A COMMON ORIGIN FOR GLOBULAR CLUSTERS AND ULTRA-FAINT DWARFS IN SIMULATIONS OF THE FIRST GALAXIES

The simulations presented here begin from two different sets of initial conditions, both of which have formed part of previously published works; Ricotti et al. (2002a, 2002b) (hereafter R02) and Muratov et al. (2013a, 2013b) (hereafter AM13). Both assume a ${\rm{\Lambda }}$CDM cosmology and represent a cubic volume of the universe, 1 ${h}^{-1}\,\mathrm{Mpc}\,$ on a side. The properties of the two sets of initial conditions are listed in Table 1.

Table 1.  The Numerical Parameters Adopted for Each of Our Simulations

Label	${\rm{\Delta }}x$	M${}_{\mathrm{DM},\mathrm{part}}$	N${}_{\mathrm{levs}}$	M${}_{* ,\min }$	Z${}_{\mathrm{crit}}$	n${}_{{\rm{H}},\mathrm{pIII}}$	f${}_{{{\rm{H}}}_{2},\mathrm{pIII}}$	E${}_{\mathrm{SN},\mathrm{pIII}}$	E${}_{\mathrm{SN},\mathrm{pII}}$	IMF	${t}_{* ,\mathrm{samp}}$	${\epsilon }_{* }$
	com. pc	(103 M${}_{\odot }$)		(M${}_{\odot })$	(Z${}_{\odot }$)	(cm−3)	(10−5)	$({10}^{51}$ erg)	$({10}^{51}$ erg)	(Pop II)	(Myr)
REFa	10.9	49.2	10	40	10−5	1.0	1.0	30	1.0	Chab.	0.1	0.1
HSFEa	10.9	49.2	10	40	10−5	1.0	1.0	30	1.0	Chab.	0.1	1.0
LSFEa	10.9	49.2	10	40	10−5	1.0	1.0	30	1.0	Chab.	0.1	0.01
AM13-REFb	10.9	5.5	9	40	10−5	1.0	1.0	30	1.0	Chab.	0.1	0.1

Notes.

^aIC with cosmological parameters: $({{\rm{\Omega }}}_{m},{{\rm{\Omega }}}_{{\rm{\Lambda }}},{{\rm{\Omega }}}_{b},{\sigma }_{8},{n}_{s})=(0.30,0.70,0.040,0.90,1.00)$. ^bIC with cosmological parameters: $({{\rm{\Omega }}}_{m},{{\rm{\Omega }}}_{{\rm{\Lambda }}},{{\rm{\Omega }}}_{b},{\sigma }_{8},{n}_{s})=(0.28,0.72,0.046,0.817,0.96)$.

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In all of our simulations the number of dark matter particles, ${N}_{\mathrm{DM}}$, is equal to the number of cells in the root level mesh ${N}_{\mathrm{root}}$, which sets the particle mass, ${M}_{\mathrm{DM}}={\rho }_{\mathrm{crit}}{{\rm{\Omega }}}_{{DM}}{V}_{\mathrm{sim}}/{N}_{\mathrm{root}}$, where ${\rho }_{\mathrm{crit}}$ is the critical density and ${V}_{\mathrm{sim}}$ is the simulation volume.

Three criteria control the refinement of the simulation mesh. Cells are refined when their gas or dark matter masses exceed the threshold values ${M}_{\mathrm{gas},\mathrm{th}}$ and ${M}_{\mathrm{DM},\mathrm{th}}$. We set ${M}_{\mathrm{DM},\mathrm{th}}$ equal to the mass of one dark matter particle and the gas mass threshold is then fixed at ${M}_{\mathrm{DM},\mathrm{th}}\times {{\rm{\Omega }}}_{b}/{{\rm{\Omega }}}_{\mathrm{DM}}$, such that the two refinement criteria are equivalent for a cell with the cosmic baryon fraction. In addition, we require that the Jeans length of the gas, ${L}_{{\rm{J}}}=20.7\,\mathrm{pc}{(T/n)}^{1/2}$, be resolved by at least five cell lengths, which satisfies the condition described by Truelove et al. (1997) required to avoid artificial fragmentation.

Pop II star formation is allowed only when

• 1.  
  The cell is maximally refined: ${\rm{\Delta }}x=10$ comoving pc (see Table 1).
• 2.  
  The flow is convergent (${\rm{\nabla }}\,.\,{\boldsymbol{v}}\lt 0$).
• 3.  
  The Jeans length of the gas can no longer be resolved by five cells (${L}_{{\rm{J}}}\lt 5\,{\rm{\Delta }}x$). For the value of ${\rm{\Delta }}x$ in our simulations ${L}_{{\rm{J}}}\lt 50$ comoving pc translates in the density threshold for star formation:

  $$\begin{eqnarray*}&&{n}_{{\rm{H}}}\gt {n}_{{\rm{H}},\mathrm{pII}}\approx 1.7\times {10}^{3}\,{\mathrm{cm}}^{-3}\left(\displaystyle \frac{T}{100\,{\rm{K}}}\right){\left(\displaystyle \frac{1+z}{10}\right)}^{2}.\end{eqnarray*}$$

  Phase diagrams show that the bulk of Pop II star formation takes place at densities ${10}^{4}\mbox{--}{10}^{5}$ cm⁻³, with maximum gas densities as high as 10⁶ cm⁻³ in a few cells. The star-forming cells have temperatures ranging $T\sim 10\mbox{--}{10}^{3}$ K and ${f}_{{H}_{2}}\sim 10 \% \mbox{--}60 \%$.
• 4.  
  We also require that the gas overdensity be at least ${\delta }_{\mathrm{gas}}\gt 2000$ and the gas temperature $T\lt {10}^{4}\,{\rm{K}}$.

If all four conditions are met, star formation starts after one dynamical time with rate:

$$\begin{eqnarray}&&\displaystyle \frac{d{\rho }_{* }}{{dt}}={\epsilon }_{* }\displaystyle \frac{{\rho }_{\mathrm{gas}}}{{t}_{\mathrm{dyn}}}\end{eqnarray} \tag{ 1 }$$

where ${\rho }_{\mathrm{gas}}$ is the gas density, ${t}_{\mathrm{dyn}}={(3\pi /32G{\rho }_{\mathrm{gas}})}^{0.5}$ is the dynamical time and ${\epsilon }_{* }$ is the fraction of gas converted into stars in ${t}_{\mathrm{dyn}}$. Note that—similarly to the recipe for Pop III star formation discussed below—we do not require the gas to be fully molecular (${f}_{{H}_{2}}=100 \%$).

Each Pop II star particle represents a stellar population with a Chabrier (2003) initial mass function (IMF). They can lose mass through stellar winds associated with massive stars and through the ejection of material by supernovae (SNe). Stellar particles emit radiation with the spectral energy distribution (SED) shown in Figure 4 of Ricotti et al. (2002a) and an overall normalization that evolves with age according to the Starburst99 model of Leitherer et al. (1999). This results in a radiative output that begins to fall off rapidly after 3 Myr and has dropped by four orders of magnitude after ∼30 Myr. Stars are formed stochastically over a typical timescale of 1 Myr and have a minimum mass of ${M}_{* ,\min }=40$ ${M}_{\odot }$. Conditions for star formation are checked every ${t}_{* ,\mathrm{samp}}=0.1$ Myr . SNe explosions begin after a time delay equal to the lifetime of an 8 ${M}_{\odot }$ star (3.4 Myr) and continue for a total of 35 Myr, with each SN generating ${E}_{\mathrm{SN},\mathrm{pII}}={10}^{51}$ erg of thermal energy and ejecting overall a mass in metals equal to 1.1% of the initial stellar mass. At each time step over this interval, the total energy and metals produced by the stellar population are deposited in the host cell.

To be eligible for Pop III star formation, cells with metallicity less than ${Z}_{\mathrm{crit}}={10}^{-5}$ ${Z}_{\odot }$ must have gas densities and molecular fractions exceeding threshold values:

• 1.  
  ${n}_{{\rm{H}}}\gt {n}_{{\rm{H}},\mathrm{pIII}}=1.0$ cm⁻³,
• 2.  
  ${f}_{{{\rm{H}}}_{2}}\,$ $\,\gt \,{f}_{{{\rm{H}}}_{2},\mathrm{pIII}}={10}^{-5}$.

When these conditions are satisfied, a particle representing a single Pop III star with mass ${M}_{\mathrm{pIII}}=40$ ${M}_{\odot }$ is formed during the next time step. Each Pop III star has a lifetime of 3.9 Myr and radiates with the same SED used for Pop II star particles, but with an increased ionizing luminosity, in agreement with the model from Schaerer (2002) for a zero metallicity star of 40 ${M}_{\odot }$. At the end of their lives, Pop III particles explode as hypernovae (Umeda & Nomoto 2003) and eject 8 ${M}_{\odot }$ of metals and ${E}_{\mathrm{SN},\mathrm{pIII}}=30\times {10}^{51}$ erg of thermal energy into their immediate surroundings. Following Muratov et al. (2013a, 2013b), we distribute metal and energy evenly over a sphere of radius 1.5 cell lengths around the star. The assumed thresholds for star formation are also listed in Table 1.

Unless otherwise specified the results shown in the next sections refer to galaxies identified at redshift z = 9. We analyze the simulations using two forms of data—log files that record global properties of the simulation at every root-level time step, and output snapshots containing the properties of the dark matter, stars, gas and radiation field throughout the simulation volume. Snapshots are written at values of the expansion factor separated by 0.01 until 0.05, and by 0.005 thereafter.

Individual galaxies are identified using a modified version of the subfind halo finder code (Springel et al. 2001). The first phase of the algorithm identifies "friends-of-friends" groups (Press & Davis 1982; Davis et al. 1985) by linking dark matter and star particles separated by less than 0.2 times the mean interparticle separation. It then finds gravitationally bound substructures (sub-halos) by iteratively unbinding particles around local density peaks. The center of each galaxy is deemed to be the potential minimum within the sub-halo. Although subfind identifies all substructures with at least 32 particles, we impose a more conservative limit of 50, consistent with Kravtsov et al. (2004) who found that the cumulative mass function converged above that resolution threshold. The minimum galaxy mass considered in the following sections is therefore $2.5\times {10}^{6}\,{M}_{\odot }$ and $2.8\times {10}^{5}\,{M}_{\odot }$ for the REF and AM13-REF simulations respectively. We stress that, while the halo mass function can be considered substantially complete down to this limit, the star formation histories of halos close to the limit will likely be affected by a lack of resolution in their progenitors. Where merger trees are required for the analysis, they are constructed by linking sub-halos in each snapshot with any progenitor in the previous snapshot that contains at least 5% of their dark matter particles.

Figure 1 shows that gas disks, although rather thick, are clearly identifiable in many of the brightest (top five ranked by stellar mass) galaxies at redshift = 9. The projected gas density is shown in two orientations for each galaxy, parallel and normal to the angular momentum vector of the gas. In the cases where a well-defined gas disk is present, this gives face-on and edge-on views.

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In Figure 2 we show the projected stellar mass density for the same five galaxies that appear in Figure 1. The mass from each star particle is spread out over nearby pixels using an SPH-like kernel⁵ enclosing 32 neighbors. The projection axes and scales of the images are the same as in Figure 1. The morphology of the stellar component of these galaxies is clearly much closer to a spheroid than that of the gas, although some flattening is apparent in the same sense.

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A straightforward way to identify stellar disks in simulated galaxies is to compute the circularity (${ \mathcal E }$) of star particle orbits (e.g., Abadi et al. 2003; Scannapieco et al. 2012). In a coordinate system where the net angular momentum of all of the galaxy's stars is in the positive z direction, circularity may be defined as:

$$\begin{eqnarray}&&{{ \mathcal E }}_{{\rm{E}}}=\displaystyle \frac{{J}_{z}}{{J}_{\mathrm{circ}}(E)},\end{eqnarray} \tag{ 2 }$$

where J_z is the z component of the star's specific angular momentum and J_circ(E) is the specific angular momentum of a star with the same binding energy on a circular orbit. For an infinitely thin, rotationally supported disk, the distribution of circularities is a δ function at ${{ \mathcal E }}_{{\rm{E}}}=1$, while a non-rotating, dispersion-dominated spheroid gives rise to a broad, symmetric distribution peaking at ${{ \mathcal E }}_{{\rm{E}}}=0$.

Figure 3 shows histograms of ${{ \mathcal E }}_{{\rm{E}}}$ for all star particles in the six brightest galaxies in our REF simulation. Most of the six distributions are consistent with non-rotating spheroids, but in two cases the mean of the distribution is positive, suggesting that the spheroid is rotating, or that a thickened disk structure is superimposed on the non-rotating spheroid. No obvious difference is apparent between metal-rich stars ([Fe/H]$\,\gt \,-1.5$, red) and metal-poor stars ([Fe/H]$\,\lt -1.5$, blue).

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Figure 4 shows the projected stellar half-mass radii (${r}_{{\rm{h}}}\,$) for all galaxies in our REF simulation with five or more star particles. To quantify the effect of different viewing angles, ${r}_{{\rm{h}}}\,$ was computed 100 times for each galaxy using random orientations. Error bars indicate the 10th and 90th percentiles of the resulting distribution for each galaxy. In contrast to Local Group dwarfs, which show a relatively clear correlation between ${r}_{{\rm{h}}}\,$ and stellar mass (e.g., Martin et al. 2008), the simulated galaxies can vary by a factor of 100 in ${r}_{{\rm{h}}}\,$ at a fixed stellar mass. The radius ${r}_{{\rm{h}}}\,$ is almost independent of viewing angle in many of the galaxies, suggesting relatively spherical distributions although, as expected, the variation increases at lower M_* where there are fewer star particles per galaxy. Note that the points clustered at the top left corner of the figure are not a numerical artifact but halos that only contain Pop III stars (colored in blue). These halos have an extended and low surface brightness stellar spheroid produced by mergers of several minihalos containing Pop III stars. However, if Pop III stars are indeed massive as we have assumed here, the present-day fossil remnants of these objects would be totally dark.

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In order to ensure that the spatial distribution and kinematics of the stars are not affected by numerical issues, we performed a series of tests. First, we estimated the stellar radii of the spheroids by defining a radius containing half of the star particles, rather than half of the stellar mass. Given that star particles in our simulations have a wide range of masses, we checked whether a few massive star particles that sank to the center of the halo could be biasing the half-mass radii to smaller values. Indeed, it makes sense that star particles, which represent a collection of stars, should be treated as an extended distribution, rather than a point mass. We found no significant differences between the half-mass radius and the radius containing half of the star particles in each case. Thus, the compact star clusters with half-mass radii of a few pc in our simulations are not a numerical artifact, but truly a compact collection of many star particles in virial equilibrium (see below).

The second test regards the kinematics of the stars. We wanted to check whether the half-mass radii are different for lighter star particles than for heavier ones. This would be the case if unphysical two-body interactions between light star particles and dark matter or massive stellar particles were to dynamically heat the system. We found that, excluding progressively more massive subsets of particles, starting with the lightest, there was no obvious effect on the calculated value of ${r}_{{\rm{h}}}\,$.

Figure 5 (left panel) shows the dark matter mass as a function of the stellar half-mass radius for all galaxies with more than five star particles in the REF simulation. The color coding shows bound objects that have only Pop II stars (red), only Pop III (blue) and both (black). In this plot we note a few interesting properties:

• 1.  
  Normal dwarf galaxies. These are bound objects with ${M}_{\mathrm{dm}}\,\geqslant \,$ ${10}^{8}\,{M}_{\odot }$ and contain both Pop II and Pop III stars.
• 2.  
  Failed dwarf galaxies or "dark galaxies". These are bound objects with ${M}_{\mathrm{dm}}\,\sim \,$ ${10}^{7}\,{M}_{\odot }$ that failed to form Pop II stars (about 13 objects). They contain only Pop III stars distributed in a relatively extended stellar halo ($\gt 50$ pc ). Their fossils today would be dark, unless low-mass Pop III stars exist.
• 3.  
  Triggered star formation and compact stellar clusters. These are bound objects with ${M}_{\mathrm{dm}}\,\leqslant \,$ ${10}^{6}\,{M}_{\odot }$ that have Pop II stars but do not have any Pop III stars (about 13 objects). Hence, either they are polluted with metals by galactic winds from nearby galaxies, or the Pop III stars were ejected. These bound objects are likely an example of "triggered" star formation induced by star formation in more massive (${10}^{8}\,{M}_{\odot }$) halos (Smith et al. 2015). Some of these objects are sufficiently compact to be candidate GCs. Their metallicity is also consistent with their identification as progenitors of today's old GCs.

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Five of the ten most compact objects (${r}_{h}\lt 7$ pc) are dark matter dominated and in halos with ${M}_{\mathrm{dm}}\,\sim \,$ ${10}^{8}\,{M}_{\odot }$. The other five are baryon dominated with ${M}_{\mathrm{dm}}\,\lt \,$ ${10}^{6}\,{M}_{\odot }$. Excluding halos without Pop II stars (that would be dark today), compact luminous objects are found to have a bimodal mass distribution.

The right panel of Figure 5 is the same as the left except the color coding shows the ratio of the kinetic to gravitational binding energy of the stars in each object. Galaxies with ${r}_{{\rm{h}}}\,$ $\sim 1\mbox{--}2$ pc and ${r}_{{\rm{h}}}\,$ $\sim 50\mbox{--}200$ pc appear to be bound ($| {E}_{K}/{E}_{W}| \lt 1$) and close to virial equilibrium ($| {E}_{K}/{E}_{W}| \sim 1/2$). Objects with intermediate radii, ${r}_{{\rm{h}}}\,$ $\sim 10\mbox{--}40$ pc , tend to be more loosely bound and further from virial equilibrium ($| {E}_{K}/{E}_{W}| \sim 1\mbox{--}2$), suggesting that these stellar systems have not yet reached an equilibrium configuration and that their half-mass radii might still be expanding as a result of gas mass loss in young star-forming clusters.

Figure 6 shows the mass-to-light ratio ${M}_{\mathrm{tot}}(\lt {r}_{{\rm{h}}})/{M}_{* }(\lt {r}_{{\rm{h}}})$, (where ${M}_{\mathrm{tot}}(\lt {r}_{{\rm{h}}})$ is the dynamical mass within ${r}_{{\rm{h}}}\,$) as a function of the total dynamical mass. The plot shows two separated set of points: those with ${M}_{\mathrm{tot}}(\lt {r}_{{\rm{h}}})/{M}_{* }(\lt {r}_{{\rm{h}}})\sim {10}^{4}$ are dark matter halos containing only Pop III stars (typically a few stars as indicated by their total stellar masses of ${M}_{* }\sim 40\mbox{--}200$ ${M}_{\odot }$). The second group of objects have either a constant mass-to-light ratio of a few (the lower horizontal branch in the plot) or a mass-to-light ratio that increases with dynamical mass from tens to a few hundreds (the upper branch in the plot). This indicates that for a fixed dynamical mass of around 10⁷ ${M}_{\odot }$, we either find dark matter dominated and low surface brightnesses objects (analogous to the ultra-faint dwarfs), or compact star clusters in which the dark matter is a subdominant component of the dynamical mass, but which still can be embedded into larger dark matter halos.

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Figure 7 is similar to Figure 4 but for three simulations adopting different values of ${\epsilon }_{* }$ (sub-grid star formation efficiency) and with color coding of the points showing the ratio of the kinetic to the gravitational binding energy of the objects $| {E}_{K}/{E}_{W}|$. Each panel shows the stellar half-mass radii as a function of stellar mass for all bound objects with more than five star particles in the LSFE (${\epsilon }_{* }=1 \%$, left panel), REF (${\epsilon }_{* }=10 \%$, center panel) and HSFE (${\epsilon }_{* }=100 \%$, right panel) simulations. The grayscale shaded regions show the stellar to dark matter mass ratio within r_h, assuming for estimating the dark matter mass within r_h an NFW profile for a halo of mass ${10}^{8}\,{M}_{\odot }$ and virial radius 2 kpc, which is representative of typical halos at z = 9 in ${(1\mathrm{Mpc})}^{3}$ volume. This is to illustrate that diffuse objects are dark matter dominated while compact objects are baryon dominated. Objects for which the ratio $| {E}_{K}/{E}_{W}|$ is greater than unity (unbound systems) are represented by filled diamond symbols, and all other objects by filled circle symbols. We find qualitatively similar bound objects in the three cases, independent of the assumed star formation efficiency (even though it changes by a factor of 100 between the simulations). The spread of the ${r}_{{\rm{h}}}\,$ distribution for a given mass (i.e., compact and low surface brightness galaxies) is found in all the simulations, indicating that feedback effects determine the interruption of star formation in the proto-star clusters. We do observe, however, a weak dependence of galaxy luminosity on ${\epsilon }_{* }$. The luminosity of dwarfs increases by about a factor of two when ${\epsilon }_{* }$ is increased by a factor of 100. In the HSFE simulation about 9–11 objects are as compact as GCs, and about 20 are similar to UFDs, with low surface brightness and ${r}_{{\rm{h}}}\,$ $\sim \,100\mbox{--}200$ pc .

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In this subsection we investigate whether the compact dwarfs resembling proto-GCs (based on morphology and kinematics) are also consistent in terms of the metallicity distribution of their stars. GCs are usually seen to have a rather uniform [Fe/H] abundance in the range [Fe/H] = −2 to −1, indicative of a single stellar population with little or no self-enrichment (e.g., Gratton et al. 2004; Carretta et al. 2009). Thus, to first-order approximation, their metallicity reflects the pre-enrichment of their host galaxy. In Figure 8 we show the distribution of star particle metallicities for six objects in the REF simulation. Note that we advect only one metallicity field, thus [Fe/H] is a proxy for the total metal enrichment from SNe. The panels are ordered based on stellar half-mass radii (from the smallest to the largest). The stellar half-mass radius, ${r}_{{\rm{h}}}$, is included as a label, along with the total stellar mass. The dashed vertical lines indicate the critical metallicity for transition from Pop III to Pop II star formation adopted in our simulations. The most compact object in the figure resembles today's old GCs: it has ${r}_{{\rm{h}}}\,$ $\approx \,2.7$ pc , ${M}_{* }\,\approx$ $5\times {10}^{5}\,{M}_{\odot }$ and a delta-function-like metallicity distribution peaked at [Fe/H] = −1.5. However, some compact stellar clusters in the simulation (for instance, the one shown in the top-right panel of Figure 8) have half-mass radii and stellar masses consistent with old GCs, but much broader metallicity distributions, with a range [Fe/H] = −4 to [Fe/H] = −1 and therefore they would be classified as ultra-compact dwarfs. These are likely formed as the result of the merger of at least three or four smaller star clusters with different metallicities. Thus, it is rather difficult to distinguish between a GC and the nucleus of an ultra-compact dwarf galaxy solely based on morphology. Indeed, some candidate proto-GCs in our simulations are similar to the nuclei of compact dwarfs because they form at the center of low-mass dark matter halos. Most simulated dwarf galaxies with sizes $\gt 10$ pc have rather broad metallicity distributions, very similar to those observed in UFDs and classical dSphs. We also find simulated dwarf galaxies with intermediate sizes (15−50 pc ) and low stellar masses, much like the faintest UFDs found around the Milky Way at distances of $\lesssim 150$ kpc . Tidal stripping of stars has been suggested as the reason for the small sizes of these observed faint dwarfs (Bovill & Ricotti 2011b), but our simulated dwarfs are already compact when they form, suggesting that this need not be the case.

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Dwarfs with low stellar masses have metallicity distributions characterized by several minor peaks, with the most prominent typically at the highest metallicity. This suggests that the stellar population is produced by in situ formation or merger of a handful of small star clusters with relatively narrow metallicity distributions. This scenario is consistent with the rather bursty and spatially segregated mode of star formation observed in most simulated galaxies. A smooth and continuous mode of star formation and self-enrichment is rarely observed in our simulations; a test of our model would be to compare it to observations of the chemical and kinematic signatures typical of star formation taking place in distinct star clusters (Bland-Hawthorn et al. 2015; Webster et al. 2016). Finally, even though the critical metallicity assumed for the transition to Pop II star formation is [Fe/H] = −5, the lowest metallicities of Pop II stars are between [Fe/H] = −4 and [Fe/H] = −3. We have run some simulations with different values of the critical metallicity (see PRG16) and have found very little change in the distribution of stellar metallicities.

Figure 9 shows the metallicities and formation times of stars in four galaxies in the REF simulation. While the star formation histories of each galaxy are complex functions of many factors, including merger history, halo mass and the local radiation field, these four halos are broadly representative of the galaxy ensemble as a whole. Each color corresponds to an independent progenitor galaxy which hosted the star particle when it formed. Progenitor colors are ordered by the stellar mass contributed, as indicated by the labels in the bottom left of each panel. Star particles were mapped to progenitor galaxies by determining their host halo in the first output after they formed and then associating each halo with a branch of the merger tree. Merger trees were constructed as described in Section 2.5. The stellar half-mass radius of each galaxy is listed in the top left corner of each panel, along with the total stellar mass. The four examples shown suggest an inverse correlation between stellar size and the fraction of stars formed late in the simulation, which is borne out in the galaxy population as a whole. Each galaxy typically contains between two and eight Pop III stars, with many progenitors forming just a single Pop III star. Multiple "tracks" are evident in many of the most massive galaxies, most obviously in the bottom right panel of Figure 9. The varying gradients of the tracks demonstrate different rates of enrichment in independent progenitor halos. We note that modifying the parameter used to distinguish between Pop III and Pop II stars (${z}_{\mathrm{crit}}\,$) affects the size of the "gap" in metallicity between the most metal-rich Pop III star and the most metal-poor Pop II star, as well as the time delay between the formation of the first Pop III star and Pop II star formation. This delay occurs when gas is enriched above ${z}_{\mathrm{crit}}\,$ before it has reached a high enough density to form Pop II stars. For most choices of the simulation's free parameters the delay has very little impact on the typical star formation and final stellar mass of the galaxy; however too large ${z}_{\mathrm{crit}}\,$ or too strong Pop III feedback can suppress Pop II star formation completely in most galaxies (see PRG16).

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In Figure 11 we show a comparison between the properties of luminous objects in our REF simulation (blue circles) in comparison to Milky Way globular clusters (shown as stars) from Harris (1996) and nearby dwarf galaxies (shown as triangles), including classical dwarfs (dSphs, dEs, dIrrs) and ultra-faint dwarfs from the McConnachie (2012) compilation. Similar plots for a compilation of GCs and compact dwarfs are shown in Kissler-Patig et al. (2006). Since our simulation stops and is analyzed at redshift z = 9, in order to compare the simulated galaxies to present-day clusters and dwarf galaxies, we assume a mass-to-light ratio $M/{L}_{V}=3$, that takes into account mass loss and passive stellar evolution over about 12 Gyrs as in Ricotti & Gnedin (2005). Several compact clusters in our simulation are recently formed, thus they may not remain bound if (i) they are not in virial equilibrium, or (ii) they contain a significant amount of gas ($\gt 50 \%$) within their half-light radius, that if expelled by SNe or photoevaporation may unbind the cluster (only a couple of objects are in this category). For those objects we evolve their half-light radii r_h(t) as in Equation (9) and their velocity dispersions ${\sigma }_{* }(t)$ as in Equation (7) until the stars becomes bound by the dark matter halo as in Equation (8). The dotted lines in the plot show the evolution of such objects, evolving from the small to the larger blue circles at the extremes of the dotted lines. Note that here we plot the line-of-sight velocity dispersion, ${\sigma }_{* ,1{\rm{D}}}$, which we simply relate to the 3D velocity dispersion as ${\sigma }_{* }=\sqrt{3}{\sigma }_{* ,1{\rm{D}}}$. Also, in order to calculate the velocity profile ${v}_{\mathrm{cir}}(r)$ of each halo we assume an NFW density profile with concentration parameter c = 4, which is appropriate for recently virialized halos. We see that many simulated objects that are initially in a region of parameter space devoid of observed objects (a narrow strip laying between GCs and UFDs) are evolving to lower ${\sigma }_{* ,1{\rm{D}}}$ and larger r_h toward the region occupied by UFDs.

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The extrapolation of observed properties to z = 0 is rather simplistic, and is not the main focus of this paper. Using N-body simulations, previous works by Bovill & Ricotti (2011a, 2011b) have looked in detail at several effects that we have neglected. They show that only small-mass halos that are rather isolated at z = 9 evolve to z = 0 without merging into larger halos and thus accrete fresh gas. A subset of these halos evolving in isolation with maximum circular velocity ${v}_{\max }\lt 20\mbox{--}25$ km s⁻¹ will not accrete fresh gas from the IGM after reionization at $z\lesssim 6\mbox{--}9$. These objects are suitable for a direct comparison to UFDs and are what have been defined as "fossil" galaxies (Ricotti & Gnedin 2005; Bovill & Ricotti 2009). Bound compact clusters may also survive intact for about a Hubble time (13.6 Gyrs) only if they are more massive than 10⁴ ${M}_{\odot }$ (e.g., Katz & Ricotti 2014). If their mass is instead smaller, the timescale for evaporation due to two-body encounters is shorter than the Hubble time. This is interesting because the faintest UFDs have ${M}_{* }\sim {10}^{3}\mbox{--}{10}^{4}$ ${M}_{\odot }$, thus they could be formed by the secular evaporation of compact clusters residing in a small-mass dark matter halo. As discussed above, the cluster will puff up until the stars become bound by the potential of the dark matter halo, as in Equation (8). Of course, GCs can be destroyed by tidal stripping and shocks while interacting with their host galaxy . Roughly, only one or two in ten GCs are expected to survive within the Milky Way over a Hubble time (Katz & Ricotti 2014).

Figure 12 shows the mass in gas within the half-light radius, ${M}_{\mathrm{gas}}$, normalized by the total baryonic mass ${M}_{* }+{M}_{\mathrm{gas}}$ as a function of r_h. In compact clusters with ${r}_{h}\lt 10$ pc the mass in gas is less than the mass in stars, while in larger objects the mass of the stars is sub-dominant with respect to the gas mass (and the dark matter mass). The points shown as diamonds are unbound objects, thus are transitioning toward larger r_h, while the circles show bound objects (bound by the baryons if ${r}_{h}\lt 10$ pc and by the dark matter for ${r}_{h}\gt 30$ pc). The figure shows that in compact clusters either $\gt 50$% of the gas has been used for star formation (i.e., the star formation efficiency in the proto-cluster is ${\epsilon }_{\mathrm{cl}}\gt 50$%), or ${\epsilon }_{\mathrm{cl}}\lt 50$% and part of the gas has been expelled by radiation and SN feedback. In the former case (${\epsilon }_{\mathrm{cl}}\gt 50$%) the cluster should remain bound while in the second case it will likely become unbound once all the gas is lost, and it will expand until $\sigma (r)={v}_{\mathrm{cir}}(r)$ (see Section 4). The figure shows that radiation and SN feedback did not clear out all the gas within the dark matter halo of most objects with ${r}_{h}\gt 30$ pc, as they are gas rich. However, the star formation rate in galaxies is self-regulated by feedback (see Section 3.5 and Figure 7) and the range of ${\epsilon }_{\mathrm{cl}}$ found in the simulation is determined by the effectivness of feedback in terminating star formation in star-forming gas clumps and by their density (the sub-grid star formation law we use converts into stars ${\epsilon }_{* }=10$% of the gas in a cell per local free-fall time). We should however note that with a spatial resolution of 1 pc, the internal structure of young compact star clusters (${r}_{h}\,\sim$ few–10 pc) is only marginally resolved. Higher-resolution simulations are needed to confirm quantitative results in the simulation. For instance, the number of bound stellar clusters may increase by increasing the numerical resolution.

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Multiple stellar populations are present in all globular clusters observed to date (e.g., Gratton et al. 2004; D'Ercole et al. 2008; Carretta et al. 2010). Figure 9 shows that when we look in detail at the stellar populations of compact star clusters, several have stars with distinct metallicities and formation times. Some compact clusters form in very low mass statellites orbiting the host dwarf galaxy (offset from center of their host), triggered by external metal enrichment from galactic winds. Others are found close to the center of their host dark matter halos, resembling in terms of their metallicity the nuclei of compact dEs. These second type of compact clusters should be able to form stars in multiple bursts while still remaining compact, due to gas fallback in the gravitational potential of the dark matter halo (Trenti et al. 2015). We do not have the necessary resolution and detailed chemical enrichment physics to carefully address the process of SN feedback, gas enrichment by AGB stars and gas fall-back, but it is an idea worth exploring with dedicated high-resolution simulations.