# The Effect of Age and Stellar Model Choice on Globular Cluster Color-to-
Metallicity Conversions

Kate Hartman Department of Physics & Astronomy, McMaster University, 1280 Main
St W, Hamilton, ON L8S 1T7, Canada  William E. Harris Department of Physics &
Astronomy, McMaster University, 1280 Main St W, Hamilton, ON L8S 1T7, Canada

###### Abstract

The photometric colors of globular clusters (GCs) act as effective proxies for
metallicity, since all normally used optical/IR color indices exhibit a
nonlinear but monotonic relation between their integrated color and their
metallicity. One color index, (g - z) or (F475W - F850LP), has been
spectroscopically calibrated in several studies, providing leverage to define
color-to-metallicity conversions for other indices. In this paper, building on
the work of Hartman et al. (2023), we study the GC color-metallicity relation
in more detail by testing the dependence of the relations on different suites
of stellar models and different assumed GC ages. Though noticeable differences
between models exist, we find that the net effect on the derived GCS
metallicity distributions is small.

Globular star clusters (656) — Metallicity (1031) — Stellar colors (1590)

††facilities: HST (WFC3)††software: Python (https://www.python.org), NumPy
(Harris et al., 2020), pandas (pandas development team, 2020), SciPy (Virtanen
et al., 2020), Matplotlib (Hunter, 2007), DOLPHOT (Dolphin, 2000)

##  1 Introduction

Globular clusters (GCs)—old, massive, dense star clusters found in all but the
least massive of dwarf galaxies (Harris, 2010; Forbes & Remus, 2018; Beasley,
2020; Eadie et al., 2022)—are powerful tracers of early galaxy growth and
chemical enrichment. Part of their appeal for observers stems from the scaling
relations that they exhibit with galaxy properties, such as the
$M_{GCS}$-$M_{h}$ relation (e.g. Blakeslee et al., 1997; Forbes et al., 2016;
Harris et al., 2017; Dornan & Harris, 2023), and from the relations between
fundamental GC properties and easy-to-observe quantities. The relation between
GC metallicity and color, for example, is monotonic (see Brodie & Strader,
2006; Peng et al., 2006; Usher et al., 2015; Harris et al., 2017; Fahrion et
al., 2020; Harris, 2023, among others), making color an attractive proxy
observable for metallicity.

Despite its usefulness, the specifics of the GC color-metallicity relation
(CMR) have proven challenging to constrain. When spectroscopy is available,
the relation is often modelled nonlinearly, as in Peng et al. (2006); Sinnott
et al. (2010); Usher et al. (2012); Fahrion et al. (2020), and Harris (2023)
(with the exception of Villaume et al., 2019). Furthermore, the relation has
been spectroscopically studied primarily for the (g - z) color index
(equivalent to the HST index (F475W - F850LP) in the Vegamag system); thus
other color indices are useful metallicity indicators but must first be
transformed to (g - z) with a color-color relation, as in Harris (2023) and
Hartman et al. (2023).

Both Harris (2023) and Hartman et al. (2023) used the PARSEC stellar models
(Marigo et al., 2017) to derive GC color-color relations. They adopted a fixed
age of 12 Gyr for their simulated clusters and fitted simple nonlinear models
to the resulting CMRs. Those studies focused on the GC metallicity
distributions as they related to host galaxy properties; while the steps taken
to construct color-color and color-metallicity relations were justifiable, the
relations were a means to an end and were not themselves investigated in
detail. Because GC integrated colors are dominated by relatively small numbers
of red-giant and subgiant stars per cluster, stellar model specifics in these
evolutionary stages could have a significant impact on the shape of GC color-
color relations, with subsequent effects carried through the CMRs.

In this paper, we take further steps towards building the CMRs for old GCs by
investigating the effect of stellar model choices on the shape and zeropoints
of the relations. For the purposes of this study, we restrict the analysis to
the particular color index (F475X-F110W) used in Hartman et al. (2023) for the
GC systems in 15 giant early-type galaxies (ETGs), with our main goal to test
the model-based transformations themselves. We will address extensions to
other indices in later work. Section 2 details the properties of our simulated
clusters, Section 3 outlines our procedures for deriving CMRs, and Section 4
looks at how the differences in CMRs affect analysis of observations. We
discuss our findings and summarize our work in Section 5.

##  2 Model GCs

We derived GC CMRs using five suites of frequently used contemporary models:
two versions of the PARSEC (PAdova and TRieste Stellar Evolution Code) models,
two versions of the BaSTI (a Bag of Stellar Tracks and Isochrones) models, and
the MIST (MESA Isochrones and Stellar Tracks) models.

First, we tested two recent versions of the PARSEC cluster simulator: CMD 3.6,
the model used in Hartman et al. (2023), and CMD 3.7, the most up-to-date set
available at the time of writing. We used the default settings from the
Osservatorio Astronomico di Padova’s online tool (CMD 3.6, CMD 3.7) for
circumstellar dust, extinction, and long period variability, and specified the
cluster age, metallicity, and mass. See Bressan et al. (2012); Chen et al.
(2014, 2015); Tang et al. (2014); Marigo et al. (2017); Pastorelli et al.
(2019), and Pastorelli et al. (2020) for details.

Next, we tested two abundance mixtures from BaSTI: solar-scaled, with
[$\alpha$/Fe] = 0.0, and alpha-enhanced, with [$\alpha$/Fe] = +0.4, both
available as drop-down options on the synthetic CMD page of BaSTI’s online
tool. We used the diffusionless grid for our solar-scaled clusters and the He
= 0.247 grid for our alpha-enhanced clusters, along with default values for
star formation rate (SFR), minimum mass, binary fraction, minimum binary mass
ratio, treatment of variable stars, and photometric error. We also specified
an SFR scale of 500,000 (this input is related to the mass of the resulting
simulated cluster). BaSTI is the only model suite in this study with an alpha
enhancement option as of the time of writing. See Hidalgo et al. (2018);
Pietrinferni et al. (2021), and Salaris et al. (2022) for details.

Finally, we tested one set of MIST models with MIST’s isochrone interpolation
online tool. We used the default settings for MIST version, stellar rotation,
extinction, and abundances. See Dotter (2016) and Choi et al. (2016), along
with Paxton et al. (2011, 2013, 2015), and Paxton et al. (2018) for details.

For each of the 5 suites of stellar models, we created a $5\times 23$ grid of
simulated clusters as they would appear in the HST WFC3 photometric system,
with a metallicity range of -2.2 to 0.0 dex in 0.1-dex steps and an age range
of 9 to 13 Gyr in 1-Gyr steps. Each simulated cluster had approximately the
same mass ($M\sim 10^{5}M_{\odot}$), a Kroupa initial mass function, and no
internal spread in metallicity [M/H]. We used these simulated cluster sets to
calculate predicted CMRs.

##  3 Deriving CMRs

In order to recreate the color to metallicity conversion construction process
from Hartman et al. (2023), we needed integrated magnitudes from simulated
clusters for each of the filters in their color indices. The BaSTI online tool
provides these automatically, producing two outputs for each model cluster: a
large file containing magnitudes and stellar properties for each simulated
star, and a small file with cluster properties and integrated magnitudes.
Conversely, the MIST online tool produces isochrones rather than simulated
clusters, and the PARSEC CMD 3.6 and 3.7 online tools’ integrated magnitude
option removes the cluster mass parameter, so we calculated integrated
magnitudes for each PARSEC and MIST cluster manually.

###  3.1 Integrated cluster magnitudes

The output files for both PARSEC CMD 3.6 and 3.7 included magnitudes in all
HST WFC3 filters for each individual star in the simulated cluster. We
converted these magnitudes to luminosities, summed them to get an integrated
cluster luminosity, and then converted that back into a magnitude for each
relevant filter.

For MIST, an output file contains for each point on the isochrone a set of
theoretical stellar parameters (e.g. mass, effective temperature, luminosity,
etc.). For the MIST models, we calculated the number of stars at each point
necessary for a $10^{5}M_{\odot}$ GC with a Kroupa IMF (Kroupa et al., 1993),
and then followed the same luminosity summing procedures that we used with the
PARSEC clusters.

Figure 1 shows (F475X - F110W) and (F475W - F850LP), the two color indices
from Hartman et al. (2023), versus metallicity for individual simulated GCs in
each of the five models. Because the MIST online tool does not produce
simulated clusters like its PARSEC and BaSTI counterparts do, those clusters
are not affected by stochasticity and do not exhibit any scatter.

Figure 1: Model GCs from the PARSEC, BaSTI, and MIST SSPs. With all five
models, we simulated GCs with metallicities ranging from -2.2 to 0.0 dex and
ages ranging from 9 to 13 Gyr, and calculated integrated magnitudes in (F475X
- F110W) and (F475W - F850LP). Top left: PARSEC CMD 3.6, the model used in
Hartman et al. (2023). Top right: PARSEC CMD 3.7, the latest version of PARSEC
available online as of the time of writing. Middle left: BaSTI’s
$\alpha$-enhanced model. Middle right: BaSTI’s solar-scaled model. Bottom
left: MIST isochrone-derived model GCs.

###  3.2 Fitting

We fitted inverse exponential equations to each set of simulated GCs in both
(F475X - F110W) and (F475W - F850LP), for a total of 10 fitted equations per
stellar model with 5 per color index. We also fitted all simulated GCs
regardless of age from each stellar model; the fitted parameters are defined
in Equation 1 and listed in Table 1. An exponential model is preferable
because unlike a quadratic model it remains monotonic even when used to
extrapolate (although significant extrapolations should still be viewed with
caution), and unlike a pair of joined linear models it assumes no sudden
change in slope. Figure 2 shows model GCs of all ages with fitted exponential
models overplotted, along with residuals for the CMRs for both color indices;
Figure 3 places all fitted models in metallicity-color space; and Figure 4
shows differences when subtracting the color-metallicity and color-color
relations of the other four models from the PARSEC CMD 3.6 color-metallicity
and color-color relations that we use as a template. (That is, for a given
value in (F475X - F110W), how much more metal-poor or metal rich, or bluer or
redder in (F475W - F850LP), is the alternate model compared to PARSEC CMD
3.6?) The synthetic color-color relations show agreement within 0.15
magnitudes among the different sets of models, and the CMRs agree within 0.3
dex.

The full set of fitted parameters, broken out by GC age, for an equation of
the form

$\text{[M/H]}=\frac{\ln{\frac{\text{color}-C}{A}}}{B}$ (1)

can be found in Appendix A.

Figure 2: Left column: model GCs with fitted exponential metallicity-color
relations overplotted. Points are colored as in Figure 1. Center column:
residuals from the (F475X - F110W)-metallicity relations. Right column:
residuals from the (F475W - F850LP)-metallicity relations. Figure 3: Fitted
exponential metallicity-color relations for the five models and the five GC
age groups. Figure 4: Differences between fitted exponential color-metallicity
and color-color relations, comparing PARSEC CMD 3.6 to the other four models
(i.e. subtracting each model from PARSEC CMD 3.6). As in previous figures,
PARSEC CMD 3.7 is shown in blue, BaSTI alpha-enhanced in orange, BaSTI solar-
scaled in red, and MIST in pink; similarly, relations based on the (F475X -
F110W) color index are shown as solid lines and relations based on the (F475W
- F850LP) index as dashed lines.

###  3.3 CMR uncertainty

The uncertainties of metallicity values derived from the CMRs presented in
this work can be estimated by multiplying a user-supplied color uncertainty
$\Delta\text{color}$ by the slope of the CMR
$\frac{d\text{[M/H]}}{d\text{color}}$:

$\Delta\text{[M/H]}=\Delta\text{color}\frac{d\text{[M/H]}}{d\text{color}}$ (2)

where

$\frac{d\text{[M/H]}}{d\text{color}}=\frac{1}{B(\text{color}-C)}$ (3)

with B and C found in Table 1 as with Equation 1. Figure 5 shows the ratio of
metallicity uncertainty to color uncertainty for each model in (F475X - F110W)
and (F475W - F850LP); because of the change in slope of the CMRs, the red ends
of the CMRs are more sensitive to metallicity than the blue ends, so the
metallicity uncertainties at the red ends are smaller.

Figure 5: Ratio of metallicity uncertainty and color uncertainty by color index for each of the five stellar models and both color indices; see Equation 2. Color is a less sensitive tracer of metallicity at the blue end of color-metallicity space than at the red end. Table 1: CMR parameters (as defined in Equation 1) for each stellar model and both color indices, derived from all five simulated GC age groups. Index | Model | A | B | C
---|---|---|---|---
(F475X - F110W) | PARSEC CMD 3.6 |  1.120 ($\pm$ 0.024) |  0.741 ($\pm$ 0.051) |  1.313 ($\pm$ 0.031)
| PARSEC CMD 3.7 |  1.173 ($\pm$ 0.037) |  0.636 ($\pm$ 0.053) |  1.258 ($\pm$ 0.045)
|  BaSTI $\alpha$-enhanced |  1.715 ($\pm$ 0.293) |  0.289 ($\pm$ 0.072) |  0.654 ($\pm$ 0.306)
| BaSTI solar-scaled |  1.581 ($\pm$ 0.307) |  0.289 ($\pm$ 0.082) |  0.655 ($\pm$ 0.320)
| MIST |  1.703 ($\pm$ 0.117) |  0.332 ($\pm$ 0.035) |  0.670 ($\pm$ 0.124)
(F475W - F850LP) | PARSEC CMD 3.6 |  0.950 ($\pm$ 0.025) |  0.678 ($\pm$ 0.052) |  1.134 ($\pm$ 0.032)
| PARSEC CMD 3.7 |  1.020 ($\pm$ 0.044) |  0.554 ($\pm$ 0.054) |  1.055 ($\pm$ 0.052)
|  BaSTI $\alpha$-enhanced |  1.632 ($\pm$ 0.319) |  0.257 ($\pm$ 0.070) |  0.434 ($\pm$ 0.330)
| BaSTI solar-scaled |  1.538 ($\pm$ 0.364) |  0.251 ($\pm$ 0.082) |  0.418 ($\pm$ 0.376)
| MIST |  1.740 ($\pm$ 0.190) |  0.251 ($\pm$ 0.038) |  0.293 ($\pm$ 0.196)

###  3.4 The GC age-metallicity relation

GC metallicity is expected on both theoretical and observational grounds to be
correlated at least slightly with age, in the sense that more metal-rich
clusters, on average, belong to a later stage of formation and are thus
younger (Leaman et al., 2013; Choksi et al., 2018; Li & Gnedin, 2020; Horta et
al., 2021). To model the correlation between GC metallicity and age, we added
an age-based correction to our color-metallicity procedure.

We calibrated the GC age-metallicity relation by combining data for Milky Way
GCs from Dotter et al. (2010), Forbes & Bridges (2010), and VandenBerg et al.
(2013), and finding a linear fit to the aggregated dataset:

$\text{age (Gyr)}=-0.744\text{[M/H]}+11.38$ (4)

Figure 6 shows the original data with Equation 4 overplotted in black. Though
there is no guarantee that GCs in other large galaxies, such as the giant
early-type galaxies studied in Harris (2023) and Hartman et al. (2023), will
follow this same mean relation, its general property that metal-richer GCs
should be younger on average is expected to hold (cf. the references cited
above). The main purpose of the present study is, instead, to explore the
sensitivity of the resulting GC metallicity distribution function to an age-
metallicity relation.

Figure 6: Age vs. metallicity for Milky Way GCs. Data are from VandenBerg et
al. (2013), Forbes & Bridges (2010), and Dotter et al. (2010); the black line
shows Equation 4, a fitted linear age-metallicity relation based on all three
datasets.

Equation 4 estimates a GC’s age based on its metallicity. To determine the
corresponding adjustment from the nominal CMRs listed in Table 1 for each
stellar model, we divided the average color index range by our model age range
(i.e. 4 Gyr) to determine an age-based color offset. After finding an
individual GC’s color offset based on its age and adding it to the measured GC
color, we applied the nominal CMR again and obtained an age-corrected
metallicity value.

Across all five stellar models, the age correction made very little difference
to the CMR (see Figure 7): all corrections were less than 0.05 magnitudes. The
age-corrected relations were slightly flatter than the original relations,
with the extreme red and blue ends of the relation moving to less extreme
metallicity values, but never leaving the color bounds set by the oldest and
youngest model clusters.

Figure 7: GC age corrections for the fitted CMRs. The corrected relations are
slightly flatter than the CMRs for specific GC ages in the PARSEC and MIST
models, and slightly more curved in the BaSTI models. In all cases, the
difference between fitted relations for individual ages and the age-corrected
relation is very small.

##  4 Application to observations and discussion

In order to compare the performance of each of our CMRs, we tested them on the
data from Hartman et al. (2023). Their sample comprised 15 massive elliptical
galaxies with densely populated GCSs, at distances ranging from approximately
60 to 110 Mpc. Because GCs appear as unresolved point sources to HST’s WFC3
camera at those distances, they extracted photometry using DOLPHOT (Dolphin,
2000) and calculated integrated (F475X - F110W) color indices for each GC in
their sample. After cleaning their sample and adjusting for completeness (see
Section 3 of Hartman et al., 2023, for details), they transformed their color
indices into metallicity values for comparison to other variables of interest.
The original images are available at MAST: 10.17909/pvve-1002 (catalog
10.17909/pvve-1002).

We followed the same analysis procedure from that work, but substituted in our
alternate CMRs before using GMM (Muratov & Gnedin, 2010) to fit double
Gaussian curves:

$N_{GCs,comp}=A_{mp}e^{{\frac{-(x-\mu_{mp})^{2}}{2\sigma_{mp}^{2}}}}+A_{mr}e^{{\frac{-(x-\mu_{mr})^{2}}{2\sigma_{mr}^{2}}}}$
(5)

where $A_{mp}$ and $A_{mr}$ are amplitudes, $\mu_{mp}$ and $\mu_{mr}$ are peak
positions, and $\sigma_{mp}$ and $\sigma_{mr}$ are peak widths for the metal-
poor and metal-rich Gaussian curves, respectively. Figure 8 shows metallicity
histograms for each of the 15 galaxies in their sample, and Figure 9 shows
double Gaussian fits to each metallicity distribution function (MDF) (standard
for GCSs; see e.g. Kim et al., 2013; Cantiello et al., 2014; Brodie et al.,
2014; Fensch et al., 2014; Escudero et al., 2015; Harris et al., 2017).

Figures 8 and 9 contain the key results from this study: the shape of the MDF
and the specifics of its Gaussian components are robust against the use of
different stellar models and assumptions about GC age. Both the MDFs and the
fitted double Gaussian curves are visually very similar to each other for each
galaxy, producing the same distribution of metallicity values within the
uncertainties. Figure 10 displays the parameter populations in box plot form
(and the double Gaussian parameters themselves can be found in Appendix B).

Figure 8: MDFs for the sample from Hartman et al. (2023), transformed from
color distribution functions using each of the five stellar models. Figure 9:
Double Gaussian fits for the MDFs from Figure 8. Figure 10: A comparison of
double Gaussian parameters from the fitted models in Figure 9. With the
exception of the outlier $\sigma_{mp}$ from the PARSEC CMD 3.7 model (see the
lack of a local minimum in that model for NGC 4839 in Figure 9) the models
produce very similar sets of double Gaussian parameters.

Visual inspection (see Figure 11) indicates that, like stellar model choice,
our built-in age correction makes no significant difference to subsequently
fitted double Gaussian parameters; this was expected based on the
sub-0.05-magnitude color changes that the correction produced. The underlying
reason for this lack of sensitivity appears to be simply that in this high 9-
to 13-Gyr age range, the positions of the stellar red giant tracks and thus
the integrated GC colors are quite insensitive to age, and it is only the
cluster metallicity that is most important. Said differently, at typical GC
ages, most of the stars still present are low-luminosity and cool; beyond an
age of 8 Gyr, the position of the red giant branch in CMDs changes so slowly
that age effects are secondary to metallicity effects.

Figure 11: A comparison of double Gaussian parameters from the 5 stellar
models with (lighter boxes) and without (darker boxes) age corrections.

Even in metallicity ranges with the most disagreement between models, the
differences in derived metallicities are no greater than $\sim 0.1$ dex in
magnitude. Based on expected color uncertainties and the corresponding
metallicity uncertainties from Equation 2 and Figure 5, for typical globular
cluster metallicities, all five models agree with each other within
uncertainty. At the GCS level, small differences in color-color
transformations are effectively washed out, as demonstrated by the box plots
in Figures 10 and 11.

##  5 Conclusions

In this study, we have compared five suites of stellar models, using each of
them in the GC color-color-metallicity conversion process from Hartman et al.
(2023) and quantifying their effects on the analysis of real GCS data. We have
also probed the effect on the CMR of adding an age-metallicity correlation
drawn from the Milky Way GCs. In all cases, regardless of stellar model choice
or presence of the age correction, there was no significant change in the
values used to characterize the MDFs derived from real data.

The data used in testing these models comes from 15 elliptical galaxies with
very similar stellar masses and GCSs with of order $10^{3}$ GCs; it would be
prudent to investigate whether the results hold for smaller GCSs. In this
work, we have studied the CMR primarily for the single WFC3 color index (F475X
- F110W); in followup work, we will extend the transformations to other HST-
based colors including the ones used in Harris (2023). In an upcoming study
with new HST multicolor imaging, the current heavy reliance on the stellar
models will be reduced through the construction of strictly empirical
transformations between color indices.

## Acknowledgments

We acknowledge financial support from the Natural Sciences and Engineering
Research Council of Canada (NSERC) through a Discovery Grant to WEH. KH thanks
Laura Greggio for her advice on working with MIST, and Veronika Dornan, Claude
Cournoyer-Cloutier, and Jeremy Karam for helpful discussions.

##  Appendix A CMR parameters

Fitted parameters for the color-metallicity relations are listed in Table 2.

Table 2: Parameters of the fitted color CMR (see Equation 1) for each stellar model, color index, and GC age. These curves appear in Figure 3. Index | Model | Age (Gyr) | A | B | C
---|---|---|---|---|---
(F475X - F110W) | PARSEC CMD 3.6 | 9 | 1.317 | 1.079 | 0.718
|  | 10 | 1.375 | 1.036 | 0.857
|  | 11 | 1.311 | 1.102 | 0.725
|  | 12 | 1.298 | 1.133 | 0.709
|  | 13 | 1.228 | 1.260 | 0.657
| PARSEC CMD 3.7 | 9 | 1.177 | 1.196 | 0.564
|  | 10 | 1.253 | 1.171 | 0.629
|  | 11 | 1.439 | 0.984 | 0.868
|  | 12 | 1.224 | 1.240 | 0.620
|  | 13 | 1.340 | 1.164 | 0.738
|  BaSTI $\alpha$-enhanced | 9 | -0.106 | 2.416 | 0.174
|  | 10 | 0.862 | 1.456 | 0.327
|  | 11 | -4.795 | 7.103 | 0.051
|  | 12 | 0.737 | 1.676 | 0.321
|  | 13 | 0.652 | 1.779 | 0.299
| BaSTI solar-scaled | 9 | 0.307 | 1.870 | 0.227
|  | 10 | 0.758 | 1.440 | 0.310
|  | 11 | 0.888 | 1.339 | 0.336
|  | 12 | 0.851 | 1.337 | 0.310
|  | 13 | 0.797 | 1.538 | 0.336
| MIST | 9 | 0.496 | 1.815 | 0.285
|  | 10 | 0.559 | 1.781 | 0.299
|  | 11 | 0.647 | 1.722 | 0.322
|  | 12 | 0.740 | 1.659 | 0.352
|  | 13 | 0.870 | 1.558 | 0.399
(F475W - F850LP) | PARSEC CMD 3.6 | 9 | 1.128 | 0.911 | 0.640
|  | 10 | 1.199 | 0.863 | 0.795
|  | 11 | 1.127 | 0.939 | 0.652
|  | 12 | 1.120 | 0.972 | 0.652
|  | 13 | 1.031 | 1.101 | 0.579
| PARSEC CMD 3.7 | 9 | 0.963 | 1.054 | 0.473
|  | 10 | 1.029 | 1.028 | 0.519
|  | 11 | 1.259 | 0.820 | 0.826
|  | 12 | 1.003 | 1.101 | 0.525
|  | 13 | 1.131 | 1.006 | 0.644
|  BaSTI $\alpha$-enhanced | 9 | -0.496 | 2.505 | 0.140
|  | 10 | 0.614 | 1.404 | 0.282
|  | 11 | -13.91 | 15.93 | 0.019
|  | 12 | 0.477 | 1.624 | 0.276
|  | 13 | 0.342 | 1.777 | 0.250
| BaSTI solar-scaled | 9 | 0.037 | 1.864 | 0.193
|  | 10 | 0.529 | 1.392 | 0.270
|  | 11 | 0.667 | 1.281 | 0.296
|  | 12 | 0.549 | 1.365 | 0.251
|  | 13 | 0.554 | 1.488 | 0.290
| MIST | 9 | 0.007 | 1.967 | 0.200
|  | 10 | 0.088 | 1.912 | 0.212
|  | 11 | 0.200 | 1.829 | 0.231
|  | 12 | 0.318 | 1.738 | 0.256
|  | 13 | 0.511 | 1.571 | 0.302

##  Appendix B Fitted double Gaussian parameters

Double-Gaussian fits obtained through GMM, for the giant galaxies studied in
Hartman et al. (2023), are listed in Table 3.

Table 3: Fitted double Gaussian parameters from Equation 5 for the Hartman et al. (2023) data transformed with our nominal color-metallicity relations. These curves appear in Figure 9. Galaxy | Model |  $A_{mp}$ (GCs) |  $\mu_{mp}$ [M/H] |  $\sigma_{mp}$ [M/H] |  $A_{mr}$ (GCs) |  $\mu_{mr}$ [M/H] |  $\sigma_{mr}$ [M/H]
---|---|---|---|---|---|---|---
NGC 57 | PARSEC CMD 3.6 | 61.0 | -1.08 | 0.48 | 119.5 | -0.03 | 0.32
| PARSEC CMD 3.7 | 61.0 | -1.08 | 0.47 | 127.8 | -0.07 | 0.30
|  BaSTI $\alpha$-enhanced | 56.2 | -1.07 | -0.52 | 110.1 | 0.06 | 0.34
| BaSTI solar-scaled | 58.4 | -1.02 | 0.53 | 115.5 | 0.06 | 0.31
| MIST | 59.0 | -1.10 | 0.49 | 123.0 | -0.06 | 0.31
NGC 410 | PARSEC CMD 3.6 | 40.5 | -0.85 | 0.66 | 39.4 | 0.02 | 0.21
| PARSEC CMD 3.7 | 41.2 | -0.84 | 0.65 | 42.8 | -0.01 | 0.19
|  BaSTI $\alpha$-enhanced | 38.9 | -0.83 | 0.70 | 35.9 | 0.13 | 0.22
| BaSTI solar-scaled | 39.1 | -0.83 | 0.68 | 38.6 | 0.11 | 0.22
| MIST | 40.7 | -0.85 | 0.65 | 41.5 | 0.00 | 0.20
NGC 533 | PARSEC CMD 3.6 | 91.6 | -0.79 | 0.68 | 61.1 | 0.04 | 0.18
| PARSEC CMD 3.7 | 94.0 | -0.79 | 0.65 | 70.9 | 0.01 | 0.17
|  BaSTI $\alpha$-enhanced | 84.9 | -0.78 | 0.72 | 58.7 | 0.13 | 0.20
| BaSTI solar-scaled | 86.9 | -0.77 | 0.69 | 62.7 | 0.13 | 0.20
| MIST | 91.8 | -0.78 | 0.67 | 63.2 | 0.02 | 0.18
NGC 741 | PARSEC CMD 3.6 | 28.0 | -1.26 | 0.32 | 41.8 | -0.34 | 0.35
| PARSEC CMD 3.7 | 28.8 | -1.20 | 0.37 | 43.1 | -0.31 | 0.31
|  BaSTI $\alpha$-enhanced | 26.9 | -1.27 | 0.33 | 39.6 | -0.29 | 0.38
| BaSTI solar-scaled | 27.6 | -1.27 | 0.30 | 40.7 | -0.30 | 0.38
| MIST | 28.2 | -1.24 | 0.34 | 42.6 | -0.33 | 0.33
NGC 777 | PARSEC CMD 3.6 | 41.0 | -1.14 | 0.46 | 94.5 | -0.09 | 0.37
| PARSEC CMD 3.7 | 42.1 | -1.09 | 0.49 | 99.4 | -0.09 | 0.34
|  BaSTI $\alpha$-enhanced | 38.6 | -1.16 | 0.48 | 87.7 | -0.01 | 0.41
| BaSTI solar-scaled | 40.7 | -1.13 | 0.47 | 91.1 | -0.01 | 0.38
| MIST | 41.3 | -1.08 | 0.52 | 94.8 | -0.08 | 0.35
NGC 1016 | PARSEC CMD 3.6 | 35.2 | -0.70 | 0.66 | 33.2 | 0.06 | 0.28
| PARSEC CMD 3.7 | 32.9 | -0.79 | 0.63 | 39.8 | -0.01 | 0.30
|  BaSTI $\alpha$-enhanced | 31.4 | -0.74 | 0.69 | 31.8 | 0.12 | 0.34
| BaSTI solar-scaled | 31.9 | -0.74 | 0.68 | 32.9 | 0.12 | 0.34
| MIST | 31.3 | -0.83 | 0.64 | 40.4 | -0.01 | 0.31
NGC 1129 | PARSEC CMD 3.6 | 107.6 | -1.00 | 0.47 | 131.6 | -0.14 | 0.32
| PARSEC CMD 3.7 | 105.8 | -1.03 | 0.44 | 146.1 | -0.20 | 0.31
|  BaSTI $\alpha$-enhanced | 95.5 | -1.04 | 0.48 | 127.3 | -0.11 | 0.37
| BaSTI solar-scaled | 102.7 | -0.95 | 0.50 | 119.5 | -0.07 | 0.34
| MIST | 102.5 | -1.05 | 0.45 | 142.0 | -0.19 | 0.32
NGC 1600 | PARSEC CMD 3.6 | 88.4 | -0.70 | 0.69 | 69.1 | -0.10 | 0.23
| PARSEC CMD 3.7 | 89.8 | -0.73 | 0.64 | 81.4 | -0.11 | 0.22
|  BaSTI $\alpha$-enhanced | 85.7 | -0.65 | 0.73 | 63.7 | 0.00 | 0.22
| BaSTI solar-scaled | 86.2 | -0.64 | 0.73 | 67.7 | -0.02 | 0.21
| MIST | 89.5 | -0.71 | 0.66 | 77.5 | -0.10 | 0.21
NGC 2340 | PARSEC CMD 3.6 | 42.9 | -1.00 | 0.49 | 55.8 | -0.06 | 0.38
| PARSEC CMD 3.7 | 46.6 | -0.91 | 0.53 | 56.0 | -0.04 | 0.32
|  BaSTI $\alpha$-enhanced | 40.5 | -0.97 | 0.54 | 49.7 | 0.05 | 0.42
| BaSTI solar-scaled | 41.4 | -0.97 | 0.52 | 51.9 | 0.03 | 0.40
| MIST | 45.0 | -0.92 | 0.55 | 51.3 | -0.03 | 0.35
NGC 3158 | PARSEC CMD 3.6 | 26.6 | -1.24 | 0.49 | 41.6 | -0.17 | 0.43
| PARSEC CMD 3.7 | 28.2 | -1.17 | 0.52 | 43.9 | -0.17 | 0.37
|  BaSTI $\alpha$-enhanced | 25.4 | -1.26 | 0.47 | 40.6 | -0.12 | 0.46
| BaSTI solar-scaled | 25.0 | -1.29 | 0.47 | 41.9 | -0.14 | 0.46
| MIST | 26.0 | -1.28 | 0.46 | 44.9 | -0.21 | 0.42
NGC 3842 | PARSEC CMD 3.6 | 30.6 | -1.37 | 0.52 | 52.2 | -0.36 | 0.43
| PARSEC CMD 3.7 | 31.8 | -1.33 | 0.50 | 54.2 | -0.36 | 0.40
|  BaSTI $\alpha$-enhanced | 29.0 | -1.39 | 0.51 | 48.6 | -0.33 | 0.48
| BaSTI solar-scaled | 34.7 | -1.14 | 0.62 | 39.4 | -0.21 | 0.42
| MIST | 28.5 | -1.48 | 0.42 | 58.3 | -0.43 | 0.44
NGC 4073 | PARSEC CMD 3.6 | 43.5 | -0.84 | 0.69 | 74.9 | -0.12 | 0.28
| PARSEC CMD 3.7 | 44.9 | -0.84 | 0.65 | 79.8 | -0.14 | 0.27
|  BaSTI $\alpha$-enhanced | 45.6 | -0.73 | 0.73 | 63.9 | -0.01 | 0.27
| BaSTI solar-scaled | 46.0 | -0.73 | 0.71 | 67.7 | -0.01 | 0.26
| MIST | 43.6 | -0.84 | 0.68 | 76.4 | -0.13 | 0.27
NGC 4839 | PARSEC CMD 3.6 | 44.6 | -1.02 | 0.55 | 45.3 | -0.09 | 0.27
| PARSEC CMD 3.7 | 45.0 | -1.00 | 0.55 | 46.9 | -0.12 | 0.25
|  BaSTI $\alpha$-enhanced | 42.0 | -1.05 | 0.54 | 46.1 | -0.03 | 0.29
| BaSTI solar-scaled | 42.2 | -1.07 | 0.51 | 51.0 | -0.06 | 0.29
| MIST | 43.8 | -1.01 | 0.56 | 45.8 | -0.11 | 0.26
NGC 4914 | PARSEC CMD 3.6 | 20.7 | -0.48 | 0.57 | 21.2 | 0.09 | 0.20
| PARSEC CMD 3.7 | 19.9 | -0.49 | 0.59 | 25.6 | 0.01 | 0.18
|  BaSTI $\alpha$-enhanced | 17.6 | -0.49 | 0.64 | 20.0 | 0.15 | 0.26
| BaSTI solar-scaled | 17.6 | -0.48 | 0.63 | 20.7 | 0.14 | 0.25
| MIST | 16.9 | -0.62 | 0.60 | 26.9 | 0.01 | 0.23
NGC 7242 | PARSEC CMD 3.6 | 124.5 | -0.55 | 0.62 | 58.7 | 0.05 | 0.13
| PARSEC CMD 3.7 | 129.8 | -0.56 | 0.60 | 53.4 | -0.57 | 0.14
|  BaSTI $\alpha$-enhanced | 118.5 | -0.50 | 0.67 | 41.7 | 0.12 | 0.14
| BaSTI solar-scaled | 122.2 | -0.48 | 0.66 | 38.9 | 0.12 | 0.13
| MIST | 128.1 | -0.56 | 0.60 | 53.1 | 0.03 | 0.14

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# Asymptotic commutativity in Jordan algebras

Albert Schwarz  A. Schwarz
Department of Mathematics
University of California
Davis, CA 95616, USA,
schwarz @math.ucdavis.edu

###### Abstract

We estimate commutators of quadratic operators $Q_{a}$ in Jordan algebras.
These estimates can be used to construct the scattering theory in quantum
fields theories formulated in terms of Jordan algebras.

Keywords Jordan algebra; quadratic operators; scattering theory

The standard algebraic approach to quantum theory is based on associative
algebras with involution. To develop scattering theory in this approach one
should fix the action of translation group on the algebra and impose some
conditions of asymptotic commutativity (in the case of bosons) or asymptotic
anticommutativity (in the case of fermions). In recently suggested geometric
approach to quantum theory ([1]-[5]) Jordan algebras are more natural than
associative algebras. It was shown in [5] that one can develop scattering
theory in the framework of Jordan algebras if quadratic operators
$Q_{a}=2R_{a}^{2}-R_{a^{2}}$ asymptotically commute. ( Here $R_{a}$ stands for
the multiplication operator: $R_{a}(b)=b\circ a.$ )

We prove asymptotic commutativity of operators $Q_{a}$ under certain
conditions. This result was used in [5] to prove the existence of (inclusive)
scattering matrix. (The present paper will be published as an appendix to
[5].)

Let us start with some remarks in the simple case when Jordan algebra $\cal B$
is obtained as a set of self-adjoint elements of associative algebra $\cal A$
with involution ∗ and the operation $a\circ b=\frac{1}{2}(ab+ba).$ We assume
that $\cal A$ is a C*-algebra, then one says that $\cal B$ is a JC-algebra. It
is easy to check that in the case at hand we have $Q_{a}=l(a)r(a)$ where
$l(a)$ and $r(a)$ are the operators of multiplication by $a$ from the left and
from the right in $\cal A.$ Using the relations
$[l(a),l(b)]=l([a,b]),[r(a),r(b)]=r([b,a])$ and
$Q_{a}Q_{b}=l(ab)r(ba),Q_{b}Q_{a}=l(ba)r(ab)$ , we obtain the estimate

$||[Q_{a},Q_{b}]||\leq 2||a||\cdot||b||\cdot||[a,b]||.$ (1)

Similarly one can estimate the commutator $[Q_{a},Q_{b}]$ in terms of
anticommutator of $a,b$ (or equivalently in terms of Jordan product $a\circ
b$):

$||[Q_{a},Q_{b}]||\leq
2||a||\cdot||b||\cdot||[a,b]_{+}||=4||a||\cdot||b||\cdot||a\circ b||.$ (2)

The proof remains almost the same: we should use the relations
$l(ab)=-l(ba)+l([a,b]_{+}),r(ba)=-r(ab)+r([a,b]_{+}).$

Let us suppose that commutative Lie group (translation group) acts on $\cal A$
by means of involution-preserving automorphisms $\alpha({\bf x},\tau)$; we use
the notation $\alpha({\bf x},\tau)a=a({\bf x},\tau)$. Then the same group acts
on $\cal B$; it transforms the operators $Q_{a}$ into the operators $Q_{a({\bf
x},\tau)}.$

We say that the elements $a,b\in\cal A$ asymptotically commute if

$[a({\bf x},\tau),b]\leq\rho({\bf x},\tau)$

where $\rho$ is small for large $\bf x$ (for definiteness we assume that that
$\rho$ tends to zero faster than any power of $||\bf x||$ and has at most
polynomial growth with respect to $\tau$)

The elements $a,b\in\cal A$ asymptotically anticommute if a similar relation
is valid for anticommutator

$[a{(\bf x},\tau),b]_{+}\leq\rho({\bf x},\tau).$

It follows from (1), (2) that in both cases operators $Q_{a},Q_{b}$
asymptotically commute:

$||[Q_{a}({\bf x},\tau),Q_{b}]||\leq 2||a||\cdot||b||\rho({\bf x},\tau).$

We would like to generalize the above relations to the case of Jordan Banach
algebras (JB-algebras). The possibility of such a generalization is prompted
by the following statements:

a) If $a,b$ are two elements of JB-algebra $\cal B$ and $a\circ b=0$ then
operators $Q_{a}$ and $Q_{b}$ commute [6], [7].

b) If $a,b$ are two elements of JB-algebra $\cal B$ and operators
$R_{a},R_{b}$ commute than the operators $Q_{a}$ and $Q_{b}$ also commute [8].

One says that $a,b\in\cal B$ operator commute iff the operators $R_{a}$ and
$R_{b}$ commute. For $JB$-algebras it was proven in [8] that the operator
commutativity of $a,b$ implies that $a^{2}$ operator commutes with $b$ and
$b^{2}$ (similarly $b^{2}$ operator commutes with $a$). This means that the
operators $Q_{a}$ and $Q_{b}$ commute.

Let us suppose that commutative Lie group (translation group) acts on $\cal B$
by means of automorphisms $\alpha(\bf x,\tau)$. The same group acts on the
operators $R_{a}$ and $Q_{a}$ transforming them into the operators
$R_{\alpha(\bf x,\tau)\it a}$ and $Q_{\alpha(\bf x,\tau)\it a}.$ We say that
$a,b\in\cal B$ asymptotically operator commute if $||[R_{\alpha(\bf x,\tau)\it
a},R_{b}]||<.$ It is natural to conjecture that in this case the operators
$Q_{a},Q_{b}$ also asymptotically commute, i. e. $||[Q_{{\alpha({\bf
x},\tau)}{\it a}},Q_{b}]||<\rho(\bf x,\tau).$ The methods of [8] are not
sufficient to prove this conjecture. However, if the pairs
$(a,b),(a^{2},b),(a,b^{2}),(a^{2},b^{2})$ asymptotically operator commute it
is obvious that $Q_{a},Q_{b}$ asymptotically commute. This statement is
sufficient for applications we have in mind.

The same statements are true in more general case when the translation group
acts by structural transformations: $\alpha({\bf x},\tau)\in Strg(\cal B)$,
where $Strg(\cal B)$ denotes the structure group (the group generated by
automorphisms and invertible operators $Q_{a}$).

Let us generalize the statement a):

Let us assume that the norm of the Jordan product $a\circ b$ of two elements
of JB-algebra $\cal B$ is $\leq\epsilon.$ Then

$||[Q_{a},Q_{b}]||\leq k(||a||,||b||)\sqrt{\epsilon}$ (3)

where $\epsilon\geq 0$ and $k$ is a polynomial function

To prove this statement we use the theorem that every JB-algebra can be
considered as a subalgebra of a direct sum of JC-algebra and purely
exceptional JB algebra. (This follows from the fact that a JB-algebra $\cal B$
is a subalgebra of JBW-algebra ${\cal B}^{**}$ and every JBW-algebra is a
direct sum of JW-algebra and purely exceptional JBW-algebra; see [8] for
definitions and formulations and [9] for details and proofs). It is obvious
that our statement is correct for a subalgebra if it is correct for the
algebra. The statement is proven already for JC-algebra (2). It remains to
prove it for simple exceptional JB-algebra (then it is true for any purely
exceptional algebra). To give the proof we modify slightly the considerations
given in [7] for $\epsilon=0.$

The only information we need about simple exceptional JB-algebra (Albert
algebra) is the equation

$a^{3}=t(a)a^{2}-s(a)a+n(a)$ (4)

satisfied by any element $a$ of Albert algebra. (Here $t,s,n$ are linear,
quadratic and cubic functions on Albert algebra.)

Notice first of all that for any JB-algebra $\cal B$

$[Q_{a},Q_{b}]=[R_{a^{2}},R_{b^{2}}]+o_{1}$

where $a,b\in\cal B$, $||o_{1}||\leq\epsilon f(||a||,||b||).$ (In [7] this
formula without the $o_{1}$ term is derived for $a\circ b=0.$ We use the same
calculations, but instead of omitting terms containing $a\circ b$ we estimate
them using $||a\circ b||\leq\epsilon.$ We apply the same procedure to the
derivation of other formulas.) The next step is the formula

$[Q_{a},Q_{b}]c=[R_{a^{2}},R_{b^{2}}]c+o_{1}c=(a^{2},c,b^{2})+o_{1}c=-2(a^{2}\circ
b,c,b)+o_{1}c$

where $c\in\cal B$ and $(x,y,z)$ stands for the associator $(x\circ y)\circ
z-x\circ(y\circ z).$

It follows from (4) that

$(a^{2}\circ b,c,b)=-s(a,b)(a,c,b)+o_{2}$

where $s(a,b)$ denotes the bilinear form, corresponding to the quadratic form
$s(a)$ and $||o_{2}||\leq\epsilon g(||a||,||b||)||c||.$ (See [7] for the
derivation in the case $\epsilon=0.$) Substituting $c=a$ we obtain

$s(a,b)(a^{2}\circ b)=o_{3}$

where $||o_{3}||\leq\epsilon h(||a||,||b||).$ We see that either
$|s(a,b)|\leq||o_{3}||^{\frac{1}{2}}$ or $||a^{2}\circ
b||\leq||o_{3}||^{\frac{1}{2}}.$ In the first case

$||[Q_{a},Q_{b}]||\leq
2||o_{3}||^{\frac{1}{2}}||a||||b||+\epsilon(g(||a||,||b||)+||o_{1}||.$

In the second case

$||[Q_{a},Q_{b}]||\leq 4||o_{3}||^{\frac{1}{2}}||b||+||o_{1}||.$

In both cases we obtain the estimate we need.

Let us consider again a commutative Lie group (translation group) acting on
JB-algebra $\cal B$ by means of automorphisms or, more generally, structural
transformations $\alpha(\bf x,\tau).$ We assume $(\alpha(\bf x,\tau)\it
a)\circ b$ is small for large $\bf x$ . (If $\cal B$ is a JC-algebra
corresponding to C*-algebra $\cal A$ this is equivalent to the asymptotic
anticommutativity of elements $a,b$ in $\cal A.$) Then it follows from (3)
that $Q_{a}$ and $Q_{b}$ asymptotically commute:

if $||(\alpha(\bf x,\tau)\it a)\circ b||\leq\rho(\bf x,\tau)$ then
$||[Q_{\alpha(\bf x,\tau)\it a},Q_{b}]||\leq k\sqrt{\rho(\bf x,\tau)}.$

Acknowledgments I am indebted to I. Shestakov for useful discussions.

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