NONLINEAR COLOR–METALLICITY RELATIONS OF GLOBULAR CLUSTERS. III. ON THE DISCREPANCY IN METALLICITY BETWEEN GLOBULAR CLUSTER SYSTEMS AND THEIR PARENT ELLIPTICAL GALAXIES

Simple linear conversion of photometric colors is frequently used for estimating metallicities for large samples of extragalactic GCs. This is a reasonable first-order assumption for obtaining mean metallicities, but for investigating the detailed structure of the MDF, including possible subpopulations, the form of the CMR must be known to higher order. However, the best empirical color–metallicity calibrations currently available (e.g., Peng et al. 2006; Lee et al. 2008; Beasley et al. 2008; Sinnott et al. 2010; Woodley & Harris 2011; Alves-Brito et al. 2011) exhibit notable observational scatter. Moreover, compared to tens of thousands of GCs in a typical giant elliptical, the calibration samples are still relatively small and sparsely populated at the high-metallicity end. Larger samples of high-quality spectroscopic metallicities are needed to establish the precise forms of CMRs. Such samples would implicitly include any correlations with age or other parameters and would provide strong constraints on the theoretical models.

In general, the slope of the dependence of a given photometric color on the logarithmic metallicity [Fe/H] will change as a function of metallicity, and several recent studies have found departures from linearity. For instance, it has been known for decades that the color of the giant branch in Galactic GCs is a nonlinear function of [Fe/H] (e.g., Michel & Smith 1984). Richtler (2006), using the observed C − T₁ versus [Fe/H] relation of Harris & Harris (2002), showed that the shape of the color distribution would differ from that of the MDF, and a non-peaked metallicity distribution could result in a bimodal color distribution. Lee et al. (2008) fitted nonlinear relations for C − T₁ color versus [Fe/H] and found that the inferred MDF changed significantly depending on the adopted relation. Blakeslee et al. (2010) fitted a quartic relation to the Peng et al. (2006) data and showed that the empirical fit produces bimodal g − z colors from unimodal MDFs.

As the observations have improved, so have the models, and these also tend to predict nonlinear CMRs (e.g., Lee et al. 2002; Paper I; Cantiello & Blakeslee 2007). This is partly, but not totally, due to improved modeling of the horizontal branch. Kissler-Patig et al. (1998b) showed that the Worthey (1994) models predict a nonlinear relation between [Fe/H] and V − I, with a wavy form qualitatively similar to that found empirically by Peng et al. (2006), although the inflection occurs at higher metallicity because the colors of these models are generally too red at the high-metallicity end (see Blakeslee et al. 2001). This is interesting because the Worthey models do not realistically model the horizontal-branch morphology, but treat it as a red clump near the giant branch with a position that varies according to age. The Lee et al. (2002) model colors clearly showed nonlinear behavior as a function of metallicity even without any horizontal-branch component, but the nonlinearity in the optical colors was more pronounced with the horizontal branch.

Despite the advances, more work is needed, especially on the behavior of the horizontal branch in extragalactic GC systems, as this is a complex multi-parameter problem. Metallicity is the primary factor governing horizontal-branch temperature, and the transition occurs in a nonlinear way at intermediate metallicities (e.g., Lee et al. 1994). However, variations in other parameters, including age, helium content, and central density, can create significant scatter in horizontal-branch morphology at a given metallicity (e.g., Sandage & Wildey 1967; Zinn 1980; Stetson et al. 1996, Sarajedini et al. 1997; Buonanno et al. 1997; Sweigart & Catelan 1998; see also the recent discussions by Yoon et al. 2008; Gratton et al. 2010; Dotter et al. 2010). There are also intricate intercorrelations among these parameters, as well as correlations with GC mass. For instance, the CMR found among the brightest GCs in external systems (e.g., Harris et al. 2006; Strader et al. 2006; Mieske et al. 2006, 2010; Peng et al. 2009) is likely the result of a mass–metallicity relation. In Galactic GCs, the presence of an extreme blue horizontal branch correlates strongly with GC mass (Lee et al. 2007) and is likely related to the presence of helium-enhanced subpopulations (e.g., Norris 2004; Lee et al. 2005a; Piotto et al. 2005; Yoon et al. 2008; Han et al. 2009; Gratton et al. 2010). More massive GCs also have smaller half-light radii and higher central densities (van den Bergh 1996). Finally, there is some evidence for correlations between age and metallicity in both the Galactic and extragalactic GC systems (Puzia et al. 2005; Beasley et al. 2008; Marín-Franch et al. 2009), but the degree of correlation must depend on the formation history of the GC system.

To summarize, the best current data indicate that optical CMRs for GC systems are nonlinear, and multiple sets of models support this general result. However, exactly how colors vary with metallicity depends on ages, the age–metallicity relations, and variations in other parameters such as α-element and helium contents. Thus, a complete picture of the multiband color–metallicity behavior in extragalactic GC systems would include an understanding of the interplay of the various stellar population parameters. Lacking more stringent observational constraints, we consider only some simple, reasonable assumptions on the standard stellar parameters, and for the present work, we use our best predicted CMRs.

2.2.1. The Milky Way

2.2.2. M31

2.2.3. Cen A and the Sombrero

2.2.4. Giant Elliptical Galaxies

In this section, we present the results of our multiband photometry for GC systems of M87 (NGC 4486) and M84 (NGC 4374). We have selected M87 and M84, giant elliptical galaxies in the Virgo Cluster, because they both have GC systems with confirmed color bimodality in g − z and are among very few elliptical galaxies with deep u-band observations available. We refer the reader to Yoon et al. (2011, hereafter Paper II) for greater details on the multiband photometry of the M87 GC system in the context of nonlinear CMRs. The archival F336W images of Hubble Space Telescope (HST)/WFPC2 and HST/WFC3 were used to obtain u_F336W for GC candidates in M87 and M84, respectively. Our u_F336W-band catalogs were matched with ACS/Wide Field Channel g_F475W- and z_F850LP-band photometry of Jordán et al. (2009). We hereafter refer to u_F336W, g_F475W, and z_F850LP mags as u, g, and z, respectively. Jordán et al. (2009) selected bona fide GCs with their magnitudes, g − z colors, and sizes. We further employed color cuts in the u-band colors to filter out contaminating sources, especially background star-forming galaxies. We used 591 GCs in M87 and 306 GCs in M84 that have reliable u, g, and z measurements in common. The samples are u-band limited.

The merit of the multiband observations is clear. Since the form of CMRs hinges on which color is used, the shape of the color distributions varies significantly depending on the colors in use. Hence a comparative analysis of the GC MDFs that are independently obtained from distributions of different colors will put the nonlinearity hypothesis to the test, as proposed in Paper I and Paper II. Among other optical colors, the u-band related colors (e.g., u − g and u − z) are theoretically predicted to exhibit the most distinctive CMRs from other preferred CMRs (e.g., for g − z), and thus the most adequate to the task. Furthermore, the u-band colors are significantly less affected by the variation in the horizontal-branch mean temperature, having less inflected, "smooth" CMRs than g − z for given ages. Therefore, for instance, the conversion from u − g color distributions to MDFs via the (u − g)–[Fe/H] relation should be more straightforward than the case of g − z. The reason why the CMRs for u-band colors are less inflected than the g − z CMR is twofolded: (1) the integrated u-band colors of main-sequence and red-giant-branch stars are smoother functions of metallicity compared to g − z and (2) the u-band colors are less sensitive to the horizontal-branch temperature variation, which is due to the fact that the blueing effect of the optical spectra with increasing horizontal-branch temperature is held back by the Balmer discontinuity where the u-band is located (Yi et al. 2004). Such properties make the u-band colors good metallicity indicators for a wide range of age, and the u-band color distributions are expected to be significantly different from distributions of other optical colors such as g − z, V − I, and C − T₁. See Paper II for a detailed discussion of the u-band colors as a tool to probe the nonlinearity of CMRs.

Figure 1 shows the observed color distributions of the M87 GC system from u, g, and z photometry and the inferred MDFs. First, Figures 1(a) and (d) present the g − z versus u and u − g versus u diagrams, respectively. Figures 1(b) and (e) present the g − z and u − g color distributions, respectively. The g − z distribution of the M87 GCs unambiguously displays two peaks around g − z = 1.0 and 1.4. In contrast, the u − g distribution for the identical sample does not appear to have clear bimodality. One may argue that the larger observational uncertainties in u-band weaken the bimodality. Table 3 shows that the typical photometric error of u − g is 2.2 times larger than that of g − z for the entire sample, but at the same time the ranges spanned by the colors are Δ(g − z) = 1.1 mag and Δ(u − g) = 2.1 mag, that is, the baseline of u − g is 1.9 times longer than that of g − z. As a result, the relative sizes of error bars are (g − z:u − g) = (1.0:1.2). In a relative sense, the errors in the two colors are quite comparable to each other. Moreover, Paper II shows that the u − z color, which has smallest relative errors, still exhibits weaker bimodality in the color distribution compared to the g − z distribution. It is, therefore, not likely that bimodality in the u − g distribution of M87 GCs is simply blurred by larger observational errors in the u-band.

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The variation in the histogram shape for different colors may suggest that the form of the CMRs varies depending significantly on the colors in use. This is shown in Figures 1(c) and (f), along with our predictions from the Yonsei Evolutionary Population Synthesis (YEPS) model.⁶ In Figure 1(c), the g − z colors are shown as a function of [Fe/H] for GCs in the Milky Way (open circles), and M49 and M87 (filled circles and triangles). The references to the observed data used in the relations are summarized in Table 1. The fifth-order polynomial fit to our model data for 13.9 Gyr GCs is overlaid (thick solid line, see Table 2). For a comparison, the straight gray line represents the linear least-squares fit to the data. Figure 1(f) is the same as Figure 1(c), but for u − g color. Open circles, blue and red filled squares represent GCs in the Milky Way, M87, and NGC 5128, respectively. The u − g colors of the GCs in the Milky Way and NGC 5128 were obtained from their U − B colors via the equation, (u − g) = 1.014 (U − B) + 1.372, derived from model data for synthetic GCs with combinations of age (10 ∼ 15 Gyr of 1 Gyr intervals) and [Fe/H] (−2.5 ∼ 0.5 dex of 0.1 dex intervals). U and B passbands are relatively close to u and g passbands, respectively, and thus U − B has responses to the horizontal-branch morphology in a way that is very similar to u − g. Therefore, U − B are a good proxy to u − g, and the U − B versus u − g relationship is best described by a linear fit over a range of ages and metallicities (Table 1). Guided by the model, the form of the observed CMRs appears to vary from the (g − z)–[Fe/H] relation to (u − g)–[Fe/H]. One may argue, however, that current data appear to be fit both by the theoretical nonlinear relations and the empirical straight relations, as there is only a weak indication that the modeled relations are actually better fits to the g − z versus [Fe/H] and u − g versus [Fe/H] data. Given the current level of observational accuracy and inhomogeneity of the data, the purpose of our study is not to determine the exact shape of color–metallicity relationships, but to investigate the consequences and implications of the possible nonlinear transformations.

We now consider the inferred GC MDFs. Figures 1(g) and (h) show the MDFs for the M87 GC system. On the one hand, Figure 1(g) presents the GC MDFs converted from g − z (red histogram) and u − g (blue histogram) colors that are based on the traditional linear color-to-metallicity conversion (thin gray lines in Figures 1(c) and (f)), and thus are just replicas of their color histograms shown in Figures 1(b) and (e). Note that both the overall shape and the peak positions do not appear to agree between the GC MDFs from the two colors. On the other hand, Figure 1(h) presents the GC MDFs converted from g − z and u − g colors that are based on the improved inflected relationship between color and metallicity (thick black lines in Figures 1(b) and (e)). In contrast to Figure 1(g), the inferred GC MDFs in Figure 1(h) are modified drastically to have a strong metal-rich peak with a metal-poor tail. The two histograms in Figure 1(h) are more consistent with each other in terms of their overall shape and peak positions than those shown in Figure 1(g). We note that our stellar population models show that, for given input parameters, the absolute quantities of output are rather subject to the choice of model ingredients such as stellar evolutionary tracts and model flux libraries; the different choices can result in up to ∼0.2 mag g − z and u − g variation among models and the inferred [Fe/H] values accordingly (up to ∼0.5 dex). Hence, one should put more weight on the relative values of inferred GC MDFs, i.e., the overall morphology of the MDFs and their unimodality. We, however, wish to emphasize that the typical GC MDF shape is obtained invariably from different colors, i.e., g − z and u − g for M87 GCs.

Figure 2 is the same as Figure 1, but for the GC system in M84. First, Figures 2(a) and (d) present the g − z versus u and u − g versus u diagrams for the M84 GCs. Figures 2(b) and (e) present the g − z and u − g color distributions, respectively. As for the possible role of observational uncertainties in weakening bimodality of the u − g color distribution, the typical photometric error of u − g is 1.7 times larger than that of g − z for the entire sample (Table 3), but at the same time the ranges spanned by the colors are Δ(g − z) = 1.1 mag and Δ(u − g) = 1.8 mag, that is, the baseline of u − g is 1.6 times longer than that of g − z. As a result, the relative sizes of error bars are (g − z:u − g) = (1.0:1.1). In a relative sense, the errors in the two colors are comparable. It is, therefore, not likely that bimodality in the u − g distribution of M84 GCs is simply blurred by larger observational errors in the u-band. Figures 2(c) and (f) are the same as Figures 1(c) and (f), respectively. Finally, Figures 2(g) and (h) show the inferred MDFs for the GC systems in M84. Again, the inferred GC MDFs in Figure 2(h) have a strong metal-rich peak with a metal-poor tail. The two histograms in Figure 2(h) show a better agreement with each other in terms of their overall shape and peak positions than those in Figure 2(g).

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In this section, we have obtained multiband colors of GCs in the two representative giant elliptical galaxies, M87 and M84, and examined their color and metallicity distributions. We have found that the distributions of different colors can be transformed into unimodal metallicity distributions that are strongly peaked with a broad metal-poor tail. The implications of the typical shape of the inferred GC MDFs and its similarity to those from chemical evolution models and field-star observations (see Section 4) will be discussed in Section 5. We note, however, that the similarity itself between the GC MDFs from multiband colors does not necessarily represent evidence that the model is correct. Whether the similar MDFs from various colors can be taken as evidence for the nonlinear-CMR scenario for the color bimodality is a sufficiently involved issue and fully explored in Paper II.

Motivated by the findings above in Section 3.1 for the individual galaxies and to avoid possible small-number statistics, we now benefit from the 100 early-type galaxies in the ACSVCS (Côté et al. 2004; Peng et al. 2006; Jordán et al. 2009). We apply the color-to-metallicity transformation scheme to ∼10,000 GCs in ACSVCS, the largest and most homogeneous photometric database of extragalactic GCs currently available. Figure 3 presents the observed color distributions and inferred MDFs of GC systems in bins of host galaxy luminosity. In Figure 3(a), we show the observed color histograms of GC systems for seven bins of host galaxy magnitude. The data are the same as in Figure 6 of Peng et al. (2006) and here we listed the data in Table 4. The histograms are normalized by the GC number at their blue peaks and multiplied by constants, C, for clarity. The magnitude bins are 1 mag wide and extend from M_B ≃ −21.5 (−22 ⩽ M_B < −21, red, C = 1.0) to ≃ −15.5 (−16 ⩽ M_B < −15, purple, C = 0.4). A Gaussian kernel of σ(g − z) = 0.05 is applied. Clearly, the histograms appear bimodal or asymmetric across the entire luminosity range, with all GC systems containing blue peaks and with more prominent red peaks in brighter hosts.

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A close scrutiny of Figure 3(a) reveals the tendency of the dip positions (and blue peak positions) in the color histograms to become progressively bluer as the host luminosity decreases. In the context of nonlinear CMRs, this can be explained if GCs in fainter galaxies are slightly younger than those in brighter galaxies. This is because at younger ages, the blue horizontal branch develops at lower metallicity (Lee et al. 1994; Yoon et al. 2008; Dotter et al. 2010) and, as a consequence, the predicted colors of the quasi-inflection points along the CMR move systematically toward the blue at younger ages. This effect is demonstrated in Figure 3(b) by an example set of the YEPS model predictions with Δt(brightest–faintest) = 3 Gyr. In this example, ages range from 10.5 Gyr (purple) to 13.5 Gyr (red) by equal intervals of 0.5 Gyr. The YEPS g − z data for 9 Gyr to 14 Gyr by steps of 0.5 Gyr are given in Table 2. The thin straight dotted line represents the linear least-squares fit to the data shown in Figures 1(c) and 2(c).

Figures 3(c)–(f) present the results of the four different color-to-metallicity transformations. Like the color distributions in Figure 3(a), the inferred GC MDFs are displayed for seven bins of M_B from M_B = −22 to −15 in steps of 1 mag. To obtain these MDFs, we converted the color of each GC into [Fe/H] using the (g − z)–[Fe/H] relations from the YEPS model. Each panel makes a different assumption on the systematic age sequence from the faintest host bin to the brightest. The modeled ages for host luminosity bins are shown in the insets of Figures 3(c)–(f), with age differences, Δt = 0, 1, 2, and 3 Gyr, respectively, between the faintest (M_B ≃ −15.5, purple) and brightest (−21.5, red) bins. The brightest, oldest (M_B ≃ −21.5, red) bin is set to be 13.5 Gyr. The color distribution shown in Figure 3(a) was transformed to MDFs via the model CMRs of the corresponding ages.

Figure 3(c) presents the case in which the age is assume to be constant at 13.5 Gyr regardless of host luminosities between M_B ≃ −15.5 (purple) and −21.5 (red). The GC MDFs for the luminous (M_B < −17) host bins (the first to fifth brightest bins) in particular are strongly peaked with a broad metal-poor tail. As Paper I suggested, the strong bimodality seen in the GC color distribution of luminous galaxies is not evident in the MDF once transformed by their wavy CMR. The MDFs in the faintest two bins, however, have broad peaks. This is likely because the less luminous galaxies primarily have blue GCs and their colors are largely bluer than the inflection point in the CMR under the constant age assumption, i.e., Δt(brightest–faintest) = 0 Gyr. Paper I mentioned that the position of the inflection is bluer for younger stellar populations and so if fainter galaxies host GCs younger than brighter counterparts, then a different CMR may apply. However, in this naive use of the single CMR, not all inferred [Fe/H] distributions appear to be unimodal.

Comparative analysis shows that the non-zero age difference of Δt(brightest–faintest) = 1–3 Gyr (Figures 3(d)–(f)) results in the GC MDFs for all the host luminosity bins that fit better with skewed Gaussian distributions with metal-poor tails. The gray dotted histogram in each panel represents the inferred MDF for the brightest (i.e., M_B ≃ −21.5) bin based on the simple straight fit in Figure 3(b) and is intended for comparison to the red solid MDF. Obviously, the strong bimodality seen in the color distributions of luminous galaxies is no longer evident in the MDFs, once transformed via the inflected color–metallicity relationship. The inferred GC MDFs for all seven host luminosity bins have the same characteristic shape—being sharply peaked with a broad metal-poor tail—across three orders of magnitude in the host galaxy mass. In addition, the mean [Fe/H] and peak position of the GC MDFs are a strong function of the host luminosity, in the sense that, for brighter host galaxies, the mean [Fe/H] increases and the peak gets redder.

Compared to their relative ages, the absolute ages of GC systems are still less certain. To test the robustness of the result shown in Figure 3 against different absolute ages, Figure 4 makes differing assumptions on the age sequence from the faintest host galaxies to the brightest. In this case, the center bin, i.e., the fourth brightest (M_B ≃ −18.5, green) bin, is set to be 13 Gyr. The modeled ages for the host luminosity bins are shown in the insets of Figures 4(c)–(f), with age differences of Δt = 0, 1, 2, and 3 Gyr, respectively, between the faintest (M_B ≃ −15.5, purple) and brightest (−21.5, red) bins. In Figure 4(b), another example set of our model prediction with Δt = 3 Gyr is shown. In this example, ages range from 11.5 Gyr (purple) to 14.5 Gyr (red) in equal intervals of 0.5 Gyr (Table 2). Figures 4(c)–(f) show that the strong bimodality is no longer evident in the inferred MDFs, and the typical shape of the GC MDFs does not depend upon differing age assignment from the faintest host galaxies to the brightest. It also holds true that the peak positions of the GC MDFs are a strong function of the host luminosity in that, for brighter host galaxies, the mean [Fe/H] increases and the peak gets redder. We also tested the stability of the results against the putative age dispersion in the examined data. The typical form of the GC MDFs persists with up to σ_t ≃ 2.5 Gyr, where the age spread among GCs in a single host luminosity bin is parameterized by a Gaussian dispersion, σ_t. The data for the inferred GC MDFs in Figures 3(d) and 4(d) (they are identical) are given in Table 5.

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If one could assume that the intrinsic shapes of the GC MDFs are similar in all host luminosity bins, our results might give a tantalizing hint of varying ages among GC systems in that GCs in more luminous parents are older than GCs in fainter galaxies. However, there is not yet any observational support that the assumption is valid. Moreover, the true shape of CMRs suggested in this study is still unproven. Hence, the purpose of our simulations should be to determine neither the absolute age of the GC systems nor their exact pattern of age sequence from the faintest bin to brightest. Nevertheless, the possible age sequence appears not inconsistent with the "galaxy downsizing" picture (e.g., Cowie et al. 1996) in which brighter galaxies are observed to form earlier. If confirmed, this can be regarded as another indication of the common characteristics shared by stellar populations of GCs and halo field stars (see the next section).

The necessity of obtaining photometry of spatially resolved field stars limits us to nearby galaxies. There are several nearby, relatively massive elliptical galaxies whose stellar MDFs have been measured, and our inferred GC MDFs can directly be compared to such stellar MDFs. Figure 5 gives a comparison of the MDFs of the ACSVCS GCs to those of resolved stars of nearby elliptical galaxies (Harris & Harris 2002; Rejkuba et al. 2005; Harris et al. 2007a, 2007b; Bird et al. 2010). The stellar MDFs of individual galaxies were obtained from color–magnitude diagrams of red giant stars whose colors are highly sensitive to their metallicities.

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In Figure 5, we plot the GC MDF shown in Figures 3(d) and 4(d) (they are identical). The data for the plot are presented in Table 5. The figure shows that the GC MDFs are similar to the MDFs of resolved constituent stars of nearby elliptical galaxies. In particular, the typical shape of the GC MDFs (colored solid line in each panel), characterized by a sharp peak with a metal-poor tail, is remarkably consistent with those of field stars in nearby galaxies. By contrast, GC MDFs obtained using the simple linear [Fe/H] versus g − z relation (gray dotted lines) do not agree with the stellar MDFs. We note that the metallicity spread of GCs tends to be broader than that of stars. As indicated in Figures 3(c)–(f) and Figures 4(c)–(f), the GC MDFs, on the whole, are broader than the stellar MDFs. This is likely due to the fact that each color histogram of the ACSVCS consists of GCs belonging to 3–20 diverse host galaxies, giving an ensemble character of each host luminosity bin. Moreover, the inferred MDFs become broader as the observational uncertainty in color is propagated to metallicity space.

In addition to the noticeable similarities found in the shape of MDFs between GCs and stars, they share a common feature in that their mean metallicity increases with increasing host luminosity. As a result, the peak positions of GCs and stars are roughly coincident in each luminosity bin—at [Fe/H] ≃ −1.0 (faint hosts) and −0.5 (bright hosts). This, however, should be taken with great caution because the mean colors of both GCs (Dirsch et al. 2003; Jordán et al. 2004; Tamura et al. 2006; Lee et al. 2008) and field stars (Harris & Harris 2002; Rejkuba et al. 2005) depend significantly on the sampled radial location in a galaxy. Their mean metallicities gradually decrease with projected radius, as seen in Figures 5(d) and (e) for the outer- and inner-halo stars of NGC 5128, respectively. Moreover, observations show that the mean GC color is bluer than stars within an elliptical galaxy as a whole (Peng et al. 2006). More importantly, GCs are on average bluer than the unresolved light of the galaxies at the same radii (e.g., Forte et al. 1981, 2009; Strom et al. 1981; Jordán et al. 2004; Tamura et al. 2006; Lee et al. 2008). For old stellar populations such as those in elliptical galaxies and GCs, the observed bluer colors imply lower metallicities in all stellar models, thus leading to the conclusion that GCs are typically more metal-poor than field stars in a galaxy. Hence it may just be a coincidence that the GC MDFs line up with the field-star MDF of the nearest elliptical galaxy. Despite these caveats, the peak and width of the MDFs of GCs and field stars are similar enough to suggest that both were formed in the same events, which built the major part of the galaxies.

It should be addressed that the [Fe/H] histogram for NGC 3379 stars (Figure 5(c)), despite the overall resemblance to the compared GC MDF, shows an excess of metal-poor stars with a bump at [Fe/H] ≃ −1.2. Harris et al. (2007b) found the metal-poor stellar halo in NGC 3379 and provided an explanation for why their earlier studies only detected a metal-rich component in the same galaxy. The NGC 3379 HST field is at a distance of 12 effective radii, which is more than twice as far as the equally small fields of view for NGC 3377 (Figure 5(b)) and NGC 5128 (Figures 5(d) and (e)). Studies of the stellar MDFs of large galaxies have been performed photometrically and concentrate on more central regions of the galaxy where the metal-rich field-star population is concentrated. The radial bias seemed to undersample metal-poor stars in the earlier studies. With the recent evidence of metal-poor halo stars emerging at great galactocentric distances, one should not assume that the stellar MDFs are unimodal at all positions. Therefore, this suggests that our interpretation should be more applicable to their main, inner parts of galaxies, less to the remote outskirts.

As envisaged by the NGC 3379 case above, each galaxy has its own evolutionary history. So a more direct way to assess the similarity between GCs and stars is to compare the MDFs of spatially resolved field stars in a galaxy with those in its own GC system. There are four galaxies (M87, NGC 5128, NGC 3377, and NGC 3379) in Figure 5, for which both the GC MDF and the stellar MDF are currently available. Figure 6 gives the comparison for M87 (the top row), NGC 5128 (the second row), NGC 3377 (the third row), and NGC 3379 (the bottom row). The stellar MDFs (gray histograms) are identical to those in Figures 5(b), (c), (d), and (f). The left-hand panels present the inferred GC MDFs (empty histograms) based on the traditional linear color-to-metallicity transformations. By contrast, the right-hand panels show the inferred GC MDFs (empty histograms) based on the inflected relations from the YEPS model (Table 2).

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First, for M87, we exploit g − z colors from the ACSVCS (Peng et al. 2006; Jordán et al. 2009) to derive the GC MDF. Since the inferred GC MDFs shown in Figure 1(h) are based on the u-band limited sub-sample, the ACSVCS g- and z-band data are more representative GC sample. Figure 6(a) shows that the MDF (empty histogram) for 1745 GCs obtained using the simple linear [Fe/H] versus g − z relation (shown in Figure 1(c)) exhibits a fundamental difference from the stellar MDF (Bird et al. 2010) measured in a similar region of the same galaxy. By contrast, in Figure 6(e), the strong bimodality is no longer present once transformed by the inflected CMR (Table 2), and the GC MDF is very similar to that of the brightest host bin in the ACSVCS. As a consequence, the MDFs of GCs and field halo stars in M87 are similar in shape and line up remarkably well with each other.

Second, for NGC 5128, Peng et al. (2004a, 2004b) presented the CTIO Blanco 4 m U-, B-, V-, and I-band photometry of the GC system. We use the B − I color distribution to derive the GC MDF because B − I is a reasonable substitute for g − z. However, our result on NGC 5128 is not affected by the choice of colors. Figure 6(b) shows the MDF (empty histogram) for 210 GCs that has strong bimodality. On the other hand, in Figure 6(f) we converted the B − I colors into [Fe/H] using the YEPS relation (Table 2). The strong bimodality seen in Figure 6(b) is not evident in the MDF, and the strongly peaked GC MDF with a broad metal-poor tail is in reasonably good agreement in shape with the stellar MDF (Rejkuba et al. 2005).

An unavoidable uncertainty in this comparison is that the field stars are sampled from only one or two projected location(s), whereas the GCs cover the wider halo. For M87, the stars were sampled in an inner region (R ≃ 10 Kpc) and the GCs were in the inner halo (≲10 Kpc). For NGC 5128, the stars were sampled in an inner region (R ≃ 20 and 30 Kpc) and the GCs were in the entire halo of the galaxy. This may partly explain the similarity of the peak positions between stellar and GC MDFs for M87, and the dissimilarity for NGC 5128. Another warning for NGC 5128 is that, although the majority of its GCs (∼90%) are known to be old (>10 Gyr; Beasley et al. 2008), there should be a certain portion of young GCs for which one needs to apply a different color versus metallicity relation to derive the GC MDF. Nonetheless, despite these sources of uncertainty, it is engrossing that the inferred GC MDFs based on the inflected color-to-metallicity transformations give better matches with the field-star MDFs of M87 and NGC 5128, compared to those based on the traditional linear relations.

Third, for NGC 3377, Cho et al. (2011) presented the HST/ACS g- and z-band photometry of the GC system as part of a deep imaging study of 10 early-type galaxies in low-density environments. Figure 6(c) shows that the MDF for 157 GCs derived from the simple linear [Fe/H] versus g − z relation has strong bimodality. On the other hand, in Figure 6(g) we converted the g − z color into [Fe/H] using the YEPS relation. The strong bimodality seen in Figure 6(c) is not evident in the MDF, and the sharply peaked MDF with a broad metal-poor tail is in reasonably good agreement in shape with the stellar MDF (Harris et al. 2007a, 2007b).

Lastly, for NGC 3379, Whitlock et al. (2003) and Rhode & Zepf (2004) carried out wide-field photometry of GCs. They verified earlier results that its GC population is quite small. Harris et al. (2007b) derived the GC MDF from Rhode & Zepf's (2004) B − R histogram for 36 GCs with an improved empirical relation, [Fe/H] = 3.13 (B − R)₀–5.04. Figure 6(d) shows the GC MDF, which is highly concentrated toward the metal-poor side. If divided at [Fe/H] = −1.2 where the metal-poor bump of the filed-star MDF is located, the metal-poor GCs outnumber the metal-rich GCs by 25–11. As a result, for the metal-rich half of the MDF, the numbers of GCs are much too small. By contrast, Figure 6(h) displays the GC MDFs based on the YEPS relation (Table 2). With the GC MDFs of only 36 clusters, it is not yet clear whether the MDFs of GCs and stars have the same shape. It is interesting to note, however, that the GC metallicities are now more evenly distributed, and the proportions of the two metallicity subgroups, when divided at [Fe/H] = −1.2, became more comparable (21:15) than those in Figure 6(d) (25:11). Interestingly, the inferred GC MDF in Figure 6(h) seems very similar to the NGC 3379 field-star MDF in the western half (farther from the galaxy center) of the ACS/WFC field (shaded histogram in Figure 14 of Harris et al. 2007b).

The new development in linking the observed GC colors to their intrinsic metallicities leads to GC MDFs that are strikingly similar in shape to MDFs of resolved field stars in nearby elliptical galaxies. Both the inferred GC MDFs and halo stellar MDFs are characterized by a sharp peak with a metal-poor tail. In this section, we proceed to compare the inferred GC MDFs to chemical enrichment models of galaxies.

In Figure 7, we compare the inferred GC MDFs for the second brightest (M_B ≃ −20.5, orange) and the brightest (M_B ≃ −21.5, red) galaxy luminosity bins with the simple closed-box model of chemical evolution (e.g., Pagel & Patchett 1975). We plotted the MDFs for this simple model using yields of [Fe/H] = −0.685 and −0.505, respectively. With the improved nonlinear relationship between color and metallicity, the general shape of GC MDFs is in remarkable agreement with that of galaxy chemical enrichment models. In contrast, the dotted line in each panel represents the inferred GC MDF for the corresponding bin based on the simple straight fit to the data.

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Despite their resemblance in the general shape, the inferred GC MDFs have fewer metal-poor GCs than the simple chemical model. This is akin to the "G-dwarf problem" in the solar neighborhood (e.g., van den Bergh 1962; Schmidt 1963). We note that the width of the chemical evolution model can be changed using various kinds of gas infall and stellar feedback, and an accreting-box model of chemical evolution yields narrower distributions that provide a better match with the inferred GC MDFs. For instance, Harris & Harris (2002) remarked that the halo of NGC 5128 (M_B = −20.9) also suffers from the G-dwarf problem in that it lacks metal-poor stars compared to the simple model. They were able to fit the field-star MDF using an accreting-box model of chemical evolution, producing a narrower distribution that is also a better match to the inferred GC MDFs in this study.

Remarkable progress has occurred over the past few decades in our understanding of extragalactic GC systems. One of the most important discoveries is that many galaxies show bimodality in their color distributions, leading to the notion that galaxies possess two distinct subpopulations of GCs.

We showed, however, that the typical GC MDF shape derived from color distributions is unimodal and characterized by a sharp peak with a metal-poor tail. If confirmed, the inferred GC MDFs may appreciably reduce the demand for the separate formation mechanisms to explain the metal-poor and metal-rich division of GCs. We warn, though, that the sample used in this study is the GC systems in the Virgo galaxy cluster obtained from the HST ACS/WFC, WFPC2, and WFC3 observations. The field of view of ACS/WFC, for example, covers galaxies' halos within R ≃ 0.6, 0.8, 7.2, 8.3, 8.9, 9.4, and 10.6 R_e (in z-band; Ferrarese et al. 2006) for the seven host luminosity bins (from the brightest bin to the faintest) of the ACSVCS, respectively. Nevertheless, the dearth of metal-poor GCs in MDFs for inner spheroids of giant (M_B ≲ −20) elliptical galaxies and nearly the entire spheroids of normal (−20 ≲ M_B ≲ −15) ellipticals suggest that the inheritance of metal-poor GCs via dissipationless accretion from dwarf satellites seems to be less significant than previously thought.

On the outskirts of giant galaxies, accretion of metal-poor GCs from low-mass satellites and/or from surrounding regions may be an important channel for a galaxy to add GCs on the metal-poor part of GC MDFs (Forte et al. 1982; Côté et al. 1998, 2002; Masters & Ashman 2010; Lee et al. 2010b). Recall, however, that one of the main consequences of the nonlinear metallicity–color relations is that their steepness at the metal-poor end naturally creates a blue peak of GCs in color space, which is a direct cause of the conventional subpopulation of blue GCs. Therefore, even for the outskirts of giant galaxies in cluster environments, whether or not accretion of metal-poor GCs is solely responsible for blue peaks of color distributions is still an open question. The strongest evidence against accretion models is that the blue peak colors are correlated tightly to the host galaxy luminosity (Larsen et al. 2001; Strader et al. 2004; Peng et al. 2006). The metal-poor relation implies that metal-poor GCs, although they formed at very high redshift and were accreted later on, already "knew" which galaxy they would ultimately belong to, and thus weakens the accretion scenario for the color bimodality. Alternatively, the nonlinear metallicity–color relation scenario (Paper I) gives cohesive explanations for the observations that the mean colors of both blue and red GCs increase progressively for more luminous host galaxies.

Further counter-evidence of the accretion model is the significant fraction of blue GCs in massive cluster galaxies in relatively lower-density regions (Peng et al. 2006) and massive field galaxies in isolation (Cho et al. 2011). In such environments, galaxies have few neighboring lower-mass galaxies, and it would be difficult to acquire many metal-poor GCs via accretion. Therefore, the accretion process seems more important when it occurs in (1) the outskirts (rather than the inner, main bodies) of (2) giant (rather than dwarf) ellipticals orbited by a large number of low-mass satellites in (3) cluster (rather than isolated) environments. In this regard, wide-field, multiband studies of GC systems in cluster and field environments are clearly needed. Wide-field photometry of nearby cluster galaxies in CTIO 4 m U-band (H. Kim et al. 2011, in preparation) and Subaru/MOIRCS NIR (S. Kim et al. 2011, in preparation), and a study of massive field galaxies in HST/ACS g and z (Cho et al. 2011) have been conducted or are in progress to further investigate our alternative scenario.

Current hierarchical models of galaxy formation in the ΛCMD cosmology predict that several thousands of small building blocks were involved for the emergence of one single massive galaxy. This is significant because the extent of complexity may leave little room for the existence of just two GC subpopulations in each massive galaxy. Indeed, the unimodal, skewed MDFs arise naturally in an aggregate of a large number of protogalactic gas clouds from its virtually continuous chemical evolution through many successive rounds of star formation. The strongly peaked unimodal GC MDFs point to GC formation with a relatively short, quasi-monolithic timescale. Remarkably, the typical GC MDF shape emerges across three orders of magnitude in host galaxy mass. This suggests that the processes of GC formation and chemical enrichment are quite universal among a variety of GC systems.

We also address the important issue of whether or not the formation of GCs is coupled with the bulk formation of the stellar population of host galaxies. Or, equivalently, does GC formation really mirror star formation in a galaxy? We have shown that the inferred GC MDFs agree reasonably well with the stellar MDFs of nearby galaxies and the MDFs produced by models of galaxy chemical evolution. The results suggest that the evolutionary histories of GC systems and their parent galaxies are strongly coupled, and thus share a more common origin and closer subsequent evolution than previously thought.

An important aspect of the GC-host galaxy co-evolution issue concerns the GC-to-star offset, in the sense that GCs are on average more metal-poor than the unresolved light of an elliptical galaxy at the same radial location (e.g., Forte et al. 1981, 2009; Strom et al. 1981; Jordán et al. 2004; Tamura et al. 2006; Lee et al. 2008). In light of the typical shape of GC MDFs proposed in this study, the lower value of the mean metallicity of GCs compared to that of field stars should not be attributed to the number excess of the metal-poor GCs. Alternatively, it is more likely that, at a given radial location, the GC MDF on the whole is shifted toward the metal-poor side with a peak at a lower [Fe/H] value, compared to the field-star MDF.

The GC-to-star offset would point to a picture in which GCs are the remnants of vigorous star burst events in the early stages of galaxy formation, and thus preferentially trace the major mode of star formation in galaxies. If so, GC formation was less prolonged than field-star formation. Recent observations show that GCs were at least an order of magnitude more massive at birth than now (e.g., Conroy 2011), and the large masses may have been the cause of the earlier truncation of their formation process than that of stars. Consequently, the chemical enrichment process of a GC system appears to have ceased somewhat earlier than that of the field stellar population in each star formation episode.

We further showed a possible age difference among GC systems, in that the GC systems in fainter galaxies are on average younger. We refer to this as "GC system downsizing." The GC system downsizing phenomenon appears to further support the similar nature shared by stellar populations of GCs and field stars. Interestingly, recent observations reveal that the GC systems in Milky Way satellites and some Milky Way GCs believed to be accreted from satellite dwarf galaxies are relatively younger than the majority of Galactic GCs (e.g., Marín-Franch et al. 2009).

Galaxy downsizing is generally defined by more prolonged residual star formation in fainter galaxies. However, the "GC system downsizing" involves the idea that the first generation of GCs in fainter hosts is created later than the counterpart in brighter galaxies and may provide further information on galaxy downsizing itself. Provided that GC systems became attached to the present host galaxies from the beginning, we can speculate that not only the first GCs but also first field stars were formed earlier in brighter galaxies than fainter galaxies. That is, massive galaxies today were likely where conditions first favored star formation, thus suggesting a prolonged epoch of galaxy formation in the universe. Combined with the fact that GCs and field stars in brighter galaxies have higher metallicities, a picture emerges in which the formation and accompanying metal enrichment of both GCs and halo stars seem to have started earlier and proceeded more rapidly and efficiently in massive galaxies, presumably in denser environments.

It is also important to note that some galaxies have substructure in their stellar MDFs and are thus more complex than a smooth transition from a metal-rich dominance to metal-poor with increasing radius. Examples include the Milky Way (Ibata et al. 2001; Majewski et al. 2003; Yanny et al. 2003; Ivezić et al. 2008) and M31 (Kalirai et al. 2006; Koch et al. 2008; McConnachie et al. 2009). The stellar MDF of the elliptical galaxy NGC 3379 (Figure 5(c)) also shows a fine substructure (Harris et al. 2007b). The remote outer halos of these galaxies are inhomogeneous in terms of surface density and show different metallicity distributions from the inner halos. This indicates that the stellar populations in the outer halos of galaxies are not well mixed, and in turn supports the build up of halo formation via satellite accretion (Forte et al. 1982; Côté et al. 1998, 2002; Masters & Ashman 2010), which can have mixing time scales of a few Gyr in these outer regions (Johnston et al. 1995).

The evidence of the satellite accretion indicates that the outskirts of galaxies do not all build up their stellar populations in one single way. For instance, the NGC 3379 observation fits into a model in which its outskirts were formed by a combination of earlier dissipative mergers and later accretion of dwarf satellites. The remote halo of NGC 3379 has been on the fringes of active, violent gas-rich mergers at the early epochs of galaxy formation, and later experienced dry accretion of stars and GCs from its metal-poor satellite dwarfs. Therefore, one should not assume that the stellar MDFs are unimodal at all positions, and our new interpretation would be more applicable to the inner, main bodies of galaxies than their remote outskirts.

Our results may be an important step forward in resolving the long-standing disagreement between GCs and field stars, and in reconstructing the history of GC systems and their parent galaxies. Although the sample used in this study (the HST ACS/WFC, WFPC2, and WFC3 photometry for the GC systems in the Virgo galaxy cluster) confines our discussion to R ≲ R_e for giant ellipticals and ≲10 R_e for normal ellipticals, our findings suggest that GC systems and their parent galaxies have shared a more common origin than previously thought, and hence greatly simplify theories of galaxy formation. The star formation and accompanying chemical evolution were virtually continuous via aggregates of a large number of protoclouds and via repeated gas-rich mergers with other galaxies, leading to the unimodal, skewed MDFs of both stars and GCs. The metal enrichment of both stars and GCs proceeded more rapidly and efficiently in massive galaxies, resulting in more metal-rich stars and GCs in those galaxies. The observed radial metallicity gradients are understood if the chemical enrichment in the dense centers was more rapid and efficient than in the less-dense outskirts. The typical GC MDF shape emerges across three orders of magnitude in the host galaxy mass, suggesting that the processes of GC formation and chemical enrichment are quite universal among various GC systems, at least for the inner, main spheroids of giant ellipticals and nearly the entire halos of normal ellipticals.

The histories of GCs and their host galaxies are reconstructed as follows.

• 1.  
  Giant ellipticals' inner, main halos. The inner, main spheroids (R ≲ 1 R_e) of today's giant elliptical galaxies (M_B ≲ −20) were first created in dense regions of the universe via dissipational mergers of a large number of protogalactic gas clouds (e.g., Searle & Zinn 1978). Their GC systems with the MDF peaks at [Fe/H] ≳ −0.7 (Figures 3 and 4) were formed together with stars in the galaxies. The GC systems from low to high metallicities (i.e., both blue and red GCs), as a whole, do not need two separate mechanisms for formation. There is strong similarity between blue and red GCs in their mass function; for the less evolved high-mass part of the mass function, these are approximately power laws with indices of −1.8 to −2. Although power-law distributions are a consequence of a variety of physical processes, their nearly identical power indices may support an identical formation history of blue and red GCs. The chemical enrichment of the galaxies was rapid, and the difference in the formation epochs of the metal-poor and metal-rich ends of MDFs should be small (≲1 Gyr), as evidenced by observations (e.g., Jordán et al. 2002). In this view, the metal-poor GCs in massive galaxies are the first generation of GCs in the universe.
• 2.  
  Normal ellipticals' entire halos. The spheroids (out to R ≲ 10 R_e) of normal galaxies (M_B ≳ −20) were created later than those of massive ones via self-collapses or mergers of protogalactic clouds that did not take part in massive galaxy formation early on. They were outside of tumultuous, dense regions, and have evolved rather independently from central massive galaxies. Their GC systems with the MDF peaks at [Fe/H] ≲ −0.7 (Figures 3 and 4) were formed together with stars in the low-mass galaxies. As a result, the first generation of both stars and GCs in low-mass galaxies are younger than those of massive galaxies, indicating a prolonged epoch of galaxy formation in the universe. This picture is supported by recent observational evidence that the GC systems in Galactic satellites and some Galactic GCs believed to be accreted from satellite dwarf galaxies are relatively younger than the majority of Milky Way GCs (e.g., Marín-Franch et al. 2009).
• 3.  
  Remote outskirts of giant ellipticals in dense environments. The remote outer halos of giant elliptical galaxies have been on the fringes of active, violent gas-rich mergers at the early epochs of galaxy formation, and later experienced dissipationless accretion of stars and GCs from its metal-poor satellite galaxies (Forte et al. 1982; Côté et al. 1998, 2002; Masters & Ashman 2010; Lee et al. 2010b) and/or from the surrounding regions (Tamura et al. 2006a, 2006b; Bergond et al. 2007; Schuberth et al. 2008; Lee et al. 2010a; West et al. 2011). Such GCs with an accretion origin tend to be of lower metallicity and on highly elongated orbits with high energy. The GCs generally favor the extended outskirts of giant galaxies (Lee et al. 1998, 2008b; Dirsch et al. 2005), although they, on highly elongated orbits, must penetrate into the inner body of galaxies. This explains the observations that the bimodality in GC colors corresponds to a change in the kinematic properties of the GCs, such that the blue GCs are dynamically hotter and there is a "bimodal" distribution of velocity dispersion. The demarcating radii between the inner, main part (the product of mergers) and the outskirts (the product of merger plus accretion) are not sharp and vary galaxy-to-galaxy depending on their individual histories of mergers and accretions. This is in line with the considerable diversity in the kinematics of the GC systems in giant elliptical galaxies (e.g., Hwang et al. 2008; Lee et al. 2010b).

Observationally, the intimately coupled histories of the GCs and constituent field stars of elliptical galaxies are encouraging in the context of mapping their metal contents, which is one of the most important yet least known properties of external galaxies. Even with the forthcoming larger telescopes, individual stars can be spatially resolved only in a few tens nearby galaxies (Tolstoy 2006). Hence, we anticipate that the estimation of GC metallicities based on their colors, equipped with a proper correction for the GC-to-star offset, will become a preferred practical method in the quest to comprehend the evolution of galaxies beyond the nearby universe. Further refinement of the exact shape of the CMRs for GCs, and the resulting implications for the formation of GC systems and galaxies, should be the topic of much future work.