CHEMICAL CONSTRAINTS ON THE CONTRIBUTION OF POPULATION III STARS TO COSMIC REIONIZATION

Our model calibration procedure is described in detail in Kulkarni et al. (2013). The model is calibrated by matching (1) the observed cosmic SFR density evolution (Hopkins & Beacom 2006), (2) the fraction of total baryon density in collapsed halos at z = 0 (Fukugita & Peebles 2004), the reionization history of the IGM as measured by (3) the electron Thomson scattering optical depth to the last scattering surface (Hinshaw et al. 2013), and (4) the hydrogen photoionization rate evolution (Meiksin & White 2004; Bolton & Haehnelt 2007; Faucher-Giguère et al. 2008; Becker & Bolton 2013).

We assume a constant star-formation efficiency parameter f_*, defined by

$$\begin{equation} \psi = f_*\left(\frac{M_\mathrm{cool}}{t_\mathrm{dyn}}\right), \end{equation} \tag{ 2 }$$

where ψ is the halo SFR, M_cool is the cold gas mass in the halo, and t_dyn is the halo dynamical time. The star-formation efficiency is then tuned to reproduce the cosmic SFR density. For our fiducial model (described below), we have f_* (Population II) =0.002 and f_* (Population III) =0.004. Star formation in halos is accompanied by SN-induced outflows, the strength of which is fixed by matching the fraction of total baryon density in collapsed halos at z = 0. There is also an additional contribution to the SFR density from short starbursts that accompany major mergers of halos, as described in the previous section.

Once the star formation efficiency is fixed, reionization depends mainly on the escape fraction of ionizing photons, f_esc, defined as the fraction of ionizing photons that escape the parent galaxy. We assume f_esc to be independent of redshift and halo mass, and calibrate it to reproduce the observed values of the electron Thomson scattering optical depth, τ_e, and the hydrogen photoionization rate, Γ_HI. It is crucial to include both these constraints as the observed hydrogen photoionization rate evolution imposes continuity on the epoch of reionization. Our fiducial model has f_esc = 0.2. Although, in our calibrated model, the IGM electron Thomson scattering optical depth τ_e is always obtained in the 1σ range of its best-fit value (0.089 ± 0.014 as given by WMAP9; Hinshaw et al. 2013), our value of τ is always closer to the upper-limit of this range. This suggests that an evolving escape fraction is necessary for fitting both τ_e and Γ_HI simultaneously (Mitra et al. 2012; Finlator et al. 2012).

Our model avoids the tension (Robertson et al. 2013) between the observed star formation at redshifts z > 4 and reionization constraints by overproducing stars at these redshifts. In other words, our normalization of the luminosity function, ϕ_*, at these redshifts is higher than the observed value. This is reflected in panel (c2) of Figure 1. In this panel, the black data points are from Bouwens et al. (2011) integrated down to −17.7 AB mag. These are compared with the model prediction of luminosity-limited SFR density. Because of the high value of ϕ_*, the luminosity-limited SFR density has a magnitude limit of just ∼ − 19 AB mag. This difficulty in reconciling high-redshift luminosity functions with the IGM reionization history has been noted before in the literature (Faucher-Giguère et al. 2008) and can potentially be overcome by introducing a redshift evolution of the ionizing photon escape fraction. However, even with this modification to the model, it is difficult to achieve multi-epoch agreement between galaxy formation models and the observed luminosity function (Weinmann et al. 2012). In this paper, we choose to work with this discrepancy and focus on the effect of chemical evolution on the traditional galaxy formation picture.

Zoom In Zoom Out Reset image size

We calibrate the model independently for each of the two Population III stellar IMFs considered in this paper. Our fiducial model assumes instantaneous mixing of metals in the ISM. We describe the results of this model in this section. We discuss the effect of relaxing the instantaneous mixing assumption in subsequent sections.

3.1.1. Population III SFR

3.1.2. Photoionization Rate

The small contribution of Population III stars to the total cosmic SFR density suggests that their contribution to reionization will also be small. This is seen in panel (b2) of Figure 1, which shows the evolution of the hydrogen photoionization rate in our model. Making the so-called local source approximation, which is valid for spectral indices typical to star-forming galaxies, the hydrogen photoionization rate is given by

$$\begin{equation} \Gamma _\mathrm{HI}(z)=(1+z)^3\int _{\nu _{912}}^\infty d\nu \lambda (z,\nu) \dot{n}_\nu (z) \sigma (\nu), \end{equation} \tag{ 3 }$$

where σ(ν) is the photoionization cross-section of hydrogen, and λ(z, ν) is the redshift-dependent mean free path of ionizing photons. We calculate the mean free path of ionizing photons as in Paper 1 by integrating over the lognormal density probability density function of the IGM and estimating the average distance between high-density, neutral regions. This method is calibrated to reproduce the incidence rate of Lyman-limit systems at low redshifts (Miralda-Escudé et al. 2000; Choudhury & Ferrara 2005; Prochaska et al. 2009). Our model agrees with the measurement of Prochaska et al. (2009). The quantity $\dot{n}_\nu (z)$ is the number density of ionizing photons in the IGM per unit time, and is given by

$$\begin{equation} \dot{n}_\nu (z) = f_\mathrm{esc}\dot{\rho }_*(z)\int dm\,\phi (m)t_\mathrm{age}(m)Q_H(m). \end{equation} \tag{ 4 }$$

The integral in this equation is over stellar masses and takes the IMF-dependence of the ionizing photon flux into account: ϕ(m) is the normalized stellar IMF, t_age(m) is the age of a star with mass m and Q_H(m) the stellar hydrogen-ionizing photon flux (in photons s⁻¹) provided by stellar evolution models (Schaerer 2002). The quantity f_esc is the escape fraction of ionizing photons that accounts for the fraction of the ionizing photons that escape into the IGM. The escape fraction is a free parameter of our model. In this paper, we assume that f_esc is constant at all redshifts. We comment on the effect of this assumption below. Our fiducial model has f_esc = 0.2

In panel (b2) of Figure 1, the data points are measurements of the hydrogen photoionization rate as deduced from the mean opacity of the hydrogen Lyα forest (Meiksin & White 2004; Bolton & Haehnelt 2007; Faucher-Giguère et al. 2008), where we note that there is disagreement between the measurements at the factor-of-two level, which likely results from different assumptions about the density distribution and thermal state of the IGM. Here we choose to fit the data by Faucher-Giguère et al. (2008), but this choice is not critically important for our main result. The solid curves in panel (b2) of Figure 1 show the model predictions corresponding to two different Population III IMFs (low-mass IMF in black and high-mass IMF in red). The model predictions agree very well with the observational measurements.⁴ As seen by the evolution of the hydrogen photoionization rate, reionization in our model is gradual. It begins at z ≳ 30 and is 90% complete by z ∼ 7. This gradual change in the ionization state of the IGM helps us simultaneously reproduce the observed electron scattering optical depth τ_e = 0.089 ± 0.014 (Hinshaw et al. 2013) and the photoionization rate data. Before reionization at z_reion = 7.5, the photoionization rate increases rapidly as UV photon sources build up. There is a sudden jump at z = z_reion when different H ii regions overlap. (This redshift is marked by the vertical dashed line in Figure 1.) This is because at this redshift, a given point in the IGM starts "seeing" multiple sources, which rapidly enhances the UV photon mean free path, thereby affecting the photoionization rate. Second, Figure 1 also shows that the contribution of Population III stars to reionization is small. This is clear from the overlap between the red and black solid curves in panel (b2), and is evident in the dashed curves, which show the contribution to Γ_HI by Population III stars. We see that the Population III contribution to the photoionization rate is subdominant over most of the reionization history. Panel (b1) of Figure 1 further highlights this by showing the fraction of the H i photoionization rate contributed by Population III star-formation, relative to the total rate. Except at the earliest stages of galaxy formation (z ∼ 50), the ratio is much less than unity. For the low-mass IMF, the ratio is less than 10% for z ≲ 20, and for the high-mass IMF, it is less than 10% for z ≲ 30.

3.1.3. Understanding the Low Population III Contribution

In the results presented above, metals injected into the ISM by SN explosions are assumed to be instantaneously mixed into the halo ISM. Instantaneous and homogeneous mixing of metals is a standard assumption in galactic chemical evolution studies (Tinsley 1980; Pagel 2009; Matteucci 2012) and is responsible for the early termination of Population III star formation in our fiducial model. Note that we are not assuming instantaneous recycling, since we directly model the delay in the synthesis of new metals due to finite stellar lifetimes. However, we have assumed instantaneous mixing, which is to say that after a SN event, newly liberated metals are instantaneously available in the ISM to influence the next generation of star-formation. In our fiducial model we have seen that the initial burst of Population III stars is sufficient to enhance the ISM metallicity beyond Z_crit in any halo mass bin over a very short time scale. When this happens, the corresponding galaxy stops forming Population III stars. Since most of the mass in the universe is contained in M_* halos, the Population III cosmic SFR density begins to decline as soon as these halos cross the Z_crit threshold. This is the crucial effect that reduces the contribution of Population III stars, which was not captured in previous models because they did not track chemical enrichment for a large population of halos in a cosmological volume. We now discuss the dependence of this result on our assumption of instantaneous mixing.

The solid lines in Figure 2 show the self-enrichment time scale as a function of halo mass for the two Population III IMFs in our fiducial model, where the low-mass IMF is shown in black and the high-mass IMF in red. We define the self-enrichment timescale as the time between the first star-formation episode of the halo and the time at which its gas-phase metallicity crosses Z_crit. In this plot, halos are labeled by their mass at z = 0; their actual mass value at the time of crossing M_min is not shown. In the low-mass IMF case (solid black), the timescale is a few times 10⁶ yr. Pair-instability SNe have higher metal output, which results in a shorter enrichment timescale by a factor of three in the high-mass IMF case (solid gray). As discussed in Section 3.1, it is this short self-enrichment timescale that restricts the contribution of Population III stars to reionization. Note that these curves have a very flat dependence on halo mass because the self-enrichment time scale depends primarily on stellar lifetimes.

Zoom In Zoom Out Reset image size

In the ISM of a galaxy, mixing of metals ejected by SNe is carried out by different mechanisms on different length scales: diffusive dispersion due to large-scale motion on kpc scales, turbulent diffusion on pc scales, and molecular diffusion on smaller scales. As a result, the self-enrichment timescale is decided by (1) the respective timescales of these mixing processes, (2) the rate of metal production by SNe, and (3) rate of interaction with the environment, via inflows and outflows of gas. In our fiducial model, the timescale of the mixing processes is assumed to be zero and therefore the self-enrichment timescale depends only on the stellar lifetimes via the SN rate. This is the reason behind the short self-enrichment timescale shown in Figure 2. We now relax this instantaneous mixing assumption.

SN-driven metal-mixing in the ISM has been studied using hydrodynamical simulations by de Avillez & Mac Low (2002), who used tracer particles in a simulation of a 1 × 1 × 20 kpc³ region of the Galactic disk and found mixing timescales between 10⁶ and 10⁸ yr (see their Figure 8) for SN rates of ∼1 to ∼100 times the Galactic SN rate. The lower end of this range is of the same order of magnitude as the self-enrichment timescale in our fiducial model. However the results of these simulations suggest that the mixing timescale could very well be two or three orders of magnitude larger.

We consider the effect of longer self-enrichment timescales on our result by implementing a "mixing function," f(t), which governs the rate at which metals are mixed in the ISM. The source term due to star formation in the metallicity evolution equation is of the form (Equation (19) in Kulkarni et al. 2013)

$$\begin{eqnarray} \dot{M}_{Z}(z) &=& \int _{m_l}^{m_u}dm\,\phi (m)\cdot \psi [t(z)-\tau (m)]\nonumber\\ &&\times mp_Z(m), \end{eqnarray} \tag{ 5 }$$

where m is the stellar mass, ψ(t) is the SFR at time t, ϕ(m) is the stellar IMF, τ(m) is the stellar age, and p_Z(m) is the mass fraction of a star of initial mass m that is converted to metals and ejected. The limits m_l and m_u define the range of stellar masses considered in the IMF. To relax the instantaneous mixing assumption, we modify Equation (5) to

$$\begin{eqnarray} \dot{M}_{Z}(z) &=& \int _{m_l}^{m_u}dm\,\phi (m)mp_Z(m)\nonumber\\ &&\times \int _{t(z)-t_\mathrm{delay}}^{t(z)}\!\!\!d\tilde{t}\,\psi [\tilde{t}(z)-\tau (m)] f(\tilde{t}), \end{eqnarray} \tag{ 6 }$$

where $f(\tilde{t})$ is a mixing function that serves to delay the mixing of metals in the ISM after a SN event. The mixing function is defined over a time duration of t_delay. In this picture, after a SN has exploded, the resulting metal mass is added to the ISM gradually over a time t_delay. Until this time, we imagine the metals to be locked up into un-mixed pockets of the ISM, which means they are not available for future star-formation. Thus, our fiducial model corresponds to the case where the mixing function is a delta function with t_delay = 0 yr. For non-zero values of t_delay, the ISM metallicity can remain below the critical metallicity Z_crit for a longer time compared to our fiducial model. Therefore we would expect that in this case, Population III star formation will continue for a longer period, and possibly impact reionization. The mixing function is determined by the complex interplay of various mixing processes in the ISM. The most conservative form of the mixing function is a constant of the form

$$\begin{equation} f(t) = \left\lbrace \begin{array}{@{}ll}t_\mathrm{delay}^{-1} & {\rm if } t < t_\mathrm{delay} \\ 0 & {\rm otherwise}, \end{array} \right. \end{equation} \tag{ 7 }$$

where t is the time since a SN explosion.⁶ Note that f(t) is normalized such that all of the SN ejecta is mixed in the ISM over the period t_delay. We adopt this simple form for the mixing function in this paper. Figures 3 and 4 show the results of our model with this mixing function and t_delay = 10¹¹ yr and t_delay = 10¹² yr, respectively. The self-enrichment timescales in these models are compared to that in our fiducial model in Figure 2 (only the high-mass Population III IMF case is shown for simplicity). Given that we have adopted a mixing timescale that is much longer than the age of the Universe (i.e., t_delay = 10¹⁰ or 10¹¹ yr), the effective result is that at z ∼ 10, where t_{z = 10} ∼ 10⁹ yr, these mixing models imply that only 1% and 0.1% of the metals produced by Population III SNe are mixed into the ISM, respectively. As we will see later, even this small amount of metals is enough to produce significant chemical feedback that influences Population III star formation. This occurs because, as shown in panel (a2) of Figure 1, a single generation of Population III SNe injects enough metals to raise the ISM metallicity of M_* halos to a Z ∼ 10⁻¹Z_☉ by z ∼ 10, if these metals are instantaneously mixed. Thus reducing this yield by a factor of ∼t_{z = 10}/t_delay ∼ 100, as in the case when t_delay = 10¹¹, is still sufficient to be in excess of the critical metallicity. For this reason, and as is also clear from the mathematical form of Equation (6), we generally expect a nonlinear dependence of the self-enrichment timescale on t_delay, and similarly the dependence of Population III star-formation on t_delay will also be nonlinear. For instance, between Figure 3 and Figure 4, changing t_delay by a factor of 10 results in small changes of factor of three in the self-enrichment timescale.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

In Figure 3 the same curves illustrating the star-formation, reionization, and enrichment history as in Figure 1 are shown, but for a model with t_delay = 10¹¹ yr. As expected, and illustrated in panel (a2) of Figure 3, halos now take longer to cross the Z_crit metallicity threshold, compared to the fiducial model (panel a2 of Figure 1, which is on the same scale). This increases the cosmic Population III SFR density, which, although still subdominant, contributes more than 10% of the total SFR density at redshifts above z ∼ 20 in the high-mass Population III IMF run (the corresponding redshift value for the fiducial model was z ∼ 30). The relative increase in Population III star-formation also manifests as an increase in the hydrogen photoionization rate, which is also enhanced relative to the fiducial model. Indeed, the contribution of Population III stars to the hydrogen photoionization rate at z = 10 is 10% for the high-mass case, whereas the contribution was less than 1% at this redshift in our fiducial model. Thus for t_delay = 10¹¹ yr, the contribution of Population III stars to reionization is subdominant but significant. Note that these numbers are slightly different for the low-mass case for the same mixing function and t_delay. In general, low mass Population III star formation lasts longer than in the high-mass IMF. This is because the metal yields of high mass Population III stars are higher than those of low mass stars, resulting in larger chemical feedback and earlier termination of Population III.

Figure 4 shows the results of the model with t_delay = 10¹² yr, for which the mixing of metal metals is even more gradual. The impact of Population III on cosmic star-formation and the ionizing photon budget is now increased in magnitude relative to the previous case shown in Figure 3. For instance, all star formation for z ≳ 10 in this model is Population III, regardless of the Population III IMF. Thus the process of hydrogen reionization is almost single-handedly carried out by Population III stars. The contribution of Population III stars to the hydrogen photoionization rate at z = 10 is about 60% for the high-mass case, and 100% for the low-mass case. Their contribution to the cosmic SFR density at z = 10 is about 20% for the high-mass case, and 80% for the low-mass case. Similar to the results in Figure 3, the Population III contribution is higher for low-mass Population III IMF because of weaker chemical feedback.

Figure 2 helps to visualize the effect of the mixing function on our model. The self-enrichment timescale in the two variant models is longer than that in the fiducial model by more than an order of magnitude. The increase, however, is still less than t_delay, as only a fraction of metals are required to increase halo metallicity beyond Z_crit.

A general lesson is that due to constraints imposed by the measurements of cosmic SFR density, the hydrogen photoionization rate, and the electron scattering optical depth, the contribution of Population III stars to reionization can be enhanced only by increasing the metal mixing timescale assumed in the model. This contribution is generally predicted to be small unless the self-enrichment timescale is ≳ 3 × 10⁸ yr.

By varying the assumptions about the Population III IMF and the metal mixing timescale, we have seen that the self-enrichment timescale can take on values between 3 × 10⁶ and 3 × 10⁸ yr, dramatically impacting the chemical feedback that eventually terminates Population III star formation. These degrees of freedom result in concomitant uncertainties on the contribution of Population III star-formation to the ionizing photon budget of 1%–100%, since all the models were able to match the star-formation and reionization observables that we considered. However, we can discriminate between these possibilities using accurate chemical enrichment observations in DLAs at post-reionization redshifts. Observations of DLAs can be used to measure gas-phase metallicities at large cosmological lookback times with high precision. Furthermore, relative abundances can still be measured accurately deep into the reionization epoch (z > 6) using metal-line transitions redward of Lyα, even though Gunn–Peterson absorption precludes the measurement of neutral hydrogen. In Kulkarni et al. (2013) we modeled the chemical evolution of DLAs, and showed how their abundance patterns can be used to constrain Population III scenarios. Here we argue that they can also constrain the contribution of Population III stars to reionization.

In our model we assigned a mass-dependent H i absorption cross-section, denoted by Σ to each halo in order to predict the expected distribution of DLA abundance ratios (see Kulkarni et al. 2013 for details). This assignment is motivated by hydrodynamical simulations (Gardner et al. 1997, 2001; Nagamine et al. 2004a, 2004b, 2007; Pontzen et al. 2008) and reproduces the observed DLA metallicity evolution (Rafelski et al. 2012), incidence rate (Prochaska et al. 2005; Noterdaeme et al. 2012), and clustering bias (Font-Ribera et al. 2012) at low redshifts (z ∼ 3) very well, and takes the form

$$\begin{equation} \Sigma (M)=\Sigma _0\left(\frac{M}{M_0}\right)^2\left(1+\frac{M}{M_0}\right)^{\alpha -2}, \end{equation} \tag{ 8 }$$

where the constants take the values of α = 0.2, M₀ = 10^9.5 M_☉, and Σ₀ = 40 kpc² at z = 3 (Pontzen et al. 2008; Font-Ribera et al. 2012). Values at other redshifts are calculated by mapping halos at these redshifts to halos z = 3 according to circular velocity (Font-Ribera et al. 2012). With this assignment, for any measurable property p (e.g., abundance ratio [M₁/M₂]) of DLAs, we can calculate the number of systems with different values of p in a sample of DLAs. This is called the line density distribution, and with Equation (8) in hand, it can be written as (e.g., Wolfe et al. 2005)

$$\begin{equation} \frac{d^2N}{dXdp}=N(M)\cdot \Sigma (M)\cdot \frac{dl}{dX}\frac{dM}{dp}\cdot (1+z)^3. \end{equation} \tag{ 9 }$$

Here, X is an absorption length element given by

$$\begin{equation} \frac{dl}{dX}=\frac{c}{H_0(1+z)^3}, \end{equation} \tag{ 10 }$$

dl = cdt is a length element, and p is the property in consideration. The halo mass is denoted by M, N(M) is the comoving number density of halos (i.e., the halo mass function), and Σ(M) is the halo cross section given by Equation (8). The quantity dM/dp in Equation (9) can be easily calculated in our model, as properties like metallicity and relative abundances are known for all halo masses. The integral of Equation (9) over all values of p is just the total line density of DLAs, dN/dX (Kulkarni et al. 2013). The average value of p in an observed sample of DLAs is given by

$$\begin{equation} \langle p\rangle =\int \frac{d^2N}{dXdp}\cdot p\cdot dp\cdot dX. \end{equation} \tag{ 11 }$$

Figure 5 shows the result of evaluating Equation (11) for p = [C/Fe] and [O/Si] in our fiducial model and its variants. It also shows the observed evolution of [C/Fe] and [O/Si] relative abundances in DLAs. The z ∼ 2–4 measurements are from Dessauges-Zavadsky et al. (2003); Péroux et al. (2007); Cooke et al. (2011) and the z > 4 measurements are from Becker et al. (2012), as compiled by Becker et al. (2012). Over a time period of about 6 Gyr (from z = 2 to z = 6), these abundance ratios are relatively constant. Furthermore, they show little scatter around the mean. Solid lines in Figure 5 show the evolution of these relative abundances in our low-mass IMF models, while the dashed lines show the evolution in the high-mass IMF case. Curves with different colors indicate different delay times.

Zoom In Zoom Out Reset image size

Our fiducial model agrees with the data for [C/Fe] and [O/Si]. In this redshift range, the mean values of the abundance ratios in this model are governed by the Population II IMF as the contribution of Population III stars is erased in all but the smallest halos. The asymptotic value of these abundance ratios toward low redshift thus simply reflects the relative yields of these elements per star integrated over the Population II IMF.

However, the variants of the fiducial model, in which Population III contribution to the cosmic SFR and photoionization rate is higher, disagree with current relative abundance measurements. We first discuss the high-mass IMF models. As described in the previous section, we considered two variants where metals are gradually mixed in the ISM over a period of t_delay = 10¹¹ yr and 10¹² yr, respectively, resulting in corresponding self-enrichment timescales of 10⁸ and 5 × 10⁸ yr. As shown in Figures 3 and 4, the first of these models has more than 20% of Population III contribution to the photoionization rate down to z ∼ 20, while in the second model the contribution is more than 20% all way down to z ∼ 10. We now see in Figure 5 that the model with t_delay = 10¹¹ yr is marginally ruled out by the existing [O/Si] data (red dashed curve in bottom panel of Figure 5), while the model with t_delay = 10¹² yr is completely ruled out. In the previous section we showed that the contribution of Population III stars to reionization could only be enhanced by increasing the metal-mixing timescale, however this results in a corresponding increase in the chemical vestiges of these Population III stars. Figure 5 shows that the DLA relative abundance measurements actually restrict the high-mass Population III contribution to reionization to be less than 10% for z ≲ 15. The model with t_delay = 10¹¹ yr acts as a kind of upper bound on the self-enrichment timescale of halos, and therefore on the role that high-mass Population III stars play in hydrogen reionization. The [O/Si] ratio is more constraining than [C/Fe] because of the significant variation in the Si yield with the IMF—the high-mass IMF produces two orders of magnitude higher Si yield, as Si is efficiently produced in massive stars due to O-burning.

The contribution of low-mass Population III stars to reionization is harder to constrain using DLA chemical data. This is because the chemical yields of low-mass Population III stars are not very different from those of the Population II stars considered in our models, as the two IMFs have similar shapes and mass ranges. (There is some difference in the yields due to differences in their metallicities.) Thus, we see in Figure 5 that the [O/Si] values in all our low-mass IMF models are consistent with the data, regardless of mixing delay and self-enrichment timescales. This is also true for the [C/Fe] abundance ratio, although the values are slightly different from the fiducial case, because the yields are different and the large mixing timescale slows dilution by Population II yields. However, even if we cannot rule out the low-mass IMF case, it is worth noting that the ionizing emissivity of the low mass Population III stars is only a factor of two higher than that of the Population II stars. Therefore, reionization by these stars is qualitatively similar than reionization by Population II stars alone.

Finally, we note that the exact behavior of the curves in Figure 5 depends on the form of the mixing function used. A mixing function different from that in Equation (7) will, in general, result in a different evolution of the mean relative abundances. However, the primary result of Figure 5 is more general—any Population III star formation activity that produces hydrogen-ionizing photons at these redshifts will also necessarily produce Population III chemical signatures, which can be constrained using measurements of abundance patterns in DLAs. In addition, DLA relative abundance measurements at z ∼ 8, using background QSOs or gamma-ray bursts,⁷ could start to constrain even the low-mass Population III IMF by witnessing the build-up of the metallicity in halos and change in relative abundances with time because of finite stellar lifetimes.