# Measuring M31 globular cluster ages and metallicities using both photometry
and spectroscopy

Christopher Usher,1 Nelson Caldwell,2 Ivan Cabrera-Ziri,3
1The Oskar Klein Centre, Department of Astronomy, Stockholm University,
AlbaNova, SE-106 91 Stockholm, Sweden
2Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA
02138, USA
3Astronomisches Rechen-Institut, Zentrum für Astronomie der Universität
Heidelberg, Mönchhofstraße 12-14, D-69120 Heidelberg, Germany
email<EMAIL_ADDRESS>
###### Abstract

The ages and metallicities of globular clusters play an important role not
just in testing models for their formation and evolution but in understanding
the assembly history for their host galaxies. Here we use a combination of
imaging and spectroscopy to measure the ages and metallicities of globular
clusters in M31, the closest massive galaxy to our own. We use the strength of
the near-infrared calcium triplet spectral feature to provide a relatively age
insensitive prior on the metallicity when fitting stellar population models to
the observed photometry. While the age-extinction degeneracy is an issue for
globular clusters projected onto the disc of M31, we find generally old ages
for globular clusters in the halo of M31 and in its satellite galaxy NGC 205
in line with previous studies. We measure ages for a number of outer halo
globular clusters for the first time, finding that globular clusters
associated with halo substructure extend to younger ages and higher
metallicities than those associated with the smooth halo. This is in line with
the expectation that the smooth halo was accreted earlier than the
substructured halo.

###### keywords:

globular clusters: general, galaxies: star clusters: general, galaxies:
stellar content, galaxies: evolution

##  1 Introduction

Globular clusters (GCs) are a powerful tool to study galaxy formation (see
reviews by Brodie & Strader 2006 and Forbes et al. 2018) as the properties of
GCs observed today reflect both the conditions of their formation but also the
physics of GC survival. Found in virtually all galaxies with stellar masses
above $10^{9}$ M⊙ (and most galaxies more massive than $10^{7}$ M⊙, Harris et
al. 2013; Eadie et al. 2022), they are significantly brighter than individual
stars, allowing them to be studied at much greater distances. The oldest GCs
serve as fossils of the earliest stages of galaxy formation while the
continuous formation of GCs until today (e.g. Martocchia et al., 2018; Bastian
et al., 2019) allows them to trace the full history of galaxy formation. GC
ages provide both an important test of theories of how GCs formed and evolved
but also can provide important constraints on how galaxies form and assemble.

In this paper we use the term GC for any star cluster older than $\sim 1$ Gyr
and young massive cluster (YMC) for any star cluster younger than this age.
This division corresponds to the transition for a massive cluster from being
dominated by formation physics to disruption physics and in stellar evolution
terms to when the red giant branch starts to make a significant contribution
to the luminosity of the cluster.

Two broad classes of models for GCs have been proposed. Motivated by the old
ages of Milky Way (MW) GCs (e.g. Janes & Demarque, 1983; VandenBerg et al.,
2013; Valcin et al., 2021) and the lack of easily observable star clusters
forming today in the MW massive enough to survive a Hubble time, the first
class of models (e.g. Peebles, 1984; Trenti et al., 2015) invoke special
conditions in the early Universe. In these models GCs form during or even
before the epoch of reionisation ($z\gtrsim 6$, lookback times $\gtrsim 13$
Gyr), often in their own dark matter halos. In the second class of models
(e.g. Elmegreen & Efremov, 1997; Elmegreen, 2010; Kruijssen, 2015), motivated
by observations of star clusters of GC mass or greater forming in starburst
galaxies today (e.g. Whitmore & Schweizer, 1995; Holtzman et al., 1996; Adamo
et al., 2020), GC formation is the natural outcome of intense star formation.
In the first class of models, all GCs in all galaxies should have formed in
the first Gyr of the Universe; in the second, the distribution of GC ages
should be broader and should vary with galaxy assembly history. Quantitative
models of GC formation (e.g. Li & Gnedin, 2014; Pfeffer et al., 2018;
Valenzuela et al., 2021; Rodriguez et al., 2023) make predictions for the age
distribution of GCs that can be compared to observations.

GC are also powerful probes of the formation and assembly history of galaxies.
Since a large star formation rate density is required to form a massive,
compact star cluster, the ages of GCs trace periods of intense star formation.
If the age and metallicity of a GC are known, the galaxy mass-metallicity
relation (see review by Maiolino & Mannucci, 2019) can be inverted to give the
likely mass of the galaxy the GC formed in (Kruijssen et al., 2019). With
multiple GCs, the mass assembly history of a galaxy can be constrained. The
presence of GCs with a significant range of metallicities at the same age
implies those GCs formed in different mass galaxies. The presence of at least
two branches of the MW GC age-metallicity relationship was used to argue that
the older branch was the in situ population and the younger branch was the
accreted population (Forbes & Bridges, 2010; Leaman et al., 2013). This
association was confirmed by the younger branch GCs having the same orbits as
accreted satellite galaxies (e.g. Massari et al., 2019) By folding in
constraints on the orbits of GCs, the merger tree of a galaxy can be
reconstructed in detail as was done for the MW by Kruijssen et al. (2020)
since earlier mergers and more massive mergers should deliver their GCs to
smaller galactocentric radii (Pfeffer et al., 2020).

Beyond the Local Group, GCs cannot be resolved into their constituent stars
and have to be studied using their integrated light. Despite their importance,
measuring reliable GC ages from their integrated light remains a challenge.
Optical photometry suffers from strong age-metallicity and age-extinction
degeneracies (e.g. Worthey, 1994; Anders et al., 2004; de Meulenaer et al.,
2014) in that old ages and higher metallicities both make a stellar population
redder as can the effects of interstellar extinction. While UV or NIR
photometry can be used to break this degeneracy, obtaining deep enough
photometry for a significant sample of GCs can be observationally expensive.
Spectroscopy can also provide ages but the quality of spectra required is
likewise observationally expensive for significant samples. While spectroscopy
does not suffer from the age-extinction degeneracy, it can suffer from a
degeneracy between the morphology of the horizontal branch and age; see
Cabrera-Ziri & Conroy (2022) for a detailed discussion.

Usher et al. (2019b) used optical photometry and spectroscopy of the near
infrared calcium triplet (CaT) to derive ages and metallicities for GCs in
three massive early-type galaxies from the SLUGGS survey (Brodie et al., 2014)
and the MW. Usher et al. (2019a) showed that the strength of the CaT is a
reliable and age-independent measure of metallicity for stellar populations
older than a couple Gyr. Usher et al. (2019b) used these metallicities as
priors when fitting the photometry with stellar population models to measure
ages. In line with previous work (e.g. Usher et al., 2015; Powalka et al.,
2016) that found that the GC colour-metallicity or colour-colour relationship
varies between galaxies, Usher et al. (2019b) found these four galaxies show
widely different distributions of GCs in age-metallicity space, suggesting
different assembly histories and disfavouring models where the majority of GCs
form in the early Universe.

As the nearest (780 kpc, Conn et al. 2012) massive (stellar mass $\sim
10^{11}$ M⊙, Tamm et al. 2012) galaxy to the Milky Way, M31 provides an
important test of techniques for studying GCs using their integrated light.
M31 is close enough that its GCs can be observed in ways that are impractical
for more distant galaxies - its GCs can be resolved into their constituent
stars using space based observations and high signal-to-noise ratio, high
resolution integrated spectroscopy can be obtained in a reasonable amount of
time - yet it is distant enough that is relatively straight forward to obtain
integrated photometry and spectroscopy of its GCs.

There is a long history of studying the metallicities and ages of GCs in M31
using their integrated light (e.g. van den Bergh, 1969; Burstein et al., 1984;
Puzia et al., 2005; Beasley et al., 2005; Galleti et al., 2009; Caldwell et
al., 2011; Sakari & Wallerstein, 2022). The superior spatial resolution of the
Hubble Space Telescope (HST) allows the star clusters of M31 to be resolved
into their individual stars. In most cases the resulting colour-magnitude
diagrams only reach the upper red giant branch (e.g. Holland et al., 1997;
Rich et al., 2005; Perina et al., 2009) but in the case of B379-G312, the
photometry reaches below the main sequence turnoff, allowing the age to be
measured directly (Brown et al., 2004; Ma et al., 2010). With shallower
photometry, lower limits on the age can be placed and the presence of a blue
horizontal branch can be used as evidence for an old stellar population (e.g.
Perina et al., 2011). While most M31 GCs seem to be old ($\gtrsim 10$ Gyr) and
M31 hosts a population of YMCs ($\lesssim 1$ Gyr e.g. Hodge 1979; Elson &
Walterbos 1988; Caldwell et al. 2009; Johnson et al. 2016), the existence of a
population of younger GCs (between 1 and 10 Gyr) has been debated (e.g.
Burstein et al., 2004; Beasley et al., 2005; Caldwell et al., 2011; Perina et
al., 2011).

In this paper we extend the analysis of Usher et al. (2019b) to M31 by using
literature spectra described in Section 2 to measure the metallicity of 290
GCs using the strength of the CaT spectral feature (Section 3). In Section 4
we use these metallicities as a prior when deriving their ages from their
optical photometry before summarising our work in 5.

##  2 Literature Data

###  2.1 Spectroscopy

We used spectra of the CaT region from Caldwell et al. (2009) and from Sakari
& Wallerstein (2016), Sakari et al. (2021) and Sakari & Wallerstein (2022).
Caldwell et al. (2009) observed over 500 star clusters in M31 using the
Hectospec mulitfibre spectrograph (Fabricant et al., 2005). The Hectospec
spectra cover 3700 to 9200 Å at a resolution of $\sim 5$ Å and a dispersion of
1.2 Å. We restricted ourselves to the subsample of 316 star clusters
determined by Caldwell et al. (2011) to be old. These Hectospec spectra all
lie within 20 kpc in projection of the centre of M31.

Sakari & Wallerstein (2016) observed 27 GCs at a range of galactocentric
distances using the Dual Imaging Spectrograph (DIS) on the 3.5 m telescope at
Apache Point Observatory and 5 GCs using the Sparsepak fibre unit (Bershady et
al., 2004; Bershady et al., 2005) and the Bench Spectrograph (Bershady et al.,
2008; Knezek et al., 2010) on the WIYN 3.5 m telescope. Sakari et al. (2021)
observed a single GC (G001-MII) with DIS and the Apache Point Observatory 3.5
m telescope and Sakari & Wallerstein (2022) observed 30 GCs in the outer halo
of M31 with same instrument and telescope. The DIS spectra cover 8000 to 9100
Å at resolution of $R\sim 4000$; the WIYN spectra cover 8300 to 8800 Å at
resolution of $R\sim 9000$. We do not include the spectrum of B457-G097 from
Sakari & Wallerstein (2016) in our analysis since Sakari & Wallerstein (2016)
suggest that they observed a different object given the substantial
differences in radial velocity ($\sim 300$ km s-1) and metallicity ($\sim 0.5$
dex) between their measurements and literature values; we do use the spectrum
of B457-G097 from Caldwell et al. (2011). Nor do we include the spectrum of
dTZZ-05 as suggested by Sakari & Wallerstein (2022) since the measured radial
velocity is more similar to a MW star than a M31 GC.

###  2.2 Imaging

We preferred the Sloan Digial Sky Survey (SDSS, York et al. 2000) $ugriz$
photometry of Peacock et al. (2010) when available. For more recently
discovered GCs in the SDSS footprint, we used the
SkyServer111http://skyserver.sdss.org/dr17 to download SDSS DR17 (Abdurro’uf
et al., 2022) model photometry for each GC. We applied the 0.04 mag correction
to bring the $u$-band onto the AB system. For halo GCs not covered by SDSS, we
used the PAndAS Canada France Hawaii Telescope MegaCam $gi$ photometry from
Huxor et al. (2014).

##  3 Calcium triplet based metallicities

We measure the metallicities of the M31 GCs using the strength of the calcium
triplet (CaT) rather than relying on the Caldwell et al. (2011) Lick index
based metallicities to both maintain commonality with Usher et al. (2019b) and
since the strength of the CaT shows little dependence on age (e.g. Vazdekis et
al., 2003; Usher et al., 2019a). We measure the strength of the CaT using the
technique of Foster et al. (2010) and Usher et al. (2012, 2019a). We use the
same Python code as in Usher et al. (2019a) to fit the observed spectra with a
linear combination of stellar templates (we use the same templates as Foster
et al. 2010; Usher et al. 2012, 2019a), normalise the continuum (using the
same parameters as in Usher et al. 2019a) and measure the combined CaT
strength of all three lines from the fitted templates using the same index
definition (that of Armandroff & Zinn 1988) as in our previous work. We refer
the interested reader to Usher et al. (2019a) for details of the measurement
process and a detailed discussion of potential systematics. To avoid
systematics due to poor sky subtraction or low signal-to-noise ratios (S/N),
we only measure spectra with S/N greater than 20 Å-1. For the Sakari et al.
spectra we also excluded a handful of spectra with higher S/N that showed
strong sky subtraction residuals.

Unfortunately, the measured strength of the CaT depends on the spectral
resolution or velocity dispersion, with the CaT strength being systematically
lower at lower spectral resolutions, especially at high metallicity (e.g.
Cenarro et al., 2001; Usher et al., 2019a). While the resolution of the Sakari
et al. spectra are high enough not to be significantly affected by this
effect, the Caldwell et al. (2011) spectra are low enough resolution to be. We
used high S/N spectra of the CaT region of 27 GCs from the WAGGS survey Usher
et al. (2017, 2019a) which span the range of MW GC metallicities ($-2.4<$
[Fe/H] $<-0.1$) to derive an empirical correction for the effects of velocity
dispersion on the measured strength of the CaT. We first smoothed each of the
WAGGS spectra to a range of velocity dispersions between 10 and 100 km s-1 and
measured the strength of the CaT on each smoothed spectra. For each WAGGS GC
we then fit a cubic spline between the measured velocity dispersion and the
measured CaT strength. We used these splines to estimate the CaT strength at a
velocity dispersion of 72.3 km s-1, the median of the velocity dispersions
fitted to the Caldwell et al. (2011) spectra, for each of the WAGGS GCs. We
then fit a cubic relationship between the CaT strength measured on the spectra
broadened to 72.3 km s-1 and the CaT strength measured on the original,
unbroadened spectra. This broadening is consistent with the line spread
function of Hectospec ($\sigma=78\pm 4$ km s-1) found by Fabricant et al.
(2013) at 8600 Å once the line spread function ($\sigma=19$ km s-1) of the
DEIMOS template spectra is accounted for. We used this relationship to correct
each of the CaT strengths measured from the Caldwell spectra. To estimate the
systematic uncertainty introduced by this correction, we repeated the
correction at 62.7 km s-1 and 86.7 km s-1, the 2.3 and 97.7 percentiles of the
velocity dispersion distribution measured from the Caldwell et al. spectra. We
used these respectively to correct the upper and lower uncertainties on the
measured CaT strength. In terms of metallicity the strength of the correction
varies from 0.09 dex at [Fe/H] $=-1$ to 0.42 dex at [Fe/H] $=0$ as seen in
Figure 1.

We used the empirical correction of Usher et al. (2019a, their equation 1) to
translate our CaT values into metallicities. The Usher et al. (2019a) relation
is based on MW and MW satellite GCs. Since CaT is sensitive to the
[$\alpha$/Fe] abundance ratio (Brodie et al., 2012; Sakari & Wallerstein,
2016; Usher et al., 2019a), if the [$\alpha$/Fe]-[Fe/H] relation of M31 GCs is
different, our [Fe/H] measurements will be biased. In Figure 1, we show a
comparison of our CaT based [Fe/H] with the [Fe/H] measured by Caldwell et al.
(2011) using Lick Fe indices and an empirical relationship before and after
performing the spectral resolution correction. The root mean squared (RMS)
difference between the two metallicities is 0.25 dex, which is in agreement
with the uncertainties. While the correction for spectral resolution improves
the agreement at the highest metallicities, small ($<0.2$ dex) systematic
differences remain, with the CaT metallicities being systematically higher
near [Fe/H] $\sim-1.5$ and systematically lower at [Fe/H] $\sim-0.5$. We note
that both the Caldwell et al. (2011) metallicities and our CaT metallicities
are based on simple empirical relationships between the strength of spectral
indices and the metallicities of MW GCs. Some level of systematic difference
is unsurprising, given that the Fe Lick indices used by Caldwell et al. (2011)
and the CaT used in this work show difference dependencies on [$\alpha$/Fe]
and the simple empirical relations used by Caldwell et al. (2011, bilinear)
and Usher et al. (2019a, linear).

In Figure 2 we show a comparison between the CaT [Fe/H] values measured from
the Sakari & Wallerstein (2016) and Caldwell et al. (2011) spectra. The
agreement between the two is excellent with a RMS difference of 0.11 dex in
agreement with the statistical and expected systematic uncertainties. In
Figure 3 we compare our CaT metallicities with those from Sakari & Wallerstein
(2022). Again we see good agreement at most metallicities although the Sakari
& Wallerstein (2022) metallicities are systematically higher at metallicities
below [Fe/H] $=-2$. Our measurement for EXT8 ([Fe/H] $=-2.71\pm 0.07$) is
closer to the value from high resolution spectroscopy (Larsen et al. 2020,
[Fe/H] $=-2.91\pm 0.04$) than the CaT based value from Sakari & Wallerstein
(2022) ([Fe/H] $=-2.27\pm 0.10$). In Figure 4 we compare our metallicities
with those based on analysis of high resolution integrated light. We see good
agreement with the optical studies of Colucci et al. (2014), Sakari et al.
(2015) and Larsen et al. (2022) as well as the near-infrared study of Sakari
et al. (2016) using either the Caldwell et al. or Sakari et al. spectra.

Figure 1: Comparison of the CaT [Fe/H] values measured from the Caldwell et
al. (2011) spectra and the [Fe/H] values measured by Caldwell et al. (2011)
using Lick Fe indices using the same spectra. The top panel shows the [Fe/H]
calculated from the CaT values uncorrected for the effects of spectral
resolution; the bottom panel shows the [Fe/H] calculated from the corrected
CaT values. The solid line in the upper panel is the spectral resolution
correction we apply to the CaT values. The median uncertainties are shown in
the lower panel. The dashed line is the one-to-one and the dotted lines show
$\pm 0.1$ dex. The metallicities measured using the two different methods
agree within 0.25 dex. Figure 2: Comparison of the CaT [Fe/H] values measured
in this work from the Sakari & Wallerstein (2016) spectra and the Caldwell et
al. (2011) spectra. The dashed line is the one-to-one and the dotted lines
show $\pm 0.1$ dex. The metallicities measured using the two different sets of
spectra agree within 0.1 dex. Figure 3: Comparison of the [Fe/H] values
measured in this work and those measured by Sakari & Wallerstein (2022) both
using the strength of the CaT and the spectra from Sakari & Wallerstein
(2022). The dashed line is the one-to-one and the dotted lines show $\pm 0.1$
dex. In general there is good agreement between the two studies except below
[Fe/H] $=-2$ where the Sakari & Wallerstein (2022) metallicities are
systematically higher. Figure 4: Comparison of the CaT [Fe/H] values with
literature [Fe/H]. The left panel shows a comparison with the Caldwell et al.
spectra and the right with the Sakari et al. spectra. The dashed line is the
one-to-one and the dotted lines show $\pm 0.1$ dex. There is generally good
agreement between our metallicitiy measurements and those from the literature
high resolution studies.

##  4 Age measurements

To determine ages we used the Monte Carlo Markov Chain (MCMC) code first
presented in Usher et al. (2019b) which we now name mcmame (Monte Carlo
Metallicity, Age, Mass and Extiction). In this analysis we sample age,
metallicity, mass and reddening posteriors of a grid of stellar population
models subject to the constraints provided by the photometry and optionally
the CaT metallicity. As in Usher et al. (2019b) we use emcee (Foreman-Mackey
et al., 2013) with a parallel-tempered ensemble sampler (Vousden et al., 2016)
with 8 temperatures, 64 walkers, 200 burn-in steps and 2560 production steps.
We use the same stellar populations models (FSPS Conroy et al. 2009; Conroy &
Gunn 2010 calculated with the MIST isochrones Choi et al. 2016 and the MILES
spectral library Sánchez-Blázquez et al. 2006), and mass priors (flat in log
mass), but use slightly different age priors (flat between 1 Myr and 15 Gyr)
and metallicity priors (flat between [Fe/H] $=-3.0$ and $+0.5$) than in Usher
et al. (2019b). Unlike in Usher et al. (2019b) where the reddening vector from
Schlafly & Finkbeiner (2011) was assumed for all ages and metallicities, we
used FSPS to calculate the age and metallicity dependent reddening vector
using the Cardelli et al. (1989) MW extinction curve. At low extinction values
this assumption is valid but for larger values it breaks down since the effect
of extinction depends on the shape of the spectral energy distribution within
a filter passband, with a population with a bluer spectrum being more effected
than a redder one. For reddening we need to adopt different priors than in
Usher et al. (2019b) given that M31 has substantial internal extinction,
unlike the massive early-type galaxies studied in Usher et al. (2019b).

As a first order approximation, we can think of most of the internal
extinction in M31 coming from a thin disc of gas and dust. The majority of old
and intermediate age GCs should be either in front of or behind this disc as
star clusters would not survive long orbiting in the same plane as massive
molecular clouds (Elmegreen, 2010; Kruijssen, 2015). Thus the prior on the
extinction should be bimodal, with one peak corresponding to the foreground
extinction and one peak corresponding to the extinction of the foreground plus
the disc of M31. In principle a GC should have equal probability of being in
front or behind of the disc. However, in a magnitude limited sample, GCs on
the far side of the disc will be fainter and more likely to fall out of the
selection.

The extinction can be modelled through various methods including from far
infrared and sub-millimetre observations (e.g. Schlegel et al., 1998;
Whitworth et al., 2019) and by modelling the colours of stars (e.g. Dalcanton
et al., 2015). Unfortunately, the distribution of gas and dust is filamentary,
with significant changes in the extinction value on comparable scales to the
radius of a GC (e.g. Bonatto et al., 2013; Saracino et al., 2019; Bonatto &
Chies-Santos, 2020). While for M31, extinction can be modelled on the scales
of $\sim 30$ pc (e.g. Whitworth et al., 2019; Dalcanton et al., 2015), for
more distant galaxies it can only be modelled on larger scales. To mimic the
low spatial resolution available for more distant galaxies, we used the
Schlegel et al. (1998) extinction maps which have a resolution 6.1 arcmin
($\sim 1.4$ kpc at the distance of M31). We modelled the contribution of the
disc of M31 to the extinction as a Gaussian with a mean provided by the value
of the Schlegel map at that location and a standard deviation equal to 50 % of
the mean. We used the median extinction from the Schlegel map of GCs outside
of the disc of M31 (0.08 mag) as the mean value of the foreground contribution
and assumed a dispersion of 0.03 mag on the foreground. In our bimodal
extinction prior we assigned equal weight to the foreground and M31 disc
components. In all cases we enforced a further constraint that the extinction
must be positive.

An example of the posterior distribution for B379-G312 is plotted in Figure 5.
This presents the best case for sampling the posterior distribution as
B379-G312 is bright, metal rich and outside of the projected disc of M31. A
more typical case is show in Figure 6, a low metallicity ([Fe/H] $=-1.60$) GC
with an apparent $g$-band magnitude equal to the median of our sample
($g=17.25$). In Table 1 we give our age, metallicity, mass and extinction
posteriors, the full version of which is provided online as Supporting
Information. Our median metallicity uncertainty is 0.10 dex and our median age
uncertainty is 2.4 Gyr.

Figure 5: Posterior distributions of metallicity, age, mass and reddenings for the M31 GC B379-G312 based on SDSS $ugriz$ photometry and a CaT based metallicity prior. In the top right, we plot the spectral energy distributions calculated by fsps for 256 points drawn at random from the posterior distribution in grey, the median colours of the posterior distribution as blue circles and the observed photometry as orange error bars. We note that B379-G312 presents the best case for fitting its photometry, as it is bright, metal rich and outside of the projected disc of M31. Figure 6: Posterior distributions of metallicity, age, mass and reddenings for the M31 GC B361-G255 based on SDSS $ugriz$ photometry and a CaT based metallicity prior. In the top right, we plot the spectral energy distributions calculated by fsps for 256 points drawn at random from the posterior distribution in grey, the median colours of the posterior distribution as blue circles and the observed photometry as orange error bars. Unlike B379-G312 plotted in Figure 5, B361-G255 presents a more typical case for fitting its photometry, due to its median brightness and metal poor nature. ID | RA | Dec | [Fe/H]${}_{\text{CaT}}$ | $A_{V\text{prior}}$ | Mass | Age | [Fe/H] | $A_{V}$ | Photo | Spectra
---|---|---|---|---|---|---|---|---|---|---
| [deg] | [deg] | [dex] | [mag] | [$\log$ M⊙] | [Gyr] | [dex] | [mag] |  |
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11)
B001-G039 | 9.9626 | 40.9696 | $-0.53\pm 0.15$ | $0.40\pm 0.20$ | $5.73_{-0.06}^{+0.04}$ | $11.4_{-3.1}^{+2.5}$ | $-0.64_{-0.11}^{+0.09}$ | $0.89_{-0.09}^{+0.10}$ | P10 | C
B002-G043 | 10.0107 | 41.1982 | $-2.16\pm 0.09$ | $0.27\pm 0.14$ | $5.10_{-0.09}^{+0.04}$ | $10.9_{-2.7}^{+2.7}$ | $-2.15_{-0.09}^{+0.09}$ | $0.29_{-0.04}^{+0.05}$ | P10 | C
B003-G045 | 10.0392 | 41.1849 | $-1.47\pm 0.11$ | $0.29\pm 0.15$ | $4.93_{-0.06}^{+0.10}$ | $4.3_{-1.2}^{+3.1}$ | $-1.47_{-0.09}^{+0.10}$ | $0.58_{-0.13}^{+0.11}$ | P10 | C
B004-G050 | 10.0747 | 41.3779 | $-0.57\pm 0.16$ | $0.24\pm 0.12$ | $5.46_{-0.03}^{+0.03}$ | $12.5_{-2.1}^{+1.7}$ | $-0.76_{-0.08}^{+0.07}$ | $0.25_{-0.07}^{+0.07}$ | P10 | C
B005-G052 | 10.0847 | 40.7329 | $-0.68\pm 0.16$ | $1.03\pm 0.51$ | $6.06_{-0.08}^{+0.03}$ | $11.9_{-4.1}^{+2.2}$ | $-0.72_{-0.12}^{+0.11}$ | $0.44_{-0.11}^{+0.15}$ | P10 | C
… | … | … | … | … | … | … | … | … | … | …
Table 1: Age and metallicity posteriors. Values provided median value of the
posterior and the 68 % confidence interval. Column (1): GC name. Column (2):
Right ascension in deg. Column (3): Declination in deg. Column (4):
Metallicity prior from the strength of the CaT in dex. Column (5): $V$-band
extinction prior from Schlegel et al. (1998) in magnitudes. Column (6): Mass
posterior in log solar masses. Column (7): Age posterior in Gyr. Column (8):
Metallicity posterior in dex. Column (9): $V$-band extinction posterior in
magnitudes. Column (10): Photometry source. P10: Peacock et al. (2010), H14:
Huxor et al. (2014), SDSS: SDSS DR17 (Abdurro’uf et al., 2022). Column (11):
Spectra source. C: Caldwell et al. (2009), S: Sakari & Wallerstein (2016);
Sakari et al. (2021); Sakari & Wallerstein (2022). The full version of this
table is provided in machine readable form in the online Supporting
Information.  Figure 7: Medians of the age posteriors versus galactocentric
distance colour coded by the extinction prior from Schlegel et al. (1998) for
GCs with spectra from Caldwell et al. (2011). GCs with young age posteriors
are found at small galactocentric distances and high possible extinction
values.

A number of GCs from the Caldwell sample show surprisingly young age
posteriors, given that the Caldwell sample of spectra were selected to be
older than a couple Gyr on the basis of their spectra between 3750 and 5400 Å
(see section 5 of Caldwell et al. 2009). The lack of young GC in both of the
Caldwell and Sakari samples is supported by the lack of GC spectra with
significant Paschen absorption which is seen in stellar populations younger
than $\sim 1$ Gyr (e.g. Usher et al., 2019a). In the case of some GCs such as
M009, the age is poorly constrained, with the posterior distribution spanning
a wide range of ages ($2.9_{-2.7}^{+8.8}$ Gyr in the case of M009 using the 16
to 84 % percentile confidence interval.). However, a number of GCs are
inconsistent with being as old as 5 Gyr at a 95 % level. As can be seen in
Figure 7, these GCs with apparently young ages are found at small
galactocentric radii and have large priors on their extinctions. Thus these
GCs are found in the crowded centre of M31, where the effects of the age-
extinction degeneracy are strongest and both photometry and spectroscopy are
most likely to suffer from systematic uncertainty.

###  4.1 Comparison with previous measurements

A wealth of studies have examined the ages of GCs in M31 using a range of
techniques. In the first class of studies, ages were constrained using
resolved star colour-magnitude diagrams. Only B379-G312 has HST imaging deep
enough to reach the main sequence turn-off and measure the age directly. For
this GC (see Figure 5) we find good agreement between our age measurement
($10.1_{-2.2}^{+2.3}$ Gyr) and the colour magnitude diagram based literature
values ($10.0_{-1.0}^{+2.5}$ Gyr Brown et al. 2004, $11.0\pm 1.5$ Ma et al.
2010 Gyr). Shallower HST imaging can be used to place lower limits on the ages
of M31 GCs from the non-detection of a main sequence or from the presence of
blue horizontal branch stars (e.g. Perina et al., 2011). Using the MIST
isochrones (Choi et al., 2016) to estimate the brightness of the subgiant
branch as a function of age and metallicity, we find no inconsistencies
between the ages and metallicities we measure and the colour magnitude
diagrams in Perina et al. (2012) for the 34 GCs in common. Of the 12 GCs from
Perina et al. (2012) with well measured HB properties, all GCs with blue HBs
are older than 11.6 Gyr except for B350-G162 ($7.6_{-2.0}^{+4.0}$ Gyr) which
is consistent with being old. Published colour-magnitude diagrams for PA06,
PA53, PA54 and PA56 (Sakari et al., 2015) and EXT8 (Larsen et al., 2021) all
reveal blue horizontal branches, consistent with the old ages and low
metallicities we measure.

Most studies of the ages of M31 GC have relied on integrated light. Some
studies have measured line indices and then fit stellar population models to
the indices (e.g. Puzia et al., 2005; Beasley et al., 2005; Caldwell et al.,
2011). Others have directly fit stellar population models to spectra (e.g.
Cezario et al., 2013; Colucci et al., 2014; Sakari et al., 2015; Chen et al.,
2016), to photometry (e.g. Ma et al., 2009; Fan et al., 2010) or a combination
there of (Wang et al., 2021).

Figure 8: Comparison of our MCMC ages with ages from Beasley et al. (2005),
Fan et al. (2010), Caldwell et al. (2011), Cezario et al. (2013), Colucci et
al. (2014), Sakari et al. (2015), Chen et al. (2016), Wang et al. (2021) and
Cabrera-Ziri et al. (in prep.). For each study we show all GCs in common with
high extinction priors ($E(B-V)>0.15$ from Schlegel et al. 1998) as orange
points and the GCs with low extinction priors ($E(B-V)<0.15$) as blue circles
with error bars. When literature age measurements are larger than 16 Gyr we
have plotted them at an age 16 Gyr. The level of agreement varies between
studies but in general we see good agreement.

In Figure 8 we show a comparison between our ages and those from literature.
In general we see better agreement for GCs with low extinction priors
($E(B-V)<0.15$ from Schlegel et al. 1998) than for GCs with higher extinction
priors which are projected onto the disc of M31. This is unsurprising given
the strong degeneracy between age and extinction for optical colours. While
this degeneracy is usually considered in studies of young star clusters, it is
often ignored in studies of old globular clusters where the observed
extinction is assumed only to be due to the Milky Way foreground. While this
assumption is acceptable in galaxy halos and in quiescent galaxies, it is not
valid in actively star forming galaxies.

We compare our ages to ages measured by fitting stellar models to spectral
indices (Beasley et al., 2005; Caldwell et al., 2011), by directly fitting
stellar population models to spectra (Cezario et al., 2013; Colucci et al.,
2014; Chen et al., 2016) and by fitting stellar population models to
photometry (Fan et al., 2010). The level of agreement varies between studies,
with the best agreement with Beasley et al. (2005), Caldwell et al. (2011) and
Colucci et al. (2014) and the worst agreement with Fan et al. (2010) and Wang
et al. (2021). Beasley et al. (2005) used both the Bruzual & Charlot (2003)
and the Thomas et al. (2003) models to convert Lick index measurements to ages
and metallicities. We find better agreement between our measurements and the
Thomas et al. (2003) based values of Beasley et al. (2005); we display the
Thomas et al. (2003) based ages in Figure 8. Fan et al. (2010) fit $UBVRIJHK$
photometry with the Bruzual & Charlot (2003) models to derive ages and
metallicities. A comparison of the Fan et al. (2010) metallicity measurements
and our own reveals that they typically over estimate metallicities which due
to the age-metallicity degeneracy causes them to underestimate the ages.
Caldwell et al. (2011) used EZAges (Graves & Schiavon, 2008) to measure ages
from Lick indices. Due to the effects of the horizontal branch at lower
metallicities, Caldwell et al. (2011) only provide ages for GCs more metal
rich than [Fe/H] $=-0.95$. Cezario et al. (2013) used full spectrum fitting
and the Vazdekis et al. (2010) stellar population models to measure ages.
Colucci et al. (2014) and Sakari et al. (2015) fitted their own stellar
population models to high resolution spectra to measure ages, metallicities
and abundances. Chen et al. (2016) measured ages using full spectral fitting
both with the PEGASE–HR (Le Borgne et al., 2004) and the MILES (Vazdekis et
al., 2010) stellar population models. The majority of PEGASE-HR based ages are
either significantly younger or older than our age measurements while the
MILES based ages show better agreement; we display the MILES based ages in
Figure 8. Wang et al. (2021) used a random forest classifier to derive ages
and metallicities from Lick indices and $ugriz$ photometry. They find ages of
6 to 8 Gyr for virtually all M31 GCs, inconsistent with our measurements and
most other literature values. Cabrera-Ziri et al. (in prep.) used alf (Conroy
& van Dokkum, 2012; Choi et al., 2014; Conroy et al., 2014; Conroy et al.,
2018; Cabrera-Ziri & Conroy, 2022) to fit spectra to measure ages,
metallicities and abundances while accounting for the possibility of a hot
horizontal branch. While we see agreement for many GCs, most of Cabrera-Ziri
et al. (in prep.) age measurements are a couple Gyr older than ours.

Population | Age | [Fe/H] | N
---|---|---|---
| [Gyr] | [dex] |
(1) | (2) | (3) | (4)
Combined | $10.7_{-0.2}^{+0.2}$ | $-1.00_{-0.01}^{+0.01}$ | 290
Projected Disc | $10.3_{-0.3}^{+0.3}$ | $-0.84_{-0.02}^{+0.02}$ | 178
Inner Halo | $10.9_{-0.4}^{+0.4}$ | $-1.22_{-0.03}^{+0.03}$ | 79
Outer Halo | $12.5_{-0.5}^{+0.5}$ | $-1.77_{-0.03}^{+0.03}$ | 28
NGC 205 | $10.8_{-1.1}^{+1.2}$ | $-1.43_{-0.06}^{+0.07}$ | 5
Smooth Halo | $12.4_{-0.8}^{+0.7}$ | $-1.8_{-0.04}^{+0.04}$ | 16
Ambiguous | $11.2_{-1.1}^{+1.2}$ | $-1.57_{-0.07}^{+0.06}$ | 5
Substructure | $11.3_{-1.0}^{+1.1}$ | $-1.98_{-0.08}^{+0.09}$ | 7
Ambiguous + Substructure | $11.3_{-0.7}^{+0.8}$ | $-1.63_{-0.07}^{+0.06}$ | 12
Milky Way | $11.7_{-0.6}^{+0.5}$ | $-1.20_{-0.04}^{+0.04}$ | 65
NGC 1407 | $11.9_{-0.3}^{+0.3}$ | $-0.60_{-0.02}^{+0.02}$ | 213
NGC 3115 | $10.9_{-0.4}^{+0.4}$ | $-0.72_{-0.03}^{+0.03}$ | 116
NGC 3377 | $8.0_{-0.4}^{+0.5}$ | $-0.80_{-0.06}^{+0.06}$ | 82
Table 2: Population age and metallicity medians from this study and from Usher
et al. (2019b). (1) Population name. (2) Median age in Gyr. (3) Median
metallicity in dex. (4) Number of GCs.

###  4.2 Age-metallicity distributions

In Figure 9 we show the age-metallicity distributions of GCs associated with
different components of M31 while in Table 2 we give the median ages and
metallicities for each of the components. For GCs projected onto the disc of
M31, which are based on the Caldwell et al. (2011) spectra and Peacock et al.
(2010) photometry and have relatively broad extinction priors, we see a wide
range of ages and metallicities. The wide range of ages is likely due to the
age-extinction degeneracy. For GCs in the inner halo of M31 - not projected on
to the disc on M31 but within 20 kpc in projection - which are based on
Caldwell et al. (2011) spectra and Peacock et al. (2010) photometry and have
relatively narrow extinction priors - we see generally old ages. The outer
halo, beyond a distance of 25 kpc in projection and based on the Sakari et al.
spectra and a mix of PAndAS and SDSS photometry, also shows old ages. Finally,
the GCs associated with the dwarf elliptical NGC 205 show low metallicities.
Four of the NGC 205 GCs show old ages but the fifth, B330-G056, shows evidence
for a younger age ($3.8_{-2.6}^{+3.9}$ Gyr).

In line with the previously observed (Galleti et al., 2009; Caldwell et al.,
2011; Sakari & Wallerstein, 2022) metallicity gradient for M31 GCs, we see the
range of metallicities shift to lower metallicities and the fraction of metal
poor GCs increase as we go out from the centre of M31. Using the commonly used
GMM code of Muratov & Gnedin (2010), the projected disc and inner halo
populations as well as the combined sample all show significant ($p\leq 0.01$
as well a negative kurtosis and well-separated peaks) evidence for bimodality
while the outer halo sample is too small to reliably use GMM. Pfeffer et al.
(2023) discusses the emergence of GC metallicity bimodality in the context of
GC formation and destruction, showing that in high mass galaxies like M31, the
relative lack of GCs at intermediate metallicities is likely due to these GCs
being preferentially disrupted rather than a deficit of cluster formation.

We also see a positive age gradient, with GCs projected onto the disc having
younger ages and the GCs in the outer halo having older ages although it is
unclear whether this is driven by the larger age uncertainties at smaller
radii. The less constraining the observations are, the more the age posterior
will resemble the prior, with our uniform prior having a median age of 7.5
Gyr. Caldwell & Romanowsky (2016) divided M31’s GCs up by metallicity into
three components. The most metal poor ([Fe/H] $<-1.5$) component and the
intermediate metallicity ( $-1.5<$ [Fe/H] $<-0.4$) component show consistent
ages ($10.9_{-0.3}^{+0.4}$ Gyr and $10.4_{-0.3}^{+0.3}$ Gyr respectively)
while the most metal rich component ([Fe/H] $>-0.4$) shows a slightly younger
median age ($8.3_{-1.6}^{+1.5}$ Gyr). We note that all but three of the 26
metal rich GCs have large reddening priors so we can not rule out that the
younger age median for the most metal rich GCs is due to their larger
uncertainties.

In the projected disc sample we see a lack of GCs with metallicities [Fe/H]
$\sim-1.2$ with median age posteriors older than 11 Gyr which are present at
higher or lower metallicities. It is unclear if this is driven by the larger
age uncertainties in this metallicity range compared to lower metallicity GCs
or a genuine lack of relatively old GCs in this metallicity range. The most
metal poor and metal rich GCs show the oldest ages, with some age posteriors
having medians older than the age of the Universe. Under the naive expectation
that the metallicity of a galaxy increase over time, the most metal poor GCs
should be the oldest. However, it is also plausible that such metal poor GCs
formed later, either in a less massive galaxy or less likely, in a galaxy that
has recently accreated a large amount of near pristine gas. We also note that
the three most metal poor GCs in M31, all found in the outer halo, lie in a
regime - more metal poor than any GCs in the Milky Way and similar to the
faintest ultra faint galaxies (e.g. Simon, 2019) \- where stellar evolution
models and stellar population models are not as well tested. The old ages of
the most metal rich GCs are more puzzling as the expectation is more metal
rich GC should be younger than GCs that formed in the same progenitor.
Although many of the lower metallicity GCs likely formed later in lower mass
progenitors, it is unlikely that no lower metallicity GCs formed in and
survived from the progenitor(s) of the metal rich GCs. We note than many of
these old GCs have large extinction priors and lie at small galactocentric
radii where the systematic and statistical uncertainty of both the spectra and
photometry is higher (see Figure 7).

If we had systematically underestimated the CaT metallicity priors we would
systematically overestimate the ages. Due to the low resolution of the spectra
used, the CaT metallicity prior is less reliable at these high metallicities.
However, Usher et al. (2019b) also saw a trend of increasing age with
metallicity for the most metal rich GCs in NGC 1407 and NGC 3115 (see Figure
11 below) suggesting this issue is not due to the lower resolution spectra.
Since the CaT metallicity is sensitive to the $\alpha$-element abundance in
general and the Ca abundance in particular, differences in the [Ca/Fe] ratio
of high metallicity M31 GCs to the MW GCs used to calibrate the CaT-[Fe/H]
relation could introduce a bias. However, measurements of the
$\alpha$-elements in M31 GCs (e.g. Colucci et al., 2014; Sakari et al., 2016;
Larsen et al., 2022) find similar ratios to MW GCs although only a handful of
near solar metallicity GCs have been studied in M31.

Additionally, if the stellar population models we utilise predict too blue of
a spectral energy distribution for a given age and metallicity, the resulting
age constraint will be biased to older ages. More subtly, the combination of
incorrectly predicted colours and poorly constrained reddening could lead to
an older age and lower reddening being a better fit than the correct age and
reddening. Again, differences in chemistry between the models and the data
could produce colour differences. The models we utilise have a scale solar
abundance pattern while GCs generally have $\alpha$-element enhanced
abundances and typical show internal spreads in the abundances of light
elements, with some stars showing enhanced He, N and Na and depleted C and O
(see review by Bastian & Lardo, 2018). Lee et al. (2009); Coelho et al. (2007)
and Choi et al. (2019) all find an $\alpha$-element enhanced population is
bluer in shorter wavelength colours such as $(u-g)$ and $(g-r)$ and redder in
longer wavelength colours like $(r-i)$ and $(i-z)$. Using models also based on
the MIST isochrones and MILES stellar library Choi et al. (2019) find that
$\alpha$-element enhanced models are a better fit to the colours of massive
early-type galaxies than scaled solar models while noting the abundances of C
and N also significantly affect the colours of an old, metal rich population.
Chantereau et al. (2018) modelled the effects of He and of enhanced N and Na
with depleted C and O populations. He enhanced populations are bluer in all
optical colours (see also Chung et al. 2017) while N and Na enhanced, C and O
depleted populations have the same $(u-g)$ colours as solar abundance models
but bluer $(g-r)$, $(r-i)$ and $(i-z)$ We note the effects of abundance
variations on colour are generally larger in older and higher metallicity
populations since the effects are stronger in cooler stars (e.g. Lee et al.,
2009). Further work is required to understand what effect different abundance
ratios have on age estimates.

We note that there are larger systematic uncertainties in comparing ages at
different metallicities than comparing ages at the same metallicity. Given the
larger systematic uncertainties at the extremes of the metallicity
distribution and the larger statistical uncertainties for GCs in the projected
disc, we do not attach any significance to the variation of the age of the
oldest GCs with metallicity.

Figure 9: Age-metallicity posterior distributions for M31 GCs based on
photometry assuming a bimodal extinction prior. The 2D histograms show the sum
of the individual age-metallicity posterior distributions while the white
points and the 1D histograms show the medians of the age and metallicity
posteriors for each GC. In the top left panel we show GCs projected on to the
disc of M31, in the top right GCs in the inner halo of M31 (projected
galactocentric distances $<20$ kpc), in the bottom right GCs in the outer halo
of M31 (projected galactocentric distances $>25$ kpc) and the bottom left GCs
associated with the M31 satelite galaxy NGC 205. For the outer halo subsample
the ages and metallicities are on the Sakari et al. spectra and a mix of SDSS
$ugriz$ and PAndAS $gi$ photometry; for the other subsamples they are based on
the Caldwell et al. (2011) spectra and the Peacock et al. (2010) SDSS $ugriz$
photometry. Figure 10: Age-metallicity distribution of outer halo GCs. In the
left panel we show the GCs associated with the smooth halo by Mackey et al.
(2019) as blue circles and GCs associated with substructure as orange squares.
GCs classified as ambiguous by Mackey et al. (2019) are denoted by open
squares. The youngest and most metal rich GCs are more likely to be associated
with substructure than the smooth halo, inline with the predictions of Hughes
et al. (2019). In the right panel we show GCs associated with the smooth halo
as black points while the GCs associated with substructure are shown as
coloured polygons. The colour and shape of these GCs corresponds to their
classification by Mackey et al. (2019) with GCs with ambiguous classifications
having open symbols. Unfortunately there are too few GCs associated with each
substructure to robustly look for different age-metallicity relationships.

For GCs in the outer halo, we used the association of Mackey et al. (2019)
between halo substructure and GCs to compare the GCs associated with the
smooth halo with those associated with substructure. As seen in the left panel
of Figure 10, although there are old and extremely metal poor GCs associated
with both the smooth halo and substructure, GCs associated with halo
substructure extend to younger ages than those associated with the smooth
halo. The situation with metallicity is more complex. Although the
substructure and smooth halo GCs cover a similar range of metallicities and
have similar medians, two of the seven GCs that are securely identified with
substructure and one of the five ambiguous GCs are more metal rich than [Fe/H]
$=-1.25$ while only one of the 16 smooth halo GCs is above this metallicity.
In Table 2 we give the medians of each of the populations. The most metal rich
smooth halo GC in our sample, PA-17, is older than the substructure GCs of
similar metallicity. That GCs with substructure and ambiguous classifications
have higher metallicities than the those classified as part of the smooth halo
has been previously noted by Sakari & Wallerstein (2022). The younger ages and
higher metallicities of some GCs associated with substructure supports the
predictions of Hughes et al. (2019) where the substructured halo has been more
recently accreted than the smooth halo. In the right panel of Figure 10 we
show the association of the substructure GCs with each halo feature;
unfortunately there are too few GCs associated with each substructure to
robustly study the age-metallicity relationship of each substructure.

Figure 11: GC age-metallicity posterior distributions for the galaxies studied
in this work (M31 and NGC 205) and the galaxies from Usher et al. (2019b). The
galaxies are ordered by stellar mass using masses from Forbes et al. (2017)
for the three SLUGGS galaxies, from Conn et al. (2012) for M31, from Bland-
Hawthorn & Gerhard (2016) for the MW and from McConnachie (2012) for the Local
Group dwarf galaxies. As in Figure 9 the 2D histograms show the sum of the
individual age-metallicity posterior distributions while the white points and
the 1D histograms show the medians of the age and metallicity posteriors for
each GC.

We can compare the age-metallicity distribution of M31 GCs with those of the
galaxies studied by Usher et al. (2019b) in Figure 11. We note that although
the technique used by Usher et al. (2019b) is the same as in this paper, the
GC selection and data quality differs between M31 and the Usher et al. (2019b)
galaxies. As noted by Usher et al. (2019a), the CaT is not a reliable
metallicity indicator at ages younger than a couple Gyr when stages of stellar
evolution other than the red giant branch or the red clump dominate the light
in the CaT spectral region. However, as noted by Usher et al. (2019b) due to
weaker sensitivity of colour to metallicity at younger ages, our method still
provides young ages for these star clusters. For the giant galaxies, M31’s GCs
are slightly younger than those of NGC 1407 and the Milky Way, similar in age
to those of NGC 3115 and older than those of NGC 3377 while being more metal
poor than those of the three SLUGGS galaxies and more metal rich than those of
the Milky Way. For the dwarf galaxies, NGC 205’s GCs show similar ages and
metallicities to the old and intermediate age GCs in the similar mass SMC.

Although we defer a detailed and quantitative discussion of M31’s formation
and assembly history to future work, we can discuss M31 GC system in a
qualitative manner similar to Usher et al. (2019b). That most of M31’s GC are
old suggests that M31 formed most of its stellar mass early, inline with the
observed star formation history of M31’s disc and spheroid (e.g. Brown et al.,
2008; Bernard et al., 2015; Williams et al., 2017). The lack of clear branches
in age-metallicity space does not have a strong implications on the assembly
history of M31 given our large age uncertainties. As can be seen in Figure 11
with similar quality data we do not see the bifurcation of the MW’s GCs into
the older in situ branch and a younger accreted branch that more precise ages
relieves (Forbes & Bridges, 2010; Leaman et al., 2013). Comparing M31 to the
galaxies studied in Usher et al. (2019b) the similarities between the age and
metallicity distributions of M31, NGC 1407 and the MW suggests some
commonality in the assembly history of these galaxies compared to the extended
formation of NGC 3377 and the bimodality of NGC 3115.

##  5 Summary

Using the strength of the near infrared CaT spectral feature we have measured
the metallicity of a large number of GCs in M31. In line with previous work
(e.g. Sakari & Wallerstein, 2016; Usher et al., 2019a), we find that the CaT
is a reliable measure of metallicity although its reliability declines at high
metallicity and at the lower spectral resolution of the Caldwell et al. (2009)
Hectospec spectra. We used these metallicities as priors when fitting optical
photometry with stellar population models to measure the ages, metallicities
and masses of the GCs. Due to the strong degeneracy between extinction and
age, the ages of GCs projected on to the disc of M31 are less reliable than
those in the halo of M31. We find good agreement between our age measurements
with most but not all literature studies. Most of the GCs projected on to the
disc and virtually all halo GCs are old (consistent with ages $>10.5$ Gyr,
formation redshifts $z>2$). The distribution of GC ages and metallicities is
similar but not identical to galaxies with a similar stellar mass. The old
ages of the majority of M31’s GCs suggest that M31 formed much of its stellar
mass early, inline with observations of the field star formation history. In
the outer halo we find the most metal rich and youngest GCs are more likely to
be associated with substructure rather than the smooth halo, in line with the
predictions of Hughes et al. (2019).

The technique used in this paper of combining photometry with a prior from
spectroscopy is likely not the best way of measuring the ages of M31 GCs where
high S/N optical spectra are available (see Cabrera-Ziri et al. in prep.).
Instead it serves as a test of a technique suitable for more distant galaxies.
Upcoming highly multiplexed red and near infrared spectrographs, such as MOONS
(Cirasuolo et al., 2014; Taylor et al., 2018) on the Very Large Telescope,
will allow large numbers of extragalactic GCs to be observed efficiently. Like
other techniques that rely on photometry, our method suffers from the age-
extinction degeneracy and will perform worse when there is not a strong prior
on extinction, such as in or near areas of active star formation. In addition,
spectra, even when only available in the red or near infrared, do provide some
age information. This technique should be refined by either fitting the
spectra with stellar population models and using that age-metallicity
posterior as a prior when fitting the photometry or by simultaneously fitting
the photometry and spectroscopy with a stellar population model.

## Acknowledgements

We thank the anonymous referee for their helpful comments which improved the
manuscript. We also wish to thank Joel Pfeffer for useful discussions. CU
acknowledges the support of the Swedish Research Council, Vetenskapsrådet.
This study was supported by the Klaus Tschira Foundation. We wish to thank
Charli Sakari for making her spectra available.

This work made use of numpy (van der Walt et al., 2011), scipy (Jones et al.,
01), matplotlib (Hunter, 2007) and corner (Foreman-Mackey, 2016) as well as
astropy, a community-developed core Python package for astronomy (Astropy
Collaboration et al., 2013). The CaT fitting code of Usher et al. (2019a)
relies on pPXF (Cappellari & Emsellem, 2004; Cappellari, 2017).

Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P.
Sloan Foundation, the U.S. Department of Energy Office of Science, and the
Participating Institutions. SDSS-IV acknowledges support and resources from
the Center for High Performance Computing at the University of Utah. The SDSS
website is www.sdss.org. SDSS-IV is managed by the Astrophysical Research
Consortium for the Participating Institutions of the SDSS Collaboration
including the Brazilian Participation Group, the Carnegie Institution for
Science, Carnegie Mellon University, Center for Astrophysics — Harvard &
Smithsonian, the Chilean Participation Group, the French Participation Group,
Instituto de Astrofísica de Canarias, The Johns Hopkins University, Kavli
Institute for the Physics and Mathematics of the Universe (IPMU) / University
of Tokyo, the Korean Participation Group, Lawrence Berkeley National
Laboratory, Leibniz Institut für Astrophysik Potsdam (AIP), Max-Planck-
Institut für Astronomie (MPIA Heidelberg), Max-Planck-Institut für Astrophysik
(MPA Garching), Max-Planck-Institut für Extraterrestrische Physik (MPE),
National Astronomical Observatories of China, New Mexico State University, New
York University, University of Notre Dame, Observatário Nacional / MCTI, The
Ohio State University, Pennsylvania State University, Shanghai Astronomical
Observatory, United Kingdom Participation Group, Universidad Nacional Autónoma
de México, University of Arizona, University of Colorado Boulder, University
of Oxford, University of Portsmouth, University of Utah, University of
Virginia, University of Washington, University of Wisconsin, Vanderbilt
University, and Yale University.

## Data Availability

The Caldwell et al. (2009) spectra are available from
https://oirsa.cfa.harvard.edu/signature_program/. The Sakari & Wallerstein
(2016), Sakari et al. (2021) and Sakari & Wallerstein (2022) spectra are
available upon request from Charli Sakari. The Peacock et al. (2010) and Huxor
et al. (2014) photometry was taken from those papers; the remaining SDSS DR17
(Abdurro’uf et al., 2022) photometry is available from
http://skyserver.sdss.org/dr17.

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