CARBON-ENHANCED METAL-POOR STARS: RELICS FROM THE DARK AGES

Within the standard cold dark matter paradigm, detailed calculations of structure formation have shown that the first stars typically formed in minihalos of mass ∼10⁶ M_☉ at redshifts 15 ≲ z ≲ 30 (Tegmark et al. 1997; Abel et al. 2002; Bromm et al. 2002). This general picture has been confirmed by more recent work, where the minimum minihalo mass that is able to produce sufficient molecular hydrogen and therefore cool efficiently to successfully host a first generation of stars is ∼2 × 10⁵ M_☉ (e.g., Yoshida et al. 2003; O'Shea & Norman 2007). We therefore start our calculations by drawing a random redshift and minihalo virial mass (M₂₀₀) from the halo mass function derived by Reed et al. (2007),⁴ with a lower-mass cut-off at ∼2 × 10⁵ M_☉ in the redshift interval 15–30. The halo concentration, c₂₀₀, is chosen by extrapolating the halo-mass–concentration relation derived by Prada et al. (2012), and is typically in the range 4–7 for the halo masses and redshift range that we consider here. Prior to the formation of the first stars, we assume that the gas confined to the dark matter minihalo obeys an isothermal density profile, which is motivated by several cosmological simulations (e.g., Alvarez et al. 2006; O'Shea & Norman 2007; Whalen et al. 2008). We assume that the initial temperature of the gas is set by the virial temperature of the minihalo scaled (by a factor of 0.6) to match cosmological simulations (see, e.g., Figure 19 of O'Shea & Norman 2007), and is of the form

$$\begin{equation} T_{\rm gas}(0) = 0.6\times 100^{1/3}\,\frac{\mu _{0} m_{\rm H}}{2\,k_{\rm B}}\,(G\,M_{\rm 200}\,H(z_{\rm coll}))^{2/3}, \end{equation} \tag{ 1 }$$

where H(z_coll) is the Hubble parameter at the minihalo collapse redshift (z_coll),⁵ and m_H, k_B, and G are the proton mass, Boltzmann constant and Newton gravitational constant, respectively. The mean molecular weight in the initially neutral primordial gas is μ₀ ≃ 1.23.

In our study, we investigate zero-metallicity massive stars in the mass range 10–100 M_☉ that would have ended their lives as Type-II core-collapse SNe. In principle, the mass function of the first stars might extend to much higher masses, which we do not consider here for the following reasons. Stars more massive than ∼100 M_☉ exhibit a pulsational instability due to the creation of electron/positron pairs in abundance (Ledoux 1941). Stars in the mass range 100–140 M_☉ experience a series of such pulsations, causing the star to eject the outermost layers before it quietly collapses to a black hole (Woosley et al. 2007). Stellar evolution models of such stars have indicated that radiative mass loss provides negligible metal enrichment (Marigo et al. 2003). However, the pulsational winds from such stars might play a role in enriching early galaxies with CNO, while producing no trace of the Fe-peak elements (Chen et al. 2014b; Woosley & Heger 2014; but see Baraffe et al. 2001). In this context, detailed yield calculations for pulsational pair-instability SNe are highly desirable (see the discussion in Section 3.2). For stars in the mass range ∼140–260 M_☉, the first instance of the pair-instability is sufficient to entirely disrupt the star (Heger & Woosley 2002; Chen et al. 2014a), leading to a pair-instability SN (PISN) with a typical explosion energy of ≳ 10⁵² erg⁶; such a high explosion energy is sufficient to evacuate the host minihalo of all its gas, thereby prohibiting the formation of a second generation of stars in this minihalo (Greif et al. 2007; Whalen et al. 2008)—evacuating the gas reservoir defeats the primary goal of our study to investigate the halos that host the second generation of stars. Stars with mass ≳ 260 M_☉ are sufficiently massive that the onset of the pair instability and the subsequent contraction and burning is unable to halt the gravitational collapse of the star, thus providing no metal enrichment (Fryer et al. 2001). In summary, metal-free stars in the mass range 10–100 M_☉ are currently considered to be the dominant source of metal enrichment for the second stars, with perhaps an additional contribution from the pulsational pair-instability SNe.

We have thus adopted the model yield calculations performed by Heger & Woosley (2010), which provide illustrative yields for 120 different stellar masses in the range 10–100 M_☉ (with some models differing in mass by just 0.1 M_☉). Each star was exploded with a piston located at the base of the oxygen shell (where the entropy per baryon ∼4 k_B), providing a final kinetic energy to the ejecta at infinity of either 0.3, 0.6, 0.9, 1.2, 1.5, 1.8, 2.4, 3, 5, or 10 in units of 10⁵¹ erg. A simple prescription of the mixing between the stellar layers is implemented by a moving boxcar filter of a set width (14 different widths are considered here). Therefore, the final set of yields for the stars considered in our study totals 16,800 models. After drawing a stellar mass from the appropriate initial mass function (IMF; see below), we randomly draw an explosion energy and level of mixing from a uniform distribution. Given that we have no handle on the mixing and explosion properties of the first stars, our assumption allows sufficient flexibility to explore all of the available parameter space. In Figure 1, we present the IMF-weighted yields for illustration purposes, where we separately plot the integrated yields for SNe that release low energy (0.3–0.6 × 10⁵¹ erg; blue curves), typical energies (0.9–1.5 × 10⁵¹ erg; green curves), high energies (1.8–3 × 10⁵¹ erg; orange curves), and hypernovae (5–10 × 10⁵¹ erg; red curves).

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The current generation of cosmological hydrodynamic simulations that follow the collapse and formation of the first stars suggest that either a single, binary, or small multiple of metal-free stars form in the center of dark matter minihalos (Turk et al. 2009; Stacy et al. 2010; Stacy & Bromm 2013; Hirano et al. 2014). We therefore assign each minihalo a single metal-free star (this assumption is relaxed in Section 2.6, where we allow small multiples of metal-free stars to form in a given dark matter minihalo), drawn from a flat stellar IMF, i.e., $dN/dm_{\star }\propto m_{\star }^{0}$, in the mass range 10–100 M_☉. This assumption is supported by several recent numerical simulations (Stacy & Bromm 2013; Hirano et al. 2014). However, the IMF of the first stars is still highly uncertain, and might instead represent a flat distribution in dN/dlog m_⋆ (Greif et al. 2011). Thus, to demonstrate how our choice of a flat IMF (in dN/dm_⋆) influences the results, we also consider the "extreme" case of a Salpeter-like IMF for the first stars, in the mass range 10–100 M_☉. For reference, the characteristic stellar masses for our chosen IMFs are 55 M_☉ (flat IMF) and 22.3 M_☉ (Salpeter-like IMF).

We also note that the initial mass function of the first stars could be very different from the mass function of stars that provide metal enrichment; there may be several windows in stellar mass whereby massive stars collapse directly to a black hole and eject no metals (Woosley & Heger 2014). If such a scenario is realized, the mass function of the first stars that eject metals may resemble a broken picket fence, rather than a power law. For this study, we assume that every star born in the mass range 10–100 M_☉ provides metal enrichment.

The first stars can have a profound impact on their environment during their lives. They emit copious quantities of H-ionizing photons that serve to ionize and thereby heat the surrounding gas to ∼10⁴ K. After the first stars turn on, the initial (isothermal) density distribution characterized by the temperature T_gas(0) becomes overpressured and causes the gas to expand. The expansion is bounded by an isothermal shock, known as a "champagne" flow, and is characterized by the self-similar solutions presented by Shu et al. (2002). The subsequent time evolution of the radial gas density distribution for a self-gravitating isothermal cloud of gas is given by

$$\begin{equation} \rho (r,t)=\frac{k_{M}\alpha (x)}{4 \pi G t^2}\quad x=\frac{r}{c_{\rm s}(0)\,t}, \end{equation} \tag{ 2 }$$

where c_s(0) = k_B T_gas(0)/μ₀m_H is the sound speed of the gas before the first stars are born. The constant k_M = M_gas/0.6 M₂₀₀ ≃ 0.17 is chosen so that the total mass of baryons inside the virial radius matches that observed in cosmological hydrodynamic simulations (see, e.g., Table 2 from O'Shea & Norman 2007). After the first stars are born, they heat the center of the cloud to a new temperature, T_gas(t), with a corresponding sound speed c_s(t) = k_B T_gas(t)/μ_ionm_H, where μ_ion = 0.59 is the mean molecular weight for fully ionized primordial gas. By numerically integrating the Shu et al. (2002) equations, the reduced central density, α(x = 0), in Equation (2) can be directly related to the ratio of the initial and final sound speeds of the gas, = c_s(0)²/c_s(t)². To minimize computational time, we generate a fine grid of α(0) values, numerically integrate the Shu et al. (2002) equations, and compute the corresponding values for . Then, given the initial and final sound speeds of the gas in each minihalo, we interpolate the α(x) values.

The boost in thermal energy that is given to the gas by a star during its life can be estimated by equating the heating rate to the cooling rate of a pure H gas (i.e., assuming thermal equilibrium in a smooth medium). The cooling rate that we use includes a contribution from recombination cooling as well as collisional excitation and ionization cooling (see Cen 1992 for the relevant fitting formulae that we have adopted herein). Assuming that recombinations balance photoionizations, the photoelectron heating rate is then simply

$$\begin{equation} \frac{\Gamma _{\rm pe}}{n_{\rm H}\,n_{e}} = \alpha _{\rm B}\,\psi k_{\rm B} T_{\rm eff}, \end{equation} \tag{ 3 }$$

where α_B is the case B recombination coefficient for H, ψ is the mean photoelectron energy, and T_eff is the effective temperature of the star. To calculate T_eff for our full range of stellar masses, we have generated the following fitting formula for a metal-free star with mass m_⋆ based on the detailed computations by Schaerer (2002):

$$\begin{equation} T_{\rm eff} = 6600\,{\rm K}\,\, \times (m_{\star }/{M}_{\odot })^{1.09 - 0.257\log _{10}(m_{\star }/{M}_{\odot })}. \end{equation} \tag{ 4 }$$

The mean photoelectron energy can then be calculated by assuming that the first stars radiated a blackbody spectrum, B_ν(T_eff), with color temperature T_eff,

$$\begin{equation} \psi k_{\rm B} T_{\rm eff} = \frac{\int _{\nu _{0}}^{\infty }[B_{\nu }(T_{\rm eff})/h\nu ]\sigma _{\nu }({\rm H})(h\nu -h\nu _{0})\,d\nu }{\int _{\nu _{0}}^{\infty }[B_{\nu }(T_{\rm eff})/h\nu ]\sigma _{\nu }({\rm H})\,d\nu }, \end{equation} \tag{ 5 }$$

where σ_ν(H) is the photoionization cross-section for H and hν₀ = 13.6 eV is the photoionization energy of H.

The gas density profile at the end of the star's life is defined by Equation (2), where the time variable corresponds to the main sequence lifetime of a Population III star. To estimate the main-sequence lifetime for the range of stellar masses we consider here, we have generated the following fitting formula, which is derived from the computations by Schaerer (2002):

$$\begin{equation} t_{\rm ms} = 1300\,{\rm Myr}\,\, \times (m_{\star }/{M}_{\odot })^{-2.407 +\, 0.5323\log _{10}(m_{\star }/{M}_{\odot })}. \end{equation} \tag{ 6 }$$

An example density profile for a 15 and 40 M_☉ star occupying a 7 × 10⁵ M_☉ minihalo that collapsed at redshift z = 24 is shown in Figure 2. The choice of these parameters allow for a direct comparison with Figures 1 and 2 from Whalen et al. (2008). Although we have not modeled the presupernova evolution using computationally expensive full three-dimensional (3D) cosmological hydrodynamic simulations, the striking agreement between our calculations and other, more expensive simulations (e.g., Abel et al. 2007; Yoshida et al. 2007; Whalen et al. 2008) demonstrates that we have captured the relevant physics of the presupernova evolution for the first stars, and the feedback they imparted on the surrounding gas. Thus, the computational simplicity of our models allows us to explore vast regions of parameter space in a fraction of the time.

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The final stage of our modeling procedure accounts for the Type II core-collapse SN blastwave at the end of a star's life. Greif et al. (2007) have shown that the blastwave evolution for the first stars is well-approximated by an energy and momentum conserving formalism. When including the host potential, however, this calculation becomes computationally expensive (Ostriker & McKee 1988).⁷ Given that the binding energy of the host minihalo is comparable to the energy released for some of our adopted models (see Section 2.2), we have instead calculated the stall radius, r_stall, which corresponds to the radius when the shock velocity is comparable to local sound speed of the gas. Specifically,

$$\begin{equation} \frac{E_{\rm sh}(r_{\rm stall})}{M_{\rm sw}(r_{\rm stall})} = \frac{k_{\rm B}\,T_{\rm gas}(r_{\rm stall})}{\mu (r_{\rm stall})\,m_{\rm H}}, \end{equation} \tag{ 7 }$$

where the kinetic energy of the shock in the thin shell approximation is given by

$$\begin{equation} E_{\rm sh}(r_{\rm stall}) = (1-f_{\rm r})\,E_{\rm exp} - \Delta W/2. \end{equation} \tag{ 8 }$$

Here

$$\begin{equation} \Delta W \,{=}\,\! \int _{0}^{r_{\rm stall}}G\,[M_{\rm dm}({<}r)\,{+}\,M_{\rm sw}(r)]\left (\!\frac{M_{\rm sw}(r)}{r^{2}}\,{-}\,4 \pi \rho (r,t_{\rm ms})\!\right)\,dr \end{equation} \tag{ 9 }$$

is the change in the total gravitational potential produced by the blastwave, and f_r = 0.7 is the fraction of the kinetic energy released by the core-collapse SN explosion (E_exp) that is radiated away in the Sedov–Taylor phase (Ostriker & McKee 1988). The dark matter mass interior to a radius r (assumed to be static) and the swept up gas mass are respectively given by

$$\begin{eqnarray} M_{\rm dm}({<}r) &=& M_{200}\,\frac{\ln (1+u) - u/(1+u)}{\ln (1+c_{200}) - c_{200}/(1+c_{200})} \end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray} M_{\rm sw}(r) &=& 4\pi \int _{0}^{r}\rho (R,t_{\rm ms})\,R^2\,dR, \qquad\quad \end{eqnarray} \tag{ 11 }$$

where u = r c₂₀₀/r₂₀₀ and r₂₀₀ is the virial radius.

The metallicity of the enriched gas depends on how efficiently the metals are mixed with the pristine material swept up by the blastwave. For relatively high-mass minihalos (∼10⁸ M_☉), the current state-of-the-art simulations that include a prescription for metal diffusion indicate that the gas is efficiently mixed within ∼300 Myr (Greif et al. 2010). For minihalos of lower mass, efficient mixing takes place within several tens of Myr (Bland-Hawthorn et al. 2011; Ritter et al. 2012). For now, we assume that the ejected metals are efficiently mixed with the swept up pristine material interior to the stall radius, such that the gas will be enriched to a metal abundance:

$$\begin{equation} {[{\rm Z/H}]} = \log _{10}\left (\frac{m_{\rm H}}{m_{\rm Z}}\frac{M_{\rm Z}}{X\,M_{\rm sw}(r_{\rm stall})}\right) - \log _{10}({{\rm Z/H}})_{\odot }, \end{equation} \tag{ 12 }$$

where m_H and m_Z are the atomic masses of hydrogen and element "Z", respectively, M_Z refers to the total ejected mass of element Z from the metal-free star, and X = 0.75 is the primordial mass fraction of H. The solar values for all elements, marked by the ☉ symbol, are taken from Asplund et al. (2009). We further consider the possibility of inefficient mixing in Section 3.2.

In order to form a second generation of stars encoded with the chemical signature of the first stellar generation, we require that the blastwave does not evacuate the gas from the host dark matter minihalo. Specifically, we assume that all minihalos where the blastwave lifts the gas to beyond the virial radius will not form a second generation of stars before merging into more massive parent systems. Thus, the only halos that will form a second generation of stars are those where r_stall < r₂₀₀. Although the retention or evacuation of gas from a minihalo can only be correctly treated within a 3D setting (Ritter et al. 2012; B. D. Smith et al. 2014, in preparation; J. S. Ritter et al. 2014, in preparation), our prescription allows us to explore large areas of parameter space that would otherwise be impossible with full 3D hydrodynamic simulations. In any case, we can test the impact of this assumption by considering a stricter limit whereby the only minihalos that retain their gas are those where the blastwave does not lift the gas beyond half the virial radius (see Section 3.2).

Recent simulations suggest that a given minihalo may form a binary or small multiple of massive metal-free stars (Turk et al. 2009; Stacy et al. 2010; Greif et al. 2012; Stacy & Bromm 2013; Hirano et al. 2014), which could reduce the chance of gas retention relative to the single star scenario (K.-J. Chen et al., in preparation). To simulate the chemical enrichment from a small multiple of Population III stars in a single dark matter minihalo, we draw one, two, three, or four stars from the stellar IMF under consideration and assume that all stars form at a redshift z_coll (i.e., the star formation occurs in a "burst"). The temperature of the H ii region heated by this small multiple is calculated as described above, but replacing B_ν(T_eff) in Equation (5) by the sum over every star in the minihalo, Σ_i B_ν(T_eff, i). The blastwave is initiated after the maximum stellar lifetime has passed (i.e., the least massive star that exploded). For multiple stars, the blastwave is followed as an "equivalent" single explosion as described in Section 2.4; specifically, the total energy of the blastwave is equal to the sum of the energy released from the stellar multiple. This assumption is the limiting case where it is most difficult for a minihalo to retain its gas. Similarly, each individual element yield is totaled for all stars in a minihalo, so that we track the total yield of, for example, carbon from the stellar population.

Thus, our calculation ignores the physics of SN kicks that might occur after the most massive star in a binary explodes (Conroy & Kratter 2012). We note that if Population III runaways are efficiently kicked from the center of the dark matter potential, they will produce off-centered explosions, which provide an enhanced gas retention (Bland-Hawthorn et al. 2011).

For the results of this paper, we are concerned with the halos that are able to retain their gas after forming the first generation of stars. To provide reliable statistics, we continuously simulate minihalos for a given IMF and multiple of stars until we have uncovered at least 10, 000 models that retain their gas.

The working definition of a "first galaxy," in its most basic form, is a dark matter halo that confines a long-lived stellar population (see, e.g., the review by Bromm & Yoshida 2011). Within our model, the first galaxies would be those that are the least disrupted by the SN explosion from the first stars, corresponding to the halos that are able to retain their gas.

There are two primary mechanisms in our models that influence the distribution of retained halo masses. Obviously, the most important factor is the energy that is available to lift the gas out of the minihalo, which largely determines the shape of the retained halo mass distribution. However, the form of the primordial stellar IMF also influences the probability that a minihalo will retain its gas, since the presupernova gas distribution strongly depends on stellar masses. For the example shown in Figure 2, a 15 M_☉ star, despite having a lower luminosity than the 40 M_☉ star, drastically reduces the central gas density of the minihalo during its relatively longer life. At a fixed minihalo mass, it is easier for an SN to evacuate the gas from the minihalos that host the longest-lived massive stars. Therefore, a minihalo of a given mass is more likely to retain its gas if the form of the primordial stellar IMF is top-heavy.

In Figure 3, we present the distribution of halo masses that retain their gas (color-coded by the average energy of the hosted SNe). For reference, we also show the halo mass function from which these models are drawn as a thick black curve. It is immediately obvious that there is a significant deficit of low-mass halos that are able to retain their gas. Given that the least disrupted minihalos are those that are able to promptly form a second generation of stars, our calculations suggest that the second stars were formed in halos of mass ∼a few × 10⁶ M_☉.

The chemistry that is born into the second stars is highly sensitive to both the stellar IMF and the energy released during the SN explosion of a metal-free star—these are the two primary mechanisms that also drive the distribution of retained halo masses. This implies that there may be a strong selection bias when interpreting the nature of the first stars by studying their chemical yields in the stellar atmospheres of the most metal-poor second generation stars in our Galaxy. The SNe that evacuated the gas from their host minihalo would have been unable to encode their metal yields in a second generation of stars. For example, there are no environments currently known where the distinct chemical signatures of PISNe (Heger & Woosley 2002) have been unambiguously detected. This could be a consequence of an observational survey bias, whereby the primary goal is to identify the most metal-poor second generation stars; PISNe are expected to enrich the surrounding medium to a metallicity of ∼1/100 solar (Salvadori et al. 2007; Karlsson et al. 2008), which is a considerably higher metallicity than that targeted by current surveys for metal-poor stars. In addition to this bias, we suggest that the high energy of their SN explosion would have evacuated the gas from their host minihalo, resulting in their chemical signature being rarely incorporated into a subsequent stellar population (see Greif et al. 2007; Whalen et al. 2008). More commonly, one should expect to find the chemical signatures from the SNe of massive stars that experienced core collapse, which typically release more than an order of magnitude less energy than PISNe. Indeed, the chemical signatures of core-collapse SNe are regularly observed in the lowest metallicity stars in the halo of our Galaxy (e.g., McWilliam et al. 1995; Cayrel et al. 2004; Lai et al. 2008; Heger & Woosley 2010; Roederer et al. 2014).

The strong link between the metal yield, explosion energy, and gas retention suggests that the second stars should exhibit a strong systematic variation in their relative element abundances. Naively, one should expect that lower atomic number nuclei (e.g., C) are overabundant relative to the higher atomic number nuclei (e.g., Fe), since the latter are synthesized in the innermost, tightly bound core of a massive star. In Figure 4, we show the distribution of [C/Fe] and [Fe/H] values for the halos that are able to retain their gas, color-coded by the average energy released by the SNe in the host minihalo. The solid and dotted contours enclose 68% and 95% of the models, respectively. We also overplot the stellar data compiled by Yong et al. (2013) and Roederer et al. (2014). We have ignored stars, with [Ba/Fe] >0.0, which are probably enriched with the s-process, from these samples. We supplement these data with three CEMP-no stars from Aoki et al. (2013), four stars from Hansen et al. (2014), and the two metal-poor halo stars reported by Caffau et al. (2011) and Keller et al. (2014). We have also included a small handful of stars from the Milky Way ultra-faint dwarf spheroidal galaxies, including three stars from Segue 1 (Norris et al. 2010; Frebel et al. 2014) and one star from Boötes I (Lai et al. 2011; Gilmore et al. 2013).

In general, the variation in [C/Fe] as a function of the Fe abundance for our models show a good agreement with the trend exhibited by the most metal-poor stars, especially given that there are no free parameters in our models. In particular, our calculations match the currently available data for the most Fe deficient CEMP stars, while simultaneously matching the Leo star (Caffau et al. 2011) and most of the "carbon-normal" stars just below [Fe/H] ≃ − 3.0. All minihalos are enriched to a total metallicity [Z/H] ≃ − 3.0, which is in good agreement with cosmological hydrodynamic simulations (e.g., Wise et al. 2012). We further note that the most Fe-deficient CEMP stars need not have formed from the products of a single SN; our models demonstrate that the most Fe-deficient CEMP stars can also be produced by a small multiple of first stars.

In Figure 4, there is also a small cluster of ∼12 "super" CEMP stars that are offset from our models toward higher [C/Fe] and higher [Fe/H]. This small handful of stars is clearly distinct from both our model calculations and from observations of other CEMP stars at a similar metallicity. In addition, these stars are found in both the Milky Way halo and satellite dwarf galaxies. One possibility is that these stars may have formed in minihalos that were enriched by early episodes of AGB nucleosynthesis (Salvadori & Ferrara 2012). A similar possibility is that these low-metallicity CEMP-no stars may have hosted an AGB binary companion and therefore represent the "low-s" extremely metal-poor counterparts of the CEMP-s population (Masseron et al. 2010). This scenario is supported by models of AGB nucleosynthesis at [Fe/H] =−3.0 which predict a [C/Fe] abundance in the range + 2.0 to + 3.0 (Campbell & Lattanzio 2008). These models are consistent with the level of enrichment observed for the super CEMP stars. Dedicated radial velocity monitoring for this sample of stars is now required to confirm if they are members of binary systems. At present, radial velocity variations for 5 of these 12 stars have already been measured (Starkenburg et al. 2014), 3 of which show evidence of binarity.⁸

Alternatively, this handful of CEMP stars could have been born out of gas that was enriched by metal-free stars exploding as Type-II SNe, as considered herein, with a boost in the CNO elements provided by stars that yield no Fe-peak elements. Several models do entertain stellar yields with no Fe-peak elements, including intermediate mass Population III stars (Chieffi et al. 2001; Herwig 2005; Campbell & Lattanzio 2008), models with rotationally enhanced mass loss from massive stars (Meynet et al. 2006; Hirschi 2007; Meynet et al. 2010), and massive stars that end their lives as a pulsational pair-instability SN (with progenitor masses in the range ∼100–140 M_☉; Woosley & Heger 2014).

On the other hand, if this cluster of stars does not represent a distinct population, their disagreement with our models might be a consequence of our assumption that the metal yields from the first stars are uniformly mixed with the swept up pristine material. Inefficient mixing has recently been proposed as a possible explanation for the elevated, constant [α/Fe] ratios seen for several stars over a broad range of metallicity in the Segue 1 ultra-faint dwarf spheroidal galaxy (Frebel et al. 2014). If mixing is considered to be less efficient than allowed by our fiducial model, the contours in Figure 4 would be inflated to account for regions of relatively higher and lower metallicity. However, inefficient mixing is unable to explain the absence of stars between the "ordinary" CEMP stars and the "super" CEMP stars. The details of the mixing process can only be addressed with more realistic 3D, cosmological hydrodynamic simulations.

We also note that the [Fe/H] abundance that we derive from our models depends on the amount of pristine H gas swept up by the blastwave, which is fixed by the energy delivered by the SN and the potential of the host minihalo. Our fiducial model (see Section 2.5) requires that the gas must not be lifted to beyond the virial radius in order for the minihalo to retain its gas. If we relax this criterion to allow models where the blastwave stalls at twice the virial radius, the distributions of the models presented in Figure 4 would be shifted toward lower [Fe/H], since "additional" pristine gas will be swept up. Similarly, if we were to enforce a stricter retention criterion, the models shift toward higher [Fe/H]. We explore this possibility in Figure 5 for the case where two stars (i.e., a Population III binary) are born per minihalo, and require that each minihalo is only able to retain its gas if the SN blastwave does not extend beyond half the virial radius. Enforcing a stricter criterion results in only a minor shift to the overall distribution of [C/Fe] and [Fe/H] values. Our retention criterion is therefore relatively insensitive to the conclusions drawn here regarding the origin of CEMP stars at low metallicity. As a final note, our models have also assumed that stellar multiplicity is independent of the minihalo mass. In principle, the multiplicity of the first stars may depend on the thermal evolution of the collapsing cloud, which in turn depends on the mass of the minihalo (e.g., Hirano et al. 2014). However, this should not significantly affect the trend of [C/Fe] with [Fe/H]; for a given IMF, our simulation results exhibit a general similarity in the [C/Fe] – [Fe/H] plane, independent of the stellar multiplicity.

The number of stars that exhibit an enhancement in carbon relative to those stars that do not is known to be much higher at lower Fe abundance (Beers & Christlieb 2005; Carollo et al. 2012; Yong et al. 2013; Carollo et al. 2014). In the extremely metal-poor regime ([Fe/H] ≲ −3.0), the dominant class of these carbon-enhanced stars are those that exhibit normal abundances of their neutron-capture elements (Aoki et al. 2007; Carollo et al. 2014). In this section, we use our suite of realistic chemical enrichment models from the first stars to estimate the fraction of CEMP stars at the lowest metallicities. In doing so, we assume that all values of the "mixing" parameter from the Heger & Woosley (2010) metal-free nucleosynthesis models are equally probable. We further assume that each minihalo that is able to retain its gas will form a second generation of stars. Using the definition proposed by Aoki et al. (2007), we define our models to be carbon-enhanced when [C/Fe] ⩾+0.7. The fraction of our models that produce carbon-enhanced minihalos as a function of the Fe abundance is displayed in Figure 6 for the two IMFs under consideration. Using the same criterion, we have calculated the observed fraction of carbon-enhanced halo stars from the data sample described in Section 3.2, excluding the four stars that have been observed in the Segue 1 and Boötes I dwarf spheroidal galaxies.

A general prediction of our models—independent of the IMF and gas mixing efficiency—is that all future stars that are discovered with an Fe abundance [Fe/H] ≲ −5.0 should almost certainly exhibit a carbon enhancement. Before drawing conclusions from the CEMP fraction at somewhat higher metallicities, we caution that some of the stars in the observational sample may not have been exclusively enriched by Population III stars; some of the CEMP stars with [Fe/H] ≲ −3.0 may have seen pollution from an extinct AGB companion star or otherwise may have condensed from gas that was also enriched by low-metallicity Population II stars. These two possibilities would respectively under- or overestimate the observed CEMP fraction relative to our models. Despite these potential caveats, our models succeed in producing a decrease in the CEMP fraction toward higher [Fe/H] that is in broad agreement with the current observational data.