The Little Engines That Could? Globular clusters contribute significantly to reionization-era star formation

Abstract

Metal-poor globular clusters (GCs) are both numerous and ancient, which indicates that they may be important contributors to ionizing radiation in the reionization era. Starting from the observed number density and stellar mass function of old GCs at z = 0, I compute the contribution of GCs to ultraviolet luminosity functions (UVLFs) in the high-redshift Universe (10 ≳ z ≳ 4). Even under absolutely minimal assumptions – no disruption of GCs and no reduction in GC stellar mass from early times to the present – GC star formation contributes non-negligibly to the UVLF at luminosities that are accessible to the Hubble Space Telescope (HST, M₁₅₀₀ ≈ −17). If the stellar masses of GCs were significantly higher in the past, as is predicted by most models explaining GC chemical anomalies, then GCs dominate the UV emission from many galaxies in existing deep-field observations. On the other hand, it is difficult to reconcile observed UVLFs with models requiring stellar masses at birth that exceed present-day stellar masses by more than a factor of 5. The James Webb Space Telescope will be able to directly detect individual GCs at z ∼ 6 in essentially all bright galaxies, and many galaxies below the knee of the UVLF, for most of the scenarios considered here. The properties of a subset of high-redshift sources with −19 ≳ M₁₅₀₀ ≲ −14 in HST lensing fields indicate that they may actually be GCs in formation.

1. INTRODUCTION

The physics of galaxy formation at high redshift persists as one of the most important questions in astrophysics. While many models exist of the ‘first galaxies’ at z ~ 20 and putative sources of cosmic reionization at z ~ 8 (e.g. Bromm & Yoshida 2011; Madau 2017), observational constraints are much more challenging, especially for the lowest luminosity sources. This is a main motivation for James Webb Space Telescope (JWST), yet even JWST will not reveal the faintest star-forming galaxies in the high-redshift Universe: the stellar fossil record of dwarf galaxies in the Local Group demonstrates the existence of galaxies in the reionization era that are at least 40 000 times fainter than JWST’s detection threshold in blank fields (Boylan-Kolchin et al. 2015; Weisz & Boylan-Kolchin 2017). The integrated ultraviolet (UV) photon output from faint but numerous galaxies is likely crucial for maintaining an ionized intergalactic medium at z ~ 6–8, as bright galaxies simply do not produce enough UV emission to maintain reionization based on our current understanding of their escape fractions (f_esc, Kuhlen & Faucher-Giguère 2012;Robertson et al. 2013; Bouwens 2016; Finkelstein 2016;Stark 2016).

While the use of the stellar fossil record for contextualizing dwarf galaxies in the reionization era is a relatively recent phenomenon, the basic idea of using the fossil record to learn about the early Galaxy and its evolution has a long and storied history (Eggen, Lynden-Bell & Sandage 1962; Peebles & Dicke 1968; Searle & Zinn 1978; Larson 1990; Bland-Hawthorn & Freeman 2000). Globular clusters, in particular, have captured the attention of observers and theorists alike as they try to unravel the mysteries of the early stages of galaxy formation. The Milky Way is surrounded by over 100 old GCs (Harris 1996), and their high stellar densities, characteristic luminosity, and old ages have been the source of intrigue and speculation for decades (for reviews, see Harris 1991; VandenBerg, Bolte & Stetson 1996; Brodie & Strader 2006; Charbonnel 2016).

The typical Milky Way GC has a present-day stellar mass of 2 × 10⁵ M_☉ and an age of 12–13 Gyr, placing its formation epoch at or near the reionization era. The sheer number density of GCs (n_GCs), coupled with their roughly coeval formation times, points to their potential importance for cosmic reionization: using n_GCs(z = 0) = 2 Mpc⁻³ (Boylan-Kolchin 2017, hereafter B17; see also Portegies Zwart & McMillan 2000) and assuming all GCs formed at high redshifts over a period of Δt, the star formation rate from GC stars is ${\stackrel{.}{\rho }}_{\star ,\text{GC}}=5\times {10}^{-4}$ ξ (Δt /1000 Myr)⁻¹ M_☉ yr⁻¹ Mpc⁻³, where ξ is the ratio of M_★ at birth to present day. This star formation rate is a non-negligible fraction of what is needed to maintain an ionized intergalactic medium at z = 6, ${\stackrel{.}{\rho }}_{\star ,\text{crit}}\approx 0.012$ M_☉ yr⁻¹ Mpc⁻³ (Shull et al. 2012, assuming a clumping factor of 3 and an escape fraction of 0.2). Specifically, if the mass in stars formed in GCs is a factor of ≳ 10 larger than their present-day stellar mass, GCs can contribute ≳ 50 per cent of the requisite ionizing flux (Ricotti 2002; Schaerer & Charbonnel 2011; Katz & Ricotti (2014)). Carlberg (2002) made the related point that the sky density of GCs is likely to be enormous: he estimated 10⁷ deg⁻² at 1 nJy (m_ab = 31.4, corresponding to M₁₅₀₀ ≈ −15.6 at z = 7).

In this paper, I consider the luminosity function (LF) of GCs in the reionization era from a theoretical perspective, confront these models with observed UVLFs, and comment on the resulting implications for GC formation models, the contribution of GCs to reionization, and the observability of high-redshift GCs with HST and JWST.

2. GLOBULAR CLUSTERS AT HIGH REDSHIFT

In the low-redshift Universe, GC systems around galaxies have a nearly-universal stellar mass function that is log-normal, with characteristic stellar mass of ~2 × 10⁵ M_☉ and log₁₀ dispersion of ~0.5–0.6 (Harris 1991; Villegas et al. 2010). Taking the known value of n_GC in the local Universe, it is possible to compute the GC UVLF at high redshifts by considering only the present-day stellar mass contained in GCs and a specified formation epoch through the use of stellar population synthesis models. One can also vary these assumptions, as several physical effects should affect GC populations through cosmic time:

• (i)GC M_★ at birth relative to present-day M_★ (ξ): ξ = 1 is an absolute minimum assumption, and most models that explain the multiple stellar populations observed in GCs (see, e.g. Gratton, Carretta & Bragaglia 2012 and Renzini et al. 2015 for recent reviews) require much larger values, ξ ~ 10–100 (e.g. Decressin et al. 2007; D’Ercole et al. 2008; Conroy & Spergel 2011; Denissenkov & Hartwick 2014; see Bastian & Lardo 2015 for a critical review of various mass-loss scenarios). Irrespective of the origin of multiple populations, evaporative losses from GCs due to collisional relaxation and mass loss from stellar evolution reduce the stellar mass of a GC over time (Spitzer 1987; Fall & Zhang 2001). Furthermore, GCs are subject to impulsive tidal shocks as they orbit through the centre of a galaxy if the time for the GC to traverse the disc or spheroid is short compared to the orbital period of GC stars. Such shocks irreversibly heat GC stars and can cause additional mass loss (Ostriker, Spitzer & Chevalier 1972; Spitzer 1987; Murali & Weinberg 1997; Gnedin, Hernquist & Ostriker 1999). The generic expectation from all models of GC evolution therefore is that ξ > 1 for old GCs.
• (ii)Fraction of GCs surviving to z = 0 (f_surv): The processes mentioned above are enough to completely disrupt some GCs. In particular, two-body relaxation will tend to disrupt all but the most extended low-mass (M_★ ≲ 10⁵ M_☉) GCs over a Hubble time, and tidal shocking should destroy some of the clusters that orbit closest to a galaxy’s centre (Chandrasekhar 1942; Spitzer 1958; Fall & Rees 1977; Binney & Tremaine 1987; see Prieto & Gnedin 2008 for estimates of the relative contributions of collisional, stellar evolution, and disruption).
• (iii)Duration of GC formation epoch (Δt): While the absolute ages of blue GCs are relatively uncertain, it is clear that (1) blue GCs are ancient, with formation times comparable to the age of the Universe, and (2) the relative age uncertainties are significantly smaller, with a dispersion of approximately 0.4–0.5 Gyr (e.g.Marín-Franch et al. 2009; VandenBerg et al. 2013). I therefore adopt Δt = 1000 Myr as my default assumption; I also consider the implications of other values of Δt. 1000 Myr is approximately the time between redshifts 10 and 4 (lookback times between 13.3 and 12.3 Gyr) in the standard cosmological model (Planck Collaboration et al. 2016). I will assume that the formation rate of globular clusters is a power-law function of time, ${\stackrel{.}{n}}_{\text{GCs}}\propto t^{-(1+\eta )}$, with η = −1 (a formation rate that is constant with time) as my fiducial model. Note that under this assumption, ${\stackrel{.}{n}}_{\text{GCs}}=n_{\text{GCs}}∕\Delta t$.

Considering only stars still present in z = 0 GCs will result in a lower limit on the contribution of GCs to high-z UVLFs. The effect of ξ is a uniform shift of the entire initial GC stellar mass function to higherM_★ values (and therefore to larger values of L_UV(t) : L → ξL), while f_surv and Δt serve to change the overall normalization of the GC UVLF at early times: ϕ → ϕ (f_surv Δt/1 Gyr)⁻¹.

Additionally, it is not clear whether the observed log-normal LF of old GCs at z = 0 is reflective of their stellar mass distribution at birth versus a result of dynamical processes operating to preferentially destroy low-mass clusters. In the former case, the lognormal shape could be a manifestation of the Jeans mass in the low-metallicity, high-z Universe (Peebles & Dicke 1968; Fall & Rees 1985) or the result of preferential disruption of low-mass proto-GC gas clouds via supernova feedback (Parmentier & Gilmore 2007; Baumgardt, Kroupa & Parmentier 2008). In the latter case, the mass spectrum of GCs at formation in the high-z Universe would resemble the spectrum of both giant molecular clouds (Solomon et al. 1987; Rosolowsky 2005; Krumholz 2015) and young star clusters forming in low-redshift systems (especially in ongoing or recent galaxy mergers;Whitmore & Schweizer 1995; Larsen 2002; Portegies Zwart, McMillan & Gieles 2010); the subsequent evolution to the observed log-normal shape is expected based on the sub-Hubble-time two-body-relaxation timescales for low-mass clusters (Fall & Zhang 2001; Prieto & Gnedin 2008). I will take the initial GC stellar mass function to be log-normal as my default assumption, as using a Schechter-like luminosity function will only result in more GCs and more UV emission at faint magnitudes. However, I explicitly consider the case of a Schechter-like initial stellar mass function and compare to the log-normal case in Appendix B.

2.1. Intrinsic luminosity functions

Assuming a log-normal form for the z = 0 GC stellar mass function, the UVLF of a cosmologically representative GC system (measured at 1500 Å in the rest frame, M₁₅₀₀) can be thought of a log-normal distribution with a mean value $\overline{M}$ that evolves with time. $\overline{M}(t)$ is fully determined by the time evolution of UV emission from a specified stellar population and the time dependence of any mass loss; based on stellar population synthesis calculations outlined in B17, I adopt

$$\overline{M}(t)=-14.5-2.5{\log }_{10}\left[\xi a{\left(\frac{t}{\text{Myr}}\right)}^{-b}\right],\text{with}$$ (1)

$$(a,b)=\{\begin{array}{cc}(1,0)\hfill & \text{if}t<5.5\text{Myr}\hfill \\ (9.17,1.3)\hfill & \text{if}5.5\le t∕\text{Myr}\lesssim 300\hfill \\ (103,1.725)\hfill & \text{if}300\le t∕\text{Myr}\lesssim 1400\hfill \end{array}\phantom{\}}\cdot$$ (2)

Fig. 1 shows the evolution of the GC UVLF over the full epoch of GC formation (see Appendix A for details of this calculation). Initially, the GC UVLF traces the log-normal form of the underlying GC mass function. As time progresses, however, two competing effects take place: the earliest-forming GCs fade passively, shifting uniformly to fainter magnitudes, and newly-forming GCs emerge, populating the initial log-normal GC UVLF curve. The full GC UVLF is a sum over populations of GCs of different ages (and, correspondingly, different mean M₁₅₀₀ values). The shape of the full GC UVLF is therefore determined by how quickly M₁₅₀₀ fades with time for a given GC: for sufficiently fast fading, only the very early stages of GC formation will be important for the UVLF, while GCs of all ages contribute roughly equally in the limit of very slow fading.

Figure 1.

Intrinsic luminosity functions of GCs (uncorrected for any internal extinction). GCs are assumed to have ξ = 1 and to form at a constant rate in time with a formation epoch duration of 1000 Myr. Left: GC UVLFs at 5.5, 50, 100, 250, 500, and 1000 Myr after the start of the GC formation era. While GCs sample a log-normal luminosity function, the combination of passive fading plus continuous GC formation results in a total GC UVLF that develops a power-law tail with a cut-off at faint magnitudes. Right: The contribution of GCs with various ages to the GC UVLF at 1000 Myr. While the youngest GCs (with ages <5.5 Myr) are the brightest, the GC UVLF is dominated by GCs formed in the previous 5.5–300 Myr.

The relative contributions to the GC UVLF at the end of the GC formation epoch (which is assumed to last 1 Gyr), for the periods t < 5.5 Myr, 5.5 < t < 300 Myr, and t> 300 Myr, are shown in the right-hand panel of Fig. 1. While the very brightest portion of the GC UVLF is dominated by the youngest GCs, it is GCs with 5.5 < t < 300 Myr that contribute the bulk of the UVLF for a wide range of luminosities, as the number of GCs in this age range is a factor of 50 larger than the newly-formed GCs. The oldest GCs contribute only to the faintest portion of the LF. This is directly related to the choice of a Kroupa (2001) stellar initial mass function (IMF), for which the luminosity of a GC evolves as L ∝ t^−b, with the time-dependence of b noted in equation (2).

Intriguingly, even though the underlying UVLF of GCs forming at any particular instant is log-normal, the overall GC UVLF can be approximated by a Schechter (1976) function for t ≫ 5.5 Myr (Fig. 2). The faint-end slope α of this Schechter-like LF is determined only by the fading of a GC’s UV luminosity with time, α ≈ −1–1/b, while the characteristic mass is given by M* = −14.5 − 2.5 log₁₀(ξ a). Since the GC UVLF is dominated by GCs with ages of 5.5 < t < 300 Myr, it is the values of a and b corresponding to this period that are appropriate: a = 9.17 and b = 1.3, yielding α ≈ −1.77 and M* ≈ −16.9 − 2.5 log₁₀ ξ;. In Appendix B, I present analogous results for GC stellar mass functions that are themselves Schechter-like at formation.

Figure 2.

Luminosity function from GCs in the fiducial model (in which the GC stellar mass function at birth is log-normal) at t= 1000 Myr (solid grey), along with Schechter function approximation (dashed black). The parameters of the Schechter function (M*, α, ϕ*) are detemined by ξ, Δt, and stellar evolution (via a, b, see equation 1).

2.2. Contributions of GCs to Φ_UV

Fig. 3 compares the intrinsic (unobscured) GC UVLF to measured global UVLFs from z = 10 to z= 4 (as compiled in section 5.2–5.3 of Finkelstein 2016). For magnitudes fainter than M₁₅₀₀ ≈ −17, there is significant uncertainty in the faint-end slope of the UVLF, particularly at z ≳ 8. The plotted results assume steeper faint-end slopes compared to some estimates and therefore can be interpreted as providing upper limits to the luminosity function of faint galaxies at early times. For the default assumptions adopted here – ξ = 1, Δt = 1 Gyr, and f_surv = 1 – GCs contribute a maximum of ≈25 per cent of the UVLF (at M₁₅₀₀ ≈ −15), with a lesser contribution for very bright magnitudes (because GCs have a maximum initial stellar mass) and very faint magnitudes (because the observed high-z UVLFs have very steep faint-end slopes) at most redshifts.

Figure 3.

The intrinsic luminosity function of globular clusters (grey-scale, with line colours identical to those in the left-hand panel of Fig. 1) during their epoch of formation. The plot assumes birth masses are equal to present-day masses (ξ = 1) and formation uniformly distributed in time over a period of Δt = 1 Gyr, approximately the time from z = 10 to z= 4. Also plotted are observed UV LFs (coloured lines) from z = 10 (dark purple) to z = 4 (light orange) as compiled in Finkelstein (2016). Horizontal arrows indicate the effects of assuming ξ = 10 (dark grey) or 100 (black), while the vertical arrow shows the shift in the globular cluster luminosity function if the formation period is 250 Myr. This would also be the shift required if f_surv = 0.25 as opposed to 1.

The inferred contribution of GCs to the global UVLF changes dramatically if we consider alternate scenarios, however, the birth masses of GCs (ξ), GC disruption (f_surv), and the duration of the GC formation epoch (Δt) all have significant effects on the contribution of GCs to the global UVLF. The horizontal location of the GC UVLF depends on ξ, while its normalization is proportional to (f_surv Δt)⁻¹. These effects are noted with arrows in Fig. 3 and indicate that ξ = 100 is extremely difficult to reconcile with observed luminosity functions.

Having established the unobscured GC UVLF, the obvious next question is: Do GCs contribute significantly to the global UVLF at high redshifts? To address this question, I need to convert from intrinsic to observed UV fluxes, i.e. to account only for UV (rest-frame 1500 Å) radiation that escapes the GC. A related quantity – the fraction of ionizing radiation that escapes the galaxy, f_esc – is perhaps the most fraught aspect of reionization modelling. Many global models of reionization assume a constant escape fraction of f_esc ≈ 0.1–0.2 (e.g. Robertson et al. 2013; Finkelstein et al. 2015; Ishigaki et al. 2017). No such consensus exists in simulations, with results ranging from fairly high to virtually negligible values of f_esc (e.g. Wise et al. 2014; Ma et al. 2015; Paardekooper, Khochfar & Dalla Vecchia 2015; Gnedin 2016; Xu et al. 2016; Anderson et al. 2017; Howard et al. 2017; Kimm et al. 2017; Trebitsch et al. 2017; Zackrisson et al. 2017). However, the escape fraction of UV radiation (hereafter f₁₅₀₀) will certainly be higher than f_esc, the escape fraction of hydrogen-ionizing radiation.

Observations of young massive clusters reveal that they are gas-free within ~5–10 Myr of their birth (Bastian, Hollyhead & Cabrera-Ziri 2014; Hollyhead et al. 2015), indicating that a combination of stellar winds, radiation pressure, and type II supernovae is highly efficient at completely removing gas from nascent GCs. It is therefore likely that f_esc for GCs depends on the time relative to t_gas-free, where t_gas-free= 10 Myr is the assumed timescale for the evacuation gas from a newly-formed GC; this value is a conservative upper limit. Furthermore, the distribution of GCs within their host haloes is significantly more extended than the size of the central galaxy (Forbes 2017; Hudson & Robison 2017), indicating that GCs typically form in regions that, while locally very dense, are globally embedded in a relatively tenuous medium from which photons can easily escape.

I will therefore consider two possibilities: f₁₅₀₀ = 1 at all times, or f₁₅₀₀(t < t_gas-free) = 0 and f₁₅₀₀(t > t_gas-free) = 1 (corresponding to no 1500 Å UV escape prior to gas evacuation and full escape afterward). Lower values of f₁₅₀₀ can be modelled by shifting M₁₅₀₀ → M₁₅₀₀ – 2.5 log₁₀(f₁₅₀₀). I consider both ξ = 1 and 10 for each f₁₅₀₀ scenario. While the f₁₅₀₀ = 0 scenario is unrealistic, it provides a firm lower limit and therefore serves a useful role in demonstrating the lowest possible contribution of GCs to UVLFs.

Fig. 4 compares the GC UVLFs computed using these parameters to the same galaxy UVLFs from Fig. 3. Black curves assume ξ = 10 and grey curves assume ξ = 1; solid and dashed lines correspond to f₁₅₀₀(t < t_gas-free) = 1 and 0, respectively. The models with ξ = 1 fall below observations, while both models with ξ = 10 exceed observed UVLFs at magnitudes accessible to HST in blank fields (M₁₅₀₀ ≲ −17; McLure et al. 2013; Schenker et al. 2013). However, I continue to consider these models with ξ = 10 for a number of reasons. First, f₁₅₀₀ and ξ can be combined into a single parameter, ξ_eff ≡ ξ f₁₅₀₀; this is the quantity that is constrained by observations and the grey (black) lines in Fig. 4 are best thought of as lines with ξ_eff = 1 (10). Second, ξ ≳ 10 is routinely invoked as a necessity in literature models of light element anticorrelations in GCs. Finally, high values of ξ provide a firm upper limit to the number of GCs that should be detectable in the high-redshift Universe. In both cases (ξ = 1 and 10), the faint-end slope of the GC UVLF is very similar to the z=4 UVLF (α≈−1.7). A clear prediction, independent of ξ, is that the GC UVLF should extend to high number densities at faint magnitudes: the LF turn-over occurs at M₁₅₀₀ ≈ −7.75 (−10.25) at the end of the GC formation epoch for ξ = 1 (10). The time evolution of the turn-over magnitude can be seen in Figs 1 and 3.

Figure 4.

Observed UV LFs (coloured lines) from z = 10 to z = 4 (as in Fig. 3, along with modelled observed GC UVLFs (thick black and grey lines). The black lines assume ξ = 10, while the grey lines assume ξ = 1; solid lines assume f₁₅₀₀(t < t_gas-free) = 1, while dashed assume f₁₅₀₀(t < t_gas-free) = 0 (here t_gas-free = 10 Myr). All cases assume f₁₅₀₀(t > t_gas-free) = 1; different values of f₁₅₀₀ can be folded into ξ_eff ≡ ξ f₁₅₀₀ (see the text for details). Models with ξ = 10 exceed observed UVLFs at levels that would likely have been already detected. Assuming that no UV radiation can escape from GCs at early times (t < t_gas−free ≈ 10 Myr) affects the bright end of ϕ(M) but leaves it unaffected at fainter magnitudes.

Fig. 5 compares the GC UVLFs to global UVLFs in ratio form in order to assess the contribution of GCs to UVLFs explicitly. Dashed lines correspond to ξ_eff = 10 and solid lines correspond to ξ_eff = 1; in each case, I assume f₁₅₀₀(t < t_gas-free) = 0 and 1 at later times. For large values of ξ_eff, a large fraction of the global UVLF comes from GCs (in fact, the GC UVLF exceeds the total UVLF at observable magnitudes for ξ_eff = 10). The contribution of GCs is substantially lower for ξ_eff = 1, particularly for magnitudes brighter than HST’s blank-field limit of M₁₅₀₀ ≈−17. At fainter magnitudes (M₁₅₀₀ ≳ −15), the contribution of GCs to the UVLF approaches or exceeds 10 per cent at all redshifts. If the faint-end slope of the galaxy UVLF is shallower than the estimates of Finkelstein (2016), the relative contribution of globular clusters would be enhanced relative to what is plotted.

Figure 5.

Ratio of GC UVLFs to global UVLFs at a variety of redshifts under the assumption that ξ = 1 (solid lines) or 10 (dashed lines), assuming f₁₅₀₀ = 0 for t < t_gas-free and 1 otherwise. If ξ = 1, then ≲1 per cent of the UVLF accessible to HST in blank fields (M₁₅₀₀ < −17) originates from GCs; by M₁₅₀₀ ≈−14, approximately 20 per cent of the UVLF comes from GCs. The GC UVLF actually exceeds the global UVLF at most redshifts at magnitudes accessible to HST if ξ = 10.

It therefore appears unavoidable that GCs contribute appreciably (at minimum) or perhaps dominantly (for high values of ξ_eff) to high-redshift UVLFs. Katz & Ricotti (2013) reached similar conclusions with a complementary approach to computing high-z contributions of GCs to UVLFs. Their derived GC UVLFs are significantly steeper than those obtained here, however, even though their assumed low-mass slope of the initial GC stellar mass function is −1.5, Katz & Ricotti (2013) find GC UVLFs that have faint-end slopes that are steeper than −2. As is shown in Appendix B, the GC UVLFs calculated here have slopes that are shallower than −2, even if the initial GC stellar mass function is assumed to have a slope of −1.9.

3. DISCUSSION

The results of the previous section indicate that GCs can contribute substantially to high-redshift UV emission. A natural question, therefore, is whether individual GCs will be detectable by future (or even current) facilities, or if only the sum of the light from multiple GCs will be visible.

Fig. 6 shows the probability distribution of M₁₅₀₀ from the integrated light in GCs for systems with 1, 10, 100, and 1000 GCs at z = 6. The plotted distributions assume ξ_eff = 1 and f₁₅₀₀(t < t_gas-free) = 0; results for other values of ξ_eff can be obtained through a straightforward rescaling (as noted on the x-axis of the figure). Changing from f₁₅₀₀(t < t_gas-free) = 0 to 1 would result in a brightward shift of 0.6 magnitudes for each distribution, as ≈40 per cent of the lifetime UV emission of each GC comes in the first 10 Myr based on the assumptions adopted here. If t_gas-free is set to 5 Myr rather than 10 Myr, then 25 per cent of the lifetime UV emission is obscured, corresponding to a brightward shift of 0.3 mag.

Figure 6.

Probability distributions for M₁₅₀₀ from GCs (integrated over a halo’s GC population) for haloes of 4 different masses at z~ 6. haloes at the threshold of GC formation [M_halo(z = 6) = 10⁹ M_☉, B17] span a wide range of possible values of M₁₅₀₀, as any individual GC has a wide range of possible masses and formation times. At increasingly high masses, and correspondingly higher numbers of GCs, the widths of the distributions shrink, as the GCs come closer to fully sampling the underlying luminosity function. All calculations for the plot assume that ξ_eff = 1 (though the results can easily be scaled to other values of ξ_eff, as is indicated on the x-axis) and f₁₅₀₀(t < t_gas-free) = 0; the probability distribution for a 10¹² M_☉ halo has been rescaled downward by a factor of 2 to decrease the dynamic range.

Whether or not any individual GC is detectable depends on how likely it is to exceed the detection threshold of a given instrument. Table 1 lists the probability that at least one GC can be directly detected in haloes of various masses at z ≈ 6 with HST or JWST. The M_halo − M₁₅₀₀ relationship is computed via abundance matching the halo mass function (computed following Sheth, Mo & Tormen 2001 using the Planck Collaboration XIII 2016 cosmology) and the observed z ~ 6 UVLF. Results are listed separately for models with ξ_eff = 1 (models 1a and 1b) and ξ_eff= 10 (models 2a and 2b). For the rarest, most massive haloes at z = 6 (M_halo ≈ 10¹² M_☉, corresponding to M₁₅₀₀ ≈ −21.6), HST will be able to detect at least 1 GC 2–20per cent of the time if ξ_eff = 1 and in virtually every case if ξ_eff = 10. JWST can detect at least one GC directly in essentially all scenarios considered here. Moving to lower-mass haloes, detection becomes increasingly unlikely. Nevertheless, JWST will detect one GC in 1.5 per cent (39 per cent) of haloes with M_halo(z ~ 6) ≈ 10¹⁰ M_☉ (corresponding to M₁₅₀₀ ≈ −15.6, which is near the faint limit of JWST detections in hypothetical deep fields) under pessimistic (optimistic) assumptions.

Table 1. Observability of GCs at high redshifts.

Model 1 assumes ξ_eff = 1, while model 2 assumes ξ_eff = 10. Model a assumes f₁₅₀₀(t < t_gas-free) = 0, while model b assumes f₁₅₀₀(t < t_gas-free = 1). The second column gives M₁₅₀₀ corresponding to the average value of the luminosity coming from all GCs in the given halo, while the third column gives the median value of M₁₅₀₀ and the symmetric 80% confidence interval about the median. Columns 4 and 5 give the fraction of such haloes that will host at least one observable GC for HST (assumed detection threshold of M₁₅₀₀ = −17) and JWST (assumed detection threshold of M₁₅₀₀ = −15).

Model	M1500(GCs)	M1500(GCs)	fobs (HST)	fobs (JWST)
Mhalo = 109 M☉, M1500 = −12.2
1a	−10.6	$-{8.7}_{-2.6}^{+2.1}$	1.0 × 10−5	1.5 × 10−3
1b	−11.2	$-{8.7}_{-2.7}^{+2.1}$	2.3 × 10−4	5.5 × 10−3
2a	−13.1	$-{11.2}_{-2.6}^{+2.1}$	3.1 × 10−3	0.034
2b	−13.7	$-{11.2}_{-2.7}^{+2.1}$	9.0 × 10−3	0.046
Mhalo= 1010 M☉, M1500= −15.6
1a	−13.1	$-{12.5}_{-1.4}^{+1.1}$	2.0 × 10−4	0.014
1b	−13.7	$-{12.8}_{-1.9}^{+1.2}$	2.6 × 10−3	0.054
2a	−15.6	$-{15.0}_{-1.4}^{+1.1}$	0.032	0.30
2b	−16.2	$-{15.3}_{-1.9}^{+1.2}$	0.092	0.39
Mhalo= 1011 M☉, M1500= −19.1
1a	−15.6	$-{15.5}_{-0.6}^{+0.5}$	1.9 × 10−3	0.14
1b	−16.2	$-{16.1}_{-0.9}^{+0.7}$	0.026	0.43
2a	−18.1	$-{18.1}_{-0.6}^{+0.5}$	0.28	0.97
2b	−18.7	$-{18.6}_{-0.9}^{+0.7}$	0.61	0.99
Mhalo= 1012 M☉, M1500= −21.6
1a	−18.1	$-{18.2}_{-0.2}^{+0.2}$	0.018	0.76
1b	−18.7	$-{18.8}_{-0.3}^{+0.3}$	0.23	1.0
2a	−20.6	$-{20.7}_{-0.2}^{+0.2}$	0.96	1.0
2b	−21.2	$-{21.3}_{-0.3}^{+0.3}$	1.0	1.0

The possibility of detecting GCs, or that some current observations may already be revealing proto-GCs, in the high-redshift Universe was anticipated by Carlberg (2002) and has recently gained significant attention (Schaerer & Charbonnel 2011; Katz & Ricotti 2013; Bouwens et al. 2017; Boylan-Kolchin 2017; Renzini 2017; Vanzella et al. 2017). In particular, the very small sizes (10s of pc) of intrinsically faint, gravitationally lensed sources at z~6 in the Hubble Frontier Fields appear consistent with GCs or star cluster complexes in formation (Kawamata et al. 2015; Bouwens et al. 2017; Vanzella et al. 2017; Kawamata et al. 2018), fig.12 of Bouwens et al. (2017) shows a population of galaxies with roughly constant sizes of 5–15 pc over the luminosity range −19.5 ≲ M₁₅₀₀ ≲ −14. GCs themselves are likely easier to detect than other objects with similar values of M₁₅₀₀, as their surface brightnesses are substantially higher (see Zick, Weisz & Boylan-Kolchin 2018 for a more detailed discussion).

Compact, UV-bright sources are also found in simulations of reionization-era galaxies (Kimm et al. 2016; Ma et al. 2017; Kim et al. 2018). The long-term stability of UV clumps is unclear: some may be clusters that are not gravitationally bound and disrupt rapidly after formation (within ~10⁷ yr), similar to star clusters in low-redshift merging systems (Bastian et al. 2005; Fall, Chandar & Whitmore 2005). If this is the case, an even higher fraction of high-redshift star formation may occur in dense, GC-like systems than the estimates in this paper imply, as these disrupting clusters would not evolve into GCs at z = 0. In fact, it may be possible to model high-z UVLFs using the tools presented here in the limit that all stars form in clusters with either log-normal or Schechter-like shapes.

There is certainly room for improvement in the modelling presented here. Perhaps the most important refinement would be to include a M_★ dependence in ξ_eff; detailed estimates of f_surv would also be highly informative. A better understanding of both absolute and relative ages for blue GCs is also essential to model their relevance in the reionization era and to constrain Δt (though it appears difficult to reconcile a later formation epoch with observed properties of z ~ 4 galaxies; Katz & Ricotti 2013). The stellar IMF is also a source of uncertainty: more top-heavy IMFs will produce more UV luminosity per unit stellar mass, changing the relationship between M₁₅₀₀ and M_★. None the less, the power of the relatively straightforward models described in this paper to directly constrain properties of GCs in the high-redshift Universe and their contributions to cosmic reionization is encouraging. Some ‘galaxies’ already detected in HST deep fields are likely to be GCs in formation; in the JWST era, observations of GCs in formation should be commonplace.

APPENDIX A: LOG-NORMAL LUMINOSITY FUNCTIONS

The component of the GCLF originating from GCs with ages in the range [t_i, t] can then be written as a time integral over the individual cluster luminosity functions:

$${\varphi }_{\text{GC}}^i(M,t)\propto {\int }_{t_i}^t\text{exp}\left(-\frac{(M-\overline{M}(t){)}^2}{\sqrt{2{\sigma }^2}}\right)f(t)\mathrm{d}t.$$ (A1)

If f(t) ∝ t^{−(1 + η)}, the resulting integral can be computed analytically (Basu, Gil & Auddy 2015):

$${\varphi }_{\text{GC}}^i(M,t)\propto \text{exp}\left(\frac{\eta B}{C}+\frac{{\eta }^2}{4C^2}\right)[E_{\eta }(t)-E_{\eta }(t_i)],$$ (A2)

$$E_{\eta }(\tau )\equiv \text{erf}\left[B+\frac{\eta }{2C}+C\text{ln}\left(\frac{\tau }{\text{Myr}}\right)\right],$$ (A3)

where

$$B=-\frac{M-M^{\ast }}{\sqrt{2{\sigma }^2}},$$ (A4)

$$C=\frac{2.5b}{\text{ln}(10)\sqrt{2{\sigma }^2}},\text{and}$$ (A5)

$$M^{\ast }=-14.5-2.5{\log }_{10}(\xi a).$$ (A6)

I will assume η = −1 (i.e. ${\stackrel{.}{n}}_{\text{GC}}=\text{constant}$). The full intrinsic GC UVLF is then a sum over component ${\varphi }_{\text{GC}}^i(M,t)$ values, with piecewise time ranges specified by equation (A2), appropriately normalized for the time evolution of ${\stackrel{.}{n}}_{\text{GC}}$.

APPENDIX B: SCHECHTER LUMINOSITY FUNCTIONS

The results of this paper generally assume that the observed log-normal luminosity function of GCs is a manifestation of the initial conditions. However, this may not be the case: an initially Schechter-like luminosity function can evolve to a log-normal one over a Hubble time.

The Schechter (1976) luminosity function is parametrized by a normalization (ϕ*), a characteristic luminosity (L*), and a faint-end slope (α):

$$\varphi (L)=\frac{{\varphi }^{\ast }}{L^{\ast }}{\left(\frac{L}{L^{\ast }}\right)}^{\alpha }\text{exp}\left(-\frac{L}{L^{\ast }}\right)$$ (B1)

Analogously to the case of the log-normal luminosity function, we can consider L* to be a function of time and compute the GC UVLF as an integral

$$\varphi (L,t)\propto \int \varphi [L,L^{\ast }(t)]\frac{\mathrm{d}t}{\Delta t}$$ (B2)

(where I have assumed ${\stackrel{.}{n}}_{\text{GC}}$ is constant, consistent with the rest of this paper; it is straightforward to generalize to a power-law dependence on time). With the time evolution of L* given by

$$L^{\ast }(t)=L_0{\left(\frac{t}{t_0}\right)}^{-b},,$$ (B3)

the luminosity function becomes

$$\varphi (L)=\frac{{\varphi }^{\ast }}{L_0}{\left(\frac{L}{L_0}\right)}^{-1-1∕b}\frac{t_0}{\Delta t}{b}^{-1}[\gamma (\lambda ,u)-\gamma (\lambda ,u_0)],$$ (B4)

where

$$\lambda =\alpha +\frac{1}{b}+1,$$ (B5)

$$u(t)=u(L,t)=\frac{L}{L_0}{\left(\frac{t}{t_0}\right)}^b$$ (B6)

and γ (a, x) is the lower incomplete gamma function (Abramowitz & Stegun 1972). As with the log-normal case, the GCLF is sum over individual components corresponding to the piecewise time ranges in equation (B2).

The faint-end slope of ϕ(L) in the limit L ≪ L₀ will be approximately −1 – 1/b if α > −1 – 1/b and α otherwise, as γ (λ, u)∝u^λ as u → 0. Notably, this faint-end slope is identical to the log-normal case for relatively shallow values of α, which is a direct result of the time evolution of L*(t) – determined by stellar evolution – in equation (B3). If the faint-end slope of the initial GC stellar mass function is steeper than −(1 + 1/b), then it is this initial faint-end slope that sets the overall slope of ϕ_GC(L).

Fig. B1 compares the GC UVLFs (after 1000 Myr of GC formation) for initial GC stellar mass functions that are Schechter-like (with faint-end slope −1.9; black line) and log-normal (grey). The input initial GC stellar mass functions are normalized such that the number of GCs brighter than $\overline{M}$ (the peak value in the log-normal stellar mass function) are equal. The resulting GC UVLFs are nearly identical for bright magnitudes, while the Schechter-like initial GC stellar mass function results in a higher amplitude for ϕ_GC(L) at fainter magnitudes.

Figure B1.

A comparison between the derived GC UVLF at the end of the GC formation epoch (after 1 Gyr) under the assumption of initially log-normal or Schechter-like GC stellar mass functions (grey and black, respectively). The GC UVLF in the case of a Schechter-like stellar mass function is itself Schechter-like, with a faint-end slope that is either the same as the underlying Schechter function (α) or, if the input Schechter function slope is shallower than −1.7, is ≈ −1.7 (see discussion below equation B5).