The Persistence of Population III Star Formation

ABSTRACT

We present a semi-analytic model of star formation in minihaloes below the atomic cooling threshold in the early Universe, beginning with the first metal-free stars. By employing a completely feedback-limited star formation prescription, stars form until the self-consistently calculated feedback processes halt formation. We account for a number of feedback processes including a metagalactic Lyman–Werner (LW) background, supernovae, photoionization, and chemical feedback. Haloes are evolved combining mass accretion rates found through abundance matching with our feedback-limited star formation prescription, allowing for a variety of Population III (Pop III) initial mass functions (IMFs). We find that, for a number of models, minihalo Pop III star formation can continue on until at least z ∼ 20 and potentially past z ∼ 6 at rates of around 10⁻⁴ to 10⁻⁵ M_⊙ yr⁻¹ Mpc⁻³, assuming these stars form in isolation. At this point LW feedback pushes the minimum halo mass for star formation above the atomic cooling threshold, where we assume that the mode of star formation changes. We find that, in most models, Pop II and Pop III star formation coexist over cosmological time-scales, with the total star formation rate density and resulting radiation background strongly dominated by the former before Pop III star formation finally ends. These molecular cooling haloesform at most ∼10³M_⊙ of minihalo Pop III stars over multiple generations during this phase and typically have absolute magnitudes in the range of M_AB = −5 to −10. We also briefly discuss how future observations from telescopes such as James Webb Space Telescope or Wide-Field Infrared Survey Telescope and 21-cm experiments may be able to constrain unknown parameters in our model such as the IMF, star formation prescription, or the physics of Pop III stars in minihaloes.

galaxies: high-redshift, dark ages, reionization, first stars, cosmology: theory

1 INTRODUCTION

The first stars to have formed in the Universe were likely very different from those observed today, and their formation was crucial to the early evolution of galaxies. It is thought that these Population III (Pop III) stars formed in metal-free gas clouds through molecular hydrogen cooling, and they were therefore much more massive (Bromm, Coppi & Larson 1999; Abel, Bryan & Norman 2002) and luminous than today’s metal-enriched stars (see Bromm 2013). Because of their high luminosity and the small size of their birth haloes, feedback may have played an important role in the formation of Pop III stars (Machacek, Bryan & Abel 2001; Shapiro, Iliev & Raga 2004; Wise & Abel 2007; O’Shea & Norman 2008), and they most likely formed in very small numbers, perhaps in isolation (Visbal, Haiman & Bryan 2017). Despite their small numbers, they must have been able to produce enough metals to eventually allow their haloesto begin forming the more traditional Population II (Pop II) stars in a reasonably short amount of time.

There have been many attempts to study the formation and properties of these Pop III stars through the use of detailed numerical simulations (e.g. Machacek et al. 2001; Wise & Abel 2007; O’Shea & Norman 2008; Maio et al. 2010; Stacy, Greif & Bromm 2012; Hirano et al. 2015; Xu et al. 2016; Sarmento, Scannapieco & Cohen 2018), analytic arguments (e.g. McKee & Tan 2008; Kulkarni et al. 2013), and semi-analytic models (e.g. Trenti, Stiavelli & Michael Shull 2009; Crosby et al. 2013a; Jaacks et al. 2017; Visbal et al. 2017). Simulations of their supernovae (Heger & Woosley 2002 and Heger & Woosley 2010) have found that such stars are able to produce a very high mass of metals that will eventually be used to cool gas more efficiently and form stars from a more traditional initial mass function (IMF). In some simulations, Pop III stars are found to form often in binaries (Turk, Abel & O’Shea 2009), which could potentially produce a cosmologically relevant abundance of Pop III high-mass X-ray binaries.

While there has been much work done to study the properties of Pop III stars, we have yet to directly observe a Pop III halo. There has been one candidate at z ∼ 7 described in Sobral et al. (2015), although there are still many other possible explanations (e.g. Agarwal et al. 2016; Bowler et al. 2017), and recent Atacama Large Millimeter Array (ALMA) observations have detected [C ii] consistent with a normal star-forming galaxy (Matthee et al. 2017). Unfortunately, since it is thought that Pop III stars form in very low mass haloesat high redshift, it may prove incredibly difficult to directly observe a Pop III source. If the mixing of metals in Pop III haloesis very inefficient, as in Sarmento et al. (2018), Pop III stars may be able to coexist with metal-enriched Pop II stars to late times, allowing for a better chance of observation. Even if these haloesare too faint to observe with the James Webb Space Telescope (JWST), however, we may be able to detect their supernovae with an instrument such as the Wide-Field Infrared Survey Telescope (WFIRST; Whalen et al. 2013), or we could potentially see their effect on the cosmological 21-cm background (Mirocha et al. 2018). All of these measurements are very sensitive to the overall shape of the Pop III star formation rate density (SFRD) and the timing of the transition to Pop II star formation.

In order to understand how Pop III haloesmake the transition to Pop II star formation, we must focus on both internal and external processes. In a particular halo, star formation produces important feedback effects such as supernovae and photoionization (see Bromm, Yoshida & Hernquist 2003; Whalen et al. 2008a,b). These effects can either limit or completely cut off Pop III star formation based on the size of the halo and its growth due to mass accretion.

But the Pop III phase is also sensitive to large-scale radiation fields generated by those stars and their successors. The evolution of Pop III star formation in the presence of global feedback driven by Pop II star formation is not well understood and is a key interest to this work. In particular, UV photons in the Lyman–Werner (LW) band emitted by Pop II stars can destroy the molecular hydrogen in Pop III clouds necessary for cooling (Haiman, Rees & Loeb 1997). This sets the minimum halo mass at which molecular hydrogen can self-shield in a halo and cool to form stars (Visbal et al. 2014). This minimum mass is a crucial physical quantity, as the halo mass function is very steep. The properties of even the Pop II haloesat these redshifts are unknown, however, so studying their effects on the formation of Pop III stars in a flexible model is of much use.

In this work, we employ an efficient and flexible semi-analytic model that allows us to model the formation of Pop III stars over a wide range of parameters and assumptions. While our model is certainly simpler than numerical simulations with similar goals, we are able to compare many different models while still self-consistently computing the effects of a wide range of feedback processes including a metagalactic LW background, supernovae, photoionization, and chemical feedback. As described below, we use many results from observations, simulations, and analytic arguments to simplify our model and justify our assumptions.

Pop III star formation is a complex process that can proceed through different channels. In this work, we will focus on the ‘classical’ process through which primordial gas remains neutral while it accretes onto a dark matter halo and then cools purely through H₂ line emission (Bromm et al. 1999; Abel et al. 2002). The resulting star-forming regions are relatively hot and thus have Jeans masses of |${\gtrsim }100 \, \mathrm{M}_\odot$|⁠, which likely leads to massive stars (though their actual masses are the subject of intense debate; e.g. Bromm et al. 1999; Abel et al. 2002; Clark et al. 2011; Hirano & Bromm 2017). In this paper, we distinguish this ‘minihalo’ mode from star formation in haloeswith virial temperatures ≳10⁴ K: in that case, the chemistry differs from the classic case (e.g. HD cooling becomes significant; Oh & Haiman 2002; Johnson & Bromm 2006; Yoshida, Omukai & Hernquist 2007b; McGreer & Bryan 2008), and the resulting stars, even if they remain metal-free, form in very different environments. Moreover, as we show below, in this regime haloesbecome more stable to supernova feedback and are likely to develop quasi-equilibrium interstellar media, inside which metal mixing will become significant. Our model will therefore focus primarily on Pop III star formation inside ‘minihaloes’ below the atomic cooling threshold. We will study how long this mode can persist through the early generations of galaxies. Several other studies consider the possibility of Pop III star formation in more massive haloes, including the mixing of metals throughout the interstellar medium (e.g. Sarmento et al. 2018).

In Section 2 we describe the properties of dark matter haloesin our model, including the halo mass function and the growth of haloes. In Section 3 we discuss our treatment of the first minihaloes that will form metal-free Pop III stars. In Section 4 we describe the properties of the Pop II haloesthat form the very first generations of metal-enriched stars. We present our results in Section 5, caveats to our model in Section 6, a discussion of their implications for observations of star formation in the early Universe in Section 7, and conclude in Section 8.

In this work, we use a flat, ΛCDM cosmology with Ω_m = 0.28, Ω_b = 0.046, |$\Omega _\Lambda = 0.72$|⁠, σ₈ = 0.82, n_s = 0.95, and h = 0.7, consistent with the results from Planck Collaboration XIII (2016). Any distances presented are shown in comoving units.

2 HALO PROPERTIES

Our model will follow the growth of a set of dark matter haloesfrom very early times to z = 6. To begin, we choose a set of haloeswith z = 6 masses from |$10^6$| to |$10^{13} \, \mathrm{M}_\odot$| in 10 000 logarithmically spaced bins. These haloesare then tracked backwards to z = 50 using the abundance matching technique described in Section 2.2 in 1 Myr time-steps. With these mass histories in hand, we then track each halo individually, applying the semi-analytic model discussed in future sections to determine the feedback-limited star formation histories of both Pop III and Pop II stars. Global quantities (to be discussed in the following section) such as the SFRD and LW background are found by averaging all haloesover the mass function.

2.1 The halo mass function

The first component of our model is the number density of dark matter haloes. We assume that all haloesthat can accrete gas will form stars, so knowing their abundance is of vital importance.

The halo mass function can be written as

\begin{equation*} n(M) = \frac{\bar{\rho }}{M} f(\sigma) \left| \frac{\text{d}\log {\sigma }}{\text{d}M}\right|, \end{equation*}

(1)

where σ is the density variance, |$\bar{\rho }$| is the matter density, and f(σ) depends on the particular form of the mass function. In general, f(σ) can be found through fits to simulations or through analytic arguments. We test the dependence of our model on the chosen mass function by examining a Sheth–Tormen mass function (Sheth & Tormen 1999) and the mass function of Trac, Cen & Mansfield (2015), which is a fit to high redshift simulations. These two choices do not lead to any significant differences in our results, especially given the wide uncertainty in the other model parameters (see Fig. 1). We employ the Trac et al. (2015) mass functions throughout the rest of this paper.

Figure 1.

Mass histories of haloesin our model. Halo growth is determined by abundance matching, where we assume haloesremain at a constant comoving number density throughout time and find their mass by comparing mass functions at each time-step (see Section 2.2). The solid and dashed lines use mass functions from Sheth & Tormen (1999) and Trac et al. (2015), respectively. The dotted line is made using the Trac et al. (2015) fit to equation (2). Haloes are initialized at the same mass at z = 6, and their masses are then tracked backwards to z = 50.

2.2 The growth of dark matter haloes

There have been many attempts to model the overall mass accretion rates of dark matter haloesfollowing simulations or analytic arguments (e.g. McBride, Fakhouri & Ma 2009; Behroozi, Wechsler & Conroy 2013; Dekel & Krumholz 2013; Goerdt et al. 2015). For example Dekel & Krumholz (2013) write the overall mass accretion rate as

\begin{equation*} \dot{M} = A M^{\mu } \left(1 + z \right)^{\beta }\!, \end{equation*}

(2)

where M is the halo mass, A is a normalization, μ ≳ 1 from simulations (see McBride et al. 2009 and Trac et al. 2015), and β = 2.5, which is consistent with analytic arguments (e.g. Neistein & Dekel 2008). By fitting to numerical simulations, Trac et al. (2015) found that μ = 1.06 for M ∼ 10⁸–|$10^{13} \, \mathrm{M}_\odot$| at z ∼ 6–10, which overlaps with the mass and redshift range we need but does not probe the full range we require.

While these accretion rates have been tested in simulations at moderate mass ranges and redshifts (z ∼ 6), it is unclear whether they are valid for the smallest haloesthat begin forming stars at very high redshift. Since the focus of our model is on Pop III star formation, we use another model for the growth of haloes.

We make the simple assumption that haloesmaintain their comoving number densities throughout time. This allows us to determine the mass histories of haloesdirectly from the halo mass function by matching their abundances given by the mass function over a range of redshifts (see Furlanetto et al. 2017 for a more detailed description of this technique). The mass histories for a number of haloesin our model found through abundance matching are shown in Fig. 1. Results with our two mass functions only differ by a factor of ∼2 at the highest redshifts, so our analysis does not depend strongly on its exact form. The dotted curves use the Trac et al. (2015) simulation fit to equation (2), which has a much larger discrepancy. However, it is only off by a factor of the order of unity for low mass haloesand does not have a large effect on persistence of Pop III star formation in minihaloes (see Section 6.1).

We note that our model for the growth of dark matter haloesonly follows the average growth, and thus we ignore the effect of mergers. Behroozi & Silk (2015) found that the majority of accretion onto dark matter haloesat high redshift occurs as smooth accretion from the intergalactic medium (IGM), though we will revisit the effects of mergers in Section 6.2.

3 PROPERTIES OF POP III HALOES

The first star-forming minihaloes were very small and likely contained only a handful of very massive, metal-free stars. These stars formed in molecular clouds that were cooled by molecular hydrogen, rather than the metal-line cooling that occurs in star-forming regions today. Correctly modelling the growth of these haloesand the time at which the first stars form is of vital importance to this work so that we can self-consistently model the feedback-limited star formation in the early Universe and the transition to more typical, Pop II star formation. In this section, we will describe the properties of these haloesand the processes by which minihalo Pop III star formation began.

3.1 The minimimum mass

In our model, we allow a new halo to begin forming stars only when it passes a minimum mass determined by the time-weighted Jeans mass (filtering mass) and the physics of H₂ cooling. Tegmark et al. (1997) found that a halo must exceed a certain fraction of molecular hydrogen in order for cooling to become efficient, which is given by

\begin{eqnarray*} f_{\text{crit, H}_2} &\approx& 1.6 \times 10^{-4} \left(\frac{1+z}{20}\right)^{-3/2} \left(1 + \frac{10 T_3^{7/2}}{60 + T_3^4}\right)^{-1} \nonumber \\ &&\times \exp {\left(\frac{512 \text{K}}{T}\right)}, \end{eqnarray*}

(3)

where T is the virial temperature of the halo and T₃ = T/10³K. In these haloes, molecular hydrogen is formed primarily through the process

\begin{equation*} \text{H} + \text{e}^- \rightarrow \text{H}^- + h\nu \end{equation*}

(4)

\begin{equation*} \text{H}^- + \text{H} \rightarrow \text{H}_2 + \text{e}^-, \end{equation*}

(5)

where free electrons catalyse the reaction. At high redshifts, H⁻ can be easily destroyed by cosmic microwave background (CMB) photons, so we must balance the formation rate with the destruction rate to get the overall H₂ fraction in a halo. Tegmark et al. (1997) found this fraction to scale with a halo’s virial temperature as

\begin{equation*} f_{\text{H}_2} \approx 3.5 \times 10^{-4} \text{ }T_3^{1.52}. \end{equation*}

(6)

Once |$f_{\text{H}_2} \gt f_{\text{crit, H}_2}$|⁠, molecular hydrogen cooling will become efficient enough to cool gas into the first star-forming molecular clouds. In our model, we assume star formation begins immediately after this criterion is met, with no delay. This occurs in very small haloeswith masses |${\sim } 10^5\, \mathrm{M}_\odot$| at the highest redshifts in our model (z ∼ 50).

After the first stars form, however, the minimum mass will instead be set by the metagalactic LW background. The LW band consists of photons in the energy range 11.5–13.6eV that photodissociate molecular hydrogen through the Solomon process (see Stecher & Williams 1967). If a halo is present in a high enough background of LW photons, it can lose all of its molecular hydrogen and no longer be able to cool gas in star-forming regions. Therefore,a self-consistent calculation of this background is required to determine the minimum masses of Pop III haloes(see Haiman et al. 1997; Holzbauer & Furlanetto 2012; Visbal et al. 2014).

Following Visbal et al. (2014) the background can be written as

\begin{equation*} J_{\text{LW}}(z) = \frac{c}{4\pi} \int _{z}^{z_{\text{m}}} \frac{\text{d}t}{\text{d}z^\prime } \left(1+z\right)^3 \epsilon (z^\prime) \text{d}z^\prime, \end{equation*}

(7)

where ε(z) is the specific LW comoving luminosity density and z_m is the maximum redshift at which LW photons can be emitted before they redshift into a Lyman line. For simplicity, we assume that all photons can be redshifted by a maximum of 4 per cent (Visbal et al. 2014) before being absorbed by a Lyman resonance of neutral hydrogen in the IGM. This can be found from a simple fit to calculations of the redshifting of these lines. We calculate ε(z) as

\begin{equation*} \epsilon (z) = \int _{M_\text{min}}^{\infty } n(M) \frac{\Omega _{\text{b}}}{\Omega _{\text{m}}} \frac{\dot{M}_\ast }{m_{\text{p}}} \left(\frac{N_{\text{LW}} E_{\text{LW}}}{\Delta \nu _{\text{LW}}} \right) \text{d}M, \end{equation*}

(8)

where M_min is the minimum mass at which a halo can form stars, |$\dot{M}_\ast$| is the star formation rate in a halo of mass M, N_LW is the number of LW photons produced per baryon in stars, E_LW is the average energy of a LW photon, and Δν_LW is the frequency range of the LW band. We take N_LW = 1 × 10⁵, which is roughly constant for massive Pop III stars (Schaerer 2002), N_LW = 9690 for Pop II stars (Barkana & Loeb 2005), E_LW = 11.9 eV, and Δν_LW = 5.8 × 10¹⁴ Hz. We compute |$\dot{M}_\ast$| individually for each halo in both the Pop II and Pop III phases.

In order to determine which haloescan form stars under a given LW background, we look to hydrodynamical simulations (e.g. Machacek et al. 2001; Wise & Abel 2007; O’Shea & Norman 2008). Visbal et al. (2014) find that the critical mass above which haloescan form stars under a given LW background can be approximated by

\begin{equation*} M_{\text{min}} = 2.5 \times 10^5 \left(\frac{1+z}{26} \right)^{-1.5} \left(1 + 6.96 \left(4\pi J_{\text{LW}} \right)^{0.47}\right). \end{equation*}

(9)

Once we calculate J_LW from our model, we can determine which minihaloes will form Pop III stars by only considering masses where M > M_min.

For a halo to obtain enough gas to begin star formation, its gas mass must exceed the time-weighted Jeans mass, known as the filtering mass (see Gnedin & Hui 1998). We enforce this in our model, only allowing stars to form in haloeswhere the gas mass exceeds the fit to the filtering mass found in Naoz & Barkana (2007). In practice, this mass is always below the minimum mass from the LW background after the first stars form, and it only exceeds the Tegmark et al. (1997) mass at redshifts much higher than we are interested. Therefore, the filtering mass does not have a noticeable impact on our results.

In this work, we neglect the potential feedback effects of X-rays, which can catalyse H₂ formation by enhancing the free electron fraction and thus act as a source of positive feedback, or, could alternatively heat gas and prevent further fragmentation and star formation (e.g. Machacek, Bryan & Abel 2003; Kuhlen & Madau 2005). The importance and sense of the feedback (i.e. positive or negative) will depend on the interplay between the LW and X-ray backgrounds, which in turn depends on the detailed properties of sources and their number density as a function of redshift (e.g. Ricotti 2016). For now, we neglect these effects and defer a more detailed consideration to future work.

We also neglect potential H⁻ photodetachment triggered by an infrared background (as discussed in Wolcott-Green, Haiman & Bryan 2017) which can halt the formation of H₂. To test this assumption, we calculate the photodetachment rate, |$k_{\text{H}^-}$|⁠, using the conversion rate presented by Agarwal et al. (2016) and our model LW backgrounds. We find that, even in the case where this effect should be most important (i.e. only Pop II star formation at lower redshifts with a high SFRD), |$k_{\text{H}^-}$| is well below the threshold for H⁻ photodetachment to dominate over the LW background (Wolcott-Green et al. 2017). We note, however, that the calculation in Wolcott-Green et al. (2017) is for haloesat the atomic cooling threshold, while our Pop III haloesare generally at lower mass. We defer a detailed calculation of this effect for lower mass haloesto the future.

3.2 The Pop III IMF

Once a halo has exceeded the minimum mass required for Pop III star formation, we begin to add the first stars. Since the gas clouds forming these stars cool primarily through molecular hydrogen, which is less efficient than metals, these kinds of Pop III stars were likely very massive (Bromm et al. 1999). The Pop III IMF is very uncertain, however, so we leave this as a parameter in our model (see Table 1). Stars are randomly sampled from the chosen IMF and placed in isolation in each star-forming region.

The simplest IMF we consider is a delta function, where all Pop III stars form at a single mass. The preferred mass is not known a priori, but one well-motivated choice is to follow McKee & Tan (2008), who find the maximum mass of a massive Pop III star by calculating the mass at which radiation pressure will stop accretion. McKee & Tan (2008) find this maximum mass to be

\begin{equation*} M_{\text{max}} \approx 145 \, \mathrm{M}_\odot \left(\frac{25}{T_3}\right)^{0.24}. \end{equation*}

(10)

If Pop III stars form in isolation in minihaloes, then it is possible that all stars eventually reach this mass. In this case, we simply set the mass of every Pop III star to this maximum mass. This case is especially interesting as it is just above the minimum mass at which pair-instability supernovae occur, so small details in the stellar masses are actually quite important.

We also allow for both lower mass and higher mass cases compared to the McKee & Tan (2008) model. In these cases we use a Salpeter-like IMF in various mass ranges. For the low-mass case, we use a minimum mass of 20 |$\, \mathrm{M}_\odot$| and a maximum mass of the McKee & Tan (2008) mass (see Bromm et al. 1999 for a discussion of the fragmentation of protostellar discs and its effect on stellar masses). As an extreme high-mass case, we use a minimum mass of 200|$\, \mathrm{M}_\odot$| and a maximum mass of 500 |$\, \mathrm{M}_\odot$|⁠. As we will discuss in Section 3.4, the choice of IMF has great implications for supernova feedback and its effects on these growing haloes.

We note that our Pop III model assumes a single star or binary system per site of star formation, with a mass drawn from the IMFs described above. This differs from other treatments (e.g. Jaacks et al. 2017; Visbal et al. 2017), which often utilize a fixed mass of star formation or a mass-dependent star formation efficiency. We compare to these similar works in Section 5.5, as well as test the dependence of our results on this assumption.

It is thought that these massive stars may often form in binaries (Turk et al. 2009), so we also investigate the case of a non-zero binary fraction. Mirocha et al. (2018) find that high-mass X-ray binary systems can produce a unique signature in the global 21-cm signal although, in this work, a non-zero binary fraction only serves to increase the overall star formation efficiency.

Table 1.

Pop III IMFs used in this work. Note that the metal yields are given as the mass of metals produced per unit mass of star formation.

IMF	Mmin	Mmax	Slope	ESN/M* [ergs / |$\, \mathrm{M}_\odot$|]	C Yield	O Yield
Low mass	20 |$\, \mathrm{M}_\odot$|	|${\sim } 145 \, \mathrm{M}_\odot$| (equation 10)	−2.35	1.58 × 1049	5.63 × 10−3	6.25 × 10−2
Mid mass	|${\sim } 145 \, \mathrm{M}_\odot$| (equation 10)	|${\sim } 145 \, \mathrm{M}_\odot$| (equation 10)	N/A	6.90 × 1049	3.13 × 10−2	3.16 × 10−1
High mass	200 |$\, \mathrm{M}_\odot$|	500 |$\, \mathrm{M}_\odot$|	−2.35	1.41 × 1049	1.41 × 10−3	4.49 × 10−2

3.3 Photoionization feedback from Pop III stars

In order to determine how many Pop III star-forming regions should form in a single halo, we appeal to photoionization feedback. Since Pop III stars were likely very massive and luminous, they would be able to ionize much of their surrounding material, making it unable to cool and form stars. Assuming a massive Pop III star emits like a blackbody with effective temperature T_eff (see Stacy et al. 2012), the number of ionizing photons produced is given by

\begin{equation*} \dot{N} = \frac{\pi L}{\sigma T_{\text{eff}}^4} \int _{\nu _{\text{min}}}^{\infty } \frac{B_{\nu }}{h\nu } \text{d}\nu, \end{equation*}

(11)

where L is the star’s luminosity, σ is the Stefan–Boltzmann constant, ν_min is the minimum photon frequency required for ionization, and B_ν is the blackbody spectrum of the star. We approximate the luminosity of the star over its lifetime as the zero-age main sequence (ZAMS) luminosity, and we calculate T_eff assuming the star emits like a blackbody. We use the ZAMS stellar radius and luminosity given by Stacy et al. (2012) as

\begin{equation*} L_{\text{ZAMS}}(M) = 1.4 \times 10^4 L_{\odot } \left(\frac{M}{10 \, \mathrm{M}_{\odot }}\right)^2, \end{equation*}

(12)

\begin{equation*} R_{\text{ZAMS}} = 3.9 R_{\odot } \left(\frac{M}{10 \, \mathrm{M}_{\odot }} \right)^{0.55}. \end{equation*}

(13)

Once a minihalo Pop III star is formed, it will begin to ionize the surrounding gas in its birth cloud. If this radiation is able to ionize all of the gas available for star formation, then no more stars will form. The size of the ionized region around a star is given by the Strömgren radius,

\begin{equation*} R_\mathrm{ S} = \left(\frac{3 \dot{N}}{4 \pi n^2_\text{H} \alpha _B}\right), \end{equation*}

(14)

where |$\dot{N}$| is given by equation (11) and n_H is the hydrogen number density. Bromm & Larson (2004) find that H₂ cooling tends to drive gas in these haloesto densities of around n_H ∼ 10⁴ cm⁻³ at T ∼ 200 K.

We then assume that the maximum number of stars allowed in a halo can be found by simply packing the halo with Strömgren spheres. Under these assumptions, we find that a single star is able to ionize most of the gas in a minihalo at the masses and redshifts relevant to Pop III star formation. Haloes above the atomic cooling threshold would be able to accommodate multiple star formation locations, but they are outside the scope of our model, as we have already assumed that they have transitioned to stable star formation. We note that a single star is still able to ionize the majority of its surrounding gas even if we utilize a more complex model for the distribution of gas in a halo, such as the disc model described in Muñoz & Furlanetto (2013).

We note that Whalen et al. (2008a) also find similar results for the destruction of H₂ in haloesforming massive Pop III stars. In their simulations, a 120 M_⊙ star is able to completely photodissociate all H₂ in a halo during its lifetime. As these stars form primarily through H₂ cooling, it is unlikely that the gas in such a minihalo could cool and form stars near another Pop III star.

It is also possible that internal LW feedback from a Pop III star could dissociate H₂ and stop star formation in a single halo (see Glover & Brand 2001). We neglect this process because ionizing feedback from the same star is already able to ionize the majority of the gas.

3.4 Pop III supernovae

When a massive Pop III star reaches the end of its life, taken to be 5 Myr in this model as stellar lifetimes do not vary appreciably at high mass (Schaerer 2002), the star will either explode in a supernova or collapse directly into a black hole. Stars with a mass between 40 |$\, \mathrm{M}_\odot$| and 140 |$\, \mathrm{M}_\odot$| and those with masses above 260 |$\, \mathrm{M}_\odot$| will not end their lives in a supernova, but they will rather collapse directly into a black hole. Stars below this range will explode in a typical core-collapse supernova, while stars in the intermediate range will end with a pair-instability supernova. We therefore only take into consideration supernovae from stars that fall into these two categories. We assume that core-collapse supernovae release a kinetic energy of 10⁵¹ ergs, while pair-instability supernovae release 10⁵² ergs (see Wise & Abel 2008; Greif et al. 2010).

Supernova feedback is particularly important in the early Universe, as the typical kinetic energy released in a supernova could be orders of magnitude larger than the binding energy of the gas in a halo, which is of the order of (Loeb & Furlanetto 2013)

\begin{eqnarray*} E_b|!\approx\!2.53 \times 10^{50}\!\left(\frac{\Omega _\text{m}}{\Omega _\text{m}(z)} \right)^{1/3}\!\left(\frac{M}{10^6 \, \mathrm{M}_\odot } \right)^{5/3}\!\left(\frac{1+z}{10}\right)\!h^{2/3}\!\text{erg}. \nonumber \\ \end{eqnarray*}

(15)

Every time a supernova explodes in a halo, we assume that the kinetic energy released is distributed throughout the remaining gas and calculate the corresponding temperature, assuming ionized, primordial gas. We then assume that the velocities of particles in this gas follow a Maxwell–Boltzmann distribution. This allows us to calculate the fraction of gas remaining in the halo and the fraction exceeding the halo’s escape velocity. We then eject this gas, adding it to a reservoir that we later allow to re-accrete onto the halo over a time-scale equal to the halo’s free fall time at the moment the gas is ejected,

\begin{equation*} t_{\text{ff}} = \frac{1}{4} \sqrt{\frac{3\pi }{2G\bar{\rho }_{\rm vir}}}, \end{equation*}

(16)

where |$\bar{\rho }_{\rm vir}$| is the average density of the halo (see Yoshida et al. 2007a for a simulation of the re-incorporation of gas into a halo at these redshifts). Note that this can be a significant fraction of a Hubble time as the highest redshifts considered here. If the velocity of the expelled gas is greater than |$\sqrt{10} v_\text{esc}$|⁠, we assume that the gas has completely escaped and will never re-accrete. We discuss the importance of this assumption on our model in Section 6.4.

This model for gas ejection has strong implications for a halo’s transition to Pop II star formation. We assume that the metals released by a supernova follow the gas, so the fraction of metals expelled is the same as the fraction of total gas expelled. This is discussed further in Section 6.3, where we investigate the fraction of metals that must be retained after a supernova in order for a halo to transition quickly.

3.5 Transitioning out of Pop III star formation in minihaloes

One very important aspect of our model is the transition between metal-free, minihalo Pop III star formation and metal-enriched Pop II star formation. The minihalo Pop III stars we have discussed are likely born in only one star-forming region per halo (see Section 3.3), whereas Pop II stars form from a more traditional, low-mass IMF (i.e. a Salpeter IMF). In order to determine at what point haloesswitch to Pop II star formation, we must look at how gas is cooled to form each type of star.

Once a halo reaches a virial temperature of T_vir = 10⁴ K, atomic line emission begins to dominate the cooling. This is, coincidentally, also the point where the gas in a halo tends to reach a binding energy of ∼10⁵¹ erg (the typical kinetic energy released in a core-collapse supernova) and becomes more stable to supernovae. At this point, we assume that enough gas will be left behind after a supernova for the halo to develop an ISM regulated by internal feedback. We then transition to the model described in Furlanetto et al. (2017). If a halo reaches this point but has yet to retain any metals, we track the first generation of stars formed after this point as a separate population of metal-free stars. As seen in Fig. 7, this mode of star formation tends to occur at significantly lower rates than both our fiducial Pop III treatment and metal-enriched Pop II star formation. Note we assume that, after this initial burst of metal-free star formation, metals immediately mix through the halo gas, so that subsequent star formation is Pop II. The transition will be much slower if mixing is inefficient, but we refer the reader to models focusing on that process (e.g. Sarmento et al. 2018).

Before a halo reaches the atomic cooling threshold, it can only transition to Pop II star formation if it has retained enough of the metals released in supernovae. Bromm & Loeb (2003) find that the most important metal transitions to consider when looking at this critical point are CII and OI. In particular, they find the metallicities at which metal line cooling will become more efficient than molecular hydrogen cooling to be [C/H]_crit ≈ −3.5 and [O/H]_crit ≈ −3.05, where [A/H] = log (N_A/N_H) − log (N_A,⊙/N_H,⊙). Once a halo’s mean metallicity surpasses one of these criteria, we then make the switch to the Pop II star-forming regime. Dust cooling may also be efficient down to metallicities as low as [O/H] ∼ −5, so we include a model using this phase of cooling for comparison (see Schneider et al. 2006 and Omukai, Hosokawa & Yoshida 2010).

We take our metal yields of Pop III stars from Heger & Woosley (2010) and Heger & Woosley (2002), which provide yields for core-collapse and pair-instability supernovae, respectively. Even though a single supernova may produce enough metals for the halo to reach the critical metallicity, this may not necessarily cause the halo to immediately begin forming Pop II stars since not all metals will be retained. As discussed in Section 3.4, if metals are ejected from the halo, we must either wait until they re-accrete, until a later period of star formation where the halo is more stable to supernovae feedback, or until enough metals are retained in the presence of supernova feedback.¹ If the binding energy of the halo is small enough that the material is never re-accreted or the halo grows enough during the re-accretion time that the metal mass is insignificant, a minihalo may ‘forget’ about its earliest periods of Pop III star formation and require another burst of star formation before it can make the transition to metal-enriched stars.

Our model assumes that the mean metallicity of the halo is the deciding factor in the transition to Pop II star formation, although this may not necessarily be the case. If metals can be more concentrated in certain regions of the halo, it is possible that Pop II stars could begin to form even if the halo does not meet our critical metallicity requirements. We discuss the effects of this assumption in Section 6.3 and leave a more detailed model of inhomogeneous mixing to a future work.

We also do not include the effects of external enrichment from nearby haloesthat could allow these massive haloesto transition more quickly and also allow low mass haloesto skip the Pop III phase altogether (see Smith et al. 2015). Jaacks et al. (2017) find that the total metals produced in their model is not enough to raise the volume averaged metallicity in their simulations above the critical metallicity, indicating that, while a single halo may be able to produce enough metals to cut off star formation locally, it does not affect global Pop III values.

4 THE PROPERTIES OF ATOMIC COOLING HALOES

Above T_vir ∼ 10⁴ K, haloesare far more stable to supernova feedback, and their cooling channel changes significantly (especially if they are able to retain metals). We therefore assume that star formation proceeds smoothly beyond this point, and we shift our stellar parameters to those appropriate to Pop II stars. In our model, we use a Salpeter IMF and break the assumption that stars form in isolation. We also stop tracking individual stars and simply prescribe the star formation efficiency in each halo. We assume a feedback-regulated star formation efficiency, as in Furlanetto et al. (2017) and Sun & Furlanetto (2016), where star formation is regulated by supernovae through either energy or momentum conservation.

We note that our treatment of star formation above the atomic cooling threshold is much simpler than that of Pop III stars in minihaloes. We are mostly concerned with the minihalo stage of halo growth, so our model for more massive haloesis only used to calculate the radiation backgrounds that may contribute to limiting Pop III star formation in minihaloes. For simplicity, we will mostly assume that star formation in these massive haloesis Pop II, and we will refer to this stage as ‘Pop II.’

In our fiducial model, we find the star formation efficiency in a halo bybalancing the kinetic energy released by supernovae with the binding energy of the halo. We follow Furlanetto et al. (2017), who parametrize the star formation efficiency (defined here as the fraction of accreting material which turns into stars) as

\begin{equation*} f_\ast \approx \frac{1}{1 + \eta \left(M_\mathrm{ h}, z\right)}, \end{equation*}

(17)

where η relates the rate at which gas is ejected due to feedback to the star formation rate, |$\dot{M}_{\text{ej}} = \eta \dot{M}_\ast$|⁠.

For energy-regulated star formation, we balance the rate at which supernova energy is released with the rate at which the binding energy of a halo is growing through newly accreted material,

\begin{equation*} \frac{1}{2}\dot{M}_\text{ej} v_\text{esc}^2 = \dot{M}_\ast \epsilon _\text{k} \omega _\text{SN}. \end{equation*}

(18)

Here, v_esc is the escape velocity from the halo, |$\dot{M}_\ast$| is the star formation rate, ε_k is the fraction of a supernova’s kinetic energy used to drive a wind and lift gas out of the halo, and ω_SN is the kinetic energy released in supernovae per unit mass of stars. With this in mind, Furlanetto et al. (2017) find η to be

\begin{equation*} \eta _{\text{E}} = 10 \epsilon _\text{k} \omega _{49} \left(\frac{10^{11.5} \, \mathrm{M}_\odot }{M_\mathrm{ h}}\right)^{2/3} \left(\frac{9}{1 + z}\right). \end{equation*}

(19)

Here, ω_SN = 10⁴⁹ω₄₉ erg |$\, \mathrm{M}_\odot ^{-1}$| and ε_k = 0.1 in our fiducial model.

We also consider the case where momentum is conserved, and we balance the star formation efficiency by comparing the momentum released in supernovae to the momentum required to eject gas from the halo. In this case, Furlanetto et al. (2017) find η to be of the form

\begin{equation*} \eta _\text{p} = \epsilon _\text{p} \pi _\text{fid} \left(\frac{10^{11.5} \, \mathrm{M}_\odot }{M_\mathrm{ h}}\right)^{1/3} \left(\frac{9}{1 + z}\right)^{1/2}, \end{equation*}

(20)

where ε_p is the fraction of the momentum released in a supernova that drives the winds (taken here to be ε_p = 0.2 in our fiducial model) and π_fid defines the momentum injection rate from stars formed from a given IMF, defined as

\begin{equation*} \dot{P} = \pi _\text{fid} \dot{P}_0 \left(\frac{\dot{m}_\ast }{\, \mathrm{M}_\odot / \text{yr}}\right). \end{equation*}

(21)

Here, |$\dot{P}_0 = 2 \times 10^{33}$| g cm s⁻² and π_fid is of the order of unity for a Salpeter IMF (see Furlanetto et al. 2017 for a more detailed discussion of this model).

In both of these cases, we impose a maximum value on the fraction of accreted mass that will form stars, f_*. In our fiducial models, we assume f_*max = 0.1 in the energy-regulated case and f_*max = 0.2 in the momentum-driven case as in Furlanetto et al. (2017). We also assume that some fraction of newly accreted gas may be virial shock heated and be unavailable for star formation. We follow Faucher-Giguère, Kereš & Ma (2011), who find the fraction of accreted material that will be able to cool onto the halo and form stars to be

\begin{equation*} f_\text{cool} = 0.47 \left(\frac{1+z}{4} \right)^{0.38} \left(\frac{M_\mathrm{ h}}{10^{12} \, \mathrm{M}_\odot } \right)^{-0.25}. \end{equation*}

(22)

This fraction is only less than 1 in the case of very massive haloesat relatively low redshifts, so we find that this process does not have a large effect on our results.

We compare our results to the observed luminosity functions of Bouwens et al. (2015) in Fig. 2.² We find reasonable agreement for our models at z = 7, although our model overpredicts the abundance of haloesat z = 10. This was seen as well in Furlanetto et al. (2017), who alleviated this discrepancy by including a redshift independent model of Pop II star formation which fits the data better. We also include this model, which is identical to the energy-regulated model at z = 8 with ε_k = 0.2 but ignores the redshift dependence in equation (19).

Figure 2.

Luminosity functions for Pop II stars in our fiducial model compared to the observed luminosity functions of Bouwens et al. (2015). We do not include the luminosity functions for Pop III stars as they are far too dim. In general, our results agree reasonably well with the data at z = 7, but the energy- and momentum-regulated models overpredict the abundance of haloesat z = 10. We therefore also include a redshift-independent model that better fits the data. Finally, we include an energy-regulated model with ε_k = 1 (i.e. all of the kinetic energy released by supernovae couples to a halo’s gas to drive a wind).

5 RESULTS

With all the components of our model in place, we now consider how our ensemble of haloesevolves through the Pop III era. In our fiducial model:

1. We assume a Salpeter IMF from 20 |$\, \mathrm{M}_\odot$| up to the limit in equation (10) (henceforth referred to as ‘low-mass’ in comparison to more extreme models; see Table1).

2. We set the gas re-accretion time equal to the halo free-fall time.

3. We assume that each star formation event generates just a single star or pair of stars, and we set the binary fraction to 0.5.

4. We assume energy regulation (equation 19) for the Pop II phase.

5. We take the Trac et al. (2015) halo mass function and set the halo growth rate via abundance matching.

We also consider a broad set of variations around these fiducial values.

Fig. 3 shows the total, minihalo Pop III, and Pop II SFRDs of our fiducial model.³ We see that Pop III star formation in minihaloes gradually rises until z∼ 25, when the LW background has grown enough to narrow the allowed mass range of Pop III haloessufficiently to begin the decline in star formation (see Fig. 4). Note how minihalo Pop III stars are able to dominate the LW background for a relatively long (z ∼ 20 instead of z∼ 25) time, even though Pop II star formation has come to dominate the SFRDs much earlier (Fig. 3). Since massive Pop III stars produce more UV photons per unit mass, small star formation rates can still create a large LW background. As this is happening, Pop II star formation continues to rise as more Pop III haloesretain enough metals to make the transition, continuing until z ∼ 25 where Pop II star formation finally begins to dominate the total star formation in the Universe. After this, minihalo Pop III star formation still continues on until z∼ 12, albeit at a much lower level than the total star formation. It is at this point that the minimum mass for Pop III star formation rises above the atomic cooling threshold, cutting off the formation of any new Pop III stars in minihaloes.

Figure 3.

SFRD of minihalo Pop III and Pop II star formation in our fiducial model. This mode of Pop III star formation ends in our model once the minimum mass for star formation rises above the atomic cooling threshold (see Fig. 8).

Figure 4.

LW background for our fiducial model. Note how the contribution from Pop III stars in minihaloes is dominant for longer than the SFRDs in Fig. 3. This is because these Pop III stars are able to produce UV photons more efficiently than their Pop II counterparts, so even a small amount of Pop III star formation can continue to produce a high LW background.

We next explore the origins of these features and their robustness in different scenarios of early star formation.

5.1 Pop III star formation in individual minihaloes

To that end, we next consider the star formation histories of single haloes. These are seen in Fig. 5, which shows the total Pop III stellar mass formed in example haloesof three different masses. The burstiness of star formation is reflected in the discrete increases in the total mass when new stars are formed. Note that the haloesthat form latest – and hence in our prescription have the smallest masses – produce a larger number of minihalo Pop III stars than their more massive counterparts, even though the latter began forming their stars much earlier. This is due to the redshift dependence of the binding energy of a halo: at fixed halo mass, E_b∝(1 + z) from equation (15). This allows a halo that forms earlier to become stable to supernovae feedback much more quickly, and it can therefore retain its metals after fewer periods of star formation.

Figure 5.

Total minihalo Pop III stellar mass for three example haloesof different masses. In general, haloesthat form later have more periods of star formation, which leads to a higher total mass produced. Masses shown are the final masses at z = 6.

Typically, minihaloes go through no more than ∼10 periods of Pop III star formation, with a delay that depends on the accretion rate of the halo. In a halo with a smaller accretion rate, it may take a few million years to accumulate enough mass to form a single cloud above the local Jeans mass if supernovae from previous episodes have cleared out the halo. In haloeswith large accretion rates, however, star formation may be able to begin again with virtually no delay. There is much stochasticity in this relationship, however, as can be seen in Fig. 6. This is due to our Monte Carlo treatment of the minihalo Pop III star formation, where stars are randomly sampled from an IMF. If the IMF produces a significant number of stars that will not end their lives in a supernova (such as in the low-mass IMF model shown here), a halo can ‘forget’ about earlier generations of Pop III star formation as any metals produced will be lost to the black hole.

Figure 6.

Total mass of Pop III stars formed in haloesby z = 6 as a function of mass. Each point corresponds to a halo in our fiducial model. Note that, since stars in our model form in isolation and will always die before the next period of star formation, this mass is not the total mass of Pop III stars at z = 6. Rather, this is the total mass that has formed in the halo, as most of the stars will have either exploded in a supernova or collapsed to a black hole.

We find that haloesin our model generally tend to become stable to supernovae shortly before reaching the atomic cooling threshold. In fact, most haloesthat transition by reaching the critical metallicity will end up crossing this threshold within the next ∼10 Myr, causing the minimum mass for Pop II star formation to be very close to the atomic cooling threshold. Our results therefore generally support assumptions made in previous works that metal-enriched star formation will only occur in haloesabove the atomic cooling threshold.

Our model allows for a minihalo that has formed Pop III stars in the past to fall below the minimum mass and cease star formation. This could, in principle, allow haloesto stay dormant after a few earlier periods of star formation. In practice, however, we find that this does not happen, as haloestend to grow faster than the minimum mass, at least until the LW background has grown large enough to completely shut off Pop III star formation in minihaloes.

5.2 The duration of Pop III star formation in minihaloes

The key question we wish to address in our models is how long Pop III star formation in minihaloes persists under a variety of physics assumptions. Fig. 7 shows this for our suite of models. We find that, generically, the SFRD increases rapidly at early times before slowing dramatically or flattening. This ‘plateau’ period is typically comparable to the age of the Universe at that time before minihalo Pop III star formation ends entirely. Extended Pop III star formation such as this is also seen in the Renaissance Simulations (see Xu et al. 2016).

Figure 7.

SFRD of minihalo Pop III stars for a variety of our models. Symbols indicate where Pop II star formation overtakes Pop III star formation. The upper panel shows our results for a low mass Pop III IMF under a variety of different assumptions for the Pop II and III star formation prescriptions. The bottom panels show a comparison between three different Pop III IMFs using energy- and momentum-regulated Pop II star formation, respectively. Note that models that employ momentum-regulated Pop II star formation will form stars more efficiently in low-mass haloes, raising the minimum mass above the atomic cooling threshold faster and transitioning from the minihalo Pop III phase sooner.

We also track the formation of the first generation of Pop III stars in atomic cooling haloes(see Section 3.5). This ‘Pop III.2’ star formation is shown for our fiducial model in Fig. 7, and we see a similar result for all other models. This mode of star formation tends to occur at least an order of magnitude below the Pop III star formation in minihaloes, although it has a much longer tail as new haloesthat have yet to form stars cross the atomic cooling threshold and become more stable to supernovae. Of course, this ratio is a result of our assumption that only the first burst of star formation in these haloesis Pop III, and the SFRD could be much larger if the metals mix inefficiently (see Sarmento et al. 2018).

The reason for the decline and end of minihalo Pop III star formation is seen in Figs 4 and 8. As the LW background begins to build up at a faster rate when haloestransition to Pop II star formation, the minimum mass rises. As it gets closer to the atomic cooling threshold, the mass range in which minihaloes are able to form Pop III stars narrows, causing the SFRD to begin to plateau. Once the minimum mass rises above the atomic cooling threshold, this mode of Pop III star formation ends.

Figure 8.

Minimum mass for Pop III star formation for a variety of our models. The atomic cooling threshold is shown as the dashed line. The top panel shows our results for the low mass Pop III IMF under a variety of different assumptions for the Pop II and III star formation prescriptions. The bottom panels compare our results for different Pop III IMFs using the energy- and momentum-regulated Pop II star formation prescriptions. Once the minimum mass crosses the atomic cooling threshold, any new haloeswill begin forming low-mass stars, even if they form out of primordial gas. This is why the Pop III SFRD vanishes so quickly in the momentum-regulated models.

5.3 The importance of Pop II stars for minihalo Pop III star formation

As seen in the bottom panels of Fig. 7, the Pop III SFRDs in models with momentum regulated Pop II star formation are quite distinct from those with energy regulated Pop II star formation. Because the star formation efficiency is higher in low mass haloes(which are very abundant at these redshifts) under momentum regulation, the LW background and minimum mass of Pop III haloesrise much more quickly. This is seen in the plots of SFRD as a much earlier transition to Pop II star formation, which generally occurs at around the same time as the plateau. Also, since we transition away from Pop III star formation once minihaloes pass the atomic cooling threshold and become more stable to supernova feedback, all new haloeswill skip this phase of star formation and begin forming stars following the Furlanetto et al. 2017 model at this point (see Fig. 8).

The IMF appears to have a more pronounced effect in the cases with energy-regulated star formation. In the case of the mid- and high-mass IMFs, the minimum mass follows the atomic cooling threshold before finally crossing it when Pop II star formation is high enough. With more massive stars, the contribution to the LW background from minihalo Pop III stars is larger, allowing these stars to more effectively regulate themselves near the atomic cooling threshold. Pop III star formation inside minihaloes does not entirely cease, however, until Pop II stars contribute enough to the LW background to completely shut off star formation by themselves.

Fig. 9 illustrates these points for a wide range of model parameters. It shows two key transition points for several of our models. The left-hand panel shows z_end, the times at which Pop III star formation in minihaloes ends for each model. The right-hand panel shows z_II, the moments at which the Pop II SFRD overtakes the Pop III SFRD in each model. Within a given Pop II model, there is very little spread in the redshift at which minihalo Pop III star formation ends. This is due to the fact that, by the time this happens, the LW background is completely dominated by Pop II stars. Fig. 8 shows this as well: for a given Pop II prescription, all the minimum mass curves cross the atomic cooling threshold at the same time, which marks the final endpoint of Pop III star formation in minihaloes. However, there is quite a bit of scatter in the maximum SFRD at this time, because that depends more sensitively on our assumptions about the Pop III stars. In fact, there appear to be two distinct ‘clouds’ for each Pop II star formation prescription in the plot. This is caused by our choice of IMF models, which separate (relatively) low- and (extremely) high-mass stars into contrasting cases. Because of the prescriptions that both form individual stars, the high-mass case produces about an order of magnitude morestellar mass per event, which directly affects the maximum SFRD. If we utilized a wider range of IMF models that uniformly spanned the relevant ranges in stellar mass, these ‘clouds’ would be connected.

Figure 9.

Maxmimum Pop III SFRDs and important redshifts for a number of different Pop II star formation prescriptions. Points correspond to the ending redshift of Pop III star formation in the left-hand panel, and crosses correspond to the redshift at which Pop II star formation overtakes Pop III in the right-hand panel. Scatter for each model is caused by varying IMFs as well as our Monte Carlo approach to model Pop III star formation. Note how there is very little scatter in the Pop III end redshift within each Pop II model. By the time Pop III star formation ends, the LW background is completely dominated by Pop II star formation, so all models typically end at the same time.

The transition redshift z_II also has a moderate amount of scatter, because it depends upon the amplitude of the Pop III SFRD, which is sensitive to our assumptions. It is apparent that, in all of our models, the minihalo Pop III era never reaches more than ∼10⁻³ M_⊙ Mpc⁻³ yr⁻¹, and in most cases much less. This is comparable to the measured SFRD at z ∼ 10 from bright galaxies (Zheng et al. 2012; Coe et al. 2013; Oesch et al. 2014; Atek et al. 2015; McLeod et al. 2015), although in our models the total Pop II SFRD is always much larger by z ∼ 10. This illustrates the difficulty of detecting the extremely faint Pop III haloes, if they exist.

Figs 7 and 9 also show that the redshift independent case (which matches the observations best at z = 10) yields a slightly more extended Pop III star formation history, as it produces a smaller rate of star formation in low mass haloes. As an extreme case, we also include in Fig. 7 a model with energy regulated Pop II star formation, but with ε_k = 1. In other words, all of the kinetic energy released by supernovae in this model is able to couple to the gas and work to lift it out of the halo. Since feedback is stronger in this case, we see a smaller Pop II star formation efficiency in haloes, and therefore a lower LW background. As a result, the minimum mass to produce Pop II stars never crosses the atomic cooling threshold. Because of this, Pop III star formation in minihaloes is able to continue on until at least z = 6. However, Fig. 2 shows that this model substantially underpredicts the observed luminosity function at z ∼ 7. We include it only to emphasize that the longevity of the minihalo Pop III phase is very sensitive to the details of Pop II star formation in low-mass haloes.

5.4 Self-regulation of Pop III star formation

In order to test the ability of Pop III stars in minihaloes to self-regulate themselves, we next consider a model in which Pop III stars do not contribute to the LW background by setting N_LW = 0 for Pop III stars. The minimum mass of Pop III haloesfor this case is shown in Fig. 10. The LW background and therefore minimum mass will be lower at early times when Pop III star formation is dominant. Without a LW background, haloesform Pop III stars earlier. But the time at which the minimum mass crosses the atomic cooling threshold is unchanged, because it is feedback from Pop II stars which eventually causes Pop III star formation to end inside minihaloes. Because of this, while Pop III stars may be able to regulate their own minimum mass at early times, global Pop III star formation will not be terminated by feedback from Pop III stars.

Figure 10.

Minimum masses for Pop III star formation for varying values of N_LW. The minimum mass is higher in the case where Pop III stars contribute to the LW background, although all cases rise above the atomic cooling threshold at the same time. This indicates that global feedback from Pop II star formation is really what ends the Pop III phase in the Universe.

5.5 Comparison to other works

Our model can be compared to similar models of Pop III star formation that use different approaches and study the effect of other parameters. Because this phase of star formation has yet to be observed, there are many uncertainties, and it is important to consider a wide range of models. We find that many of our results generally agree with the numerous other models of similar nature, although there are some distinct differences due to our specific assumptions.

For example Jaacks et al. (2017) run a hydrodynamical simulation where Pop III supernova remnants are ‘painted’ onto the fluid with their properties calibrated from simulations and analytical arguments. They assume a fixed mass of ∼500 M_⊙ per star formation event once a gas particle passes the thresholds of n = 100 cm⁻³ with T ≤ 10³ K. They find, like us, that Pop III star formation is not a self-terminating process. However, we come to this conclusion differently. They find that haloesare simply not able to produce enough metals to raise the volume-averaged metallicity of their box above the critical metallicity required for Pop II star formation. In our models, we find instead that Pop III stars in minihaloes never form rapidly enough to raise the LW background high enough to completely cut off star formation in haloesbelow the atomic cooling threshold on their own. We must instead wait until Pop II star formation begins to dominate. We also find that, even if minihalo Pop III stars do not contribute at all to the LW background, star formation will still end at the same time (see Section 5.4). They also find a maximum SFRD of around 10⁻³ M_⊙ yr⁻¹ Mpc⁻³, however, which is about an order of magnitude higher than the SFRDs found in most of our models.

In order to fully compare with their results, we include a model that also forms a fixed mass of 500 M_⊙ per star formation episode. We note that, even under this assumption, we still find an SFRD about an order of magnitude smaller than theirs. This model is shown in Figs 7 and 8. In this case, the minimum mass is higher before Pop III star formation ends, because more stars form and the LW background is larger. However, the redshift at which minihalo Pop III star formation ends is virtually unchanged, because by that point Pop II star formation dominates by far. Thus this model can lead to a slightly larger Pop III SFRD in minihaloes (though still about an order of magnitude smaller than that of Jaacks et al. 2017) but does not affect our major conclusions. Haloes in this model tend to have a similar number of minihalo Pop III star formation episodes as haloesin our fiducial models, as the increased star formation results in more energy injected into the system from supernovae. While more metals are produced, they are much more easily ejected out of the halo, causing haloesto be unable to transition due to reaching the critical metallicity. Thus, these haloestend to transition away from minihalo Pop III star formation simply by reaching the atomic cooling threshold.

The differences between our models and those of Jaacks et al. (2017) are mainly caused by different assumptions about the Pop II SFRD. Their models have systematically lower star formation in the Pop II phase than ours (their equation 21), causing a lower LW background and therefore minimum mass. They also assumed lower values of N_LW for both Pop II and Pop III stars (by factors of 5 and 10, respectively), exaggerating the decline in the LW background.

Visbal et al. (2017) apply a semi-analytic model to N-body dark matter simulations, allowing them to take into account any processes that require spatial information such as clustering and mergers. They allow stars to form at a specific fraction of a halo’s baryonic mass, fiducially taken to be 10⁻³ (we find ∼5 × 10⁻⁴ for the total Pop III stellar mass formed in a minihalo in our fiducial, low-mass model). They find SFRDs consistent with ours, although their models only run to z∼ 20 so it is difficult to compare any results that rely on feedback from Pop II stars, such as the duration of Pop III star formation in the Universe. They do find that the effects of external metal enrichment may be important only if metals are allowed to travel far from their original haloes. This is similar to Jaacks et al. (2017), who find that small haloesthat are externally enriched exhibit much lower metallicities than more massive haloeswhich are internally enriched by their own star formation. While we do not include the effects of external enrichment in our model, we note that it would work to transition haloesfaster, potentially turning the plateau seen in many of our SFRDs into a more gradual decline.

Trenti & Stiavelli (2009) use a semi-analytic model similar to ours with dark matter haloesfrom the Press–Schechter formalism (Press & Schechter 1974). They find SFRDs comparable to ours in cases where Pop III stars form in isolation. They also study the importance of Pop III stars forming in haloesabove the atomic cooling threshold. This form of star formation was only important when Pop III stars were allowed to form with efficiencies much larger than what is found in our models, even in the cases where multiple generations of Pop III stars can form per halo.

Crosby et al. (2013b) carry out a semi-analytic model based on dark matter haloesfrom numerical simulations. They use a similar model for the ejection of metals produced by Pop III supernovae as in this work, although they find, in contrast to us, that haloesare generally able to retain enough metals to transition to Pop II star formation after the first supernova. This is because they assume a substantially smaller fraction of a supernova’s total energy is used to drive outflows from a halo. Their SFRDs are therefore lower than our fiducial model by about an order of magnitude.

Maio et al. (2010) use a detailed set of numerical simulations spanning a wide range of parameters to study the onset of metal-free Pop III star formation and the transition to metal-enriched Pop II stars. They find star formation rates densities that plateau to values comparable or less than ours for their range of models. Similar to us, they also explore the effects of different Pop III IMFs which may drastically change the relative number and energy of their supernovae. They find that high mass IMFs generally result in a lower SFRD as these stars will produce more supernovae quicker and enrich their haloesfaster. As shown in the bottom panels of Fig. 7, we find the opposite result as higher mass stars will be able to blow out metals easier due to their higher supernova energies, therefore allowing Pop III star formation in minihaloes to persist for longer.

Many of these models make the assumption that only one Pop III star forms per halo before making the transition to Pop II star formation. We include a model with this feature for comparison (see Fig. 7). Since haloesin our fiducial model generally have ∼10 periods of Pop III star formation, the resulting SFRD is lower by around an order of magnitude.

6 CAVEATS

In this section we describe some of the simplifications of our model and their consequences.

6.1 Mass growth rates

In Fig. 1 we compare not only halo growth from our abundance matching technique, but also growth histories from the Trac et al. (2015) fit to equation (2). In this case, haloesthat will end up with the same mass at z = 6 will be less massive by a factor of the order of unity at z = 50. These haloeswill therefore take slightly longer to cross the minimum mass and form their first Pop III star. For example in models using these accretion rates, the first Pop III star will form at z ∼ 40, compared to z∼ 45 in our fiducial models. Once Pop III star formation begins to plateau, however, the two models become very similar. Because this assumption does not affect Pop II star formation, the LW background is the same at later times, causing minihalo Pop III star formation to end at the same time in both cases.

6.2 Mergers

Our fiducial model assumes that haloesgrow primarily through smooth accretion from the IGM. While Behroozi & Silk (2015) have shown that this is the primary source of growth for haloesat high redshift, it is possible that mergers could also play an important role. In our model, the primary way in which mergers could change our results is if combining the metals produced in two merging haloesallowed the halo to transition to the Pop II phase sooner than it would have on its own. However, we find that only a narrow range of minihaloes are able to form Pop III stars at any given time. Thus, with the exception of major events, only the larger mass progenitor would have been capable of creating stars in the past, so the smaller halo would have no metals to contribute. Even if it did, merging two haloescurrently forming Pop III stars would still not cause a halo to exceed the critical metallicity unless one of the haloeshad already transitioned, as both the haloes’ gas and metals would be mixed. Because of this, we neglect the effect of mergers in our model. Nevertheless, we plan to investigate the effects of mergers in more detail in the future.

6.3 Metal retention and the critical metallicity

In order to test the importance of our Maxwell–Boltzmann treatment of the ejected gas as discussed in Section 3.4, we include an alternate prescription in which we fix the fraction of metals left behind after supernova feedback. Since Pop III stars are able to produce such high masses of metals (see Table 1), we find that even if only a very small fraction (∼5 per cent) of metals remain inside the halo after a supernova event, haloeswill immediately transition to Pop II star formation. This is shown in Fig. 11, where a halo that retains 1 per cent of its metals is able to stay in the Pop III phase for multiple episodes of star formation, while a halo with 5 per cent will transition much more quickly. This has the biggest effect in models that use a Pop III IMF where every star ends its life in a supernova. In our low- and high-mass models, for example some fraction of stars will directly collapse into a black hole, adding no metals to the halo itself. In this case, no matter how high the metal retention fraction is, a minihalo could go through many periods of Pop III star formation if it happens to have formed a number of stars in mass ranges that do not produce supernovae.

Figure 11.

Star formation histories of a |$10^{10} \, \mathrm{M}_\odot$| halo with various metal retention fractions. Note that the results shown here are for the ‘mid’ IMF in order to ensure every star will supernova and release metals.

As discussed in Section 3.5, we also include a case where dust cooling is efficient. In this model, haloescan transition to Pop II star formation at a much lower metallicity. We find that this causes individual haloesto transition quicker as they do not need to retain as many metals after a supernova. This shows up in Fig. 7 as a lower total SFRD, although the timing of the Pop III phase remains the same.

6.4 Gas re-accretion

In our fiducial model, we assume that ejected gas re-accretes after one free-fall time, t_ff, calculated at the time the gas is ejected. In order to test this assumption, we ran models where this time was set to 1/10t_ff and 10t_ff. In each case, we found results almost identical to our fiducial model. This indicates that re-accreted material is not important for the transition to Pop II star formation, and haloesthat transition before the atomic cooling threshold do so when they become more stable to supernovae feedback and are able to retain metals produced in current periods of star formation. This is because haloesgrow very quickly during the early phases of the Pop III era, so they quickly transition from being so fragile that they are completely blown apart by a supernova (with no re-accretion) to being able to retain a fair fraction of their metals.

6.5 Photoionization heating

Although we consider photoionization feedback inside each source’s halo, assuming that it limits each halo to a single star-forming region at any given time, we do not consider the effects of the photoionization on the gas surrounding the halo. Because the excess energy from ionizing photons typically heats gas to ∼10⁴ K, the resulting Hii regions will be much hotter than the average IGM. Even if the gas recombines, it will retain excess entropy, which will reduce the rate at which gas accretes onto the host halo (Oh & Haiman 2003). For simplicity, we ignore the potential of photoheating to suppress accretion onto the small minihaloes in which our Pop III stars form, which amounts to assuming that most of the accretion occurs through dense filaments that self-shield from the stellar radiation. If photoheating does suppress accretion, Pop III haloeswill experience longer delays between star formation episodes. In the most extreme case, accretion would halt until the haloessurpass a virial temperature of ∼10⁴ K, which we have shown is also approximately the point at which they become stable to supernova feedback. At that point, star formation will likely proceed similarly to our Pop II prescription.

7 OBSERVATIONAL IMPLICATIONS

7.1 Observing Pop III haloes directly

Unfortunately, the luminosities of Pop III minihaloes in our model are very small and well below the capabilities of any current telescopes. We find that the absolute magnitude of these haloescan vary between M_AB ∼ −5 for the lower mass Pop III models to ∼−10 for the higher mass IMFs. Our models with a fixed mass of Pop III stars are only slightly brighter, reaching M_AB ∼ −10.5. While these haloesare faint, though, they are actually quite abundant. Fig. 12 shows the number density of Pop III minihaloes for a variety of models. In the cases where Pop III haloesare around for the longest, their abundance is actually comparable to that of Pop II galaxies (Fig. 2). Unfortunately, Pop III minihaloes in our model are far too dim to be detected by any forthcoming instruments, and they would likely require the use of lensing or an even more advanced generation of telescope to detect.

Figure 12.

Number densities of Pop III haloesfor a variety of models. This is calculated as the number of haloesthat are currently forming Pop III stars (i.e. haloesin the allowed mass range that have formed a Pop III star in the past and have yet to make the transition to Pop II star formation.).

Observations of the luminous Ly α emitter CR7 by Sobral et al. (2015) have indicated the potential presence of a Pop III halo at z = 6.6 with a stellar mass of |${\sim } 10^7 \, \mathrm{M}_\odot$|⁠. In order to find a halo with these properties in our model, we would have to break our single star-forming region assumption, as it is not possible for us to reach this mass with only a handful of Pop III stars. We would also have to allow haloesto form massive Pop III stars in haloesabove the atomic cooling threshold, as it would otherwise be impossible for a halo to form such a high mass in stars (see Fig. 8). Recently, however, ALMA observations of this object have detected [C ii] consistent with a normal star-forming galaxy, so it is unlikely that CR7 actually contains 10⁷ M_⊙ in Pop III stars (Matthee et al. 2017).

7.2 Pop III supernova rates

While it is very unlikely that we will be able to directly observe a Pop III minihalo in the near future, it may be possible to observe their supernovae. When a Pop III star with a mass between |$140 \, \mathrm{M}_\odot$| and |$260 \, \mathrm{M}_\odot$| reaches the end of its life, it will likely explode in a pair-instability supernova. If Pop III haloesform many of their stars in this range (as in our model using McKee & Tan 2008 masses or the high-mass, Salpeter-like IMF), then it may be possible to observe them with JWST or WFIRST. In particular, Whalen et al. (2013) find that Pop III supernovae will be detectable out to z = 30 for JWST and z = 20 for WFIRST, which is in the redshift range in which our model produces the most supernovae. The Pop III supernova rates from various models are shown in Fig. 13. At z = 20, an event rate of ∼10⁻⁶ Mpc⁻³ yr⁻¹ translates to ∼3 events per year per square degree per unit redshift. Thus, provided minihalo Pop III stars produce luminous supernovae, these events may be within the reach of large-scale surveys.

Figure 13.

Pop III supernovae rates for three of our fiducial models with energy-regulated Pop II star formation. Note that the low mass model does not produce pair-instability supernovae. Whalen et al. (2013) find that these supernovae can be detected by JWST and WFIRST out to z = 30 and z = 20, respectively, which is where our calculated supernovae rates begin to flatten. The case with energy-regulated Pop II star formation with ε_k = 1 is shown as an example of a model which produces Pop III supernovae out to at least z = 6.

7.3 The 21-cm global signal

Pop III stars will also affect the surrounding IGM through their radiation fields. The sky-averaged (‘global’) 21-cm signal is a particularly appealing tracer of the IGM as it is sensitive not just to the ionization state, but the temperature and Ly α intensity as well. Its sensitivity to the thermal history of the IGM opens up the possibility that X-rays from Pop III remnants could leave traces of their existence, in addition to the impact of Pop III stars themselves.

In Mirocha et al. (2018), we indeed find that the remnants of Pop III stars have a unique impact on the signal. While in general the addition of new sources of X-rays reduces the contrast between otherwise cold neutral regions and the CMB and thus weakens the 21-cm background, Pop III minihaloes also give rise to a characteristic asymmetry due to the generic rise and fall of the Pop III SFRD from minihaloes. In contrast, models neglecting Pop III sources tend to be quite symmetrical (Mirocha et al. 2017).

8 CONCLUSIONS

We have presented a simple, semi-analytic model investigating the formation of Pop III stars in minihaloes in the early Universe and the subsequent transition of their haloesto the more traditional Pop II star formation. Our model works by combining the results of a number of numerical simulations and analytic arguments with our self-consistent calculations of important feedback processes such as a meta-galactic LW background, supernovae, photoionization, and chemical feedback. From our results, we conclude that the SFRD of minihalo Pop III stars increases rapidly as structure formation generates more haloesat very high redshifts, until the stellar population increases enough to generate a substantial LW background, which slows the rate of star formation relative to halo formation. However, because Pop III star formation is limited in each halo by chemical feedback, minihalo Pop III stars are never able to self-terminate globally by generating a dominant LW background. Instead, more massive galaxies forming Pop II stars are ultimately responsible for choking off Pop III star formation in minihaloes.

More specifically:

1. Depending on our choice of Pop II star formation prescription, Pop III stars can continue to form in minihaloes at a low level for an extended period of time, in principle until z ∼ 6 at rates of around 10⁻⁴ − 10⁻⁵ M_⊙ yr⁻¹ Mpc⁻³. In general, models with efficient star formation in low mass galaxies (i.e. our momentum-regulated model) will cut off minihalo Pop III star formation much earlier by raising the minimum mass required for the Pop III star formation to occur. Alternatively, inefficient star formation in low mass galaxies (i.e. our energy-regulated model) will allow Pop III star formation in minihaloes to last longer.

2. The key parameters driving our results are the Pop II star formation prescription and the Pop III IMF. Secondary effects are the binary fraction, halo mass function, and LW yield of Pop III stars.

3. Supernova feedback is the most important feedback process in a single halo, because efficient expulsion of metals allows Pop III star formation to persist in single minihaloes for several generations. On a cosmological scale, the LW background dictates the halo masses at which Pop III stars can form, and it is responsible for stopping the formation of new minihalo Pop III stars once it causes the minimum mass to exceed the atomic cooling threshold where haloesare also more stable to supernova feedback.

4. If metal mixing is inefficient in Pop III haloes, metal-free star formation may persist in haloeswith masses above the atomic cooling threshold. They may form from a different IMF, however, as the cooling channels will be different than in the minihaloes that are the focus of this work.

5. While it may be possible to observe the presence of Pop III stars in minihaloes through their supernovae or through cosmological 21-cm experiments, it is very unlikely that we will be able to directly observe a Pop III halo in the near future. Our model produces Pop III minihaloes with magnitudes in the range of M_AB = −5 to −10 depending on our assumptions of the IMF, which is well below the capabilities of any current instruments.

ACKNOWLEDGEMENTS

We thank the referee for his or her careful reading and constructive comments. This work was supported by the National Science Foundation through awards AST-1440343and 1636646, by NASA through award NNX15AK80G, and was completed as part of the University of California Cosmic Dawn Initiative. In addition, this work was directly supported by the NASA Solar System Exploration Research Virtual Institute cooperative agreement number 80ARC017M0006. We also acknowledge a NASA contract supporting the ‘WFIRST Extragalactic Potential Observations (EXPO) Science Investigation Team’ (15-WFIRST15-0004), administered by GSFC. We acknowledge support from the University of California Office of the President Multicampus Research Programs and Initiatives through award MR-15-328388.

Software: matplotlib (Hunter 2007) and numpy (Van Der Walt, Colbert & Varoquaux 2011).

Footnotes

1

Note that we do not assume that the metals have thermalized; rather, we assume that they are entrained in the outflow from the halo.

2

We note that other groups have produced luminosity functions at these redshifts (e.g. McLure et al. 2013; Oesch et al. 2013; Schenker et al. 2013; Bowler et al. 2015; Finkelstein et al. 2015) that are generally consistent with our results. The Finkelstein et al. (2015) data have a lower amplitude than our results and those of Bouwens et al. (2015), although their shape is similar so this does not affect the model significantly as the amplitude is degenerate with our assumptions on the star formation efficiency. See Mirocha, Furlanetto & Sun (2017) and Mason, Trenti & Treu (2015) for more detailed comparisons.

3

The fluctuations in these curves are caused by our finite number of mass bins. Since the mass range in which haloesare actively forming Pop III stars is relatively narrow, individual haloesturning their star formation on or off are seen as features in the SFRD.