Chemical constraints on the contribution of Population III stars to cosmic reionization
Girish Kulkarni, Joseph F. Hennawi, Emmanuel Rollinde, Elisabeth Vangioni
aa r X i v : . [ a s t r o - ph . C O ] O c t Draft version February 26, 2018
Preprint typeset using L A TEX style emulateapj v. 5/2/11
CHEMICAL CONSTRAINTS ON THE CONTRIBUTION OF POPULATION III STARS TO COSMICREIONIZATION
Girish Kulkarni , Joseph F. Hennawi , Emmanuel Rollinde , Elisabeth Vangioni Draft version February 26, 2018
ABSTRACTRecent studies have highlighted that galaxies at z = 6–8 fall short of producing enough ionizingphotons to reionize the IGM, and suggest that Population III stars could resolve this tension, becausetheir harder spectra can produce ∼ × more ionizing photons than Population II. But this argumentdepends critically on the duration of the Population III era, and because Population III stars formfrom pristine gas, in turn depends on the rate of galactic enrichment. We use a semi-analytic modelof galaxy formation which tracks galactic chemical evolution, to gauge the impact of Population IIIstars on reionization. Population III SNe produce distinct metal abundances, and we argue that theduration of the Population III era can be constrained by precise relative abundance measurements inhigh- z damped Ly α absorbers (DLAs), which provide a chemical record of past star-formation. We findthat a single generation of Population III stars can self-enrich galaxies above the critical metallicity Z crit = 10 − Z ⊙ for the Population III-to-II transition, on a very short timescale t self − enrich ∼ yr, owing to the large metal yields and short lifetimes of Population III stars. This subsequentlyterminates the Population III era, hence they contribute &
50% of the ionizing photons only for z &
30, and at z = 10 contribute < z . .
10% at z = 10. Future abundance measurements of z ∼ Subject headings:
Cosmology: dark ages, reionization, first stars — galaxies: evolution — galaxies:ISM — stars: Population III INTRODUCTION
Reionization of the intergalactic medium (IGM) isa watershed event in the history of the universe, andis tightly coupled to the problem of galaxy forma-tion at high redshift. The primary evidence for hy-drogen reionization comes from observation of Gunn-Peterson troughs (Gunn & Peterson 1965) in spectra ofhigh-redshift quasars (Fan et al. 2006b; Mortlock et al.2011). Other pieces of evidence are in the angularpower spectrum of polarization anisotropy of the CMB(Larson et al. 2011; Hinshaw et al. 2012), evolution ofthe IGM temperature (Hui & Haiman 2003; Becker et al.2011; Raskutti et al. 2012), and evolution of the lumi-nosity function of Lyman- α emitters (Ouchi et al. 2010).For a more complete discussion of observational con-straints on reionization see, e.g., review by Fan et al.(2006a). A general picture that emerges from these ob-servations and a broad class of theoretical models is thatH i reionization was a gradual process that lasted forhundreds of Myr from z ∼
20 to z ∼
6, and that star-forming galaxies most likely provided the required ion-izing photons (Choudhury & Ferrara 2006; Mitra et al.2011; Kuhlen & Faucher-Gigu`ere 2012; Robertson et al.2013).Given this evidence, several recent studies have usedaveraged radiative transfer models to ask whether the Max Planck Institute for Astronomy, K¨onigstuhl 17, 69117Heidelberg, Germany; [email protected] Institut d’Astrophysique de Paris, UMR 7095, UPMC, ParisVI, 98 bis boulevard Arago, 75014 Paris, France observed populations of galaxies at high redshift pro-duce enough ionizing photons to reionize the IGM(Robertson et al. 2010). Kuhlen & Faucher-Gigu`ere(2012) studied constraints from measurements of thehydrogen photionization rate, Γ HI , from the (post-reionization) Lyman- α forest (Faucher-Gigu`ere et al.2008; Becker & Bolton 2013) and the requirement thatΓ HI should evolve in a continuous manner through theepoch of reionization. They concluded these two con-ditions require either a rapid evolution in the ioniz-ing photon escape fraction, f esc , or an extrapolation ofthe galaxy luminosity function to extremely faint lu-minosities ( M UV ∼ −
10 or L ∼ . L ∗ ). Severalindependent studies have come to similar conclusions(Fontanot et al. 2012; Mitra et al. 2012; Finlator et al.2012; Raskutti et al. 2012; Haardt & Madau 2012;Alvarez et al. 2012; Robertson et al. 2013). This worksuggests that (1) f esc is at least ten times higher at z = 10than at z = 4, and/or (2) there is a large population ofundetected faint galaxies that produces the lion’s shareof the total ionizing flux, and/or (3) new galactic sources,such as mini-quasars or Population III stars, are activeat high redshift and assist star-forming galaxies in reion-izing the IGM.Owing to their primordial composition, Population IIIstars have harder spectra and thus emit more hydrogen-ionizing photons. A cluster of Population III stars (with100–260 M ⊙ Salpeter IMF) produces an order of mag-nitude more hydrogen-ionizing photons than a clusterof Population II stars (with 0.1–100 M ⊙ Salpeter IMF)with the same total mass (Schaerer 2002). Thus, if thePopulation III star formation rate is high enough, theircontribution to the total ionizing photon budget could besignificant. In this paper, we study the impact of Popula-tion III stars on reionization using a semi-analytic modelof galaxy formation that tracks galactic chemical evolu-tion and is fully coupled to the evolution of the thermaland ionization state of the IGM. Tracking the chemicalevolution is crucial for understanding the contributionof Population III stars, which in our model form whenthe gas-phase metallicity of interstellar media (ISM) ofgalaxies is below a critical metallicy Z crit . It is thesestars in the very first galaxies that presumably initiatedthe process of reionization as they were likely the firstsources of hydrogen-ionizing photons. As the formationof these stars depends on the metallicity of the gas out ofwhich they form, the total constribution of these stars tothe cosmic star formation rate (SFR) density, and hencethe ionizing photon budget, depends on the chemical evo-lution of their environment, which is precisely what ourmodel aims to calculate.The role of Population III stars in cosmic chemicaland ionization evolution has been studied previously.Faced with a high value of the Thomson scattering op-tical depth to the last scattering surface, τ e , reportedby the first-year WMAP results τ e = 0 .
17 (Spergel et al.2003), which suggested that reionization occured veryearly, several studies considered the ionizing emissiv-ity of Population III stars (Venkatesan et al. 2003;Wyithe & Loeb 2003; Cen 2003; Sokasian et al. 2004;Yoshida et al. 2004), and others considered reionizationscenarios driven by Population III stars (Daigne et al.2004; Choudhury & Ferrara 2005; Daigne et al. 2006;Rollinde et al. 2009; Mitra et al. 2011). Due to thewidely different methods and assumptions of these works,it is is not straigthforward to compare their results.Nonetheless, there are some common features in their re-sults. In all of these models, reionization is initiated byPopulation III stars, which dominate the cosmic stellarcontent for some time. Eventually, however, the Popula-tion III SFR is reduced due to chemical enrichment of thestar-forming gas, and the process of reionization is com-pleted by Population II stars. A general conclusion ofthese studies was that, under conservative assumptionsregarding various feedback processes (chemical, star for-mation, photoionization), Population III stars could con-tribute significantly to hydrogen reionization at z & This is a significant drawback, becausethe time-scale over which the ISM of these galaxies is en-riched by the first generation of Population III stars di-rectly regulates the cosmic Population III star formationrate and hence the contribution of Population III stars toreionization. Indeed, Wise et al. (2012) recently used ra-diation hydrodynamics simulations to argue for just thiskind of feedback. These authors found that for high-massPopulation III stars (Chabrier IMF with M char = 100M ⊙ ), which produce pair-instability supernovae, just onesupernova is enough to enrich the parent halo to a metal- Models of Population III star formation that implement chem-ical evolution exist in the literature (e.g., Salvadori et al. 2007).These have proven to be very useful in studying, e.g., the extremelymetal-poor stars in the Galactic halo. But these models do not cal-culate IGM reionization history. licity of 10 − Z ⊙ and prevent further Population III starformation. However, these authors did not study the im-plications of this chemical feedback on the contributionof Population III star-forming haloes to ionizing photonbudget and reionization. This would require simulatingstar formation and chemical feedback for a cosmic ensem-ble of haloes with a wide range of masses, and couplingthese to the IGM via radiative transfer while consideringvarious observational constraints on reionization.To summarize, previous work aiming to under-stand the contribution of high-redshift faint galaxiesor Population III star-forming galaxies to reionization(e.g. Venkatesan et al. 2003; Choudhury & Ferrara 2005;Robertson et al. 2013) did not consider the detailedphysics of galaxy formation, with chemical evolution andPopulation III star formation, while models that includedthese effects (Salvadori et al. 2007; Wise et al. 2012) didnot couple galaxy formation with IGM reionization. InKulkarni et al. (2013) we presented a a semi-analyticmodel of galaxy formation that tracks the chemical evo-lution of galaxies as well as the thermal and ioiniza-tion evolution of the IGM, and used this model to ar-gue that measurements of relative abundances in high-redshift Damped Ly α systems can place interesting con-straints on the Population III IMF. In this paper, we usethis model to study the impact of Population III star-formation on reionization. Our model improves uponprevious work on this subject in three important ways.First, it accounts for chemical evolution within a halo indetail, as part of a semi-analytical model of galaxy for-mation, taking into account stellar lifetimes, and inflowsand outflows. This lets us study the Population III-to-IItransition in haloes of various masses. Second, it useshalo mass assembly histories from cosmological simu-lations and fully couples galaxy formation to the ther-mal and ionization evolution of the IGM. This also letsus consider halo-mass-dependent effects like photoioniza-tion feedback. Third, we consider a range of Popula-tion III IMFs that presumably bracket the true IMF. Achemical evolution model with these three features, cou-pled self-consistently with thermal and ionization evolu-tion of the IGM, provides a useful framework to studythe contribution of Population III stars to reionization. MODELING THE COUPLED EVOLUTION OF GALAXIESAND THE IGM
We use a model of galaxy formation and IGM evolutiondescribed in Kulkarni et al. (2013). Here we highlight thethe main features of the model, and refer the reader tothat paper for additional details.1. Average mass assembly histories of dark matterhaloes are obtained from fitting functions cali-brated to cosmological simulations (Fakhouri et al.2010) for a number of logarithmically spaced halomasses. These assembly histories are a function ofhalo mass at z = 0.2. Baryonic evolution is implemented for each halomass value. In simplest terms, this assumes thata halo (1) accretes baryons through cosmologi-cal accretion, (2) forms stars from any gas con-tained in the halo for sufficiently long duration, and(3) ejects baryons via supernova powered outflows.This lets us calculate various properties—such asmetallicity and star formation rate—as functionsof halo mass. Global averages, such as the cosmicSFR density, are calculated by integrating over allhalo masses.3. Gas content of a halo is influenced by gas inflow dueto cosmic accretion, star formation, stellar massloss, and gas outflow due to supernova feedback.Gas inflow is calculated as being proportional tothe dark matter acretion rate as given by the massassembly history. Outflow rates are calculated ac-cording to the stellar IMF employed, by compar-ing the kinetic energy released by supernovae withthe depth of the halo potential well. We also ac-count for stellar lifetimes and mass loss from exist-ing stars.4. We assume that star formation rate, ψ , in ahalo tracks the total amount of cold gas, M cool ,which is determined by defining a metalicity de-pendent cooling radius (Kauffmann et al. 1999;Springel & Hernquist 2003).5. Metal content of a halo is calculated by account-ing for nucleosynthetic yields by integrating overthe Population II and Population III stellar IMFs.Halo metallicity is diluted by inflows from themetal-poor IGM and is enhanced by stellar nucle-osynthesis. We do not assume instantaneous recy-cling in this calculation, i.e., we fully account forthe delay in the production of metals due to finitestellar lifetimes. In this work, our fiducial modelassumes instantaneous and homogeneous mixing ofmetals in the ISM. But we also consider variationsof the model which relax this assumption.6. We implement Population III stars using a criticalmetallicity argument Z crit , with our fiducial valueas Z crit = 10 − Z ⊙ . When the ISM metallicity in ahalo becomes larger than Z crit , Population III starformation stops, and new stars form according toa Population II IMF.7. We consider two different Population III IMFs(Turk et al. 2009; Dopcke et al. 2011): 1–100 M ⊙ Salpeter and 100–260 M ⊙ Salpeter. These are se-lected to represent two extreme possibilities. (Wediscuss this choice below.) The Population II IMFis kept constant at 0.1–100 M ⊙ Salpeter.8. Stellar lifetimes are taken from Maeder & Meynet(1989) and Schaerer (2002). Metal yieldsare taken from Heger & Woosley (2002) andWoosley & Weaver (1995). Population II SEDs aresynthesised using starburst99 (Leitherer et al.1999; V´azquez & Leitherer 2005) with respectivemetallicities. Synthetic spectra of Population IIIstars are taken from Schaerer (2002).9. We model the thermal, ionization, and chemicalevolution of the IGM by implementing an inho-mogeneous IGM with a lognormal density distri-bution. We calculate the evolution of volumefilling factor of H ii regions, Q HII ( z ), accordingto the method outlined by Miralda-Escud´e et al. (2000). Reionization is said to be complete whenall low-density regions are ionized (we denote thecorresponding redshift of reionization by z reion .).The corresponding IGM thermal evolution is de-terming by solving the thermal evolution equation(Chiu & Ostriker 2000; Hui & Haiman 2003), ac-counting for photoheating and Compton and re-combination cooling. The minimum mass of star-forming regions is calculated from the IGM temper-ature using the Jeans criterion (Barkana & Loeb2001). Chemical evolution of the IGM is calcu-lated by accounting for outflows from haloes of allmasses.Before discussing our results, we briefly comment onthe Population III IMFs considered in this work. TheIMF of Population III stars is poorly understood. Severalearly studies predicted that Population III stars have acharacteristic mass of a few hundred M ⊙ (e.g., Abel et al.2002). In recent years, this prediction has come down inthe range of 20–100 M ⊙ (e.g., Turk et al. 2009). Re-cently, Dopcke et al. (2011) argued that in the absenceof metal-line cooling, dynamical effects can still lead tofragmentation in proto-stellar gas clouds. In their simu-lations, this resulted in Population III stars with massesas low as 0.1 M ⊙ (this picture predicts the existence ofmany Population III stars surviving in the Galaxy tillthe present day.). Given our current poor understand-ing of the Population III IMF, in this work we chooseto work with two different IMFs: 1–100 M ⊙ Salpeter(“low-mass IMF”) and 100–260 M ⊙ Salpeter (“high-massIMF”). The former has metal yields from AGB stars andcore-collapse supernovae, while the latter has yields frompair-instability supernovae. RESULTS
We now present the results of our model. In Section3.1, we describe the results of our fiducial model which as-sumes instantaneous and homogeneous mixing of metalsin the ISM, and show that it predicts a very small Popula-tion III contribution to reionization. The main reason forthis is chemical feedback, which we study in Section 3.2.We then study the conditions under which Population IIIstars could contribute significantly to reionization, bymodelling delayed mixing of metals in the ISM, therebyreducing chemical feedback. In Section 3.3 we look atthe constraints from chemical enrichment measurementsof high redshift DLAs and argue that these rule out sig-nificantly delayed mixing, and hence any significant con-tribution from high-mass Population III stars.
Population III Contribution to Reionization
Our model calibration procedure is described in de-tail in Kulkarni et al. (2013). The model is calibratedby matching (1) the observed cosmic SFR density evo-lution (Hopkins & Beacom 2006), (2) the fraction oftotal baryon density in collapsed haloes at z = 0(Fukugita & Peebles 2004), and the reionization his-tory of the IGM as measured by (3) the electronThomson scattering optical depth to the last scatter-ing surface (Hinshaw et al. 2012) and (4) the hydrogenphotoionization rate evolution (Meiksin & White 2004;Bolton & Haehnelt 2007; Faucher-Gigu`ere et al. 2008;Becker & Bolton 2013).We assume a constant star-formation efficiency param-eter f ∗ , defined by ψ = f ∗ (cid:18) M cool t dyn (cid:19) , (1)where ψ is the halo star formation rate, M cool is thecold gas mass in the halo, and t dyn is the halo dynamicaltime. The star-formation efficiency is then tuned to re-produce the cosmic SFR density. For our fiducial model(described below), we have f ∗ (Pop. II) = 0 .
002 and f ∗ (Pop. III) = 0 . z = 0.Once the star formation efficiency is fixed, reioniza-tion depends mainly on the escape fraction of ionizingphotons, f esc , defined as the fraction of ionizing photonsthat escape the parent galaxy. We assume f esc to be in-dependent of redshift and halo mass, and calibrate it toreproduce the observed values of the electron Thomsonscattering optical depth, τ e and the hydrogen photoion-ization rate, Γ HI . It is crucial to include both these con-straints as the observed hydrogen photoionization rateevolution imposes continuity on the epoch of reioniza-tion. Our fiducial model has f esc = 0 .
2. Although, inour calibrated model, the IGM electron Thomson scat-tering optical depth τ e is always obtained in the 1 σ rangeof its best-fit value (0 . ± .
014 as given by WMAP9;Hinshaw et al. 2012), our value of τ is always closer to theupper-limit of this range. This suggests that an evolvingescape fraction is necessary for fitting both τ e and Γ HI simultaneously (Mitra et al. 2012; Finlator et al. 2012).We calibrate the model independently for each of thetwo Population III stellar IMFs considered in this paper.Our fiducial model assumes instantaneous mixing of met-als in the ISM. We describe the results of this model inthis section. We discuss the effect of relaxing the instan-taneous mixing assumption in subsequent sections. Population III SFR
Fig. 1 shows the results of our fiducial model. Panel(c2) shows the predicted total (Population II + Pop-ulation III) cosmic SFR density evolution for the twodifferent Population III IMFs (low-mass IMF in blackand high-mass IMF in red). The red data pointsare observational measurements from a compilation byHopkins & Beacom (2006), in which consistent dust ob-scuration corrections, SFR calibrations, and IMF as-sumptions are applied to ultraviolet and far-infrareddata. Our model predictions are in good agreement withthe data regardless of what Population III IMF is as-sumed. Indeed, the total SFR density in the two cases isnearly identical (the red and black solid curves overlap),because the contribution of Population III star forma-tion to the total SFR is very low over most of the cos-mic history. The Population III SFR density evolutionis shown separately by the dashed curves. The corre-sponding panel (c1) shows the fraction of the total SFRproduced by Population III star formation. We find thatthe contribution of Population III stars to the total SFRdensity is low (less than 10%) throughout most of the ageof the Universe. It is greater than 10% for less than 100Myr ( z > z ∼ z <
8, all haloes capable of forming stars(i.e., above the minimum mass for star-formation) havealready enriched themselves above the critical metallic-ity, and hence the Population III SFR is zero at these latetimes. Thus, although Population III stars initiate theepoch of reionization, they quickly relinquish their domi-nant role by enriching their environment. We discuss thereason behind this below.We note that the star formation history shown inFig. 1, together with our chemical evolution model, isalso consistent with the observed mass-metallicity rela-tions (Erb et al. 2006; Maiolino et al. 2008) at redshifts2.3 and 3.7 (see Kulkarni et al. 2013 for details). Thecorresponding IGM metallicity is also consistent with ob-servational estimates (Schaye et al. 2003; Simcoe et al.2004). We also note here that the difference between thetwo Population III SFR density curves (dashed curves)in panel (c2) of Fig. 1 can be understood from the differ-ent metal yields for the two Population III IMFs (recallthat the Population II IMF is the same in both cases).The high-mass IMF has a larger metal yield. As a re-sult, in this case haloes are enriched beyond Z crit earlieras compared to the low-mass IMF, and Population IIIstar formation is terminated earlier. This is reflected inthe reduced level of Population III SFR density for thehigh-mass IMF. Photoionization rate
The small contribution of Population III stars to thetotal cosmic SFR density suggests that their contribu-tion to reionization will also be small. This is seen inpanel (b2) of Fig. 1, which shows the evolution of thehydrogen photoionization rate in our model. Making theso-called local source approximation, which is valid forspectral indices typical to star-forming galaxies, the hy-drogen photoionization rate is given byΓ HI ( z ) = (1 + z ) Z ∞ ν dνλ ( z, ν ) ˙ n ν ( z ) σ ( ν ) , (2)where σ ( ν ) is the photoionization cross-section of hy-drogen, λ ( z, ν ) is the redshift-dependent mean free pathof ionizing photons. We calculate the mean free pathof ionizing photons as in Paper 1 by integrating overthe lognormal density PDF of the IGM and estimat-ing the average distance between high-density, neutralregions. This method is calibrated to reproduce theincidence rate of Lyman-limit systems at low redshifts(Miralda-Escud´e et al. 2000; Choudhury & Ferrara 2005;Prochaska et al. 2009). Our model agrees with the mea-surement of Prochaska et al. (2009). The quantity ˙ n ν ( z )is the number density of ionizing photons in the IGM perunit time, and is given by˙ n ν ( z ) = f esc ˙ ρ ∗ ( z ) Z dm φ ( m ) t age ( m ) Q H ( m ) . (3)The integral in this equation is over stellar masses andtakes the IMF-dependence of the ionizing photon fluxinto account: φ ( m ) is the normalised stellar IMF, t age ( m )is the age of a star with mass m and Q H ( m ) the stellarhydrogen-ionizing photon flux (in photons s − ) provided Figure 1.
Results for our fiducial model. Panels (a1) and (a2) show the metallicity evolution for three particular halo masses forillustration. The long-dashed curve in panel (a1) shows the evolution of M min in cosmological H ii regions. Panel (b1) and (b2) show theevolution of the hydrogen photoionization rate and the contribution to it from Population III stars for the two models considered. Thecorresponding panel (b1) plots the ratio of the two quantities. In this panel, the black curve corresponds to the low-mass IMF model andthe red curve to the high-mass IMF model. Panels (c1) and (c2) show the evolution of the cosmic SFR density in the model and thecontribution from Population III stars. Vertical dashed lines in all panels show the redshift of reionization. by stellar evolution models (Schaerer 2002). The quan-tity f esc is the escape fraction of ionizing photons thataccounts for the fraction of the ionizing photons thatescape into the IGM. The escape fraction is a free pa-rameter of our model. In this paper, we assume that f esc is constant at all redshifts. We comment on the effect ofthis assumption below. Our fiducial model has f esc = 0 . α forest (Meiksin & White 2004; Bolton & Haehnelt 2007;Faucher-Gigu`ere et al. 2008), where we note that thereis disagreement between the measurements at the factor-of-two level, which likely results from different assump-tions about the density distribution and thermal stateof the IGM. Here were choose to fit the data byFaucher-Gigu`ere et al. (2008) but this choice is not crit-ically important for our main result. The solid curves inpanel (b2) of Fig. 1 show the model predictions corre-sponding to two different Population III IMFs (low-massIMF in black and high-mass IMF in red). The model pre-dictions agree very well with the observational measure-ments. As seen by the evolution of the hydrogen pho-toionization rate, reionization in our model is gradual. Itbegins at z &
30 and is 90% complete by z ∼
7. Thisgradual change in the ionization state of the IGM helpsus simultaneously reproduce the observed electron scat-tering optical depth τ e = 0 . ± .
014 (Hinshaw et al. Note that we have ignored the ionizing photon contributionfrom quasars in our model, which are expected to only contributesignificantly for z < z reion = 7 .
5, the photoionization rate increasesrapidly as UV photon sources build up. There is a suddenjump at z = z reion when different H ii regions overlap.(This redshift is marked by the vertical dashed line inFig. 1). This is because at this redshift, a given point inthe IGM starts “seeing” multiple sources, which rapidlyenhances the UV photon mean free path, thereby affect-ing the photoionization rate. Secondly, Fig. 1 also showsthat the contribution of Population III stars to reioniza-tion is small. This is clear from the overlap between thered and black solid curves in panel (b2), and is evidentin the dashed curves, which show the contribution to Γ HI by Population III stars. We see that the Population IIIcontribution to photoionization rate is subdominant overmost of the reionization history. Panel (b1) of Fig. 1 fur-ther 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 earli-est stages of galaxy formation ( z ∼ z .
20, and for the high-mass IMF, it isless than 10% for z . Understanding the Low Population III Contribution
We now discuss the reason behind the small contribu-tion of Population III stars to reionization in our model.This contribution depends on three parameters of themodel: (1) the efficiency with which cold gas is con-verted into stars, (2) the escape fraction of ionizing radi-ation, and (3) chemical feedback, quantified by the timescale over which a halo enriches itself beyond the criti-cal metallicity Z crit and stops the formation of Popula-tion III stars. These three factors are not independent ofeach other: a higher efficiency of star formation reducesthe self-enrichment time scale of a halo, and a higher f esc requires us to reduce the star formation efficiency ifwe are to satisfy the observational constraints on reion-ization. In our model, the constraints from Γ HI , τ e andthe cosmic SFR density completely fix the star formationefficiency and the escape fraction. Therefore, as we as-sume instantaneous metal mixing in our fiducial model,the self-enrichment time scale is also fixed.Panel (a2) of Fig. 1 shows the metallicity evolution ofthree different haloes in our model, which helps one un-derstand the small contribution of Population III starsto reionization in our model (for simplicity, we show re-sults only for the high-mass IMF in this panel, but thelow-mass IMF gives similar trajectories). The horizontaldashed line in this panel shows the critical metallicity Z crit , which we always take to be 10 − Z ⊙ . The mass as-sembly history of these haloes is shown in the correspond-ing panel (a1); their masses at z = 0 are about 10 M ⊙ ,10 M ⊙ , and 10 M ⊙ respectively. The dashed curve inthe panel (a1) shows the evolution of the minimum massof star-forming haloes, M min , which is set according tothe Jeans criterion, which in turn depends on the ther-mal evolution of the IGM (see, e.g., Rorai et al. 2013,for a recent discussion). To be specific, here we considerhaloes that collapse in cosmological H ii regions, and the M min evolution shown corresponds only to these regions(but the discussion could be easily generalized to H i regions). In these regions, M min is roughly constant at ∼ M ⊙ as this evolution is only determined by theJeans scale corresponding to the characteristic tempera-ture of the IGM at T ∼ K. In each halo’s assemblyhistory, star formation begins when its mass crosses thethreshold set by M min . This is manifest in the metallicityevolution history of each halo as shown in panel (a2). Itis seen that the halo metallicities increase mostly mono-tonically. Additionally, in each case, the initial burstof Population III stars is sufficient to rapidly enhancethe ISM metallicity beyond Z crit . This metallicity evo-lution is in good agreement with the hydrodynamic sim-ulations of Wise et al. (2012). It is exactly this promptenrichment of the ISM of high-redshift galaxies whichcauses an early cut-off in the Population III SFR. Therange of redshifts in panel (c2) that show sub-dominantPopulation III SFR, arises from only the lowest massstar-forming halos, which only recently crossed the min-imum mass threshold. However, the vast majority ofstar-forming halos at higher masses have already beenchemically self-enriched by this time, and have stoppedforming Population III stars, which in turn reduces theoverall contribution of Population III stars to the pho-toionization rate. As previously mentioned above, apartfrom the star formation efficiency assumed in the model,the rapid enrichment of galaxies is a result of the as-sumption that metals are instantaneously mixed in theISM gas. We discuss the assumption in detail in the nextsection.Finally, we briefly note that haloes down to a masslimit of M min ∼ M ⊙ contribute to the ionization Some non-monotonicity seen in panel (a2) of Fig. 1 is due tothe behaviour of M min at the highest redshift, which can restrictgas inflow into haloes at certain times, affecting the metallicity. Figure 2.
Halo self-enrichment time scale in our model, definedas the time between the first star-formation episode of the haloand the time at which its gas-phase metallicity crosses Z crit . Itquantifies the time scale of Population III star formation in a givenhalo. Halo mass at z = 0 serves as a label for different haloes. Thesolid curves show the time scale for the fiducial models. Othercurves show the time scale for models with delayed enrichment. flux in Eqn. (3). At z ∼ M = −
10 in our model, whichagrees well with the very faint minimum galaxy mag-nitude, to which the UV luminosity function of Lymanbreak galaxies (LBGs) must be extrapolated to in or-der to reionize the universe (Kuhlen & Faucher-Gigu`ere2012; Fontanot et al. 2012; Robertson et al. 2013). Fur-thermore, the halo masses which we deduce for thesefaint galaxies M min ∼ M ⊙ agree well with massesdeduced using abundance matching (Trenti et al. 2010;Kuhlen & Faucher-Gigu`ere 2012). This indicates that inour fiducial model, it is the faint galaxies that producethe bulk of the ionizing photons that reionized the IGM,and not Population III stars. The Halo Self-Enrichment Timescale
In the results presented above, metals injected into theISM by supernova explosions are assumed to be instanta-neously mixed into the halo ISM. Instantaneous and ho-mogeneous mixing of metals is a standard assumption ingalactic chemical evolution studies (Tinsley 1980; Pagel2009; Matteucci 2012) and is responsible for the earlytermination of Population III star formation in our fidu-cial model. Note that we are not assuming instantaneousrecycling, since we directly model the delay in the synthe-sis of new metals due to finite stellar lifetimes. However,we have assumed instantaneous mixing, which is to saythat after a supernova event, newly liberated metals areinstantneously available in the ISM to influence the nextgeneration of star-formation. We have seen in our fidu-cial model, that the initial burst of Population III stars issufficient to enhance the ISM metallicity beyond Z crit inany halo mass bin over a very short time scale. When this Figure 3.
Same as Fig. 1 but for a model in which metal enrichment in ISM is gradual with a delay time of t delay = 10 yr. Thisincreases the self-enrichment time scale of haloes to about 10 yr. Consequently, the contribution of Population III stars to the cosmic starformation rate and to the hydrogen photoionization rate is enhanced relatived to the fiducial model. See text for details. Figure 4.
Same as Figs. 1 and 3 and but for a model in which metal enrichment in ISM is gradual with a delay time of t delay = 10 yr,i.e., an order-of-magnitude slower that in Fig. 3. This further increases the self-enrichment time scale of haloes to about 4 × yr. As aresult, the contribution of Population III stars to the cosmic star formation rate and to the hydrogen photoionization rate is high enough tocomplete hydrogen reionization. Note that in this model, the redshift of reionization is slightly different for the two Population III IMFs;we only show the low-mass IMF case for simplicity. happens, the corresponding galaxy stops forming Popu-lation III stars. Since most of the mass in the universe iscontained in M ∗ haloes, the Population III cosmic SFRdensity begins to decline as soon as these haloes cross the Z crit threshold. This is the crucial effect that reduces thecontribution of Population III stars, and that was notcaptured in previous models because they did not trackchemical enrichment for a large population of haloes in acosmological volume. We now discuss the dependence ofthis result on our assumption of instantaneous mixing.The solid lines in Fig. 2 show the self-enrichment timescale as a function of halo mass for the two Population IIIIMFs in our fiducial model, where the low-mass IMFis shown in black and high-mass IMF in red. We de-fine the self-enrichment time scale as the time betweenthe first star-formation episode of the halo and the timeat which its gas-phase metallicity crosses Z crit . In thisplot, haloes are labelled by their mass at z = 0; theiractual mass value at the time of crossing M min is notshown. In the low-mass IMF case (solid black), the timescale is a few times 10 yr. Pair-instability SNe havehigher metal output, which results in a shorter enrich-ment time scale by a factor of 3 in the high-mass IMFcase (solid gray). As discussed in section 3.1, it is thisshort self-enrichment time scale that restricts the contri-bution of Population III stars to reionization. Note thatthese curves have a very flat dependence on halo massbecause the self-enrichment time scale depends primarilyon stellar lifetimes.In the ISM of a galaxy, mixing of metals ejected by su-pernovae is carried out by different mechanisms on differ-ent length scales: diffusive dispersion due to large-scalemotion on kpc scales, turbulent diffusion on pc scales,and molecular diffusion on smaller scales. As a result,the self-enrichment time scale is decided by (1) the re-spective time scales of these mixing processes, (2) therate of metal production by supernovae, and (3) rate ofinteraction with the environment, via inflows and out-flows of gas. In our fiducial model, the time scale of themixing processes is assumed to be zero and therefore theself-enrichment time scale depends only on the stellarlifetimes via the supernova rate. This is the reason be-hind the short self-enrichment time scale shown in Fig. 2.We now relax this instantaneous mixing assumption.Supernova-driven metal-mixing in the interstellarmedium has been studied using hydrodynamical simula-tions by de Avillez & Mac Low (2002), who used tracerparticles in a simulation of a 1 × ×
20 kpc region of theGalactic disk and found mixing time scales between 10 and 10 yr (see their Fig. 8) for supernova rates of ∼ ∼
100 times the Galactic supernova rate. The lower endof this range is of the same order of magnitude as theself-enrichment time scale in our fiducial model. But theresults of these simulations suggest that the mixing timescale could very well be two or three orders of magnitudelarger.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 mixedin the ISM. The source term due to star formation in themetallicity evolution equation is of the form (Eqn. 19 in Kulkarni et al. 2013)˙ M Z ( z ) = Z m u m l dm φ ( m ) · ψ [ t ( z ) − τ ( m )] × mp Z ( m ) , (4)where m is the stellar mass, ψ ( t ) is the star formationrate at time t , φ ( m ) is the stellar IMF, τ ( m ) is the stellarage, and p Z ( m ) is the mass fraction of a star of initialmass m that is converted to metals and ejected. Thelimits m l and m u define the range of stellar masses con-sidered in the IMF. To relax the instantaneous mixingassumption, we modify Eqn. (4) to˙ M Z ( z ) = Z m u m l dm φ ( m ) mp Z ( m ) × Z t ( z ) t ( z ) − t delay d ˜ t ψ [˜ t ( z ) − τ ( m )] f (˜ t ) , (5)where f (˜ t ) is a mixing function that serves to delay themixing of metals in the ISM after a supernova event.The mixing function is defined over a time duration of t delay . In this picture, after a supernova has exploded,the resulting metal mass is added to the ISM graduallyover a time t delay . Until this time, we imagine the met-als to be locked up into un-mixed pockets of the ISM,and so they are not available for future star-formation.Thus, our fiducial model corresponds to the case wherethe mixing function is a delta function with t delay = 0 yr.For non-zero values of t delay , the ISM metallicity can re-main below the critical metallicity Z crit for a longer timecompared to our fiducial model. Therefore, we wouldexpect that in this case, Population III star formationwill continue for a longer period, and possibly impactreionization. The mixing function is determined by thecomplex interplay of various mixing processes in the ISM.The most conservative form of the mixing function is aconstant of the form f ( t ) = (cid:26) t − if t < t delay , (6)where t is the time since a supernova explosion. Notethat f ( t ) is normalised such that all of the supernovaejecta is mixed in the ISM over the period t delay . Weadopt this simple form for the mixing function in this pa-per. Figs. 3 and 4 show the results of our model with thismixing function and t delay = 10 yr and t delay = 10 yr respectively. The self-enrichment time scales in thesemodels is compared to that in our fiducial model in Fig. 2(only the high-mass Population III IMF case is shownfor simplicity). Given that we have adopted a mixingtimescale which is much longer than the age of the Uni-verse, i.e., t delay = 10 or 10 yr, the effective resultis that at z ∼
10, where t z =10 ∼ yr, these mix-ing models imply that only 1% and 0.1% of the metalsproduced by Population III SNe are mixed into the ISM,respectively. As we will see below even this small amount This form of the mixing function is the most conservative in thesense that all time instances from 0 to t delay are treated equally. Asa result, metals are mixed into the ISM at a constant rate and, inthe limit of an exactly closed system, the metallicity would increaselinearly. of metals is still enough to produce significant chemicalfeedback that influences Population III star formation.This results because as shown in panel (a2) of Fig. 1, asingle generation of Population III SNe inejcts enoughmetals to raise the the ISM metallicity of M ∗ halos toa Z ∼ − Z ⊙ by z ∼
10, if these metals are instanta-teously mixed. Thus reducing this yield by a factor of ∼ t z =10 /t delay ∼ t delay = 10 ,is still sufficient to be in excess of the critical metallicity.For this reason, and as is also clear from the mathemat-ical form of Eqn. 5, we generally expect a non-linear de-pendence of the self-enrichment timescale on t delay , andsimilarly the dependence of Population III star-formationon t delay will also be nonlinear. For instance, betweenFig. 3 and Fig. 4, changing t delay by a factor of 10 resultsin small changes of factor of 3 in the self-enrichment timescale.In Fig. 3 the same curves illustrating the star-formation, reionization, and enrichment history as inFig. 1 are shown but for a model with t delay = 10 yr. Asexpected, and illustrated in panel (a2) of Fig. 3, haloesnow take longer to cross the Z crit metallicity threshold,compared to the fiducial model (panel a2 of Fig. 1, whichis on the same scale). This increases the cosmic Popula-tion III SFR density, which, although still subdominant,contributes more than 10% of the total SFR density atredshifts above z ∼
20 in the high-mass Population IIIIMF run (the corresponding redshift value for the fidu-cial model was z ∼ z = 10 is 10% for the high-mass case, whereas thecontribution was less than 1% at this redshift in our fidu-cial model. Thus for t delay = 10 yr, the contribution ofPopulation III stars to reionization is subdominant butsignificant. Note that these numbers are slightly differ-ent for the low-mass case for the same mixing functionand t delay . In general, low mass Population III star for-mation lasts longer than for the high-mass IMF. Thisis because the metal yields of high mass Population IIIstars is higher than those of low mass stars, resultingin larger chemical feedback, and earlier termination ofPopulation III.Fig 4 shows the results of the model with t delay = 10 yr, for which the mixing of metal metals is even moregradual. The impact of Population III on cosmic star-formation and the ionizing photon budget is now in-creased in magnitude relative to the previous case shownin Fig. 3. For instance, all star formation for z &
10 inthis model is Population III, regardless of the Popula-tion III IMF. Thus the process of hydrogen reionizationis almost single-handedly carried out by Population IIIstars. The contribution of Population III stars to thehydrogen 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 = 10is about 20% for the high-mass case, and 80% for thelow-mass case. Similar to the results in Fig. 3, the Pop-ulation III contribution is higher for low-mass Popula-tion III IMF because of weaker chemical feedback.Fig. 2 helps in visualising the effect of the mixing func-tion on our model. The self-enrichment time scale in the two variant models is longer than that in the fiducialmodel by more than an order of magnitude. The in-crease, however, is still less that t delay , as only a fractionof metals are required to increase halo metallicity beyond Z crit .A general lesson from above is that due to constraintsimposed by the measurements of cosmic SFR density, thehydrogen photoionization rate, and the electron scatter-ing optical depth, the contribution of Population III starsto reionization can be enhanced only by increasing themetal mixing time scale assumed in the model. Thiscontribution is generally predicted to be small unless theself-enrichment time scale is & × yr. Chemical Enrichment Constraints on theContribution of Population III to Reioinization
By varying the assumptions about the Population IIIIMF and the metal mixing timescale, we have seen thatthe the self-enrichment timescale can take on values be-tween 3 × –3 × years, dramatically impacting thechemical feedback which eventually terminates Popula-tion III star formation. These degrees of freedom resultin concomitant uncertainties on the contribution of Pop-ulation III star-formaiton to the ionizing photon budgetof 1–100%, since all of the models were able to matchthe star-formation and and reionization observables thatwe considered. However, we can discriminate betweenthese possibilities using accurate chemical enrichmentobservations in damped Ly α absorbers (DLAs) at post-reionization redshifts. Observations of DLAs can be usedto measure gas-phase metallicities at large cosmologicallookback times with high precision. Furthermore, rel-ative abundances can still be measured accurately deepinto the reionization epoch ( z >
6) using metal-line tran-sitions redward of Ly α , even though Gunn-Peterson ab-sorption precludes measurement of neutral hydrogen. InKulkarni et al. (2013) we modeled the chemical evolutionof DLAs, and showed how their abundance patterns canbe used to constrain Population III s cenarios. Here weargue that they can also constrain the contribution ofPopulation III stars to reionization.In our model, we assigned a mass-dependent H i ab-sorption cross-section, denoted by Σ to each halo in or-der to predict the expected distribution of DLA abun-dance ratios (see Kulkarni et al. 2013 for details). Thisassignment is motivated by hydrodynamical simulations(Gardner et al. 1997, 2001; Nagamine et al. 2004a,b,2007; Pontzen et al. 2008) and reproduces the observedDLA metallicity evolution (Rafelski et al. 2012), in-cidence rate (Prochaska et al. 2005; Noterdaeme et al.2012), and clustering bias (Font-Ribera et al. 2012) atlow redshifts ( z ∼
3) very well, and takes the formΣ( M ) = Σ (cid:18) MM (cid:19) (cid:18) MM (cid:19) α − , (7)where the constants take the values of α = 0 . M =10 . M ⊙ , and Σ = 40 kpc at z = 3 (Pontzen et al.2008; Font-Ribera et al. 2012). Values at other red-shifts are calculated by mapping haloes at these red-shifts to haloes z = 3 according to circular velocity(Font-Ribera et al. 2012). With this assignment, for anymeasurable property p (e.g., abundance ratio [M /M ])of DLAs, we can calculate the number of systems with0 Figure 5.
Evolution of mean values of [C/Fe] and [O/Si] DLA relative abundances in all models considered in this paper. Data pointsshow observed DLA relative abundances from a compilation by Becker et al. (2012). Right panel shows low-mass IMF models, left panelhigh-mass IMF models. Our fiducial model (black curves) is consistent with the observational measurements. The high-mass IMF modelwith gradual enrichment with t delay = 10 yr is only marginally consistent with the [O/Si] data, while that with t delay = 10 yr is ruledout by the data. This constraints the Population III contribution to reionization. Note that the low-mass IMF models are harder to ruleout, as discussed in the text. different values of p in a sample of DLAs. This is calledthe line density distribution, and with Eqn. (7) in hand,it can be written as (e.g., Wolfe et al. 2005) d NdXdp = N ( M ) · Σ( M ) · dldX dMdp · (1 + z ) . (8)Here, X is an absorption length element given by dldX = cH (1 + z ) , (9) dl = cdt is a length element, and p is the property inconsideration. The halo mass is denoted by M , N ( M ) isthe comoving number density of halos (i.e., the halo massfunction), and Σ( M ) is the halo cross section given byEqn. (7). The quantity dM/dp in Eqn. (8) can be easilycalculated in our model, as properties like metallicity andrelative abundances are known for all halo masses. Theintegral of Eqn. (8) over all values of p is just the totalline density of DLAs, dN/dX . The average value of p inan observed sample of DLAs is given by h p i = Z d NdXdp · p · dp · dX. (10)Fig. 5 shows the result of evaluating Eqn. (10) for p =[C/Fe] and [O/Si] in our fiducial model and its vari-ants. It also shows the observed evolution of [C/Fe] and[O/Si] relative abundances in DLAs. The z ∼ z > z = 2 to z = 6), these abundance ratios are rel-atively constant. Furthermore, they show little scatteraround the mean. Solid lines in Fig. 5 show the evolu-tion of these relative abundances in our low-mass IMFmodels, while the dashed lines show the evolution in thehigh-mass IMF case. Curves with different colors indi-cate different delay times.Our fiducial model agrees with data for both [C/Fe]as well as [O/Si]. In this redshift range, the mean val-ues of these abundance ratios in this model are governedby the Population II IMF as the contribution of Popula- tion III stars is erased in all but the smallest haloes. Theasymptotic value of these abundance ratios towards lowredshift thus simply reflects the relative yields of theseelements per star integrated over the Population II IMF.However, the variants of the fiducial model, in whichPopulation III contribution to the cosmic SFR and pho-toionization rate is higher, disagree with current rela-tive abunance measurements. We first discuss the high-mass IMF models. As described in the previous sec-tion, we considered two variants where metals are grad-ually mixed in the ISM over a period of t delay = 10 yr and 10 yr respectively, resulting in correspondingself-enrichment time scales of 10 and 5 × yr. Asshown in Figs. 3 and 4, the first of these models hasmore than 20% Population III contribution to the pho-toionization rate down to z ∼
20, while in the secondmodel the contribution is more than 20% all way downto z ∼
10. We now see in Fig. 5 that the model with t delay = 10 yr is marginally ruled out by the existing[O/Si] data (red dashed curve in bottom panel of Fig. 5),while the model with t delay = 10 yr is completely ruledout. In the previous section we showed that the contri-bution of Population III stars to reionziation could onlybe enhanced by increasing the metal-mixing time scale,however this results in a corresponding increase in thechemical vestiges of these Population III stars. Fig. 5shows that the DLA relative abundance measurementsactually restrict the high-mass Population III contribu-tion to reionization to be less than 10% for z .
15. Themodel with t delay = 10 yr acts as a kind of upper boundon the self-enrichment time scale of haloes and there-fore on the role that high-mass Population III stars playin hydrogen reionization. The [O/Si] ratio is more con-straining than [C/Fe] because of significant variation inthe Si yield with IMF: the high-mass IMF produces twoorders of magnitude higher Si yield as Si is efficientlyproduced in massive stars due to O-burning.The contribution of low-mass Population III stars toreionization is harder to constrain using DLA chemicaldata. This is because chemical yields of low-mass Popu-lation III stars are not very different from those of Popu-lation II stars considered in our models, as the two IMFshave similar shapes and mass ranges. (There is some1difference in the yields due to difference in their metal-licities.) Thus we see in Fig. 5 that the [O/Si] values in allour low-mass IMF models are consistent with the data,regardless of mixing delay and self-enrichment timescale.This is also true for the [C/Fe] abundance ratio, althoughthe values are slightly different from the fiducial case, be-cause 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 isworth noting that the ionizing emissvity of low mass Pop-ulation III stars is only a factor of two higher than that ofPopulation II stars. Therefore reionization by these starsis qualitatively similar to reionization by Population IIstars alone.Finally, we note that the exact behaviour of the curvesin Fig. 5 depends on the form of the mixing function used.A mixing function different from that in Eqn. 6 will ingeneral result in a different evolution of the mean rela-tive abundances. However, the primary result of Fig. 5 ismore general: any Population III star formation activitythat produces hydrogen-ionizing photons at these red-shifts will also necessarily produce Population III chemi-cal signatures, which can be constrained using measure-ments of abundance patterns in DLAs. Also DLA rel-ative abundance measurements at z ∼
8, using back-ground QSOs or GRBs, could start to constrain even thelow-mass Population III IMF by witnessing the build-upof the metallicity in halos and change in relative abun-dances with time because of finite stellar lifetimes. CONCLUSION
In this paper, we have demonstrated that Popula-tion III stars contribute very little to the cosmic SFRdensity and to the reionization history. This is becausethe halos dominating the cosmic star-formation rate athigh-redshift rapidly (time scales of only ∼ yr) self-enrich to the critical metallicity, terminating Popula-tion III star formation. We quantify this rapid self-enrichment by defining a halo self-enrichment time scale,which is ∼ yr in our fiducial model. This time scaleis set by stellar lifetimes, and is almost independent ofhalo mass. Although this rapid self-enrichment occursat different redshift in halos of different masses, the neteffect is to reduce the Population III star formation andcontribution to the hydrogen photoionization rate to lessthan 1% of the total at z = 10 in our fiducial model.Previous studies did not uncover this rapid chemicalfeedback of Population III star-formation, because theydid not implement a self-consistent chemical enrichmentmodel, as we have done here. Since our fiducial modelassumes instantaneous mixing of metals in the ISM, westudied how relaxing this assumption impacts our results.By slowing the rate at which metals mix into the ISM,we found that the termination of Population III star-formation can be delayed, thus increasing their contribu-tion to reionization. However, mixing delay times andthe resulting self-enrichment timescales cannot be arbi-trarily long. This is because relative metal abundancepatterns in DLAs retain the chemical signatures of Pop-ulation III SNe, thus providing a chemical record of thePopulation III star-formation history. Indeed, we findthat halo self-enrichment timescales significantly longerthan 10 yr produce abundance patterns that are signif-icantly different from those observed in DLAs at z .
6, and are thus ruled out. As a result, the maximum alloweddelay time implied by existing observations restricts thefractional contribution of high-mass Population III starsto the ionization rate to be .
10% at z = 10. Con-straints on low-mass Population III are weaker, becausethere elemental yields are very similar to Population IIstars. However, the ionizing emissivity of low-mass Pop-ulation III stars does not significantly differ from Pop-ulation II stars, and so they do little to ease the ten-sion between reionization constraints and observationsof star-forming galaxies at high redshift.One possible way in which our chemical constraints canbe evaded is by having an UV photon escape fractionclose to 100% for Population III stars, thus dramaticallyenhancing the impact of Population III on reionizationfor a given amount of Population III star-formation (andcorresponding heavy element production). However, inour fiducial model, the contribution of Population IIIstars to the hydrogen photoionization rate is low pre-dominantly because their contribution to the cosmic SFRdensity is extremely low throughout the epoch of reion-ization. As a result, even an escape fraction of 100%does not increase their contribution to reionization be-yond that of the Population II stars. Additionally, it isnot clear what physical process could lead to such a dra-matic increase of the escape fraction at higher redshifts(Ferrara & Loeb 2013).Our work suggests that Population III stars proba-bly do not resolve the tension between reionization con-straints and the paucity of ionizing photons implied bythe observed population of star-forming galaxies at highredshift. Looking forward, our model also predicts thatrelative abundance measurements in the highest redshift( z ∼ ACKNOWLEDGEMENTS
We acknowledge useful discussions with Nishita De-sai, Kristian Finlator, Katherine Inskip, Khee-GanLee, J. Xavier Prochaska, Alberto Rorai, and RaffaellaSchneider. JFH acknowledges generous support from theAlexander von Humboldt foundation in the context ofthe Sofja Kovalevskaja Award. The Humboldt founda-tion is funded by the German Federal Ministry for Edu-cation and Research.
REFERENCESAbel, T., Bryan, G. L., & Norman, M. L. 2002, Science, 295, 93Alvarez, M. A., Finlator, K., & Trenti, M. 2012, ApJ, 759, L38Barkana, R., & Loeb, A. 2001, Phys. Rep., 349, 125Becker, G. D., & Bolton, J. S. 2013, ArXiv e-printsBecker, G. D., Bolton, J. S., Haehnelt, M. G., & Sargent,W. L. W. 2011, MNRAS, 410, 1096Becker, G. D., Sargent, W. L. W., Rauch, M., & Carswell, R. F.2012, ApJ, 744, 91Bolton, J. S., & Haehnelt, M. G. 2007, MNRAS, 382, 325Cen, R. 2003, ApJ, 591, L5Chiu, W. A., & Ostriker, J. P. 2000, ApJ, 534, 507Choudhury, T. R., & Ferrara, A. 2005, MNRAS, 361, 577—. 2006, MNRAS, 371, L55Cooke, R., Pettini, M., Steidel, C. C., Rudie, G. C., & Nissen,P. E. 2011, MNRAS, 417, 1534Daigne, F., Olive, K. A., Silk, J., Stoehr, F., & Vangioni, E. 2006,ApJ, 647, 7732