Early Structure Formation and Reionization in a Cosmological Model with a Running Primordial Power Spectrum
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Citation Yoshida, Naoki, Aaron Sokasian, Lars Hernquist, and Volker Springel. 2003. “Early Structure Formation and Reionization in a Cosmological Model with a Running Primordial Power Spectrum.” The Astrophysical Journal 598 (1): 73–85. https:// doi.org/10.1086/378852.
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ΛCDM k n hl al tutr om irrhclyi h S oe,qiesimilar quite model, RSI the in hierarchically forms structure early While esuyhg esitsrcuefrainadrinzto naΛCD a in reionization and formation structure redshift high study We ycryn u aitv rnfrcluain,w losuyreion study also we calculations, transfer radiative out carrying By AL TUTR OMTO N EOIAINI COSMOLOGIC A IN REIONIZATION AND FORMATION STRUCTURE EARLY with 3 a-lnkIsiu ¨rAtohsk Karl-Schwarzschi f¨ur Astrophysik, Max-Planck-Institut n A 1. T ≈ E tl mltajv 04/03/99 v. emulateapj style X INTRODUCTION ∼ 1 seLt ito19 o a for 1999 Riotto & Lyth (see 1 avr-mtsna etrfrAtohsc,6 adnSt Garden 60 Astrophysics, for Center Harvard-Smithsonian 10 omlg:hoy-eryUies eoiaino h Universe the of reionization - Universe early - cosmology:theory 6 2 M ainlAtooia bevtr aa,Mtk,Tko18 Tokyo Mitaka, Japan, Observatory Astronomical National IHARNIGPIODA OE SPECTRUM POWER PRIMORDIAL RUNNING A WITH cl-nain est fluc- density scale-invariant ⊙ < z mn-ao”cmae oteΛD oe.Teaudneo primor of abundance The model. ΛCDM the to compared “mini-halos” ) ak Yoshida Naoki 5 eeal eutn nasalrttlTosnotcldphthan depth optical Thomson total smaller a in resulting generally 15, n 1 , a at- has , 2 ao Sokasian Aaron , okrSpringel Volker oapa nApJ in appear To ABSTRACT and 1 dsr ,D870Grhn e M¨unchen, Germany bei Garching D-85740 1, ld-str. a CMmdlwt h pcrlindex power spectral best-fit the a 2003). with gives al. model alone et ΛCDM data Peiris primordial law 2003; WMAP the the al. of et Although “running” (Spergel a spectrum power with model cosmological nstoyPoe(MP aa h d aayred- galaxy 2dF Lyman- the and survey, the data, Microwave Interestingly, shift (WMAP) Wilkinson first-year Probe scales. density the Anisotropy small of of analysis to amplitudes spec- combined large the relative of on the “running” fluctuations This modulates scale. index length tral of function a be ercnie ftepioda oe pcrmhsamild a has spectrum power ( primordial tilt the if reconciled be h utainamplitude fluctuation the tutr omto Znnr&Blok2002). Bullock & (Zentner formation structure ml cls(oosy&Tre 95 ansa tal. et Hannestad 1995; Turner & 2002). nega- on (Kosowsky power with scales linear models reduced small scale, progressively large predict running a tive on normalization given a bevtos(rf ta.19,20;MDnl tal. low- et a 1 McDonald scales favor on consistently 2002; model 2002) amplitude 1998, Hamilton & Gnedin al. 2000; et (Croft observations h pcrlidxo h rmrilpower, primordial the of index spectral The upesn ierpwro ml,glci osub- to galactic small, on power linear Suppressing < n 1 3 . asHernquist Lars , hogotterdhf ag 5 range redshift the throughout 6 τ z e )o eaiernig( running negative a or 1) ∼ ∼ et abig MA02138 Cambridge reet, riaiyhg htneiso rate emission photon high ordinarily ucino eghsae eadopt We scale. length of function a r bevtos ecryothigh out carry We observations. ure zto yselrppltosformed populations stellar by ization ,teecp rcino ultraviolet of fraction escape the 7, ino rmrilgscod where clouds gas primordial of tion ro antd at magnitude of er 0 la ugse ytecombined the by suggested as el . o pia et nerdb the by inferred depth optical son nvreudrteassumption the under universe M o esnbemdl fearly of models reasonable for 1 edlyi h omto epoch formation the in delay le o-ev nilms function mass intial top-heavy a -58 Japan 1-8588, dlidctsta reionization that indicates odel yt h tnadpower-law standard the to ly 1 α σ 8 ∼ k < oetosrain aosa favors observations forest 0 = [Mpc LMODEL AL . > z ,teLyman- the 9, dn/d 5between 15 − 1 < z < ] ∼ < ln nthe in 0 hc can which 10, < k dial n n 18 a even may , ≈ α ) For 0). forests and 1 2 galactic scales may offer a resolution to the often-claimed (Yoshida et al. 2003a), we found that non-equilibrium ef- problems of Cold Dark Matter (CDM) models (e.g. Moore fects associated with dynamical heating play a critical role et al. 1999). It could alleviate apparent discrepancies in regulating the supply of molecular gas, contrary to the between the observed structure of dark halos and predic- assumptions commonly made in semi-analytic studies of tions from CDM models (Zentner & Bullock 2003). On this process. Hence, it is necessary to follow the dynamics the other hand, too much reduction of small-scale power of structure formation explicitly to reliably estimate the may result in a conflict with other observations. Dalal & abundance of early stars. It is also important to specify Kochanek (2002) and Chiba (2002) argue that the abun- the source location accurately when carrying out radia- dance of dark matter substructure in several galactic lens tive transfer caculations. Unlike semi-analytic methods, systems inferred from observed flux anomalies is consis- direct hydrodynamic simulations enable us to locate plau- tent with predictions from the standard ΛCDM model sible star-forming sites robustly in a cosmological volume. (see however Bullock & Zentner 2002 for an attempt to As we discuss later in the present paper, the distribution constrain some variant models). In models with reduced of the sources strongly affects the overall topology of the small scale power, nonlinear objects with characteristic ionized regions. 6 9 mass 10 − 10 M⊙ form late, and hence reionization due We show that the reduced small scale power in the to photons from stellar sources formed in low mass sys- RSI model causes a considerable delay in the formation tems is expected to occur late (Somerville, Bullock & Livio epoch of low mass halos and the primordial gas clouds 2003). To the contrary, the measurement of TE polariza- within them compared to a ΛCDM counterpart. In view tion by WMAP suggests a large Thomson optical depth of this result, it appears unlikely that massive stars in τe =0.17 ± 0.04, implying that reionization took place as mini-halos can contribute significantly to reionization in early as zreion ∼ 17 (Kogut et al. 2003). cosmologies with a running spectral index, at least for Somerville et al. (2003) argue that models like the RSI those with parameters similar to what is inferred from the one we consider here are inconsistent with the inferred joint WMAPext + 2dFGRS + Lyman-α forest analysis early reionization epoch. Yoshida et al. (2003b) studied (Spergel et al. 2003). To study the impact of ordinary early structure formation in a warm dark matter model (“Population II”) stars formed in galaxies and protogalax- and showed that suppressing the linear power even on very ies, we carry out ray tracing calculations of reionization by small scales (k > 100 Mpc−1) can considerably affect the stellar sources. For an initial mass function (IMF) similar formation of high redshift star-forming gas clouds, making to that in local galaxies, reionization can be completed at early reionization by massive, metal-free stars unlikely to sufficiently early times to account for the Lyman-α optical occur early on. Chiu, Fan & Ostriker (2003) argue that a depths measured in quasars at z > 6, but in the case of the large optical depth τe > 0.1 may require exotic radiation RSI model only if a large fraction of ionizing photons are sources other than a normal stellar population. Avelino able to escape from galaxies. Furthermore, because reion- & Liddle (2003) also conclude that running of the primor- ization occurs late in this context, the Thomson optical dial power systematically delays the reionization epoch. depth is too small to be compatible with that suggested Overall, these theoretical studies suggest that, if the large by the WMAP measurements. Thomson optical depth observed by the WMAP satellite Finally, we consider the possibility that the production is confirmed, the standard power-law (n ≈ 1) ΛCDM rate of ionizing photons could be boosted in galaxies and model is favored over other models with reduced small protogalaxies with a top-heavy IMF. In this case, Thomson scale power. Thus, the WMAP results appear to suggest optical depths approaching that favored by the WMAP a somewhat contradictory cosmological model. Since the analysis are possible, but only if the parameters describ- major reason to favor power-law ΛCDM in this context ing these stars are pushed to extreme values. is the inferred early epoch of reionization, it is clearly im- The paper is organized as follows. We first describe the portant to study and compare quantitatively details of the cosmological simulations we use to study the abundance reionization process in the two cosmologies, the RSI model of high-redshift star-forming systems in §2. We present and a power law ΛCDM universe. the simulation results in §3. In §4, we discuss the possible In the present paper, we study early structure for- effect of the finite size of our simulation volume. In §5, mation in RSI and ΛCDM universes using cosmologi- the abundance of primordial gas clouds and implications cal N-body/hydrodynamic simulations. By combining are described. There, we also give a brief description of these dynamical models with radiative transfer calcula- the basic relation between reionization epoch and the to- tions, we compute the epoch of reionization by various tal Thomson optical depth. In §6 we show the results of stellar sources in the two cosmologies and compare the radiative transfer calculations in which we consider ordi- Thomson optical depth to that measured by the WMAP nary stellar populations in galaxies and protogalaxies as satellite. radiation sources. A summary and discussion are given in In particular, at high redshifts, z ∼> 15, we consider the §7. implications of an early population of massive, metal-free stars (the “first stars”), which have been hypothesized to 6 form in “mini-halos” (∼ 10 M⊙). We refer to the sites where these stars can form as “primordial gas clouds;” i.e. dense regions of cold gas which accumulate through the action of cooling by molecular hydrogen. In order to iden- tify these primordial gas clouds, our simulations follow the non-equilibrium chemical reactions of a chemically pristine gas and molecular hydrogen cooling. In an earlier study 3
moderates the overcooling problem present in earlier for- 6 10 mulations of SPH (Yoshida et al. 2002; see also Pearce et al. 1999, Croft et al. 2001). 4 10 The simulations follow the non-equilibrium reactions of LCDM − + + ++ nine chemical species (e , H, H , He, He , He , H2, 2 + − 10 RSI H2 , H ) using the reaction coefficients compiled by Abel
] et al. (1997). We use the cooling rate of Galli & Palla 3 100 (1998) for molecular hydrogen cooling. Further simula- tion details are found in Yoshida et al. (2003a,b). The 3 10-2 simulations employ 2 × 324 particles in a comoving vol-
P(k) [Mpc ume of 1 Mpc on a side. We work with Λ-dominated cos- mologies with matter density Ω =0.3, cosmological con- 10-4 m stant ΩΛ = 0.7 and the Hubble constant at the present −1 −1 -6 time h = 0.7 in units of 100km s Mpc . We set the 10 baryon density to Ωb = 0.04 and the normalization pa- -8 rameter to σ8 =0.9. The transfer functions are computed 10 by the Boltzmann code of Sugiyama (1995) for the adopted 0.001 0.01 0.1 1 10 100 1000 -1 cosmology and initial conditions are realized as described k [Mpc ] in Yoshida, Sugiyama & Hernquist (2003). The primor- dial power spectrum for the ΛCDM case is assumed to 1000.0 be P (k) ∝ k, whereas that of the RSI model is given by P (k) ∝ kns with
100.0 LCDM 1 dns k ns = ns(k0)+ ln . (1) 2 d ln k k0 (k) 2 10.0 RSI ∆ −1 Specifically, we choose k0=0.05 Mpc , ns(k0)=0.93, and dns/d ln k=-0.03, as indicated by the WMAPext + 2dF- GRS + Lyman-α analysis (Spergel et al. 2003). Note that 1.0 simulation box we adopt the same normalization, σ8 = 0.9, for both the power law ΛCDM model and the Running Spectral Index model. Hereafter, we refer to the conventional power law 0.1 ΛCDM model simply as the “ΛCDM model,” and to the 0.1 1.0 10.0 100.0 1000.0 other as the “RSI model”. k [Mpc-1] Figure 1 shows the linear power spectra P (k) for the two models. In the bottom panel, we plot the variance Fig. 1.— The linear power spectra for the power law ΛCDM per unit logarithmic interval in k, ∆2(k)= k3P (k)/(2π2), model (solid line) and for the RSI model (dashed line). In the bot- 2 to better describe the difference between the two models tom panel we plot the variance ∆ (k) linearly extrapolated to the −1 present epoch for the two models. Note the difference in the plotted in the range 0.1 < k[Mpc ] < 1000. Since the input range in k between the top and the bottom panels. The arrow in power spectra are normalized at a length scale larger than the bottom panel indicates the length scales our simulations probe. the simulation boxsize, the amplitudes of the density fluc- tuations realized in the RSI simulation are smaller than in the ΛCDM simulation on all length scales. Hence, the 2. density fluctuations on large scales (k ∼ 10) in the RSI THE N-BODY/HYDRODYNAMIC SIMULATIONS model should evolve similarly to those in ΛCDM but with We use the parallel Tree-PM solver GADGET2 com- a slight delay in time. We note that, while the boxsize of bined with smoothed particle hydrodynamics (SPH) in our our simulations is sufficiently large to ensure that the fun- simulations. Our treatment of SPH employs the conserva- damental fluctuation mode in the simulations remains in tive entropy formulation of Springel & Hernquist (2002), the linear regime at the early epochs we simulate, it may which offers several distinct advantages over previous ver- not sample a fair volume of the Universe. We will discuss sions of SPH. In particular, because the energy equation finite boxsize effects in §4. is written with the entropy as the independent thermo- Figure 2 shows the projected gas distribution at z = 22 dynamic variable, as opposed to the thermal energy, the (top panels) and at z = 17 (bottom panels) for the two ’pdV ’ term is not evaluated explicitly, reducing noise from models. We choose these output redshifts such that the smoothed estimates of e.g. the density. By including terms density fluctuation on the largest length scale in the RSI involving derivatives of the density with respect to the par- model at z = 17 resembles that of the ΛCDM model at ticle smoothing lengths, this approach explicitly conserves z = 22. As expected, the RSI model appears as a delayed entropy (in regions without shocks), even when smooth- ΛCDM model on these scales. Also, rich small-scale struc- ing lengths evolve adaptively, avoiding the problems noted ture develops early on only in the ΛCDM model, as can by e.g. Hernquist (1993). Furthermore, this formulation be inferred from the linear power spectrum (Figure 1). 4
LCDM RSI 105 105 LCDM z=22 RSI z=22 104 104
103 103 M M 10 10 102 102 dN/dlog dN/dlog 101 101
100 100
10-1 10-1 104 105 106 107 108 109 104 105 106 107 108 109 z=22 z=22 Msun/h Msun/h 5 105 10
LCDM z=17 4 RSI z=17 104 10
3 103 10 M M 10 10 2 102 10 dN/dlog dN/dlog 1 101 10
0 100 10
-1 10-1 10 4 5 6 7 8 9 104 105 106 107 108 109 10 10 10 10 10 10 Msun/h Msun/h z=17 z=17 Fig. 3.— The mass function for the ΛCDM model (left) and for Fig. 2.— The projected gas density distribution at z = 22 (top the RSI model (right) at z = 17 and z = 22. The solid line is the panels) and z = 17 (bottom panels) for the ΛCDM (left) and RSI Press-Schechter mass function for the ΛCDM model, and the dashed (right) models. line is that for the RSI model.
The halo abundance in the RSI model appears to devi- 3. HALO ABUNDANCE ate slightly from the PS mass function. Although a simi- Since the typical masses of the halos that host early lar feature is also seen in the ΛCDM model, the deviation 5 7 primordial gas clouds are 5 × 10 − 10 M⊙ (Tegmark et from the PS mass function is more clearly noticeable for al 1997; Machacek et al. 2001; Yoshida et al. 2003a), the RSI model. We discuss this issue together with finite the abundance of such small mass halos is an important boxsize effects in the next section. quantity to compare. In order to quantify the difference between the two models, we measure the mass function 4. of dark matter halos. Whereas locating dark halos in MASS VARIANCE pure N-body simulations is relatively simple using con- It is well known that structure in CDM models grows ventional techniques such as the friends-of-friends (FOF) hierarchically; i.e., smaller mass objects form earlier and algorithm, there are a few subtleties in identifying and as- merge to form bigger objects. While this can be easily signing properties to halos in N-body/hydrodynamic sim- inferred from the variance ∆2(k) for the ΛCDM model, ulations. Since our main objective is to compare the halo which has larger amplitudes on smaller length scales (see abundance between the two models, we avoid ambiguities Figure 1), the situation is less clear for the RSI model. by adopting a simple procedure, as follows. As Figure 1 shows, the variance in the RSI model ap- We locate halos by running a FOF groupfinder with link- pears approximately constant over a range of scales 10 < ing parameter b =0.2 on the dark matter particles in our k[Mpc−1] < 1000, indicating that nonlinear objects with simulations. We then assign the total mass to each group widely different masses could form nearly at the same as the number of linked member particles multiplied by the epoch. It is also worth mentioning that it is not clear scaled particle mass mpart = mdm + mgas, where mdm is whether the Press-Schechter theory can be used for such the dark matter particle mass and mgas is the gas particle models. For the comparison shown in Figure 3, we cal- mass. We compute the Press-Schechter (PS) mass function culated the PS mass function by formally inputting the to which we input the linear power spectrum for the dark linear power spectrum for the RSI model. Although the matter component. (Note that, for our simulations, sepa- agreement shown in Figure 3 is reasonably good, it is not rate transfer functions are used for the dark matter and gas trivially expected. The flatness of the variance of the RSI components in setting-up the initial conditions [for details, model deserves particular attention also for another rea- see Yoshida, Sugiyama & Hernquist 2003].) We compare son; the hierarchical nature of the formation of dark mat- the mass function of the halos identified in this manner at ter halos has a strong influence on the formation of primor- z = 17 and z = 22 in Figure 3∗. For both models, the dial gas clouds (Yoshida et al. 2003a). Hence it is impor- 6 measured mass function is overall well fitted by the PS tant to verify whether or not small mass (∼ 10 M⊙) non- mass function (the solid line for the ΛCDM model and the linear objects in the RSI model grow hierarchically early dashed line for the RSI model). Again, the RSI model at on. For this purpose, it is more appropriate to work with z = 17 resembles the ΛCDM model at z = 22 with the a properly defined mass variance rather than the power abundance of low-mass halos being appreciably small. spectrum. ∗ −1 Only in Figure 3 and Figure 11, we use units that include the Hubble constant h, to be consistent with those commonly used; h M⊙ for mass and (h−1Mpc)−3 for abundance. 5
LCDM 1000 LCDM ( LCDM kcut ) RSI RSI
( RSI kcut ) )
crit 100 10 (M) σ
N halo (>M 10
1 1 103 104 105 106 107 108 15 20 25 30 M [Msun] redshift
5 Fig. 4.— The mass variance computed from the power spec- Fig. 5.— The number of halos with mass greater than 7 × 10 M⊙ tra linearly extrapolated to the present epoch. The thick solid line per cubic mega-parsec (comoving) volume. The solid line is the an- is for the ΛCDM model, and the thick dashed line is for the RSI alytic estimate obtained from the Press-Schechter mass function for model. The thin lines show the effective mass variance as defined the ΛCDM model, and the dashed line is for the RSI model. in equation (3) for each model. We compute the mass variance as slightly smaller than σ(M) on the relevant mass scales 5 6 (∼ 10 −10 M⊙), and thus a systematic delay in the char- 1 acteristic formation epoch (or a slightly lower abundance σ2(M)= P (k)W 2(kR)k2dk, (2) 2π2 Z at a given epoch) is expected. This may account for the small discrepancy in the mass function plotted in Figure where the top-hat window function is given by W (x) = 3. Since we use linearly extrapolated power spectra to 3 2 3(sin(x)/x − cos(x)/x ). Figure 4 shows σ(M) for the compute σ(M) and σeff (M), the precise effect of a finite two models. In the plotted range, the mass variance in the boxsize remains still uncertain. The apparent difference RSI model monotonically increases toward smaller mass shown in Figure 4 should be interpreted as qualitatively scales, indicating that less massive objects form at ear- representing the finite boxsize effect. Note that the RSI lier epochs; i.e., structure formation is still expected to model is more affected by the finite boxsize because the be “bottom-up”. Note that the RSI model adopted in relative fractional power on large scales is bigger than in the present paper predicts an effective slope of the power the ΛCDM model (see Figure 1). Systematic studies using spectrum smaller than -3 for k ≫ 200. On very small larger box simulations are clearly needed to address this length scales (k > 1000), or equivalently on low mass issue further. scales (M < 1000M⊙), structure formation should then By constructing a halo merger history using 30 simu- proceed differently from the “bottom-up” picture. Owing lation outputs dumped frequently between 17
must contribute to reionization for this model to be com- 0.30 patible with the WMAP results. Furthermore, it is impor- tant to establish whether or not ordinary stellar sources 0.25 in galaxies in the RSI model can account for reionization early enough to even be consistent with measurements of the Lyman-α optical depth in quasars at z > 6. 0.20 WMAP 6. REIONIZATION BY STARS IN GALAXIES
e In the previous section, we have shown that low-mass
τ 0.15 mini-halos and primordial gas clouds which can harbor massive, metal free stars (the “first stars”) do not form in 0.10 sufficient numbers at high redshift in the RSI model to ac- count for reionization at z ∼> 15. As we emphasize in §5.2 0.05 and Figure 7, reionization must occur early if the WMAP measurement of the Thomson optical depth is accurate. 7 9 The number of larger mass (∼ 10 − 10 M⊙) halos is 0.00 also predicted to be small for the RSI model. Thus, reion- 0 5 10 15 20 ization by stars in systems that are the likely progenitors z REION of galaxies will be less efficient than for a ΛCDM model, for similar assumptions about the stellar populations. In Fig. 7.— The optical depth to Thomson scattering to zreion when abrupt and complete reionization at zreion is assumed. this section, we study reionization by stars in galaxies and 7 proto-galaxies with total masses larger than ∼ 4×10 M⊙, Putting xe = 1 for an interval between z1 and z2, we where the gas can cool rapidly by atomic line transitions. obtain the contribution as The nature of the stars forming early in these environ- 1/2 z2 ments is uncertain. It is possible, for example, that the 2 σT ne(0)c 3 ΩΛ IMF will be top-heavy, at least initially, perhaps increas- dτ | 1 2 = (1 + z) + . (6) e z →z 2 1/2 3 (ΩmH0 ) Ωm ing the production rate of ionizing photons per unit stellar z1 mass compared to a conventional IMF inferred for stellar Consider a simple case where abrupt and complete populations in the local Universe. reionization occurs at z. Using the WMAP results for the In what follows, we begin by considering star formation 2 2 cosmological parameters, Ωbh =0.0224 and Ωmh =0.135, in halos in the RSI model by assuming that the stellar and assuming a plasma of primordial abundance with, for population is similar to that locally and is unevolving. example, fully ionized hydrogen and singly ionized helium, Namely, we adopt a Salpeter (1955) IMF and normalize we obtain the total optical depth to redshift z: the star formation rate (SFR) so that it reproduces the Kennicutt (1998) law. For simplicity, we refer to this de- z 3 1/2 scription as a “Population II” SFR. In §6.4 we examine the dτe ≈ 0.0023 [1 + z] + 2.7 − 1.93 . (7) overall effect of relaxing these assumptions. However, it is Z0 h i already clear that the RSI model will have difficulty ac- Figure 7 shows the total optical depth against reioniza- counting for the WMAP estimate of the Thomson for any tion epoch at z = zreion. The WMAP result is indicated reasonable assumptions about star formation simply be- at τe =0.17 (solid line) with a 1-σ range 0.17 ± 0.04 (dot- cause structure forms too late in this scenario to produce ted line). The actual reionization history could be more a large τe, as suggested by Figure 7. complicated, but Figure 7 provides the useful insight that To study how reionization proceeds in the RSI model, we a large Thomson optical depth, say τe > 0.13, can be run multi-source radiative transfer simulations using the achieved only if reionization takes place at an early epoch, technique of Sokasian et al. (2001, 2003a,b). In order to zreion ∼> 14. locate radiation sources and to compute the IGM density Our simulation of early structure formation in the RSI field, we utilize existing outputs of a large volume ΛCDM model showed that there are only 5 gas clouds at z = 15 simulation from Springel & Hernquist (2003b), normaliz- in the simulated volume. As Yoshida et al. (2003b) ing the star formation rate expected for the RSI model demonstrate, approximately 100 very massive stars must using the analytic description of Hernquist & Springel be turned on within ∼1 recombination time in a (1 Mpc)3 (2003), which can be applied to any cosmology. volume to cause complete reionization. Owing to the The star-formation rate in the RSI model strong radiative feedback from the first stars, the num- 6.1. ber of stars formed in the primordial gas clouds is likely to First, we compute the “generic” star-formation rate for be limited to one (Omukai & Nishi 1999; Oh et al. 2001). the RSI model using the analytic model of Hernquist & Hence the number of “first stars” in the RSI model is far Springel (2003). Their approach describes the formation too small at z > 15 to reionize the Universe. The epoch of of dark matter halos using an analytic halo mass function complete reionization is certainly much later than z ∼ 15, (Sheth & Tormen 1999; 2002), and employs simplified pre- implying a small total optical depth τe ≪ 0.13. scriptions for the cooling of gas and star-formation which In summary, primordial gas cloud formation which relies are commonly used in semi-analytic models of galaxy for- on molecular hydrogen cooling in low-mass, “mini-halos” mation (Kauffmann et al. 1999; Springel et al. 2001; appears to be extremely inefficient at z > 15 in the RSI Yoshida et al. 2002). This analytic formalism has been model and thus other sources, such as early protogalaxies, tested in detail by comparing its predictions to those from 8
choices p = 1.5, q = 3.05, r = 1.85, α = 0.01, β = 0.057, 3 1.0000 andρ ˙∗(0) = 0.0176M⊙/yr/Mpc . In Figure 8, we compare the SFR density estimated in this manner for the two cosmologies. The RSI model pre- dicts a significantly smaller SFR than the ΛCDM model, 0.1000 in qualitative agreement with the semi-analytic estimates ] -3 of Somerville et al. (2003). We use the forms for the SFR
Mpc LCDM density in our ray-tracing calculatiosn only within the re- -1 yr 0.0100 stricted redshift range 5 SFR [M 6.2. Source luminosities 0.0010 The second step is to prescribe the luminosity of the sources (“galaxies”) located in the simulations. Specif- ically, we use the outputs from Run Q5 of Springel & Hernquist (2003b), which simulates a cube of 14.3 Mpc on 0.0001 a side. We first calibrate the SFRs of individual sources 5 10 15 20 redshift identified in the simulation, and then convert these into luminosities. As in Sokasian et al. (2003a), we use the de- Fig. 8.— The star-formation rate density in galaxies (“Popula- rived SFR (equation 8) as a reference to match the total tion II”) as a function of redshift for the ΛCDM (solid line) and RSI SFR of the sources. We carry out the calibration proce- (dashed line) models. dure after selecting sources in the simulation outputs in the following two ways. cosmological simulations that include hydrodynamics and star formation (Springel & Hernquist 2003b). A thorough 6.2.1. Dim source model description of the model is given by Hernquist & Springel (2003) and references therein. In this approach, we identify all the halos found in each In particular, Hernquist & Springel (2003) show that output as “galaxies” and assign luminosities to them based under rather general circumstances the star-formation rate on the actual star-formation rates which are computed in density is well-approximated by the simple form a dynamically consistent way in the cosmological simula- tion. The minimum mass of the sources is set by the reso- p 7 −1 χ lution of the simulation, which is Mmin =4.3×10 h M⊙ ρ˙ (z) =ρ ˙ (0) , (8) ∗ ∗ 1+ α(χ − 1)q exp(βχr) in our case. The SFRs of the sources with masses close to the resolution limit are corrected in the same manner where as in Sokasian et al. (2003a, §4.2.1). We then scale the 2/3 SFRs of all the galaxies by multiplying by a constant fac- H(z) χ(z) ≡ . (9) tor CL(z) such that the total SFR within the simulated H0 volume matches that for the RSI model obtained from equation (8). Note that the scaling factor C (z) is kept In equation (8), the factors β and r are fixed by the under- L constant for all the galaxies within one simulation output, lying cosmological power spectrum, the choices for p and but it is time-dependent. q reflect the interplay between radiative cooling in halos and the expansion of the Universe, and the normalizing 6.2.2. Massive halo model constantsρ ˙∗(0) and α depend on the rate at which cold, dense gas is converted into stars. Since the halo abundance itself is lower in the RSI model The form of equation (8) is motivated by two compet- than in the ΛCDM cosmology, it may be more appropri- ing effects which regulate star formation. At sufficiently ate to populate a smaller number of objects as radiation high redshifts, collapsed gas is dense and cooling times are sources in the RSI model. Motivated by this, we first com- short so that star formation is only limited by the grav- pute the number density of halos with virial temperature 4 itational growth of massive halos. At low redshifts, the greater than 10 K in which the gas can cool by atomic cooling time in halos increases owing to the decline in the line transitions. The threshold mass is given by mean density of the Universe by cosmic expansion. In this −3/2 regime, the supply of star-forming gas is throttled by the 8 1+ z M = 10 M⊙. (10) expansion of the Universe. min 11 Based on the multiphase description of star-forming gas described by Springel & Hernquist (2003a), Hernquist & Then we use the halo mass function to compute the ex- Springel (2003) show that the appropriate values for the pected abundance of halos with mass greater than Mmin. parameters in equation (8) for the ΛCDM model employed Let us denote this number by Nsource(z). In the simulation here are p = 2, q = 3, r =7/4, α =0.012, β =0.041, and output produced at z, we select Nsource(z) halos in order 3 ρ˙∗(0) = 0.013M⊙/yr/Mpc . of their mass. Luminosities are assigned to the selected For the RSI cosmology, we use the RSI linear power sources in proportion to their mass, L ∝ Mhalo such that spectrum as input and follow the procedure described in the total SFR summed over the selected sources matches Hernquist & Springel (2003). We find that the resulting the global SFR given by equation (8), with the parameter SFR density is again well-fitted by equation (8) for the values appropriate for the RSI model. 9 6.2.3. Escape fraction 1.0 Finally, the ionizing flux from the sources is modulated by the “escape fraction,” fesc, which describes the proba- Tvir4 sources bility that individual photons can escape from the galaxy 0.8 in which they were produced into the surrounding inter- LCDM (f_esc = 0.2) galactic medium (IGM). Sokasian et al. (2003a) studied 0.6 RSI (f_esc = 0.6) reionization by stellar sources in the ΛCDM universe by adopting several values for fesc. Based on a detailed com- parison with the observational results of the z = 6.28 0.4 quasar (SDSS 1030.10+0524) in Becker et al. (2001), they conclude that fesc = 0.2 is most consistent with the observed Lyman-α flux transmittance. We take this ionized volume fraction 0.2 fesc =0.2 model in Sokasian et al. (2003a) as our fiducial case for the ΛCDM model. For the RSI model, our zeroth 0.0 order requirement is that reionization should be completed 6 8 10 12 14 by z = 7. After several experiments, we found that set- redshift ting fesc > 0.6 is necessary for the RSI model to satisfy this condition. We thus set fesc = 0.6 as our basic choice for Fig. 9.— The ionized volume fraction for the ΛCDM model with the RSI model and compare the results with the fiducial fesc = 0.2 (solid line) and for the RSI model with fesc = 0.6 (dashed line), for “Population II” stars. ΛCDM model. 0.10 6.3. Numerical results for “Population II” star formation We compute the total volume filling factor of the ionized 0.08 LCDM (f_esc = 0.2) medium Q from the results of our ray-tracing calcula- HII (f_esc = 0.6) tions. Figure 9 shows the resulting ionized volume frac- RSI tion, and Figure 10 shows the Thomson optical depth τe 0.06 as a function of redshift. In the figures, we show only the e results for our “massive-halo model” of source selection τ for the RSI model. Comparing the results for the dim- 0.04 source model and the massive-halo model, we found that there is no substantial difference between the two in ei- 0.02 ther τe(z) or QHII(z). We note, however, that the overall morphology of the ionized regions looks somewhat differ- ent between the two cases, because there are more sources 0.00 in the dim-source model that reside in less dense regions. 0 2 4 6 8 10 12 14 Nevertheless, for global quantities such as the total optical redshift depth, the two models are found to be quite similar. Fig. 10.— The Thomson optical depth computed from the outputs On the other hand, the results for the ΛCDM and the of our simulations for the ΛCDM model with fesc = 0.2 (solid line) RSI models differ substantially. As Figure 9 shows, the and for the RSI model with fesc = 0.6 (dashed line) for “Population ionized volume fraction in the RSI model remains small II” stars. until z ∼ 9, and then rapidly increases close to unity at z ∼ 7. At later epochs, z ∼< 9, the abundance of high photon supply does not matter in the total optical depth 9 mass (∼ 10 M⊙) halos rapidly increases, and the total as long as the IGM is kept almost fully ionized thereafter, star-formation (and hence the photon production) is dom- as implied by Figure 7. (Note that here we discuss only inated by such large halos. This can also be understood in hydrogen reionization.) terms of the halo mass function at z = 7 shown in Figure Our fiducial ΛCDM case with fesc =0.2 predicts a larger 11. By this time, the mass functions for the two models are total optical depth than the RSI model with fesc = 0.6 already quite similar, with the difference being only a con- by ∆τe ∼ 0.01. The difference mostly comes from the stant factor of roughly two (compare with the much larger fact that QHII is still quite small at z > 9 in the RSI difference at z = 17 at smaller mass scales shown in Fig- model. Interestingly, none of the “conventional” scenarios ure 3). Namely, the RSI model quickly catches up with the we have considered for “Population II” star formation re- ΛCDM model in the abundance of halos around z ∼ 7−11 produces the claimed high optical depth from WMAP of (see also the increasing SFR in Figure 8), which causes the 0.13 < τe < 0.21 (1-σ; Kogut et al. 2003). It is clear that rapid increase in the ionized volume fraction as seen in Fig- these models are inconsistent with the WMAP measure- ure 9. Reionization is nearly completed (QHII > 0.95) at ment and thus may need, for example, additional radiation z = 8 in the ΛCDM model, whereas it occurs at z ≈ 7.3 sources to ionize the IGM early on. in the RSI model. However, as shown in Figure 10, the In related work, we have shown that an early generation resulting total optical depth τe =0.055 for the RSI model of massive, metal-free stars in mini-halos supplemented by is still small, owing to the small QHII at earlier epochs. “ordinary” star formation at lower redshifts to account for We emphasize that a large Thomson optical depth can be the spectra of high redshift quasars can yield a Thomson achieved only if reionization occurs early (§6). In other optical depth approaching that of the WMAP measure- words, after reionization is completed, the additional ment in a ΛCDM cosmology (Yoshida et al. 2003b, 10 3 10 0.20 2 LCDM 10 RSI boost RSI 0.15 101 M 0 e 10 10 τ 0.10 10-1 dN/dlog 0.05 10-2 10-3 z=7 0.00 0 5 10 15 20 10-4 redshift 107 108 109 1010 1011 1012 Msun/h Fig. 12.— The Thomson optical depth for the RSI model in which the photon emission rate is “boosted” according to equation (11). The filled square is the WMAP result with its 1-σ error bar. Fig. 11.— The Press-Schechter mass function at z = 7. −4 −1 −3 ∼ 5 × 10 M⊙yr Mpc at z = 15. Assuming a con- Sokasian et al. 2003b), along the lines suggested by stant conversion factor of 1053s−1 ionizing photons per Haiman & Holder (2003) and Cen (2003). However, it −1 star-formation rate of 1 M⊙yr for an ordinary stel- does not appear that a similar conclusion will obtain for lar population with solar metallicity (Madau, Haardt & the RSI model because of the low abundance of mini-halos Rees 1999), we obtain a total rate of ∼ 5 × 1049 ionizing at high redshifts, as we found in the case of a warm dark photons per second per cubic comoving 1 Mpc volume matter model with particle mass 10 keV (Yoshida et al. at z = 15 for our fiducial RSI model. If all the formed 2003b). stars have a mass of 300M⊙, the star-formation rate of −4 −1 −3 6.4. Star formation in galaxies with boosted photon ∼ 5×10 M⊙yr Mpc corresponds to the formation of emission 5 such stars every three million years within the volume. The calculation of Bromm et al. (2001) suggests that The results described above indicate that the RSI model a total emission rate of 2.5 × 1051 ionizing photons per is incompatible with a high Thomson optical depth like second, a 50 times higher photon emission rate, can be that suggested by the WMAP measurements. A remain- maintained in this case, assuming such very massive stars ing loophole in this argument is that the stellar population are continuously formed. (Note that these massive stars in galaxies at early times may have differed from that in are typically short-lived, with a main-sequence lifetime of our underlying model so that a larger number of ionizing about two million years.) We thus conclude that a 100 photons were produced from galaxies at z > 10 than we as- times higher photon production rate than assumed in our sumed above (e.g. Ciardi, Ferrara & White 2003; Wyithe original RSI model serves as a conservative upper limit. & Loeb 2003; Sokasian et al. 2003a). We first discuss We examine the overall effect of such an additional pho- such possibilities quantitatively, and then show numerical ton supply by “boosting” the photon emission rate of the results for a model that enhances the photon production sources in the RSI massive-halo model. Sokasian et al. rate in galaxies. (2003a) found that boosting the photon emission rate at For a Salpeter-type power-law IMF, the hydrogen ioniz- high redshifts progressively shifts the reionization epoch ing photon emission rate of zero-metallicity stars is larger (see their Figure 13). Motivated by their result, we em- than that of a Population II (i.e. having metals) coun- ploy a simple prescription which increases the emission terpart only by about 50% (Thumlinson & Shull 2000; see rate of individual sources according to Leitherer et al. 1999 for low-metallicity cases.). The stellar IMF in the first galaxies may have had even an extreme ˙ ˙ top-heavy shape, with massive stars exclusively forming Nph,eff. = exp(0.53 (z − 6)) × Nph, (11) out of a primeval gas (Schwarzschild & Spitzer 1953; Mat- suda, Sato & Takeda 1969; Yoshii & Saio 1986; Larson where N˙ ph and N˙ ph,eff. are the original and the 1998; Abel, Bryan & Norman 2002; Bromm, Coppi & Lar- boosted photon emission rates, respectively. With this son 2002). Very massive Population III stars are efficient parametrization, the ionizing photon emission rate of the emitters of UV photons, with the hydrogen ionizing pho- galaxies steeply increases toward higher redshift, modeling ton emission rate per stellar mass being up to 10-20 times either a transition of the stellar IMF or a contribution from higher than for ordinary stellar populations (Bromm, Ku- massive Population III stars or both. This rather extreme dritzki & Loeb 2001; Schaerer 2002). It is thus conceivable model requires effectively more than a 10 times higher pho- that, in early galaxies, the photon emission rate per stellar ton emission rate at z > 10, and more than 100 times mass was ∼ 10 times higher than that of the “Population higher at z > 15. We carry out radiative transfer simula- II” case we considered above. tions for the RSI model with this luminosity “boost”. Fig- More specifically, the predicted star-formation rate for ure 12 shows the resulting τe(z). The reionization epoch the RSI model shown in Figure 8 is is found to be shifted to earlier times, z ∼ 10, than in 11 the case considered in §6.3, and consequently τe is greatly z > 6 we hypothesized as coming from metal-free stars increased compared with the fiducial RSI model. How- cannot be maintained for a sufficiently long time, and the ever, the total optical depth for this boost model reaches reionization history in the RSI model will be just as shown τe ∼ 0.12 at z = 0, still lower than the 1-σ range of the in Figure 9, with a small total optical depth (τe < 0.06) WMAP result. as shown in Figure 10. 7. In the end, we are left with a contradiction: the run- SUMMARY ning of the primordial power spectrum favored by the Our numerical simulations of early structure formation WMAPext + 2dF + Lyman-α analysis and the high show that primordial gas cloud formation in low-mass Thomson optical depth measured by the WMAP satel- “mini-halos” is very inefficient at z > 15 in the RSI model, lite appear to be inconsistent with one other. We have making it unlikely that the “first stars” contribute signif- discussed various possibilities to resolve the conflict, and icantly to reionization in this scenario. Using radiative conclude that a rather radical solution or a fine-tuned com- transfer calculations, we also show that reionization by bination, if any is possible, of the proposed resolutions are ordinary (“Population II”) stellar sources in galaxies is necessary. Models with a weaker running of the primor- completed late in the RSI model if we employ conven- dial power spectrum will alleviate the discrepancy slightly, tional models of star-formation. In order to be compatible but the overall results would not be changed. If the high with the epoch of the end stage of reionization inferred Thomson optical depth is confirmed and the running of from high redshift quasar observations, the photon escape the primordial power spectrum turns out to be real, it may > fraction from galaxies must be large, fesc ∼ 0.6. We found be necessary to invoke exotic radiation sources other than that the resulting total Thomson optical depth in this case stellar populations or some ionization mechanisms. Such is τe ≈ 0.055, in apparent conflict with the recent WMAP scenarios include decaying particles (Sciama 1982; Dodel- measurements of CMB polarization. son & Jubas 1992; Hansen & Haiman 2003) or formation In order for the total optical depth to be as large as of early mini-quasars (Eisenstein & Loeb 1995; Sasaki & τe > 0.1, reionization must be completed before z ∼ 10 Umemura 1996). (Figure 7). Ciardi et al. (2003) and Wyithe & Loeb (2003) The prospects for observationally resolving this dis- suggest that employing an effectively top-heavy IMF may crepancy in the near future appear bright. Data from be necessary to cause early reionization. Cen (2003) argues planned CMB polarization experiments, by the continued that a large Thomson optical depth could be obtained if operation of WMAP, and post-WMAP observatories such the “Population III era” is prolonged so that massive stars as Planck will pin down a precise value for τe. Lyman- continuously form until quite low redshift. If a large vol- α forest observations exploiting a large sample of SDSS ume of the IGM could be kept chemically pristine until low quasars will place a tighter constraint on the matter power redshift (z ∼ 10) due to the very low global star-formation spectrum on large scales, as will the SDSS galaxy redshift rate in the RSI model, the stellar IMF in galaxies may be survey. In the longer term, it may be possible to map top-heavy (Schneider et al. 2002; Mackey, Bromm & Hern- out the evolution of reionization from redshifted 21 cm quist 2003). Unfortunately, little is presently known about emission using instruments such as the Square Kilometer star-formation in a primordial gas in high mass halos –the Array† or the Low Frequency Array‡ (e.g. Madau et al. “first galaxies”– and clearly needs to be addressed using 1997; Tozzi et al. 2000; Iliev et al. 2002; Ciardi & Madau numerical simulations as well as theoretical modeling (Lar- 2003). For example, Furlanetto et al. (2003) show that son 1998; Omukai 2001; Oh & Haiman 2002). We have frequency fluctuations can be used to distinguish between explored the possibility of enhancing the photon emission models with a different number of reionization epochs. rate by employing a simple prescription. The model pa- Analyses of these various high-precision data promise to rameters were chosen such that the photon emission rate provide a more complete picture of the matter density is close to a plausible maximum bound. Our analysis has distribution in the early Universe over a wide range of shown that, whereas the extreme boost model does pre- scales and its relationship to the formation of stars and dict an earlier reionization epoch zreion ∼ 10 and a larger galaxies. total optical depth τe ∼ 0.12, the result is still marginally inconsistent with the WMAP data. We thank James Bullock, Volker Bromm, and Ravi Recent numerical simulations by Bromm, Yoshida & Sheth for helpful comments on the earlier draft. NY ac- Hernquist (2003) show that early Population III super- knowledges support from the Japan Society of Promotion novae quickly pollute the surrounding IGM with metals. of Science Special Research Fellowship. This work was If many massive population III stars are formed in “mini- supported in part by NSF grants ACI 96-19019, AST 98- halos” and explode as hyper-energetic supernovae, the 02568, AST 99-00877, and AST 00-71019. The simulations global metal-enrichment of the IGM is quickly achieved were performed at the Center for Parallel Astrophysical to such a degree that a transition of the stellar IMF is Computing at the Harvard-Smithsonian Center for Astro- caused at a very early epoch (Yoshida, Bromm & Hern- physics. quist 2003). Then, the large photon production rate at †http://www.usska.org ‡http://www.lofar.org 12 REFERENCES Abel, T., Anninos, P., Norman, M. L., & Zhang, Y. 1997, New Matsuda, T., Sato, H. & Takeda, H. 1969, Prog. Theor. 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