Nonlinear Color–Metallicity Relations of Globular Clusters. VIII. Reproducing Color Distributions of Individual Globular Cluster Systems in the Virgo and Fornax Galaxy Clusters

Sang-Yoon Lee; Chul Chung; Suk-Jin Yoon

doi:10.3847/1538-4365/aaecd4

1. Introduction

Most galaxies harbor a system of globular clusters (GCs) that closely traces the formation history of its host galaxy. GC systems predominantly exhibit bimodal color distribution functions (CDFs), and that phenomenon has been a major topic in the field of the extragalactic GC research (e.g., Ostrov et al. 1993; Zepf & Ashman 1993; Gebhardt & Kissler-Patig 1999; Kundu & Whitmore 2001; Larsen et al. 2001; Peng et al. 2006; Lee et al. 2008; Jordán et al. 2009; Blakeslee et al. 2010; Faifer et al. 2011; Forbes et al. 2011; Foster et al. 2011; Liu et al. 2011; Blom et al. 2012; Chies-Santos et al. 2012; Cho et al. 2012; Forte et al. 2012; Kim et al. 2013a; Usher et al. 2013; Cantiello et al. 2014; Kartha et al. 2014, 2016; Richtler et al. 2015; Cho et al. 2016; Harris et al. 2017, see also West et al. 2004; Brodie & Strader 2006 for reviews and references therein). Understanding the origin of the bimodal CDFs of extragalactic GC systems should offer valuable constraints on the evolutionary paths taken by their host galaxies.

The key assumption of GC formation scenarios in attempts to explain GC color bimodality is the existence of two GC subgroups with distinct mean metallicities. These explanations invoke different origins for metal-poor and metal-rich subgroups, including (a) the metal-rich population is the product of major merging between two gas-rich spiral galaxies (Toomre & Toomre 1972; Ashman & Zepf 1992; Zepf & Ashman 1993; Whitmore & Schweizer 1995; Miller et al. 1997); (b) the lower-mass galaxies with metal-poor GCs are accreted onto a massive galaxy (Muzzio et al. 1987; Côté et al. 1998, 2002; Hilker et al. 1999); and (c) multiphase dissipational collapse leads to the discrete metallicity groups of GCs (Harris & Pudritz 1994; Forbes et al. 1997; Harris et al. 1999; Santos 2003). A modern way to describe the formation of early-type galaxies and their GC systems, such as the two-phase formation scenario (Forbes et al. 2011; Park & Lee 2013; Lee & Jang 2016; Beasley et al. 2018, see also Oser et al. 2010), is more in line with (b). Current hierarchical galaxy formation models in the ΛCDM cosmology show that tens of thousands of small (proto-)galaxies have been involved in making one single galaxy. This is important because the great degree of complexity seems to leave little room for the existence of just two GC subpopulations in each galaxy.

An alternative explanation was proposed by Yoon et al. (2006, hereafter Paper I), in which the nonlinear metallicity-to-color conversion creates GC color bimodality even from a broad, unimodal metallicity distribution function (MDF), without invoking two distinct GC subgroups within one galaxy. Paper I incorporates a realistic treatment of core helium-burning, horizontal-branch stars in stellar population modeling, and finds that nonlinearity is greatly enhanced by the inclusion of such stars. Yoon et al. (2011b, Paper II) and Yoon et al. (2013, Paper IV) demonstrated that the GC CDFs vary systematically for GC samples in M87 (Paper II) and M84 (Paper IV), respectively. Using their theoretical color–metallicity relations (CMRs), they reproduced the CDF morphologies with different filter combinations (u − g, u − z, and g − z), which are in good agreement with the observations. Yoon et al. (2011a, Paper III) demonstrated that by applying nonlinear color-to-metallicity conversions, the inferred GC MDFs have skewed, broad shapes and are remarkably similar to the MDF shapes of halo field stars in their host galaxies, implying common evolutionary histories between GCs and halo stars. Kim et al. (2013, Paper V) and Kim & Yoon (2017, Paper VII) showed that the diverse morphology of absorption-line index (e.g., Hβ and Mg2) distributions for M31 GCs (Paper V) and NGC 5128 GCs (Paper VII) can readily be reproduced by nonlinear "index"–metallicity relations, exactly analogous to the nonlinear CMRs. Chung et al. (2016, Paper VI) further extended these results to the infrared Calcium II Triplet index (CaT) and proposed the nonlinear CaT–metallicity relation as the origin of observed bimodal CaT distributions of GCs in 12 early-type galaxies.

In this paper of the series, we attempt to reproduce quantitatively color distributions of individual GC systems in the ACS Virgo Cluster Survey (ACSVCS; Côté et al. 2004) and ACS Fornax Cluster Survey (ACSFCS; Jordán et al. 2007). Our model shows that the majority (∼70%) of early-type galaxies have old (∼13 Gyr) and coeval GCs. The remaining (∼30%) galaxies are readily reproduced by including additional intermediate-age GCs. The paper is organized as follows. Section 2 describes the observational data set that we examined. Section 3 presents our simulated CDFs of individual GC systems and derives the best-fit parameters through the Kolmogorov–Smirnov (K-S) analysis. In Section 4, we examine the derived parameters for the GC systems as functions of host galaxy luminosity. Section 5 investigates the GC systems whose derived ages are younger than the majority. Our interpretation of the GC CDFs from the viewpoint of galaxy formation is given in Section 6. Finally, we conclude in Section 7.

2. Data

The data used herein are from the ACSVCS (Jordán et al. 2009) and ACSFCS (Jordán et al. 2015) GC catalogs, which obtained GC photometry for a total of 143 early-type galaxies in the F475W and F850LP filters (hereafter g and z) using HST/ACS imaging. The ACSVCS and ACSFCS provide the deepest and most homogeneous photometric catalogs of extragalactic GC systems so far. The galaxies span a wide luminosity range (−15.1 < M_B < −22.3). The morphologies of the GC CDFs show great diversity. To avoid the small number statistics, we examine the galaxies with a number of observed GCs,³ N_GC, greater than 50, and our sample consists of 78 galaxies (56 and 22 galaxies in the Virgo and Fornax clusters, respectively). A caveat is that the field of view (202'' × 202'') of the ACS/WFC only takes in the GCs of the inner region for large galaxies, and thus the GC lists for large galaxies do not fully represent their entire populations of GCs above the detection limit.

3. Reproducing CDFs of Individual GC Systems

3.1. Modeling of GC Color Distributions

Our main goal is to test the nonlinear-CMR scenario for the GC color bimodality of early-type galaxies. The theoretical g − z CMRs are based on the Yonsei Evolutionary Population Synthesis (YEPS) model. The YEPS model is described in detail in Chung et al. (2013, 2017). To simulate GC CDFs, we generate the g − z color distributions of 10⁶ model GCs with various ages, mean [Fe/H] $(\langle [\mathrm{Fe}/{\rm{H}}]\rangle )$ , and dispersion (σ([Fe/H])). The metallicity spread of a GC system is assumed to be a Gaussian normal distribution. For the transformation from MDFs to CDFs, we use the fifth-order polynomial fit to the model data. For a realistic comparison, we apply the photometric error based on the observed magnitude–error relations and luminosity functions (Villegas et al. 2010) of the GC system of interest. Figure 1 presents our CDF models according to the change of the input parameters. Table 1 summarizes the input parameters of our simulations.

**Figure 1.** Monte Carlo simulations of g − z color distributions for 10⁶ model GCs. Left two columns: the simulated GC CDFs according to the change of the mean metallicity and age. In the left column, the g − z CMRs (black solid lines) and the input MDFs are shown. The mean metallicity spans from $\langle [\mathrm{Fe}/{\rm{H}}]\rangle =-1.2$ (blue) to $\langle [\mathrm{Fe}/{\rm{H}}]\rangle =-0.2$ (red) with equal [Fe/H] intervals of 0.2 dex. The dispersion of MDFs is taken as a fixed value, σ([Fe/H]) = 0.55. In the right column, the same color code is used for the histogram of the CDFs. Right two columns: the simulated GC CDFs according to the change of the σ([Fe/H]) and age. In the left column, the mean metallicity of the MDFs is taken as a fixed value, $\langle [\mathrm{Fe}/{\rm{H}}]\rangle =-0.8$ . The dispersion of the MDFs spans from σ([Fe/H]) = 0.45 (blue) to σ([Fe/H]) = 0.65 (red) by equal σ([Fe/H]) intervals of 0.05 dex. In the right column, the same color code is used for the histogram of the CDFs.
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**Figure 1.** Monte Carlo simulations of g − z color distributions for 10⁶ model GCs. Left two columns: the simulated GC CDFs according to the change of the mean metallicity and age. In the left column, the g − z CMRs (black solid lines) and the input MDFs are shown. The mean metallicity spans from $\langle [\mathrm{Fe}/{\rm{H}}]\rangle =-1.2$ (blue) to $\langle [\mathrm{Fe}/{\rm{H}}]\rangle =-0.2$ (red) with equal [Fe/H] intervals of 0.2 dex. The dispersion of MDFs is taken as a fixed value, σ([Fe/H]) = 0.55. In the right column, the same color code is used for the histogram of the CDFs. Right two columns: the simulated GC CDFs according to the change of the σ([Fe/H]) and age. In the left column, the mean metallicity of the MDFs is taken as a fixed value, $\langle [\mathrm{Fe}/{\rm{H}}]\rangle =-0.8$ . The dispersion of the MDFs spans from σ([Fe/H]) = 0.45 (blue) to σ([Fe/H]) = 0.65 (red) by equal σ([Fe/H]) intervals of 0.05 dex. In the right column, the same color code is used for the histogram of the CDFs.
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Table 1. Input Parameters of Simulated Color Distribution Functions

Parameter	Range	Grid Interval
The age of a GC system, t (Gyr)	8.0 ∼ 15.0	0.1
The mean [Fe/H] of a GC system, $\langle [\mathrm{Fe}/{\rm{H}}]{\rangle }_{\mathrm{GC}}$ (dex)	−1.80 ∼ 0.80	0.05
The dispersion of [Fe/H] distribution, σ([Fe/H]) (dex)	0.45 ∼ 0.65	0.05

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3.2. The Kolmogorov–Smirnov Test

To determine the best-fit parameters, the two-sample K-S test is applied to the observed and modeled CDFs within the parameter space of age, $\langle [\mathrm{Fe}/{\rm{H}}]\rangle$ , and σ([Fe/H]) (Table 1). Figure 2 shows an example of the best model CDF, the observed CDF, and the K-S statistics for the case of VCC 1316 (M87) that has the largest GC sample. The right contour plot shows the p-value from the K-S statistics when σ([Fe/H]) = 0.55 dex. One can take a probability of p > 0.05 for the observed and modeled CDFs being drawn from the same underlying distributions.

**Figure 2.** Monte Carlo simulation of the g − z color distribution for the case of the VCC 1316 (M87) GC system. Top left panel: the (g − z)–[Fe/H] relation and assumed GC metallicity distributions of our best model. Second left panel: the g − z color distribution of 10⁶ model GCs, which is transformed from the assumed MDF using the theoretical relation shown in the left top panel. Third left panel: the observed g − z color histogram for 1745 GCs belonging to VCC 1316. The black line is a smoothed histogram with a Gaussian kernel with σ(g − z) = 0.06. Bottom left panel: the K-S test for the model CDF against the observed CDF. The blue and black curves represent the cumulative function of the best modeled and the observed CDFs, respectively. The difference between the red lines shows the K-S statistic D. Right panel: the p-value contour plot of the K-S probability in the age–[Fe/H] plane with the color scale from white to black for increasing K-S probability. The red star marks the location of the highest probability.
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3.3. Four Different Types of CDFs

Figure 3 presents our best model CDFs and the observed CDFs for the whole galaxy sample (78 ACSVCS/FCS galaxies) in order of N_GC. With the exception of FCC 21, the p-values of the best-fit models for all the GC systems are greater than 0.05, implying the modeled CDFs reproduce the observed CDFs successfully. Table 2 presents the GC simulation result, along with the basic information on the host galaxies.

Table 2. Simulation Results for the ACS Virgo and Fornax Cluster Survey Galaxies

				Galaxy										GC
Name	Other Name	N_GC	M_B	Morphological	Age	$\langle [\mathrm{Fe}/{\rm{H}}]\rangle$	σ([Fe/H])	D	p-value	f_red	p(χ²)	p(DD)	p(kurt)	CDF
			(mag)	Type	(Gyr)	(dex)		(K-S)	(K-S)	(GMM)	(GMM)	(GMM)	(GMM)	Type
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)	(11)	(12)	(13)	(14)	(15)
VCC 1316	M87, N4486	1745	−21.6	E0	12.6	−0.65	0.55	0.020	0.468	0.57	<0.010	<0.010	<0.010	1a
VCC 731	N4365	907	−21.4	E3	12.5	−0.75	0.45	0.026	0.572	0.53	<0.001	<0.001	<0.001	1a
VCC 1978	M60, N4649	807	−21.4	S0₁(2)	12.9	−0.65	0.55	0.026	0.667	0.57	<0.001	<0.001	<0.001	1a
VCC 1226	M49, N4472	765	−21.9	E2/S0₁(2)	13.0	−0.65	0.55	0.028	0.568	0.58	<0.001	<0.001	<0.001	1a
VCC 798	M85, N4382	507	−21.3	S0₁(3) pec	9.3	−0.75	0.55	0.023	0.956	0.31	<0.001	0.168	0.031	1b
VCC 763	M84, N4374	506	−21.2	E1	12.8	−0.90	0.50	0.026	0.895	0.37	<0.001	<0.001	0.003	1a
VCC 1632	M89, N4552	456	−20.4	S0₁(0)	9.3	−0.60	0.60	0.029	0.819	0.51	<0.001	0.089	0.106	1b
VCC 881	M86, N4406	367	−21.3	S0₁(3)/E3	9.6	−0.90	0.50	0.024	0.981	0.14	0.001	0.464	0.835	2b
VCC 1903	M59, N4621	308	−20.2	E4	11.6	−0.75	0.50	0.033	0.893	0.57	0.013	0.003	0.075	1a
VCC 1231	N4473	254	−19.9	E5	8.1	−0.70	0.55	0.035	0.911	0.38	<0.001	0.279	0.108	1b
VCC 1535	N4526	244	−20.6	S0₃(6)	12.4	−0.80	0.65	0.035	0.916	0.49	<0.001	<0.001	<0.001	1a
VCC 1199		228	−15.7	E2	13.0	−0.90	0.60	0.026	0.998	0.38	<0.001	<0.001	<0.001	1a
VCC 1192	N4467	213	−16.1	E3	13.2	−1.00	0.65	0.048	0.706	0.31	<0.001	<0.001	0.024	1a
VCC 2000	N4660	197	−19.1	E3/S0₁(3)	13.4	−1.15	0.60	0.033	0.981	0.13	<0.001	0.366	0.994	2a
VCC 1154	N4459	192	−19.9	S0₃(2)	12.6	−0.90	0.45	0.028	0.998	0.36	0.015	0.028	0.061	1a
VCC 1062	N4442	179	−19.6	SB0₁(6)	12.8	−0.85	0.50	0.027	0.999	0.42	<0.001	<0.001	0.006	1a
VCC 369	N4267	179	−19.4	SB0₁	12.1	−0.80	0.45	0.035	0.976	0.55	0.002	0.003	0.053	1a
VCC 1030	N4435	176	−19.4	SB0₁(6)	12.5	−0.85	0.50	0.040	0.936	0.50	<0.001	<0.001	0.002	1a
VCC 1327	N4486A	173	−18.2	E2	13.3	−1.10	0.65	0.040	0.936	0.25	<0.001	0.603	0.564	2a
VCC 759	N4371	172	−19.5	SB0₂(r)(3)	12.6	−0.95	0.50	0.042	0.922	0.41	0.001	0.013	0.008	1a
VCC 685	N4350	167	−19.2	S0₁(8)	12.5	−1.00	0.55	0.033	0.992	0.27	<0.001	0.526	0.358	2a
VCC 1297	N4486B	152	−16.8	E1	13.3	−1.05	0.60	0.040	0.966	0.29	<0.001	0.404	0.094	2a
VCC 1664	N4564	146	−19.1	E6	11.4	−0.75	0.55	0.037	0.988	0.66	0.236	<0.001	0.493	1a
VCC 1279	N4478	138	−19.1	E2	10.7	−1.05	0.45	0.043	0.958	0.39	0.348	0.398	0.288	1b
VCC 1692	N4570	136	−19.4	S0₁(7)/E7	12.8	−1.05	0.65	0.055	0.787	0.33	<0.001	<0.001	0.016	1a
VCC 2095	N4762	134	−20.0	S0₁(9)	10.3	−0.95	0.50	0.051	0.863	0.46	0.251	0.211	0.319	1b
VCC 698	N4352	119	−17.9	S0₁(8)	10.3	−1.10	0.45	0.049	0.936	0.06	<0.001	<0.001	0.995	2b
VCC 1242	N4474	116	−18.5	S0₁(8)	12.9	−0.90	0.45	0.046	0.962	0.14	0.008	0.673	0.759	2a
VCC 1025	N4434	104	−18.8	E0/S0₁(0)	14.5	−1.50	0.65	0.045	0.983	0.04	<0.001	0.542	1.000	2a
VCC 1938	N4638	101	−19.2	S0₁(7)	8.6	−1.15	0.60	0.032	1.000	0.11	0.003	0.517	0.934	2b
VCC 2092	N4754	92	−19.7	SB0₁(5)	9.1	−0.75	0.60	0.035	1.000	0.43	0.108	0.148	0.035	1b
VCC 944	N4417	91	−19.0	S0₁(7)	9.1	−0.95	0.65	0.049	0.977	0.38	0.009	0.010	0.060	1b
VCC 1178	N4464	90	−17.7	E3	12.2	−1.00	0.45	0.048	0.982	0.31	<0.001	0.422	0.657	1a
VCC 1475	N4515	86	−17.9	E2	14.5	−1.50	0.65	0.054	0.956	0.05	0.186	<0.001	0.884	2a
VCC 1883	N4612	83	−18.6	RSB0_1/2	9.0	−0.90	0.50	0.043	0.998	0.14	0.052	0.646	0.768	2b
VCC 1146	N4458	82	−18.2	E1	10.8	−0.70	0.45	0.085	0.577	0.82	0.032	0.748	0.754	1b
VCC 828	N4387	80	−18.6	E5	12.9	−1.20	0.60	0.046	0.995	0.20	<0.001	0.689	0.911	2a
VCC 778	N4377	74	−18.7	S0₁(3)	8.4	−0.85	0.60	0.058	0.962	0.14	0.022	0.697	0.688	2b
VCC 1720	N4578	71	−18.9	S0_1/2(4)	12.2	−0.95	0.65	0.060	0.952	0.40	0.004	0.011	0.015	1a
VCC 1431	I3470	71	−16.7	dE0,N	14.7	−1.35	0.65	0.044	0.999	0.09	0.004	0.658	0.993	2a
VCC 1087	I3381	68	−16.9	dE3,N	8.7	−1.10	0.45	0.061	0.959	0.07	0.009	<0.001	0.988	2b
VCC 1619	N4550	66	−18.6	E7/S0₁(7)	10.4	−0.95	0.60	0.052	0.992	0.37	0.054	0.264	0.058	1b
VCC 1283	N4479	66	−17.9	SB0₂(2)	12.7	−1.05	0.45	0.051	0.994	0.28	0.082	0.026	0.459	2a
VCC 1913	N4623	65	−18.1	E7	8.4	−0.90	0.45	0.051	0.995	0.14	0.151	0.759	0.870	2b
VCC 784	N4379	64	−18.4	S0₁(2)	13.1	−0.90	0.45	0.055	0.989	0.24	0.086	0.028	0.405	2a
VCC 1545	I3509	63	−16.3	E4	12.3	−1.25	0.50	0.047	0.999	0.06	0.001	0.665	0.993	2a
VCC 1125	N4452	62	−17.9	S0₁(9)	8.9	−1.15	0.45	0.057	0.985	0.06	0.077	<0.001	0.950	2b
VCC 355	N4262	62	−18.7	SB0_2/3	12.6	−0.95	0.65	0.072	0.890	0.34	<0.001	0.205	0.095	1a
VCC 1303	N4483	61	−18.1	SB0₁(5)	13.5	−1.55	0.60	0.056	0.990	0.07	<0.001	0.687	0.999	2a
VCC 1910	I809	60	−17.0	dE1,N	9.8	−0.95	0.65	0.062	0.972	0.16	0.007	<0.001	0.621	2b
VCC 1407	I3461	60	−15.8	dE2,N	14.6	−1.25	0.65	0.061	0.974	0.09	0.002	<0.001	0.983	2a
VCC 1630	N4551	57	−18.3	E2	12.6	−0.95	0.55	0.047	0.999	0.40	0.271	0.464	0.374	1a
VCC 1250	N4476	54	−18.5	S0₃(5)	14.1	−1.30	0.60	0.091	0.749	0.02	<0.001	0.459	1.000	2a
VCC 437	U7399A	50	−16.8	dE5,N	12.8	−1.40	0.65	0.065	0.982	0.10	<0.001	<0.001	0.995	2a
VCC 1321	N4489	50	−18.2	S0₁(1)	13.2	−1.10	0.50	0.058	0.995	0.19	0.004	0.312	0.794	2a
VCC 856	I3328	50	−17.0	dE1,N	13.1	−1.10	0.60	0.056	0.997	0.17	0.002	0.084	0.890	2a
FCC 213	N1399	1075	−21.0	E0	11.4	−0.60	0.55	0.027	0.399	0.64	<0.010	<0.010	<0.010	1a
FCC 21	N1316	647	−22.3	S0₃(pec)	14.8	−1.10	0.45	0.068	0.005	0.12	0.242	0.560	0.416	2a
FCC 167	N1380	424	−20.4	S0/a	11.8	−0.70	0.45	0.039	0.540	0.71	<0.001	<0.001	0.052	1a
FCC 219	N1404	380	−20.7	E2	12.4	−0.75	0.55	0.043	0.476	0.55	<0.001	<0.001	<0.001	1a
FCC 276	N1427	362	−19.7	E4	9.0	−0.85	0.50	0.034	0.802	0.35	<0.001	<0.001	0.003	1b
FCC 147	N1374	320	−19.6	E0	12.7	−0.85	0.45	0.030	0.927	0.46	<0.001	<0.001	<0.001	1a
FCC 184	N1387	306	−19.2	SB0	12.6	−0.60	0.55	0.055	0.303	0.65	<0.001	<0.001	<0.001	1a
NGC 1340	N1344	280	−20.4	E5	8.9	−1.15	0.55	0.038	0.797	0.17	<0.001	0.086	0.940	2b
FCC 47	N1336	276	−18.1	E4	8.8	−0.95	0.50	0.042	0.712	0.33	<0.001	<0.001	0.228	1b
FCC 83	N1351	274	−19.2	E5	12.9	−1.00	0.50	0.036	0.868	0.34	<0.001	0.022	0.203	1a
FCC 202	N1396	232	−16.3	d:E6,N	12.5	−0.85	0.60	0.035	0.937	0.51	<0.001	0.003	<0.001	1a
FCC 63	N1339	231	−18.8	E4	8.0	−0.85	0.50	0.025	0.999	0.30	<0.001	0.179	0.129	1b
FCC 190	N1380B	156	−18.1	SB0	13.5	−1.30	0.55	0.030	0.999	0.09	<0.001	0.448	1.000	2a
FCC 249	N1419	155	−18.2	E0	9.4	−1.15	0.45	0.053	0.775	0.01	0.002	<0.001	0.999	2b
IC 2006	ESO 359-G07	132	−19.4	E	8.1	−0.70	0.55	0.037	0.993	0.42	0.004	0.065	0.019	1b
FCC 148	N1375	87	−18.0	S0(cross)	12.5	−1.10	0.45	0.058	0.925	0.34	<0.001	0.608	0.778	1a
FCC 255	ESO 358-G50	80	−17.8	S0₁(6),N	13.7	−1.20	0.50	0.045	0.996	0.05	<0.001	<0.001	0.999	2a
FCC 170	N1381	71	−18.8	S0(9)(boxy)	13.9	−1.30	0.55	0.054	0.983	0.02	0.004	0.775	1.000	2a
FCC 177	N1380A	70	−18.4	S0₂(9)(cross)	13.4	−1.20	0.45	0.056	0.976	0.11	<0.001	0.163	0.987	2a
FCC 143	N1373	63	−17.2	E3	13.5	−1.30	0.45	0.081	0.789	0.10	<0.001	0.697	1.000	2a
FCC 153	ESO 358-G26	60	−18.6	S0₁(9)	13.2	−1.40	0.65	0.051	0.997	0.13	<0.001	0.448	0.993	2a
FCC 182		59	−16.6	SB0 pec	13.4	−1.40	0.55	0.064	0.964	0.14	<0.001	<0.001	0.964	2a

Note. (1) Galaxy VCC and FCC number. (2) Other name. (3) The total number of observed GCs. (4) Galaxy B-band magnitude (Glass et al. 2011). (5) Morphological classification obtained from Côté et al. (2004) and Jordán et al. (2007). (6) The best-fit age. (7) The best-fit mean [Fe/H]. (8) The best-fit dispersion of MDF. (9) K-S statistic D. (10) K-S probability. (11) Red GC fraction in g − z color distributions from GMM analysis. (12)–(14) p-values from GMM analysis. p(χ²), p(DD), and p(kurt) are based on the likelihood-ratio test, separation of the peaks, and kurtosis, respectively. The lower p-values suggest more significant bimodality. (15) GC CDF type.

Download table as: ASCIITypeset images: 1 2

A closer look at Figure 3 suggests that the GC CDFs can be classified into four different types based on two factors. The first factor is the number ratio of the blue and red GCs, which have a large effect on the overall shape of the CDFs. To derive the number ratio of the blue and red GCs, we apply the Gaussian Mixture Modeling (GMM) code by Muratov & Gnedin (2010). We classify the CDFs into Types 1 and 2 using the red GC fraction (f_red) in the g − z color distributions, and f_red = 0.3 divides the entire galaxy sample into halves. The second factor is a model-derived parameter, the best-fit age, which enables more detailed classification of the CDFs. The best-fit age used for dividing the subgroups, a and b, is 11 Gyr. We suspect that the galaxies classified as the subgroup b (i.e., t_best-fit < 11 Gyr) possess additional intermediate-age GCs, weakening the bimodality made by the underlying old GCs. Table 3 gives the classifying criteria and the mean values of the model-derived parameters for each type.

Table 3. Classification of Color Distribution Function Types and Mean Values of the Simulated Parameters for Each Type

Type	Selection Criteria	N_Gal	Age	$\langle [\mathrm{Fe}/{\rm{H}}]\rangle$	σ([Fe/H])	N(S0)/N(E)
			(Gyr)	(dex)
(1)	(2)	(3)	(4)	(5)	(6)	(7)
Type 1a	t_best-fit > 11 Gyr, f_red > 0.3	28	12.5 ± 0.09	−0.84 ± 0.026	0.54 ± 0.013	1.00
Type 1b	t_best-fit ≤ 11 Gyr, f_red > 0.3	13	9.3 ± 0.27	−0.83 ± 0.038	0.54 ± 0.017	0.86
Type 2a	t_best-fit > 11 Gyr, f_red ≤ 0.3	26	13.5 ± 0.13	−1.22 ± 0.035	0.56 ± 0.015	1.60
Type 2b	t_best-fit ≤ 11 Gyr, f_red ≤ 0.3	11	9.1 ± 0.18	−1.03 ± 0.038	0.51 ± 0.022	1.20

Note. (1) GC CDF type. (2) Selection criteria. f_red is the red GC fraction in the g − z color distributions. In the GMM model, the homoscedastic case is adopted to derive f_red. (3) The number of galaxies belonging to each type. (4) The mean value of the best-fit ages. (5) The mean value of the best-fit $\langle [\mathrm{Fe}/{\rm{H}}]\rangle$ . (6) The mean value of the best-fit MDF dispersions. (7) The number ratio of lenticular galaxies to elliptical galaxies.

Download table as: ASCII Typeset image

Figure 4 shows the representative GC systems for the four GC CDF types (Types 1a, 1b, 2a, and 2b). Type 1a is the most common GC CDF type and has a clear bimodal distribution with well-developed two peaks. Most luminous galaxies (M_B ≲ −19) favor this type, and VCC 1316 (M87) and VCC 1226 (M49) show the representative GC CDFs of this type. For the case of Type 1b, two distinct peaks are not prominent. Compared to Type 1a, the distance between two peaks is shorter and the dip is less clear and shallower. VCC 798 and VCC 1632 show the typical GC CDF shapes of this type. This type is also found preferentially in luminous (M_B ≲ −19) galaxies. On the other hand, Type 2a is the second most common GC CDF type and has a dominant blue peak along with a broad red mound (e.g., VCC 1297) or a red tail (e.g., VCC 1303). This type is typically found in less luminous (M_B ≳ −19) galaxies. For the case of Type 2b, their CDFs have a blue peak but do not have additional structures such as a red mound or a red tail as seen in Type 2a. The blue peak is less cuspy than Type 2a. Thus, the overall shape of this type is a fat, skewed unimodal distribution. This type is also found preferentially in less luminous (M_B ≳ −19) galaxies.

The lower part of Figure 4 shows the p-value contours for the four types in the age– $\langle [\mathrm{Fe}/{\rm{H}}]\rangle$ plane for different assumptions of σ([Fe/H]). The contour plots for the whole galaxy sample are given in Figure 7 in the Appendix. The contours of the K-S statistics are often stretched along the age axis, with narrow width along the [Fe/H] axis. This indicates that our K-S test is able to place a stronger constraint on the mean metallicity of given GC systems than the age. From the p-value contours, one can find two important elements that pin down the age of a GC system. The first element is the strength of the bimodality of the color distribution. Our GC CDF model predicts that the positions of two peaks and a dip are highly sensitive to the age of the GC system (see Figure 1). The dip position in color corresponds to the quasi-inflection point along the CMR (Paper I) and becomes redder with increasing age. As a result, the K-S statistics better determine the age if an observed CDF has clear bimodality and a well-shaped dip. The other element is the total number of observed GCs. The larger the sample sizes, the more compact the p-value contours. The Type 1a galaxies with high luminosity are well suited to these conditions and exhibit compact p-value contours compared to other types.

When it comes to Type 2a, the sharp blue peak is one of the most distinguishing features of this type. Since the sharp blue peak is generated by the steep slope of old CMRs, the contour converges to an old age (11 ∼ 15 Gyr). Some Type 2a CDFs that have only a feeble red tail show diagonal contours in the age range of 12 ∼ 14 Gyr (e.g., VCC 1025, VCC 1475, VCC 1431, VCC 1303, and FCC 170), undergoing the age–metallicity degeneracy (see Figure 7 in the Appendix). Weak red tails occur when the mean metallicity is so low ( $\langle [\mathrm{Fe}/{\rm{H}}]\rangle \lt -1.3$ ) that only a few red GCs are present. In such cases, the quasi-inflection point of the CMR hardly affects the CDF shapes and thus accuracy in age dating becomes low. For instance, the 12 Gyr, $\langle [\mathrm{Fe}/{\rm{H}}]\rangle =-1.3$ model and the 14 Gyr, $\langle [\mathrm{Fe}/{\rm{H}}]\rangle =-1.8$ model have similar GC CDFs, with a sharp blue peak with a feeble red tail. This uncertainty in age, however, does not affect our type classification. Note also that Type 2a galaxies are relatively less luminous galaxies and have a small number of GCs, thus the confidence contours are rather broad.

While Types 1a and 2a have the characteristics of the old CDF models (i.e., clear bimodality of Type 1a and a sharp blue peak of Type 2a), Types 1b and 2b do not have such distinct morphological features. Thus, Types 1b and 2b have a relatively younger best-fit age (8 ∼ 11 Gyr), and the age variation due to the different choice of σ([Fe/H]) is larger than that of Types 1a and 2a. In other words, there is quite a bit of degeneracy between three parameters (age, $\langle [\mathrm{Fe}/{\rm{H}}]\rangle$ , and σ([Fe/H])) for Types 1b and 2b. Even with similar N_GC, the p-value contours are less compact compared to Types 1a and 2a, indicating that the assumption of old and coeval GCs may not hold for Types 1b and 2b. We attempt to address the nature of Types 1b and 2b in Section 5.

4. The Derived Parameters as Functions of Host Galaxy Properties

4.1. Best-fit Parameters and Galaxy Luminosity

Figure 5 shows the best-fit age, $\langle [\mathrm{Fe}/{\rm{H}}]\rangle$ , and σ([Fe/H]) of the entire GC systems as functions of the host galaxy luminosity. The best-fit age does not show a clear correlation with the host luminosity. The distribution of the best-fit age can be broken into two groups at 11 Gyr, above which the g − z CMRs are highly inflected (see Figure 1). A detailed explanation for the galaxies with a model-derived age less than 11 Gyr is given in Section 5. Unlike the best-fit age, the best-fit $\langle [\mathrm{Fe}/{\rm{H}}]\rangle$ shows a tight correlation with the host galaxy luminosity. The derived M_B– $\langle [\mathrm{Fe}/{\rm{H}}]\rangle$ slope for the whole galaxy sample is −0.094 ± 0.015, while the slope excluding the galaxies with t_best-fit < 11 Gyr is −0.108 ± 0.019. The best-fit σ([Fe/H]) distribution has no obvious correlation with the host luminosity.

We note that the ACSVCS and ACSFCS data have a limitation in radial coverage for large galaxies due to the small field of view of ACS/WFC. For instance, ACS/WFC covers only the inner region (∼1 R_eff) of VCC 1316 (M87), the largest galaxy in our sample. The mean metallicity of GCs is known to be higher in the inner region (e.g., Geisler et al. 1996; Kundu et al. 1999; Forte et al. 2001; Harris 2009; Blom et al. 2012; Forbes & Remus 2018). Thus, for some very large galaxies the mean metallicity of observed GCs is determined to be higher than their mean metallicity for all the GCs. However, most of our sample galaxies are fairly covered by the ACS/WFC field of view (see Figure 2 in Wang et al. 2013) and that limitation does not have a significant effect on the overall trend of the M_B– $\langle [\mathrm{Fe}/{\rm{H}}]\rangle$ relation. Moreover, observations of the color bimodality of GC systems in early-type galaxies reveal that the red GC fraction gets higher with host galaxy luminosity. The faintest galaxies in ACSVCS on average have a 15% fraction of red GCs, and the fraction increases to 60% for the brightest galaxies (Peng et al. 2006). Hence, the main driver behind the M_B– $\langle [\mathrm{Fe}/{\rm{H}}]\rangle$ relation is most likely this changing fraction of red GCs rather than the field of view bias of ACS/WFC.

4.2. Best-fit Parameters and Galaxy Morphological Type

In the top panel of Figure 5, the best-fit age derived from our GC color distribution model shows no difference depending on the morphological types (E versus S0) of the host galaxies. About 30% among both E and S0 galaxies are identified as t_best-fit < 11 Gyr; 33% for E galaxies and 29% for S0 galaxies. In the middle panel, the best-fit $\langle [\mathrm{Fe}/{\rm{H}}]\rangle$ of both E and S0 galaxies increases as the host galaxy luminosity increases. The slope of the relation for the E galaxies (−0.079 ± 0.018) is slightly shallower than that of the S0 galaxies (−0.113 ± 0.027). In the bottom panel, the mean σ([Fe/H]) of both E and S0 galaxies shows almost the same value (∼0.54).

Chies-Santos et al. (2011) estimated the ages of the GC systems in 14 E/S0 galaxies from the (g − k) versus (g − z) diagram, and found that their S0 galaxies, compared to the E galaxies, have preferentially younger blue GCs. The discrepancy in the GC age versus galaxy morphology relations between Chies-Santos et al. and ours seems to be due to the different choices of age estimation method. Moreover, the galaxy morphological classification depends fairly on references even for such well-studied nearby galaxies. For instance, VCC 1978 (M60), a galaxy without young GCs in both studies, is classified as E in the Third Reference Catalogue of Bright Galaxies (RC3⁴ ) (de Vaucouleurs et al. 1991) but as S0 in the Revised Shapley-Ames Catalog of Bright Galaxies (RSA⁵ ) (Sandage & Tammann 1981). VCC 2000 (NGC 4660), a galaxy with young GCs in Chies-Santos et al. but old GCs only in ours, is classified as E in the RC3 and RSA as well as Emsellem et al. (2007) and Cappellari et al. (2007), but as S0 in Kormendy et al. (2009).

5. GC Systems with ${t}_{\mathrm{best} \mbox{-} \mathrm{fit}}\lt 11\,\mathrm{Gyr}$

5.1. Dilution of Bimodality

In Figure 5, the GC systems can be divided into two groups; old (11 ∼ 15 Gyr) and relatively younger (<11 Gyr) systems. The best-fit ages of the GC systems in 24 galaxies (out of 78 galaxies) are estimated to be younger than 11 Gyr. The model fails to reproduce FCC 21's observed CDF. Thus, a total of 25 galaxies (∼30%) among our sample do not fit into our old (>11 Gyr) and coeval assumption. There are two ways to interpret these GC systems. First, one can simply accept the result as it is: that GCs in each galaxy are younger than 11 Gyr, with coeval formation epochs. Indeed, the match between the younger model CDFs and the observed CDFs is fairly good (p-value > 0.05). But there is another possibility: the GC systems have newly formed GC populations along with the underlying old GCs. There are many galaxies possessing young or intermediate-age GC populations suggested by spectroscopy (e.g., Goudfrooij et al. 2001; Larsen et al. 2003; Strader et al. 2003; Beasley et al. 2008; Woodley et al. 2010; Ko et al. 2018; Sesto et al. 2018) and UV or IR photometry (e.g., Puzia et al. 2002; Sohn et al. 2006; Hempel et al. 2007; Chies-Santos et al. 2011; Georgiev et al. 2012; Trancho et al. 2014). Even with young or intermediate-age GCs, however, the galaxies are dominated by old (≳10 Gyr) GCs. While our old systems (Types 1a and 2a) show typical CDF shapes, the younger systems (Types 1b and 2b) show broad unimodal shapes in their CDFs (see Figures 3 and 4). We suspect that such "diluted" bimodality is the result of contamination of underlying, old CDFs (i.e., Types 1a and 2a) by young or intermediate-age GCs.

5.2. Notes on Galaxies Presumably Containing Younger GCs

One of the most common sources of new stellar populations is mergers. Peculiar morphological structures (e.g., tidal tails, shells, kinematically decoupled cores, and isophotal twist) can be interpreted as a remnant of recent mergers (e.g., Toomre & Toomre 1972; de Zeeuw & Franx 1991; Barnes 1992; Mehlert et al. 1998; Naab & Burkert 2003; Li et al. 2004; Moore et al. 2004; Smith et al. 2008; Hoffman et al. 2009, 2010; Knierman et al. 2012; Torres-Flores et al. 2012). In the following, we briefly discuss the observational work in the literature on galaxies with t_best-fit < 11 Gyr, focusing on possible vestiges of their recent mergers and/or star formation.

VCC 798 (NGC 4382). The morphological feature of this lenticular galaxy shows that it experienced a recent merger (Ko et al. 2018). This galaxy shows possible dust patches and likely hosts a stellar disk (Ferrarese et al. 2006). Terlevich & Forbes (2002) estimated the age of VCC 798 based on the Hβ and [MgFe] indices. Its luminosity-weighted age, 1.6 Gyr, points to recent star formation. Ko et al. (2018) estimated the spectroscopic ages of 20 GCs with the Gemini Multi-Object Spectrograph. Among their sample GCs, 11 GCs are classified as an intermediate-age population (∼3.7 Gyr).

VCC 1632 (NGC 4552). This lenticular galaxy is well known for its active galactic nucleus (Cappellari et al. 1999) and nuclear outflow activity (Machacek et al. 2006b). Machacek et al. (2006a) found a gas-stripping feature in the galaxy's interstellar medium with a Chandra observation.

VCC 881 (NGC 4406). This lenticular galaxy has a dust trail in the central region that is considered to be due to ram-pressure-stripping by its companion VCC 882 (Elmegreen et al. 2000). The Hα feature produced by a collision with NGC 4438 (Kenney et al. 2008) and a ram-pressure-stripped tail against the Virgo intracluster medium (Randall et al. 2008) are also reported. Park et al. (2012) estimated the spectroscopic age of eight GCs with the Faint Object Camera and Spectrograph on the Subaru telescope. The estimated mean age of eight GCs is reported to be 9.7 Gyr.

VCC 1231 (NGC 4473). This elliptical galaxy is well known for its kinematically distinct components, namely double peaks in the velocity dispersion map. The galaxy shows peculiarities in both its kinematical and photometrical data, which could originate from a recent merger (Pinkney et al. 2003). The spectroscopic result by Koleva et al. (2011) shows that the central part of this galaxy consists of younger and more metal-rich stellar populations compared to the outer part. Alabi et al. (2015) suggested that GCs follow peculiar stellar kinematics, which is evidence of a gas-rich major merger event.

VCC 1279 (NGC 4478). This elliptical galaxy contains a kinematically decoupled core (Halliday et al. 2001; Emsellem et al. 2007) and a nuclear stellar disk (Morelli et al. 2004, 2010; Ledo et al. 2010). The nuclear disk is younger (∼6 Gyr), more metal-rich ([Z/H] ∼ 0.4), and less α-enhanced ([α/Fe] ∼ 0.2) than the main body of the host galaxy, supporting a prolonged star formation history (Morelli et al. 2010).

VCC 2095 (NGC 4762). This edge-on lenticular galaxy shows an unusual internal structure that consists of four distinct components (Wakamatsu & Hamabe 1984; Hamabe & Wakamatsu 1989; Kormendy & Kennicutt 2004). Wakamatsu & Hamabe (1984) presumed that the four components are the bulge, bar, lens, and outer ring. The galaxy shows an extremely small bulge-to-total light ratio (B/T = 0.13 ± 0.02) as a S0 galaxy. Kormendy & Bender (2012) suggested that this late-type S0 galaxy bridges the gap between S0 and spheroidal galaxies.

VCC 698 (NGC 4352). This lenticular galaxy has both small-scale and large-scale stellar disks (Ferrarese et al. 2006). Based on the spectroscopic data, the bulge appears to be younger and more metal-rich than the outer part (Johnston et al. 2014).

VCC 1938 (NGC 4638). This edge-on lenticular galaxy is a disk-dominated system (Barway et al. 2007). Kormendy & Bender (2012) viewed it as a "missing link" between S0s and spheroidals like VCC 2095. They suggested that the galaxy was created from a late-type galaxy by dynamical heating due to surrounding galaxies.

VCC 1883 (NGC 4612). This lenticular galaxy has a bar, a ring, and an isophotal twist. The bar shows misalignment with the main body of the galaxy (Ferrarese et al. 2006).

VCC 778 (NGC 4377). This lenticular galaxy shows an isophotal twist due to the misaligned bar. There are three spiral galaxies that are regarded as a background group (Ferrarese et al. 2006). Thus, there is a possibility that its GC color distribution is contaminated by GCs in the background galaxies.

VCC 1087 (IC 3381). The luminosity-weighted age of this dwarf elliptical galaxy is estimated to be 5 ∼ 7 Gyr (Beasley et al. 2006; Michielsen et al. 2008; Toloba et al. 2012).

VCC 1619 (NGC 4550). This lenticular galaxy harbors two counter-rotating disks (Coccato et al. 2013) and shows emission lines (Sarzi et al. 2006). The galaxy also has a peculiar dust distribution that looks like a spiral arm structure (Wiklind & Henkel 2001).

VCC 1125 (NGC 4452). This is an edge-on lenticular galaxy. Kormendy & Bender (2012) showed that the galaxy has an extremely low pseudo-bulge-to-total luminosity ratio (=0.017 ± 0.004) and a warped outer disk that seems to stem from gravitational encounters. They also showed that the galaxy consists of five stellar components like VCC 2095. The luminosity-weighted mean age using higher-order Balmer absorption lines is ∼5 Gyr (Caldwell et al. 2003).

VCC 1910 (IC 809). This dE or Sph, N type galaxy (Kormendy et al. 2009) shows a possible disk feature (Lisker et al. 2006). The luminosity-weighted mean age based on spectroscopy is 7.5 ∼ 9 Gyr (Michielsen et al. 2008; Toloba et al. 2014).

FCC 21 (NGC 1316). This is the only galaxy for which our coeval model does not mimic the CDF. The galaxy is lenticular and well known for its recent merger feature (Schweizer 1980; Horellou et al. 2001) and the presence of intermediate-age GCs (Goudfrooij et al. 2001; Sesto et al. 2018). The tidal tails, loops, and arms are extended out to 54'' and lots of dust structures are visible in the central region (Schweizer 1980).

FCC 276 (NGC 1427). This elliptical galaxy has a kinematically decoupled core in the central region (∼7'') (Scott et al. 2014).

NGC 1340 (a.k.a. NGC 1344). This elliptical galaxy shows a clear shell structure in the V-band image (Huang et al. 2013).

FCC 249 (NGC 1419). This E0 galaxy has anomalously bright K_S-band surface brightness fluctuation (Liu et al. 2002), which can be induced by recent star formation.

Among 24 galaxies with t_best-fit < 11 Gyr plus FCC 21, seven galaxies (VCC 944, VCC 1146, VCC 1913, VCC 2092, FCC 47, FCC 63, and IC 2006) lack the observational data in the literature needed to grasp the characteristics of their morphologies or stellar populations.

5.3. Two-component Model for the GC Systems with ${t}_{\mathrm{best} \mbox{-} \mathrm{fit}}\lt 11\,\mathrm{Gyr}$

Given that most galaxies with t_best-fit < 11 Gyr (namely, Type 1b and 2b galaxies) seem to have experienced recent mergers and/or star formation, we hypothesize that their "diluted" bimodality is due to contamination of old GC CDFs (i.e., Types 1a and 2a) by young (or intermediate-age) GCs. We further surmise that Types 1b and 2b are respectively nothing but Types 1a and 2a with additional young GCs. In order to test whether additional younger GC populations are the origin of Types 1b and 2b CDFs, we have performed two-component (old GCs + young GCs) model simulations.

Figure 6 shows the simulated CDFs for VCC 798 and VCC 1938, the representative examples of Types 1b and 2b, respectively. As mentioned in Section 5.2, Ko et al. (2018) estimated the spectroscopic ages and metallicities of 20 GCs in VCC 798 (M85). Out of 20 GCs, 11 are classified as young GCs and their mean age is measured to be 3.7 ± 1.9 Gyr with $\langle [\mathrm{Fe}/{\rm{H}}]\rangle$ = −0.26. We adopt these values in modeling the young GCs of VCC 798, as well as VCC 1938 (due to lack of spectroscopic data on VCC 1938 GCs). The young GCs on top of the old GCs tend to weaken bimodality (of Type 1a) and reduce the sharpness of blue peaks (of Type 2a), bringing the theoretical predictions in good agreement with the observations. It is highly likely that Types 1b and 2b are the results of adding a young or intermediate-age GC population to Types 1a and 2a CDFs, respectively. For better assessment of young or intermediate-age GC populations, further spectroscopic observations for the GC systems are badly needed.

6. Discussion

Among 78 early-type galaxies (with N_GC > 50) in ACSVCS and ACSFCS, 53 galaxies (∼70%) have GC color distributions that can be well reproduced by old (t_best-fit > 11 Gyr) and coeval model CDFs (Types 1a and 2a). We have shown that the bimodal color distributions of Types 1a and 2a are attributed to the nonlinear metallicity-to-color conversion of GCs. The difference in Types 1a and 2a is simply due to the difference in the mean GC metallicity, in that Type 1a is more metal-rich than Type 2a. On the other hand, our experiment has suggested that for ∼30% of the sample, young or intermediate-age GCs contaminate the color distribution of underlying old GCs, diluting the color bimodality (of Type 1a) or broadening blue peaks (of Type 2a). We suggest that Types 1b and 2b are respectively nothing but Types 1a and 2a with additional young or intermediate-age GCs.

Our simulations show that for most Type 1b and 2b galaxies, young or intermediate-age GCs in a galaxy comprise ∼20% of underlying old GCs in number. This is ∼17% (=20/120) with respect to the total number of (old + young) GCs. If ∼30% of galaxies host ∼17% of young GCs, the number fraction of the young GCs is as low as ∼5% of the total GCs of the entire galaxy sample. This is why the ACSVCS CDFs for GCs in the seven bins of host galaxy luminosity (Peng et al. 2006) are successfully reproduced under the assumption of no young GCs but old, coeval GCs (see Figure 3 in Paper I).

Our methodology usually classifies CDFs as Type 1b or 2b if there are additional young or intermediate-age GCs comprising more than ∼10% of old GCs. For additional GCs less than ∼10%, our classification scheme is less reliable. For instance, VCC 731 (NGC 4365), which is reported to host an additional GC population (Larsen et al. 2003; Brodie et al. 2005; Kundu et al. 2005; Blom et al. 2012), is classified as Type 1a in our model (Table 2). The observed CDF shows that its dip is slightly shallower than other Type 1a galaxies. The best-fit σ([Fe/H]) and age of this galaxy are 0.45 and 12.5 Gyr, respectively, but if σ([Fe/H]) is set to >0.5, K-S statistics prefer a younger (<11 Gyr) CDF model (see Figure 7). This phenomenon is caused by the fact that both the smaller σ([Fe/H]) and the fill-in of additional GCs alike make the dip of a CDF shallower. Thus, some Type 1a galaxies whose best-fit σ([Fe/H]) values are smaller than 0.5 might contain some (<10%) young or intermediate-age GCs.

There exist some GC systems that exhibit CDFs with two strong two peaks with only a few (if any) GCs in between. Good examples are NGC 3115 (Brodie et al. 2012) in isolation and NGC 1387 and NGC 1404 near NGC 1399 in the Fornax cluster (Kim et al. 2013a). Their CDFs are also characterized by red GCs that are more abundant than other galaxies with similar luminosity. Our sample contains some galaxies showing an unusually high red GC fraction for their luminosities and $\langle [\mathrm{Fe}/{\rm{H}}]\rangle$ (e.g., VCC 1146, FCC 167, FCC 184 = NGC 1387, and FCC 219 = NGC 1404). The strong bimodality with few GCs in between and/or the unusually high red GC fraction may point to metal-rich GCs created by heavy star formation after metal-poor GCs were generated earlier on. This seems consistent with the prediction by the two-phase galaxy formation scenario, in that in low-mass galaxies, star formation is often prolonged at a significant level toward lower redshift (see Figure 6 in Oser et al. 2010).

Even with the upcoming large telescopes, it is extremely challenging for spectroscopic observations to quantify the number of young or intermediate-age GCs with respect to underlying, old GCs. We hence anticipate that systematic analyses of the "dilution" effect of GC CDFs, equipped with precise knowledge on the shape of the GC CMRs, will become a vital tool for examining the star formation histories of external galaxies beyond the Virgo and Fornax clusters of galaxies.

7. Conclusion

We have simulated individual g − z CDFs for GC systems of 78 early-type galaxies in Virgo and Fornax clusters using the YEPS model (Chung et al. 2013). The GC CDFs are from the GC catalogs by ACSVCS (Jordán et al. 2009) and ACSFCS (Jordán et al. 2015). The main results of this study are as follows:

1.
The HST g − z color distributions of the majority (∼70%) of GC systems are naturally reproduced by the nonlinear metallicity-to-color conversion under the simple assumption of the presence of old (>11 Gyr), coeval GCs. We refer to them as Types 1a and 2a. The variation of the GC CDFs stems from systematic differences in the mean metallicity of GC systems, in that more luminous galaxies (mostly Type 1a) possess more metal-rich GC systems than less luminous galaxies (mostly Type 2a).
2.
The other GC systems (∼30%) show CDFs that can be best reproduced by the two-component (old GCs + young GCs) model. We refer to them as Types 1b and 2b. Most of the galaxies which host GC systems with t_best-fit < 11 Gyr show signs of mergers and/or star formation. These galaxies most likely have additional young or intermediate-age GCs, which alter the color distributions of underlying, old GCs. Types 1b and 2b are suggested to be the variants of Type 1a and 2a, respectively.
3.
There is a strong, positive correlation between the mean metallicity and the host galaxy luminosity, as also reported by other studies (e.g., Peng et al. 2006). The best-fit age and metallicity dispersion of GCs show no obvious correlation with the host galaxy luminosity.

S.J.Y. acknowledges support from the NRF of Korea to the Center for Galaxy Evolution Research (No. 2017R1A5A1070354). This work was partially supported by the KASI–Yonsei Joint Research Program (2018).

Appendix:

This Appendix presents the impact of differing σ([Fe/H]) on the determination of the best-fit age and $\langle [\mathrm{Fe}/{\rm{H}}]\rangle$ . Figure 7 shows the p-value contour plots for the whole galaxy sample.