observed one of ZL0.55 (Harris et al. 2006). Although the models with steeper mmaxMh relations can show flatter ZL relations, the simulated dispersions in MV for given metallicity bins (in particular, for [Fe/H]>−1.5) are too large to be consistent with observations. These suggest that the models with mmaxMh relations still miss some important ingredients of GC formation. We discuss these problems in our forthcoming papers (Bekki et al., in preparation).

Figure E1

The same as Fig. 11 but for the model in which maximum GC masses (mmax) in haloes are assumed to correlate with their host halo masses.

Figure E1

The same as Fig. 11 but for the model in which maximum GC masses (mmax) in haloes are assumed to correlate with their host halo masses.