Fossil H ii regions: self-limiting star formation at high redshift

Abstract

Recent results from the Wilkinson Microwave Anisotropy Probe (WMAP) satellite suggest that the intergalactic medium (IGM) was significantly reionized at redshifts as high as z∼ 17. At this early epoch, the first ionizing sources probably appeared in the shallow potential wells of mini-haloes with virial temperatures T_vir < 10⁴ K. Once such an ionizing source turns off, its surrounding H ii region Compton cools and recombines. None the less, we show that the ‘fossil’ H ii regions left behind remain at high adiabats, prohibiting gas accretion and cooling in subsequent generations of mini-haloes. This greatly amplifies feedback effects explored in previous studies, and early star formation is self-limiting. We quantify this effect to show that star formation in mini-haloes cannot account for the bulk of the electron scattering opacity measured by WMAP, which must be due to more massive objects. We argue that gas entropy, rather than IGM metallicity, regulates the evolution of the global ionizing emissivity and impedes full reionization until lower redshifts. We discuss several important consequences of this early entropy floor for reionization. It reduces gas clumping, curtailing the required photon budget for reionization. An entropy floor also prevents H₂ formation and cooling, due to reduced gas densities: it greatly enhances feedback from ultraviolet photodissociation of H₂. An early X-ray background would also furnish an entropy floor to the entire IGM; thus, X-rays impede rather than enhance H₂ formation. Future 21-cm observations may probe the topology of fossil H ii regions.

galaxies: formation, intergalatic medium, cosmology: theory, large-scale structure of Universe

1 Introduction

The high optical depth τ= 0.17 ± 0.04 detected by the Wilkinson Microwave Anisotropy Probe (WMAP) satellite has lent greater credence to the notion of an early period of star formation and reionization, z_r= 17 ± 8 (Kogut et al. 2003; Spergel et al. 2003). If indeed the first stars formed at high redshift z∼ 20, they are expected to form in mini-haloes¹ with shallow potential wells, in which H₂ cooling is dominant (Abel, Bryan & Norman 2000; Bromm, Coppi & Larson 2002). More massive haloes with T_vir > 10⁴ K, in which collisional ionization and line cooling can operate, are expected to be very rare at these redshifts. Modulo the effects of ultraviolet (UV) feedback on H₂ formation and cooling, stars forming in such haloes could therefore play a dominant role in an early reionization epoch. A great deal of effort has gone into assessing the impact of UV feedback on H₂ cooling, and the counter-veiling results of positive feedback effects such as an early X-ray background (e.g. Ciardi, Ferrara & Abel 2000; Haiman, Abel & Rees 2000; Machacek, Bryan & Abel 2001; Ricotti, Gnedin & Shull 2001).

The main point of this paper is that the population of mini-haloes is likely to be considerably sparser than previously assumed. This is because mini-halo formation is strongly suppressed even inside the fossil H ii regions of dead ionizing sources. Although such H ii regions recombine and cool by Compton scattering with cosmic microwave background (CMB) photons, they cannot cool back to the temperature of the undisturbed intergalactic medium (IGM). Strong Jeans mass filtering takes place (Gnedin 2000), and subsequent mini-haloes will no longer be able to accrete gas due to the smoothing effects of finite gas pressure. Thus, once any patch of the Universe is ionized, it can no longer host any more mini-haloes, even if it subsequently cools and recombines. In effect, the birth of the first stars leads to the demise of the mini-halo population: only one generation of stars can form within these shallow potential wells.

While many authors have noted and commented on the Jeans mass filtering in actively ionized regions of the IGM after full reionization (Bullock, Kravtsov & Weinberg 2000; Benson et al. 2002; Somerville 2002), the Jeans filtering in fossil H ii regions after ionizing sources have turned off has received little attention. An exception is the recent work by Ricotti, Gnedin & Shull (2001, 2002a, b), who consider the effect of a cosmic UV field on the H₂ abundance. They find that when an H ii front sweeps by a fluid element that has already contracted to a high density, it can result in an overall increase in the H₂ abundance after the UV field is turned off. In this paper, we consider the behaviour of the background gas near the mean IGM density inside fossil H ii regions around the first ionizing sources. We show that the entropy floor established in the fossil H ii regions prevents this gas from contracting and building up dense cores of mini-haloes (on scales much smaller than possible to address in cosmological simulations). We argue that this will strongly suppress the mini-halo population, with the following interesting consequences.

• (i)

  Impact of mini-halo population on reionization. Since the recombination time is shorter than the Hubble time at high redshift, t_rec≪t_H, and the ionizing sources, expected to be massive stars, have short lifetimes, t_MS∼ 3 × 10⁶ yr ≪t_H, many generations of star formation are required to keep a given patch of the IGM ionized. However, once the first generation of stars born in mini-haloes dies out, subsequent generations will not be able to form in mini-haloes in a previously reionized patch of IGM, even after the patch cools and recombines. Previous semi-analytical work on the reionization history has already included photoionization feedback in actively ionized regions of the IGM (Haiman & Loeb 1997, 1998; Haiman & Holder 2003; Wyithe & Loeb 2003), by excluding new sources in shallow potential wells from forming in these regions. Here we show that the additional exclusion of ionizing sources from the fossil H ii regions produces a strong additional modification of the reionization history, both quantitatively and qualitatively (see Fig. 16 below). Specifically, the feedback in fossil H ii regions inevitably produces a non-monotonic reionization history with an early peak of partial reionization, followed by recombination and eventual full reionization. This is qualitatively similar to the reionization history derived by Cen (2003) and Wyithe & Loeb (2003), although the physical reason for the non-monotonic evolution in the present work is different (entropy injection, rather than a Population III to II transition caused by a universal metallicity increase). Regardless of the extent of UV feedback effects on H₂ production and cooling, the majority of stars which reionized the Universe were hosted by more massive haloes able to survive Jeans mass filtering, T_vir > few × 10⁴ K.

• (ii)

  Photon budget for reionization. Dense gas that collects in mini-haloes can form a considerable sink of ionizing photons by boosting the overall effective gas clumping factor C_II=〈n²_e〉/〈n_e〉² (Haiman, Abel & Madau 2001; Barkana & Loeb 2002; Shapiro et al. 2003). The clumping factor C_II increases rapidly at low redshift as structure formation progresses; thus, despite the higher mean gas density n∝ (1 +z)³ at high redshift, the recombination time t_rec= 1/(αn_eC_II) does not evolve strongly from z= 10 to 20. However, if the Universe is filled up with fossil H ii regions at high redshift, the photon budgets required to subsequently ionize it and to keep it ionized are much lower, since gas clumping in mini-haloes is strongly suppressed.

• (iii)

  Source clustering. Since pre-heating boosts the threshold mass required for efficient gas cooling and star formation, it increases the mean bias of the early protogalaxy population by a factor of a few (e.g. see fig. 2 of Oh, Cooray & Kamionkowski 2003). This is likely to increase the clustering amplitude of background fluctuations due to these faint unresolved early protogalaxies, such as the free–free background due to ionized haloes (Oh 1999; Oh & Mack 2003), Sunyaev–Zel'dovich fluctuations due to high-redshift H ii regions and supernovae (SNe) winds (Knox et al. 1998; Oh et al. 2003; Santos et al. 2003), and the infrared background due to stellar emission (Santos, Bromm & Kamionkowski 2002; Magliocchetti, Salvaterra & Ferrara 2003; Haiman, Spergel & Turner 2003). The increase in clustering bias boosts the amplitude of rms fluctuations by a factor of several. This may be necessary, for instance, if thermal SZ fluctuations from high-redshift supernovae are to account for the small-scale CMB anisotropies observed by BIMA and CBI (Oh et al. 2003).

• (iv)

  Star formation history/metallicity/initial mass function. Initially, each mini-halo is expected to harbour a single massive, metal-free star (no fragmentation is seen in numerical simulations by Abel et al. (2000) and Bromm et al. (2002). Because of the pre-heating, no further mini-haloes can form in the entire comoving volume reionized by this mini-halo. As a result, the first generation of stars is expected to form as a population of single, isolated stars. Star formation ensues again only once more massive haloes with deeper potential wells aggregate. These rare, high-density peaks are likely to coincide with the highly biased regions where the first isolated stars had already formed. At the time of formation of deeper potential wells, such sites were inevitably already polluted with metals, unless the first stars collapsed directly to black holes without associated metal production. As a result, the transition from Population III to II (metal-free to normal) stellar populations is associated with the halo mass scale. This is in contrast to scenarios (Cen 2003; Wyithe & Loeb 2003) where star formation can continually proceed in lower-density peaks, which are far from the initial sites of star formation and still contain relatively pristine gas (allowing metal-free star formation to continue to relatively low redshifts). Given the factor of ∼10–20 difference in the ionizing photon production efficiency per unit mass between Population III and II stars (Tumlinson & Shull 2000; Bromm, Kudritzki & Loeb 2001b; Schaerer 2002), this will have important consequences for the redshift evolution and topology of reionization.

• (v)

  21-cm observations. It has been suggested that mini-haloes will be observable at the redshifted 21-cm line frequency both in emission (Iliev et al. 2002a, b), and absorption (Furlanetto & Loeb 2002). If mini-halo formation is strongly suppressed, these two windows on small-scale structure during the cosmological Dark Ages will disappear. 21-cm observations of mini-haloes therefore provide an interesting probe of the topology of fossil H ii regions: if mini-haloes are seen, that comoving patch of the IGM has never been ionized. This is complementary to other observations such as Gunn–Peterson absorption, which only probe the instantaneous ionization state.

Figure 16.

The evolution of the volume filling fraction F_{H ii} of ionized hydrogen in reionization models with different assumptions concerning feedback on mini-halo formation. The upper and lower panels assume constant clumping factors of C= 1 and 10, respectively. In both panels, the dashed curves show the volume filling fraction of the fossil H ii regions. The thick solid curves show the evolution of F_{H ii} assuming that mini-haloes cannot form inside fossil H ii regions. The upper solid curves assume that mini-haloes are only excluded from forming inside active ionized regions. Finally, the lower solid curves ignore the contribution to F_{H ii} from any mini-haloes.

The rest of this paper is organized as follows. In Section 2, we compute the ‘entropy floor’ due to early mini-haloes. In Section 3, we calculate the effects of this entropy floor on the gas density profile in mini-haloes. In Section 4, we discuss and quantify several important effects of this entropy floor for reionization, and in Section 5 we explicitly calculate the effect on the reionization history. We summarize our findings and discuss their implications in Section 6. Throughout this paper, we adopt the background cosmological parameters as measured by the WMAP experiment (Spergel et al. 2003, tables 1 and 2), Ω_m= 0.29, Ω_Λ= 0.71, Ω_b= 0.047, h= 0.72 and an initial matter power spectrum P(k) ∝kⁿ with n= 0.99 and normalization σ₈= 0.9.

2 Entropy Floor Due to Early Reionization

2.1 Entropy floor in Fossil H ii regions

Early reionization introduces an ‘entropy floor’ in the intergalactic medium that impedes gas accretion and cooling in haloes with T_vir < 10⁴ K, which cannot excite atomic cooling. Early reionization even introduces Jeans smoothing effects in haloes with T_vir > 10⁴ K, since the strength of the accretion shock is weaker in pre-heated gas, and a deeper potential well is required to collisionally ionize the gas. The overall effect is to introduce a substantial core in the gas density profile. The effect is similar to the presumed pre-heating of the IGM by galactic winds or AGN outflows at low redshift. There, the finite entropy of the gas introduces a core in group and cluster gas density profiles and is thought to be responsible for the deviation from self-similarity in the observed cluster L_X–T_X relation (for a recent review, see Rosati, Borgani & Norman 2002). Entropy is more fundamental than gas temperature in low-redshift clusters and high-redshift mini-haloes: in both cases, entropy is conserved, since t_cool≫t_H. We exploit this analogy, and deliberately employ language and techniques from the well-studied low-redshift case to study high-redshift mini-haloes. Here, we calculate the expected level of the ‘entropy floor’ due to early reionization.

Reionization is likely to be a highly stochastic process where sources ionize a patch of the IGM and then fade; the fossil H ii region subsequently cools and recombines. What is the final temperature a fossil H ii region can cool to? At the high redshifts of interest, Compton cooling off the CMB is by far the dominant source of gas cooling in the low-density IGM (e.g. at z= 19 and T= 10⁴ K, t_C∼ 0.1t_coolδ, where δ is the gas overdensity and t_cool is the atomic cooling time-scale). The final gas temperature is therefore determined by the competition between Compton cooling and hydrogen recombination: after the gas recombines, it ‘decouples’ from the CMB, and Compton cooling is no longer efficient. The Compton cooling time-scale is independent of density and temperature:  

(1)

whereas the recombination time  

(2)

is shorter for higher-density gas. Thus, denser gas ‘decouples’ from the CMB earlier and freezes out at higher temperatures. For the redshifts of interest, t_rec(δ= 1) > t_C, allowing for substantial cooling below ∼10⁴ K.

We can study the temperature evolution of a cooling and recombining parcel of gas T(t, T_i, δ, z) as a function of time t (or redshift interval δz), initial temperature T_i, gas overdensity δ_i and redshift z. In Fig. 2 (below) we show the dependence of the final temperature on these parameters by solving the full set of chemical evolution equations for primordial gas. We neglect H₂ chemistry as a trace UV flux is sufficient to photodissociate H₂ at these low IGM densities. Our fiducial patch has an initial temperature T_i= 20 000 K, cools at z= 19, for a time interval t= 0.6t_H(z= 19) (corresponding to roughly δz≈ 7), and lies at an overdensity of δ= 1; each panel shows the dependence on one of the variables holding the others fixed. We do not take into account time-dependent variation of the gas density arising from infall on to non-linear structures or Hubble expansion, which would provide adiabatic heating/cooling of the gas; however, the effects of this may be roughly estimated by considering some mean overdensity δ of the gas during its evolution. The final temperature is almost independent of T_i, as long as t≫t_C and the gas was initially fully ionized. However, the final temperature declines at higher redshifts and low overdensities δ, since t_C/t_rec∝ (1 +z)⁻¹δ, and the gas cools faster than it recombines.

Figure 2.

Dependence of the final entropy K=T/n^2/3 on redshift z and overdensity δ. Points correspond to the analytic solution from equations (33) and (35). As in Fig. 1, our fiducial patch has T_i= 1, δ= 1, z= 19, t= 0.5t_H(z= 19). The dependence on T_i and t are the same as in Fig. 1. The gas retains more entropy at lower redshift, since Compton cooling is less efficient. However, the final entropy shows only a weak dependence on overdensity δ: the gas cools out at higher temperatures at higher densities, and the increased temperature and density roughly cancel.

In the Appendix, we develop an analytic expression for T(t, Ti, δ, z) taking into account only Compton cooling and recombination which is remarkably accurate; it is given by the points in Fig. 1. This allows one to estimate the final temperature quickly without evolving the coupled differential equations. It deviates from the true answer significantly only at high overdensity and/or low redshift, when atomic line cooling becomes comparable to Compton cooling. In any case, for metal-free gas, line cooling alone can only cool the gas down to ∼5000 K. The only processes that could alter the final entropy of the gas significantly are H₂ cooling and metal line cooling. These are more efficient than Compton cooling only in regions of high overdensity, in the halo core, and so do not affect the initial entropy floor.

Figure 1.

The parameter dependence of the final temperature of a parcel of isochorically cooling gas. The solid lines indicate the results from the full non-equilibrium chemistry code, while the points indicate the approximate analytic solution from equations (33) and (35), which is in excellent agreement. Our fiducial patch has an initial temperature T_i= 20 000 K, cools at z= 19, for a time interval t= 0.6t_H(z= 19) = 1.9 × 10⁸ yr (corresponding to roughly δz≈ 7), and lies at an overdensity δ= 1. The gas cools most effectively at high redshift (when the Compton cooling time is short) and at low overdensity (when it decouples from the CMB at late times). For z= 19 and δ= 10, (which is perhaps characteristic of gas being accreted on to haloes) the gas remains at a few × 10³ K.

Once cooling becomes ineffective, the gas evolves adiabatically. In accordance with convention, we shall refer to the quantity  

(3)

as the ‘entropy’ of the gas, even though it is not strictly the thermodynamic entropy S∝ log (K). This is a useful convention because K is conserved when the gas evolves adiabatically. Thus, it is conserved during Hubble expansion, and during accretion on to haloes (provided the accretion shock is weak, gas will be accreted isentropically). In Fig. 2, we show the dependence of the final entropy K on redshift (the dependence on T_i, t follow simply from Fig. 1). The gas has significantly lower entropy at higher redshift, since Compton cooling is more efficient. However, the final entropy at a given redshift depends only weakly on the overdensity δ. Denser gas remains at higher temperature, since it recombines faster than it cools, and the combination of higher temperature and higher density roughly cancel K∝Tδ^−2/3. We can therefore ignore the weak δ dependence of K, and assume that it depends only on z. The entropy floor is thus roughly independent of the details of structure formation and gas density distribution.

It is useful to define a parameter that compares the IGM entropy floor to the entropy generated by gravitational shock heating alone. Let us define the quantity , where K₀=T_vir/n(r_vir)^2/3, and n(r_vir) = (Ω_b/Ω_m)ρ_NFW(r_vir)/(μm_p)[and ρ_NFW is the NFW (Navarro, Frenk & White 1997) dark matter density profile]. This is the entropy due to shock heating alone at the virial radius; the justification for this will become clearer in the ensuing section. As increases, the Jeans smoothing effects due to finite IGM entropy become increasingly more significant. This parameter will be used extensively in the following sections, and it is useful here to obtain a sense of what values of are expected. In Fig. 3, we plot for a T_vir= 9000 K halo as a function of redshift, using K_IGM(z) shown in Fig. 2. Since , and a T_vir= 9000 K halo is about the most massive that would still not experience atomic cooling, the solid line depicts a lower limit on for all mini-haloes. Appropriate values for for smaller haloes can be read off simply from the scaling. We see that for most haloes, ; although K_IGM(z) falls at high redshift, this is somewhat offset by the fact that typical potential wells are much shallower, and thus K₀ also falls rapidly (as can be seen by the dashed lines, which depict for 2σ and 3σ fluctuations).

Figure 3.

The evolution of the dimensionless entropy parameter as a function of redshift, assuming the gas cools at the mean IGM density δ= 1. Here K₀ is the gas entropy at the virial radius due to shock heating alone, and K_IGM is the entropy of the fossil H ii regions as computed in Fig. 2. The solid line describes a T_vir= 9000 K halo, the most massive in which atomic cooling is still not important. It therefore defines a lower limit on . The entropy parameter can be simply scaled () for haloes with lower virial temperatures. The dashed lines show for 2σ and 3σ haloes at that redshift; the vast majority of mini-haloes have .

Typical values for are illustrated further in Fig. 4, where we show the mass-weighted fraction of mini-haloes that have entropy parameters less than a given , given by  

(4)

where is the mass corresponding to for a given K_IGM(z), M_J(z) is the cosmological Jeans mass (e.g. see Barkana & Loeb 2002), M_u(z) is the mass corresponding to a T_vir= 10⁴ K halo, and dn/dM is the Press–Schechter mass function. Virtually all haloes have in fossil H ii regions at all redshifts of interest (z≲ 20), and median values of are much higher. We shall soon see that such haloes are subject to very substantial Jeans smoothing effects.

Figure 4.

The mass-weighted fraction of mini-haloes that have entropy parameters , for z= 10, 15, 20, assuming K_IGM(z) shown in Fig. 2. Virtually all haloes have , and the median value of is substantially higher. Thus, the vast majority of haloes accrete gas isentropically.

2.2 Entropy floor due to X rays

Reionization by X-rays (Oh 2001; Venketesan et al. 2001) would produce a warm (few ×100–1000 K), weakly ionized IGM with an entropy floor similar to that in fossil H ii regions. Such X-rays could arise from supernovae, AGN or X-ray binaries, or in more exotic models with decaying massive sterile neutrinos (Hansen & Haiman 2003). The Universe is optically thick to all photons with energies  

(5)

where x_{H i} is the mean neutral fraction, and we have assumed σ_ν∝ν⁻³. Thus, all energy radiated below E_thick will be absorbed. The relative efficiency of UV photons and X-rays in setting an entropy floor deserves a detailed separate study; here we discuss some salient points. UV photons are an ‘energetically extravagant’ means of producing an entropy floor. Most of the energy injected by UV photons is lost to recombinations and Compton cooling at high redshift; we see in Fig. 1 that the gas typically Compton cools to T_floor∼ few × 100 K at the redshifts of interest. Thus, only ∼10⁻² of the heat injected by UV photons is eventually utilized in setting the entropy floor; all energy expended in heating the gas above T_floor is ‘wasted’ (it is dumped into the CMB, where it may eventually be observable as a spectral distortion, Fixsen & Mather 2002). On the other hand, in the weakly ionized gas produced by X-rays, both the recombination time and the Compton cooling time are longer than the Hubble time. In particular,  

(6)

and almost none of the entropy injected is lost to cooling. Furthermore, a larger fraction of the X-ray energy goes toward heating rather than ionization [in general, a few ×0.1 of the energy of the hot photoelectron created by an X-ray goes toward heating; this fraction quickly rises toward unity as the medium becomes progressively more ionized (Shull & van Steenberg 1985)]. Thus, if ε_bol(X-ray)/ε_bol(UV) ≥ 10⁻² (where ε_bol is the comoving emissivity), X-rays could be comparable or even more effective than UV photons in setting an entropy floor. The relative emissivities of UV and X-rays is unknown, but for supernovae could be as high as (Oh 2001)  

(7)

where ƒ_esc is the escape fraction of ionizing UV photons from the host halo and ƒ_X is the fraction of supernova explosion energy that goes into soft X-rays, either through inverse Compton scattering of CMB photons by relativistic electrons (Oh 2001), or free–free emission from the hot SN remnant.

To make a quick estimate, let us (fairly conservatively) assume that ƒ_X∼ 3 per cent of the explosion energy of a supernova goes into soft X-rays. Of this, ƒ_heat∼ 50 per cent of the energy goes into heating; the rest goes into secondary ionizations and atomic excitations. A supernova releases E_Z∼ 0.5 MeV in explosion energy per metal baryon, relatively independent of metallicity (a Population III ‘hypernova’ produces ∼100 times more energy, but also ∼100 times more metals than a standard type II SN). X-ray heating thus results in an entropy floor , or  

(8)

where is the mean metallicity of the Universe. This is comparable to the entropy of fossil H ii regions, but has a much larger filling factor, of the order of unity. A metallicity of corresponds to roughly ∼1 ionizing photon per baryon in the Universe, both for Population II and III stars. As a result of recombinations, the filling factor of ionized regions will of course be considerably less than unity. However, the filling factor of fossil H ii regions, which is the relevant quantity, will also be less than unity by a factor of ƒ_overlap, which represents the overlap of H ii regions with fossil H ii regions that have recombined. Energy spent in reionizing fossil H ii regions is ‘wasted’ in terms of establishing an entropy floor. This factor is likely to be large because early galaxy formation is highly biased, and higher-mass haloes (which can resist feedback effects) will be born in regions already pre-reionized by earlier generations of mini-haloes. A fossil H ii filling factor of the order of unity with little overlap can only be achieved if the formation of mini-haloes is highly synchronized (see the discussion in Section 5).

The mean free path of X-ray photons generally exceeds the mean separation between sources, becoming comparable to the Hubble length for ∼2 keV photons. The entire Universe is thus exposed to a fairly uniform X-ray background, and the entire IGM acquires a uniform entropy floor, with an amplitude that scales with the amount of star formation, as in equation (8). Even regions far from sites of star formations, which have never been engulfed in an H ii region, will be affected. In contrast, in pure UV reionization scenarios, there are large spatial fluctuations in entropy, which depend on the topology of reionization, and the redshift at which a comoving patch was last ionized.

Because of the relative uncertainty of the amplitude of the X-ray background, we use the entropy floor associated with fossil H ii regions in the rest of this paper. The mass fraction of affected mini-haloes scales with the filling factor of fossil H ii regions. This is therefore a minimal estimate; the filling factor could approach unity if X-rays are important.

We now consider the effects of a finite entropy floor on mini-halo gas density profiles.

3 Gas Density Profiles

Once Compton cooling and radiative cooling become inefficient, the gas evolves adiabatically. We can therefore compute static equilibrium density profiles, and see how they change as a function of the entropy floor. The models we construct are in the spirit of Voit et al. (2002), Tozzi & Norman (2001), Oh & Benson (2003) and Babul et al. (2002), which match observations of low-redshift cluster X-ray profiles well.

Naively, one might assume that only gas that remains at temperatures comparable to the virial temperatures of mini-haloes would suffer appreciable Jeans smoothing. Since in many cases the IGM can cool down to a few × 100 K, one might assume that this level of pre-heating would have negligible effects on the density profile of gas in mini-haloes. This is false: the important quantity is not the temperature but the entropy of the gas. Since gas in the IGM is heated at low density, it has comparatively high entropy. Gas at mean density that is heated to temperatures  

(9)

(where δ is the overdensity of the gas in the mini-halo in the absence of pre-heating) will have an entropy in excess of that acquired by gravitational shock heating alone. Its temperature will therefore exceed the virial temperature after infall and adiabatic compression. As the level of pre-heating increases, gas at progressively larger radii in the halo undergoes Jeans smoothing effects. We now calculate this in detail.

We first construct the default entropy profile of the gas without pre-heating. We assume that in the absence of heating or cooling processes, the gas distribution traces that of the dark matter, an Ansatz which is indeed observed in numerical simulations (e.g. Frenk et al. 1999) (the dark matter is assumed to follow the NFW, Navarro et al. 1997, profile). This assumption becomes inaccurate at the very centre of the halo, where finite gas pressure causes the gas distribution to be more flattened and less cuspy than the dark matter density distribution. In particular, even in the absence of pre-heating the IGM has a finite temperature after decoupling from the CMB and cooling adiabatically: T_min(z) ≈ 2.73(1 +z_d)[(1 +z)/(1 +z_d)]², where the matter–radiation decoupling redshift z_d≈ 150. This gives rise to a finite entropy floor:  

(10)

independent of redshift. Thus, there will be a finite core in the gas density profile even in the absence of pre-heating. In regions where gas traces the dark matter, hydrostatic equilibrium gives the entropy profile due to shock heating as  

(11)

The final entropy profile is therefore K(r) = max[K_min, K_shock(r)]. Note that the mean molecular weight is μ= 0.59 for fully ionized gas and μ= 1.22 for fully neutral primordial gas; in this paper we are dealing with the case where the ionization fraction is small and therefore use the latter figure. This results in virial temperatures that are higher by a factor of ∼2.

What is the effect of pre-heating on the entropy profile? If the infall velocity does not exceed the local speed of sound, then the gas is accreted adiabatically and no shock occurs; the gas entropy K_IGM is therefore conserved. If gas infall is supersonic, then the gas is shocked to the entropy K_shock computed in equation (11). Tozzi & Norman (2001) found that the transition between the adiabatic accretion and shock heating regime is very sharp. To a very good approximation K(M) = max(K_shock(M), K_IGM); with this new entropy profile we can compute density and temperature profiles. In fact, for most high-redshift mini-haloes the ‘pre-heating’ entropy exceeds the shock entropy even at the virial radius, so the gas mass is accreted isentropically. This occurs when the entropy floor exceeds  

(12)

where the gas overdensity δ∼ 50 at the virial radius in the absence of pre-heating is a weak function of the NFW concentration parameter c. Although c depends weakly on the collapse redshift, for simplicity we shall assume in this paper that c= 5 for all mini-haloes.

We have compared our entropy profiles calculated using equation (11) with the prescription in Tozzi & Norman (2001) (and also adopted by Oh & Benson 2003). In this method an accretion history for a halo is prescribed via extended Press–Schechter theory; this allows one to compute the strength of the accretion shock and thus the gas entropy (using standard Rankine–Hugoniot jump conditions) for each Lagrangian mass shell. The two methods agree extremely well; we therefore use equation (11) for both speed and simplicity.

Given an entropy profile, from hydrostatic equilibrium the density profile of the gas is then given by  

(13)

where . The temperature profile can then be determined from .

The solution of this equation requires a boundary condition which sets the overall normalization, here expressed as P(r_vir). There is some ambiguity in this choice. The often used boundary condition M_g=f_BM_halo at r=r_vir is unphysical as it does not take into account the suppression of accretion due to finite gas entropy. We make the following choice. Let us define P_shock(r_vir) ≡ (Ω_b/Ω_m)ρ(r_vir)/(μm_H)k_BT_vir, the pressure at the virial radius due to shock heating alone. For K_shock(r_vir) > K_IGM, the final conditions at the virial radius are not strongly affected by the entropy floor, since the shock boosts the gas on to a new adiabat. We therefore set P(r_vir) =P_shock(r_vir). The boundary condition must change when K_IGM > K_shock(r_vir), when accretion takes place isentropically, and the entropy floor is fundamental in determining the gas pressure. In this case, P(r_vir) (which is essentially a constant of integration) is chosen so that as r→∞. The latter boundary condition implicitly assumes that hydrostatic equilibrium prevails beyond the virial radius. This is questionable, but is arguably reasonable: the gas speed of sound c_s= (γK_IGMρ^γ−1)^1/2 in pre-heated gas is much higher than in cold gas, which is essentially in free-fall. The sound-crossing length-scale over which hydrostatic equilibrium can be established, L_sc≈c_st_H, is  

(14)

which is approximately five times larger than the virial radius for a ∼6000-K halo at the same redshift. Indeed, by definition L_sc > r_vir for Jeans smoothing effects to be important.

Having established the appropriate boundary conditions, we can now compute detailed gas profiles. The entropy, density, pressure and temperature profiles as a function of are shown in Fig. 5. These profiles are of course universal and independent of the halo mass once is set, if the weak dependence of the NFW concentration parameter c with mass is ignored. As the entropy floor increases, the central pressure and density decline, while the central temperature increases. We see that reasonable values of the entropy floor produce dramatic effects on the gas density profile of the halo, smoothing it out considerably.

Figure 5.

The dimensionless entropy , pressure , temperature , and density , as a function of radius . K₀, P₀, T₀ are the values of these quantities at r_vir without pre-heating, while , the mean baryonic density. The entropy profile uniquely specifies , independent of halo mass or redshift.

We now use these density profiles to compute how the accreted gas fraction ƒ_g≡ (M_g/M_halo)/(Ω_b/Ω_m) scales with the entropy parameter . This is shown as the dark solid line in Fig. 6. Again, because of the self-similarity of the problem, this plot is valid for mini-haloes of all virial temperatures at all redshifts, provided is appropriately rescaled. It is reassuring to see that ƒ_g is continuous at , when we switch from one boundary condition to another. This need not have been the case, and gives us confidence that we handle the transition to isentropic accretion correctly. We see that realistic levels of the entropy floor (as computed in Section 2) cause a substantial depression in gas fractions in mini-haloes.

Figure 6.

The gas fraction within the virial radius ƒ_gas= (M_g/M_halo)/(Ω_b/Ω_m) as a function of the entropy parameter . The solid line indicates our fiducial boundary conditions; the dotted line corresponds to the Gnedin (2000) fit to numerical simulations; while the dashed line corresponds to the Bondi accretion prediction (valid only for isentropic accretion, ). All three show good agreement. The self-similarity of this problem implies that the computed gas fractions applies to mini-haloes of all virial temperatures, provided is appropriately scaled. For illustrative purposes, appropriate values of for a mini-halo of T_vir= 5000 K at z= 20, 13 are shown (see Fig. 3).

It is interesting to compare our derived gas fractions with other estimates. For the case where K_IGM > K_shock(r_vir), accretion takes place isentropically. The halo will therefore accrete gas at roughly the adiabatic Bondi accretion rate (e.g. Balogh, Babul & Patton 1999):  

(15)

The total accreted gas mass is then roughly , where ƒ is some unknown normalization factor, which takes into account the fact that the total halo mass is not constant but was lower in the past (and hence that the gas accretion rate was lower in the past). Another estimate which is a good fit to the results of hydrodynamic simulations is (Gnedin 2000)  

(16)

There is only one free parameter: M_1/2, the mass of the halo in which the gas mass fraction ƒ_g= 0.5. Gnedin (2000) shows that M_1/2 is well approximated by the ‘filtering mass’M_F. However, M_F depends on the unknown thermal history of the IGM. To make a self-consistent comparison, we compute M_1/2 with the density profiles computed using our fiducial boundary conditions. Interestingly, we find that M_1/2 roughly corresponds to the halo mass when accretion begins to take place isentropically .

The results are shown in Fig. 6. All three estimates agree well (note that the normalization ƒ of the Bondi accretion prediction is a free parameter; the plot shown is for ƒ= 0.3). The widely used Gnedin (2000) fitting formula predicts even lower gas fractions (and as we will see, clumping factors) at high entropy levels . On the other hand, the slope of the relation for our boundary conditions agrees very well with the Bondi accretion prediction in this regime. Our boundary conditions therefore yield fairly conservative estimates of the effects of pre-heating. Moreover, unlike these other estimates, we are able to compute detailed density profiles, which is crucial for some of our later calculations.

There are two other quantities which are of particular interest when computing density profiles. One is the central density ρ_c, which affects the ability of mini-haloes to form H₂ in the face of UV photodissociation. Another is the gas clumping of the halo, defined as  

(17)

where the angled brackets indicate a volume-averaged quantity,  

(18)

Note that C_halo≥ 1 always. The clumping factor C_halo plays a central role in determining the photon budget required for reionization; we evaluate the global clumping factor in the next section. In Fig. 7, we show the effect of increasing the entropy parameter on the central density and the clumping factor. Both decline rapidly with . We shall use these two results in the following sections.

Figure 7.

The effect of increasing the entropy parameter on the central gas density (bottom panel), and gas clumping C^1/2_halo (top panel), as defined in equation (17). The solid lines correspond to our fiducial boundary conditions, while the dashed lines correspond to adjusting P(r_vir) to reproduce the gas fractions from the Gnedin (2000) fit to numerical simulations (the results for the Bondi accretion prediction are almost identical to the solid curve). Both and C^1/2_halo decline rapidly as increases.

We now use these gas density profiles to compute global effects of mini-halo suppression.

4 Global Effects of Mini-Halo Suppression

4.1 Suppression of collapsed gas fraction and gas clumping

An entropy floor suppresses the fraction of gas which is bound within mini-haloes. As we discuss in Section 4.4, the mean 21-cm emission from mini-haloes is directly proportional to this global gas fraction; if it is strongly suppressed the signal will be unobservable. The global collapsed gas fraction is  

(19)

where is the halo gas fraction as in Fig. 6, and ρ_b is the comoving baryon density. This is shown in the bottom panel of Fig. 9 (below). Curves are shown for no pre-heating K_IGM=K_min, fixed values of the entropy floor K_IGM= 1, 10 eV cm² (which correspond to the IGM settling to some temperature at a given redshift and remaining at constant entropy thereafter), and the redshift-dependent entropy K_IGM(z) shown in the top panel of Fig. 2, which reflects the increasing entropy of gas that Compton cools at late epochs. For realistic values of K_IGM, the collapsed gas fraction is suppressed by one to two orders of magnitude.

Figure 9.

The evolution of the collapsed gas fraction in mini-haloes (lower panel) and the gas clumping factor (upper panel) with redshift. The results are shown for no pre-heating (with K_IGM=K_min), and constant values of entropy K_IGM= 1, 10 eV cm² and the redshift-dependent entropy K_IGM(z) shown in the top panel of Fig. 2. Both the global gas fraction in mini-haloes and gas clumping are strongly suppressed for realistic values of the entropy floor. In particular, C→ 1 for realistic values of K_IGM, greatly reducing the photon budget required for reionization.

It is interesting to plot the effect of pre-heating on the baryonic mass function. This can be computed simply as  

(20)

We show this in Fig. 8, at z= 10, 15, 20 for K_IGM(z) as in the top panel of Fig. 2, and for a fixed entropy floor K_IGM= 3 eV cm². As expected, the effect of pre-heating is most drastic at low masses. Note that the baryonic mass function is very similar at all redshifts for K_IGM(z); this can also be seen in the bottom panel of Fig. 9, where the collapsed fraction in mini-haloes does not change appreciably with redshift.

Figure 8.

The mass function of baryons, as in equation (20), for z= 10, 15, 20. Dark solid curves show the mass function for K_IGM=K_min, light solid curves are for redshift-dependent entropy K_IGM(z) as in the top panel of Fig. 2, and dashed curves are for a fixed entropy floor K_IGM= 3eVcm². Note that the baryonic mass function is very similar at all redshifts for K_IGM(z). For T_vir > 10⁴ K, an entropy floor has no effect on the mass function, which reverts back to the dark solid curves (we do not extend the light solid and dashed curves above the corresponding halo masses).

Another quantity of great interest is the global gas clumping factor C=〈n²〉/〈n〉². Gas clumping shortens the recombination time t_rec≈ 1/(αnC) and thus increases the total number of photons per baryon required to achieve reionization. It is given by  

(21)

where f_V is the collapsed fraction by volume of mini-haloes, C_IGM≈ 1 is the clumping factor of the IGM, is the halo clumping factor as calculated in the previous section and shown in the top panel of Fig. 7, and V_halo= (4π/3)r³_vir is the volume of a mini-halo. The factor (t_evap/t_H) deserves some explanation. Mini-haloes only contribute to recombinations when they are photoionized. However, once they are exposed to ionizing radiation, they will be evaporated, essentially in the sound crossing time of photoionized gas (Shapiro, Raga & Mellema 1998; Barkana & Loeb 1999). As in Haiman et al. (2001), we therefore set the evaporation time t_evap=r_vir/(10 km s⁻¹), and weight the halo clumping factor by the duty cycle t_evap/t_H.

The clumping factor in equation (21) is a lower limit to the total gas clumping; there will be an additional contribution from larger scales as well. However, the clumping due to mini-haloes is dominant at the high redshifts of interest.

Our result is shown in the top panel of Fig. 9. For the no entropy floor K_IGM=K_min case (top curve), gas clumping is much lower at high redshift, since fewer mini-haloes have collapsed. Once a patch of IGM is reionized early at high redshift, then an entropy floor is established and subsequent gas clumping is suppressed. Thus, early star formation reduces the total photon budget required to achieve full reionization. We see that gas clumping is reduced by an order of magnitude even for the low entropy levels ∼1 eV cm² associated with reionization at high redshift z∼ 20. For the redshift-dependent entropy K_IGM(z) shown in the top panel of Fig. 3, mini-halo clumping is strongly suppressed, and C≈ 1.

4.2 Suppression of H₂ formation

The formation of H₂ molecules at high redshift have long been thought to be critical to gas cooling and the earliest episodes of star formation (Couchman & Rees 1986). It has also long been shown that H₂ molecules are fragile, and subject to photodissociative feedback from an early UV background in the Lyman–Werner bands of H₂ (Haiman, Rees & Loeb 1997). This negative UV feedback on the H₂ abundance has since been studied by numerous authors (e.g. Ciardi et al. 2000; Haiman et al. 2000; Machacek et al. 2001; Ricotti et al. 2001, 2002a, b).

An important point is that H₂ formation and cooling in mini-haloes will be further suppressed relative to the no-pre-heating case. This is because the finite entropy of the gas allows it to resist compression; thus, the gas is at considerably lower density. Collisional processes with cooling times t_cool∝ 1/n will be suppressed relative to radiative processes that photodissociate H₂. It has been recognized that gas profiles should have a central core rather than a cusp (e.g. Shapiro, Iliev & Raga 1999) and that lower central densities will affect H₂ chemistry (Haiman, Rees & Loeb 1996; Tegmark et al. 1997). However, the fact that pre-heating can produce a much larger, lower-density core than hitherto considered, and the subsequent implications for H₂ formation and cooling, has not been explored.

It has been shown that H₂ formation is enhanced in fossil H ii regions due to increased electron fraction there (Ricotti et al. 2001). Although H₂ formation is very slow in the low-density IGM, a high initial electron abundance will persist when weakly ionized gas is accreted on to non-linear structures, and aid H₂ formation at that point (Oh 2000). The relevant question is whether the gas can reach sufficiently high densities so that the two-body H₂ formation and cooling processes (catalysed by the enhanced electron fraction) dominate over H₂ photodissociation. Ricotti et al. (2002b) have indeed found that a net positive feedback occurs only inside fluid elements that are already dense when they are photoionized. Here we start from gas that is near the IGM density, and ask whether this gas can achieve sufficiently high densities (by contracting on small scales, below the resolution of cosmological simulations) for a similar net positive effect on the H₂ abundance. We can conservatively take into account the H₂ enhancement by assuming the maximum initial abundance of H₂, x ∼ 10⁻³; this hard upper limit is independent of density or ionization fraction and is due to ‘freeze-out’ (Oh & Haiman 2002 hereafter OH02). The H₂ abundance can only exceed this if three-body processes are important, which only takes place at very high density n > 10⁸ cm⁻³.

In OH02, we showed that the minimum temperature T_min a parcel of gas can cool to depends almost exclusively on t_cool/t_diss∝J_UV/n (see fig. 6 of OH02). This scaling behaviour is only broken when the gas approaches high densities n > 10⁴ cm⁻³ (at this point, the cooling time becomes independent of density and t_cool/t_diss∝J_UV). However, the latter regime is never reached in pre-heated gas, which is at much lower densities (n∼ 10⁴ cm⁻³ corresponds to δ∼ 6 × 10⁶ at z= 19). In subsequent discussion, we shall assume the dependence of T_min on J_UV/n shown in fig. 6 of OH02.

Thus, in the presence of a radiation field J_UV, the gas must be at a minimum density n_crit to cool down to a temperature T_min. We have already calculated the maximum central density n_c of gas in a halo given an entropy parameter , as in Fig. 7. If the central density is less than the critical density, n_c < n_crit, then none of the gas in the halo can cool down to T_min. For a given entropy parameter , there is therefore a minimum radiation field J_UV above which no gas can cool down to T_min. We plot this in the top panel of Fig. 10. This plot is valid for all haloes at all redshifts provided and J_UV are both rescaled appropriately [note that since n∝ (1 +z)³, to keep J_UV/n constant, J_UV∝ (1 +z)³]. To get a sense of typical values of J_UV, the radiation field corresponding to n_γ ionizing photons per baryon in the Universe is J_UV≈ (h_Pc/4π)n_γn_b(1 +z)³/10⁻²¹ erg s⁻¹ cm⁻² Hz⁻¹ sr⁻¹≈ 10n_γ[(1 +z)/16]³ (where n_b is the comoving baryon number density and h_P is the Planck constant).

Figure 10.

Top panel: the critical radiation field J_UV required to suppress gas cooling below some temperature T, as a function of entropy parameter . An entropy floor greatly reduces the radiation field required to photodissociate H₂, by several orders of magnitude. Note that J_UV∝ (1 +z)³; the curves shown are for a halo at z= 14. Bottom panel: the fraction of gas in a halo able to cool below temperature T, for J_UV= 10⁻³ (lines), and J_UV= 10⁻¹ (dots connected by lines), at z= 14. Beyond a critical value of , the cooled gas fraction plummets dramatically.

We see that an entropy floor greatly reduces the radiation field required to prevent gas cooling. For reasonable values of , the reduction can be as much as four orders of magnitude. Thus, even if there is only a very weak radiation field, in the presence of an entropy floor effective H₂ cooling and star formation will be quenched. Note also that the cooling time exceeds the Hubble time for large values of . This regime is reached only when gas densities are so low that even a very weak radiation field can dissociate H₂: t_cool∼t_diss∼ 2 × 10⁸ (J_UV/10⁻⁴)⁻¹ yr.

Without an entropy floor, the density profiles of mini-haloes at a given redshift are self-similar; thus, roughly comparable fractions of their gas can cool down to T_min and will be available as fuel for star formation. In the presence of an entropy floor, the fraction of gas with n > n_crit is greatly reduced, and much less gas can cool. Moreover, the self-similarity is broken: shallower potential wells (which have higher ) are more strongly affected by an entropy floor, and their central gas densities are much lower. Using our halo density profiles, we can calculate the fraction of gas above the critical density, ƒ_gas=M_b(>n_crit)/[(Ω_b/Ω_m)M_halo]. We plot this in the lower panel of Fig. 10. At some critical value of , the fraction of gas that can cool plummets dramatically. This value of corresponds roughly to when the low-density core falls below the critical density.

Note that we assume static rather than evolving density profiles. In reality, as gas cools, densities should increase as the gas loses pressure support, making the gas more resistant to H₂ photodissociation (Machacek et al. 2001). However, the dynamical time t_dyn≈ 10⁸(n/1 cm⁻³)^−1/2 yr is typically much longer than the photodissociation time t_diss≈ 2 × 10⁴J⁻¹_UVƒ⁻¹_shield yr (where the factor ƒ_shield takes into account self-shielding; see the discussion in Section 4.2.3 of OH02 and references therein). So our use of the initial density profile to estimate time-scales should be approximately valid.

We can also compute the critical radiation field required to suppress H₂ cooling down to some temperature T as a function of redshift. We plot this in Fig. 11 for a T_vir= 8000 K halo; such a halo represents an upper mass limit to the mini-halo population. Smaller haloes, which have lower central densities for a given entropy floor, will require a weaker radiation field for H₂ cooling to be suppressed. This is shown for four different entropy floors: K_IGM(z) as shown in Fig. 2, no pre-heating K_IGM=K_min, and fixed entropy floors K_IGM= 1, 10 eV cm². Gas cooling in fossil H₂ regions is very easily suppressed; the radiation field required is two to four orders of magnitude lower than required in the no-pre-heating case.

Figure 11.

The critical radiation field J_UV required to suppress gas cooling below T= 1000 K, as a function of redshift z. The curves are shown for a T_vir= 8000 K halo, which is an upper limit in mass to the mini-halo population: smaller mini-haloes will require a weaker radiation field for cooling to be suppressed. The curves are shown for four different entropy floors: no pre-heating K_IGM=K_min, an evolving entropy K_IGM(z) as shown in Fig. 2, and fixed entropy floors K_IGM= 1, 10 eV cm². Points are also shown for 2σ and 3σ haloes at each redshift, assuming K_IGM(z) (the points end when T_vir < 1000 and >8000 K, respectively). Typical entropy floors at each redshift reduce the critical radiation field required for H₂ suppression by two to four orders of magnitude, compared with the no-pre-heating case.

In a recent high-resolution simulation, Yoshida et al. (2003) quantified the effects of the feedback (not taking pre-heating into account), and found that UV photodissociation is effective in suppressing cooling in small haloes if the Lyman–Werner flux is above J₂₁ > 10⁻², but is ineffective in larger haloes that can self-shield against the LW photons. Our new results, which take an entropy floor into account, indicate that a LW radiation field which is two orders of magnitude lower that this can suppress H₂ formation inside fossil H ii regions. Gas cannot cool even in larger haloes close to T_vir= 10⁴ K, in contrast with models where the effects of an entropy floor are neglected.

One possible caveat is that since fossils are likely to be metal-enriched, gas cooling is dominated by metal line cooling, rather than H₂ cooling. We argue that this is unlikely, for two reasons. (i) Metal line cooling only dominates H₂ cooling above a critical metallicity Z > 10⁻³Z_⊙ (Hellsten & Lin 1997; Bromm et al. 2001a). This level of metal enrichment corresponds to ∼1ƒ⁻¹_Z ionizing photon per baryon in the Universe (where f_Z is the volume filling factor of metals). Thus, metal line cooling becomes significant only at late times. (ii) After an ionizing source turns off and explodes as a supernova, the metal-polluted region is much smaller than the fossil H ii region (Madau, Ferrara & Rees 2001). Most of the volume is therefore still of pristine composition, and undergoes the entropy floor suppression we have described. The metal-polluted region lies close to the high-density peak where the very first stars formed, where in any case, more massive haloes T_vir > 10⁴ K will collapse. To summarize: metal line cooling is therefore unlikely to spoil our assumption of adiabaticity in most of the volume of the fossil H ii region. The exception is at the most highly biased density peaks, where metals can be thought of as effectively increasing the star formation efficiency.

The net result is that entropy injection greatly boosts the negative feedback from early star formation: the entropy floor in reionized regions results in low-density cores in the centre of haloes, in which H₂ is easily photodissociated by a weak external UV radiation field.

4.3 Negative feedback from X-rays

It is often argued that X-rays boost H₂ production and cooling in mini-haloes, by penetrating deep into the dense core and increasing the free-electron fraction, which is critical for gas phase H₂ production. Thus, X-rays are thought to exert a positive feedback effect, counter-acting photodissociation by UV radiation (Haiman et al. 1996, 2000). We show here by fairly general arguments that X-rays, in fact, exert a strong negative feedback effect, due to the entropy that they inject into the IGM. This prevents gas from compressing to sufficiently high density to produce H₂ and cool efficiently, and far outweighs any positive feedback from increasing the free-electron fraction.

For the sake of definiteness, we consider an intrinsic source spectrum of the form  

(22)

where hν_L= 13.6 eV and J_UV/J_X=ƒ_break= 1–100 is the spectral break at the hydrogen Lyman edge; note that ƒ_break≥ 1 always. A spectrum of the form J_ν∝ν⁻¹ is characteristic of the mean spectra of quasars (Elvis et al. 1994), as well as inverse Compton emission from high-redshift SNe (Oh 2001). It is distinctive in that νJ_ν= constant, i.e. there is equal power per logarithmic interval (though our arguments can be generalized to other spectra). Note that the actual ionizing spectrum at any given point in the IGM will be much harder, since photoelectric absorption hardens the spectrum away from the source.

What is the heating associated with this radiation field? As we argued in Section 2.2 above, all energy in the X-ray radiation field with E_L < E < E_thick (as defined in Section 5) will be absorbed by the IGM, since the Universe is optically thick at these frequencies. The radiation field may be subdivided into two components. The ‘mean field’ consists of photons with E > E_overlap, where E_overlap is defined by  

(23)

i.e, photons with E > E_overlap (typically, E_overlap∼ 100 eV) have a mean free path greater than the mean separation between sources, so that a homogeneous ionizing background is established. The ‘fluctuating field’ consists of photons with E < E_overlap; this component is dominated by radiation from a single source, and is subject to large Poisson fluctuations. We are interested in the heating due to the ‘mean field’, which can lead to an entropy floor even outside the H ii regions of ionizing sources. It is  

(24)

where ƒ_heat is the fraction of energy of the hot photoelectron created by an X-ray which goes into heating. We have conservatively set ε_ν≈ (4πJ_ν)/l_H; this underestimates ε_ν for a given J_ν. For the spectrum we have chosen, the result is only logarithmically sensitive to the integration bounds; we set ln(ν_thick/ν_thin) ≈ 2.3. For steeper spectra, J_ν∝ν^−α where α > 1, the lower integration limit becomes important.

Over a Hubble time t_H, the internal energy density of the gas becomes  

(25)

This make sense: since some fraction ƒ_heat of the radiation field goes directly into heat, ƒ_heat4π(νJ_ν) ≈Uc. Thus, X-rays will heat the IGM to a temperature  

(26)

There is therefore a direct relation between the X-ray radiation field and the entropy floor:  

(27)

As we argued in Section 2.2 above, the IGM evolves adiabatically for x_e < 0.1[(1 +z)/15]^5/2. What level of the radiation field does this correspond to? For x_e≤ 0.1, approximately of the injected energy goes toward performing ionizations; the number of photoionizations is simply E_photon/(37 eV). Thus, we obtain  

(28)

and so for J_X < 5, the IGM evolves adiabatically [note also that since t_rec≈ 10t_H(x_e/10⁻³)⁻¹(T/100 K)^−0.7[(1 +z)/15]^−3/2, we can ignore recombinations].

Given the entropy floor from equation (27), we can now calculate density profiles of haloes. In particular, we can calculate the maximum central density of a halo from the entropy floor induced by a given X-ray background. As before, from the central density of a halo we can calculate the critical level of the LW flux J_UV required to suppress H₂ cooling down to some temperature T_min. This critical flux is shown as a function of ƒ_X=J_X/J_UV in Fig. 12, for a halo of T_vir= 5000 K at z= 19. The curves may be simply rescaled for haloes of different mass at different redshifts by translating the x-axis: ƒ_X∝T_vir(1 +z)⁵. The effect of X-rays increases for less massive haloes (which are less able to withstand an entropy floor) at lower redshifts (when gas densities are lower). Also shown are the maximum value of the X-ray background ƒ_X≈ 1 for the hardest sources, and the typical minimum value of the X-ray background required for positive feedback to be effective ƒ_X≈ 0.1 (Haiman et al. 2000). The shaded region therefore denotes the X-ray fluxes thought to produce positive feedback effects. We see for these large values of the X-ray background, X-rays cannot exert a positive feedback effect. The required X-ray background is so large that it would produce an unacceptably large entropy floor, which prevents the formation of dense regions where H₂ can withstand photodissociation. Instead, X-rays produce an overall negative feedback effect. In fact, values of ƒ_X≈ 0.1–1 reduce by almost two orders of magnitude the value of the UV background required to photodissociate H₂ and prevent it from cooling down to low temperatures.

Figure 12.

The UV background J_UV required to suppress cooling down to some temperature T_min, as a function of ƒ_X=J_X/J_UV. The curves plotted are for a T_vir= 5000 K halo at z= 19; the results may be rescaled to other haloes by rescaling the x-axis: ƒ_X∝T_vir(1 +z)⁵. Also shown as dashed and dotted lines are the maximum value of ƒ_X≈ 1 in the hardest sources, and the minimum typical value of ƒ_X≈ 0.1 required for X-rays to exert a positive feedback effect; the shaded region therefore shows the possible region whereby X-rays were assumed exert a positive feedback effect. Instead, we see that such a strong X-ray background exerts a negative feedback effect. By inducing a low-density core in the halo, an X-ray background in the required zone reduces by one to two orders of magnitude the minimum UV flux required to suppress cooling, compared with the case where ƒ_X→ 0.

The only way of circumventing these difficulties is if X-rays turn on suddenly at some epoch. Haloes which have already collapsed will be unaffected by the entropy floor, but will undergo the usual positive feedback from X-rays. However, the subsequent generation of mini-haloes will be suppressed by the entropy floor.

4.4 21-cm observational signatures

At present, the only observational probes proposed for small-scale structure at high redshift such as mini-haloes are 21-cm observations with future radio telescopes such as the Square Kilometer Array (SKA)² and the Low Frequency Array (LOFAR).³ It has been proposed that mini-haloes could be detected statistically in emission (Iliev et al. 2002a, b) and individually in absorption along the line of sight to a high-redshift radio source (Furlanetto & Loeb 2002). If an entropy floor exists, such signatures will be strongly suppressed. 21-cm signatures of mini-haloes therefore provide an interesting indirect probe of reionization history: if mini-haloes are detected in large numbers at a given redshift, it would imply that the filling factor of fossil H ii regions is still small at that epoch. In particular, a comoving patch where mini-haloes are seen has likely never been ionized before (which is a much stronger constraint than merely being neutral at the observed epoch). This could prove a very useful probe of the topology of reionization.

Mini-haloes are too faint to be seen individually in emission, and can only be detected statistically through brightness temperature fluctuations. Provided T_S≫T_CMB (where T_S is the spin temperature), the 21-cm flux S is independent of the spin temperature, and depends only on the H i mass, S∝M_{H i}. Thus, S(haloes)/S(IGM) is simply equal to the collapsed gas mass fraction in mini-haloes, which is always less than unity. The mini-halo signal is much smaller if an entropy floor exists, since the collapsed gas fraction declines rapidly: the bottom panel of Fig. 9 can simply be read off as S(haloes)/S(IGM). Thus, the IGM dominates 21-cm emission. Mini-haloes dominate only when: (i) the IGM spin temperature has not yet decoupled from the CMB {the critical thermalization radiation field J₂₁ such that T_S→T_α≈T_K, where T_α is the colour temperature of the radiation field, is J₂₁≈ 5[(1 +z)/10] (Madau, Meiskin & Rees 1997)} (ii) the filling factor of fossil H ii regions is small. The high WMAP optical depth, which implies significant reionization at high redshift, pushes the latter constraint to high redshift and thus low radio frequencies v_obs= 93[(1 +z)/15] MHz, making this a very challenging observation.

We now turn to absorption signatures. How does the mini-halo 21-cm absorption signature scale with the IGM entropy? The IGM 21-cm optical depth is  

(29)

See Madau et al. (1997) for expressions for the spin temperature T_S(J₂₁, n_{H i}, T_K; we do not reproduce them here. We use a fit to the results of Allison & Dalgarno (1969) for the collisional coupling coefficient C₁₀ for T_K < 1000 K, with a C₁₀∝T^−0.33_K extrapolation for higher temperatures. The mini-halo 21-cm optical depth along an impact parameter α to the halo centre is (Furlanetto & Loeb 2002)  

(30)

where A₁₀= 2.85 × 10⁻¹⁵ s⁻¹ is the spontaneous emission coefficient, b(r)²= 2k_BT_K/m_p is the gas Doppler parameter, r²=R²+α², and v(ν) =c(ν−ν₀)/ν₀. Unlike Furlanetto & Loeb (2002), we do not attempt to model the velocity field of the infall region, which we ignore; the total observed optical depth is taken to be τ_tot=τ_halo+τ_IGM. The velocity field in the pre-heated gas will differ significantly from the Bertschinger self-similar solution they assume, since gas pressure retards accretion; the infall region optical depth will be smaller because of the reduced gas column. Since even in their case the contribution of the infall region to 21-cm absorption is small (see their fig. 2, where the equivalent width drops drastically for α > r_shock), our neglect is justified.

In Fig. 13, we show the optical depth as a function of observed frequency, for different levels of the entropy parameter . The values of shown correspond to IGM temperatures of T_IGM= 70, 700, 7000 K; since fossil H ii regions at z= 10 cannot significantly cool below 7000 K (see Fig. 1), the value of is most appropriate. We see that the optical depth contribution from mini-haloes falls drastically with increasing . Also shown is the optical depth for J₂₁= 10. The radiation field drives T_S→T_K, reducing the 21-cm optical depth τ∝T⁻¹_S.

Figure 13.

Mini-halo optical depth profiles as a function of observed frequency for a 2 × 10⁶ M_⊙h⁻¹ halo at z= 10, all for impact parameters of α= 0.3r_vir. The values of shown correspond to IGM temperatures of T_IGM= 70, 700, 7000 K; since fossil H ii regions at z= 10 cannot significantly cool below 7000 K, the value of is most appropriate. Solid lines correspond to J₂₁= 0; dashed lines correspond to J₂₁= 10. The entropy floor makes the mini-halo contribution to the 21-cm absorption undetectable.

Since τ_IGM also depends on , the most relevant quantity for detecting a mini-halo is the observed equivalent width:  

(31)

which measures the 21-cm absorption due to the mini-halo in excess of that due to the IGM. It falls rapidly with and is extremely small for the most likely value of . Figs 13 and 14 can be compared with figs 1 and 2 of Furlanetto & Loeb (2002) (although note that in Fig. 13, we plot τ as a function of observed rather than intrinsic frequency). An entropy floor greatly reduces the cross-sectional area over which an observable absorption signal may be detected. If one detects significant mini-halo absorption over a large comoving patch along the line of sight to a radio source, one can place an upper limit on the entropy floor there. This in turn places an upper limit on the high-redshift X-ray background (from equation 8), and a lower limit on the redshift at which that patch was first reionized (since fossils from higher redshift have lower entropy, as in Fig. 2).

Figure 14.

The observed 21-cm absorption equivalent width of a 2 × 10⁶ M_⊙h⁻¹ halo at z= 10, for J₂₁= 0, 1, 10, as a function of . Solid lines are for an impact parameter of α= 0.3r_vir, while dashed lines are for α= 0.7r_vir. The equivalent width falls rapidly as the entropy floor rises; for the expected value of at this redshift, equivalent widths are very small.

The chief driver of high-redshift 21-cm proposals has of course been to observe the IGM itself in 21-cm emission (Madau et al. 1997; Tozzi et al. 2000; Ciardi & Madau 2003). We note that if the filling factor of fossil H ii regions is large, conditions are very favourable for such observations. Provided T_S→T_K in the IGM, the only condition for the IGM to be seen in 21-cm emission against the CMB is for T_K≫T_CMB(z_obs). This is certainly satisfied in fossil H ii regions. It was previously thought that recoil heating from Lyα photons could heat neutral regions (Madau et al. 1997), but a detailed calculation (Chen & Miralda-Escude 2003) shows the heating to be insufficient. In that case, a high-redshift X-ray background would be required to heat neutral regions. Such concerns are moot if a period of early reionization took place (as seems to be indicated by WMAP observations), followed by recombination: large tracts of warm, largely neutral gas would exist. Note, however, that brightness temperature fluctuations can only be detected if the contribution from unresolved radio point sources (Tozzi et al. 2000; Oh & Mack 2003) can be successfully removed by using spectral structure in frequency space.

5 Effects on Global Reionization Scenarios

5.1 General considerations

Next we address the effect of the suppression of mini-halo formation on the global reionization history. The importance of the effect depends on: (i) the synchronization of the formation of mini-haloes t_sync relative to the typical lifetime t_ion of the ionizing source, which determines the fraction of mini-haloes subjected to feedback and (ii) on the recombination time t_rec relative to t_sync, which determines how long the ionized regions last and thus the fraction of active (as opposed to fossil) ionized regions at any given time.

In the limit of t_sync≪t_ion, the fossil H ii regions would appear only after all the mini-haloes had already formed, and hence the feedback would have no effect on the total amount of reionization by mini-haloes. In the absence of any other feedback effects, the IGM could then, in principle, be fully reionized by mini-haloes, given a high enough ionizing photon production efficiency (although would subsequently recombine).

In the opposite limit, t_sync≫t_ion, the contribution of mini-haloes to reionization will be strongly suppressed. We can obtain a rough estimate for the maximum fraction of the IGM that can be ionized by the mini-haloes. During the lifetime of its resident ionizing source, each mini-halo produces an ionized volume V_{H ii}, which will correspond to the total number N_γ of ionizing photons injected into the IGM, i.e. , where is the mean hydrogen density. To a good approximation, recombinations can be ignored in this phase, since massive stars have lifetimes shorter than the recombination time, t_rec≈ 3 × 10⁷[C(1 +z)/25]⁻³ yr (see Fig. 15). On the other hand, the recombination time is shorter than the Hubble time, t_rec/t_hub≈[C(1 +z)/11]^−3/2, so in general, at high redshift z > 10, and for clumping factors C > 1, the ionized volume will recombine once its driving source turns off.

Figure 15.

The evolution of comoving radius of the ionized region around a single 10³ M_⊙ metal-free star that turns on at z= 25, and is assumed to emit 40 000 ionizing photons in ∼3 × 10⁶ yr. The upper and lower solid curves show solutions that include recombinations with C= 1 and 10. The ionized volume shrinks [or equivalently, the ionized fraction is decreased by the factor (R_S/R_S,max)³] after the source turns off at z∼ 24. The corresponding dashed curves show the radii of the fossil H ii regions; these are assumed to stall at the maximum radius of the H ii sphere and stay constant thereafter.

As argued above, for the purpose of the mini-halo suppression, we may nevertheless simply imagine that the H ii region never recombines. Ignoring recombinations inside the ionized region, this is equivalent to setting the size of the fossil H ii region to be the maximum (comoving) size of the active H ii region, reached at the time when the ionizing source turns off.

The evolutions of the radii of the active and fossil H ii regions are illustrated in Fig. 15 for a single 10³ M_⊙ metal-free star that turns on at z= 25. We follow the expansion of the ionization front R_S by solving the standard differential equation, taking into account ionizations, recombinations and the Hubble expansion (e.g. Shapiro & Giroux 1987; Cen & Haiman 2000). We assume a constant clumping factor of C= 1 (upper solid curve) or C= 10 (lower solid curve). We assume further that the metal-free star emits 40 000 ionizing photons per stellar proton at a constant rate for ∼3 × 10⁶ yr (Tumlinson & Shull 2000; Bromm et al. 2001b; Schaerer 2002). Once the ionizing source turns off, the solid curves show the formal solution for the evolution of the ionization front.⁴ The dashed curves show the size of the fossil H ii region.

Next we can consider the evolution of the volume filling factor of the fossil H ii regions from the ensemble of mini-haloes, and define the epoch z_f when the filling factor of these fossils reaches unity. As argued above, no new mini-haloes can form at z < z_f. At this epoch, the global ionized fraction will be smaller than unity, because each fossil H ii region has already partially recombined. Thus the global ionized fraction will be , assuming that the sources typically turned off a time t_sync ago, and is the average recombination time over the interval t_sync. For example, assuming that the fossil H ii regions overlap at z_f= 20, with the typical sources born at z= 25, and the recombination time evaluated at z= 22.5 (with C= 1), we find , and a maximal ionized fraction of ∼30 per cent.

5.2 Models for the reionization history

To refine the above considerations, next we compute the evolution of the global ionized fraction using a semi-analytical model adopted from Haiman & Holder (2003, hereafter HH03). For technical details, the reader is referred to that paper. Here we only briefly summarize the main ingredients of the model, and describe the modifications we made to include the suppression of the formation of new mini-haloes inside fossil H ii regions. Feedback in actively ionized regions has already been considered in previous reionization models (Haiman & Loeb 1997, 1998; Haiman & Holder 2003; Wyithe & Loeb 2003). The main purpose of the new calculations presented here is to demonstrate that adding the additional feedback in the fossil H ii regions has an important further impact on the reionization history.

In this model, we follow the volume filling fraction F_{H ii} of ionized regions, assuming that discrete ionized Strömgren spheres are being driven into the IGM by ionizing sources located in dark matter haloes. For simplicity, we ignore helium in this work. The dark matter halo mass function is adopted from Jenkins et al. (2001). We consider two distinct types of ionizing sources, located in haloes with virial temperatures of 10²≲T_vir≲ 10⁴ K (mini-haloes) or T_vir≳ 10⁴ K (large haloes). We assume that in mini-haloes, a fraction ƒ_*= 0.005 of the baryons turn into metal-free stars (Bromm, Coppi & Larson 1999; Abel et al. 2000; Abel, Bryan & Norman 2002; Bromm et al. 2002), which produce N_γ= 40 000 ionizing photons per baryon, and ƒ_esc= 100 per cent of these photons escape into the IGM. The product ε≡ƒ_*ƒ_escN_γ= 200 determines the size of the ionized regions around mini-haloes. For large haloes, we adopt ƒ_*= 0.1, ƒ_esc= 20 per cent and N_γ= 4000, for a total efficiency ε= 80. In contrast with HH03, here we do not distinguish haloes with 10⁴ < T_vir < 2 × 10⁵ K from T_vir > 2 × 10⁵ K (see Dijkstra et al. 2003 for the reason why this distinction is probably unimportant).

As in HH03, we then follow the evolution of the global ionized fraction F_{H ii}(z) by summing the ionized volumes surrounding the individual dark matter haloes. We exclude the formation of mini-haloes in actively ionized regions (but assume larger haloes are impervious to photoionization feedback). This is accomplished by multiplying the rate of formation of new mini-haloes by the suppression factor 1 −F_{H ii}, as in equation (8) in Haiman & Holder (2003) (see also equation 11 in Wyithe & Loeb 2003 and section 3 in Haiman & Loeb 1997 for similar treatments of the actively ionized zones). The evolution of F_{H ii} in this model is shown by the upper thin solid curve in Fig. 16 (with C= 1 in the upper panel, and C= 10 in the lower panel). As the figure shows, in the absence of other feedback effects (see the discussion in Haiman et al. 2000; Haiman 2003 for other feedback effects), the mini-haloes could ionize the IGM in full by z∼ 14(C= 1), or nearly fully by z∼ 10 (if clumping is assumed to be more significant, C= 10).

Next we compute F_{H ii} in the same model, except that we now also exclude the formation of new mini-haloes in fossil H ii regions. This is easily accomplished in practice: the suppression factor (1−F_{H ii}) for the formation rate of mini-haloes is replaced by a factor (1−F′_{H ii}). Here the fossil filling factor F′_{H ii} is computed the same way as the ionized fraction F_{H ii}, except that the individual ionized regions are assumed to follow the dashed curves from Fig. 15 rather than the solid curves. In Fig. 16, we show the evolution of the volume filling factor F′_{H ii} of the fossil H ii regions (dashed curves), and the reionization history F_{H ii}(z) as the thick solid curves. Finally, for reference, we show F_{H ii}(z) with mini-haloes completely excluded (ε= 0) as the lower thin solid curves.

As apparent from Fig. 15, the exclusion of new mini-halo formation from the fossil H ii regions causes a significant suppression of the total ionized fraction that can be reached by mini-haloes. Under the rather optimistic set of assumptions described by the thick solid curve in the upper panel, the maximum ionized fraction that can be reached is ∼40 per cent. In reality, clumping is unlikely to be unity in the immediate vicinity (≲100 kpc) of the ionizing sources (Haiman, Abel & Madau 2001).

The total electron scattering optical depth attributable to mini-haloes (the appropriately weighted integral between the thick solid curve and the lower thin solid curve) is τ= 0.07 and 0.014 in the C= 1 and 10 cases, respectively. This makes it unlikely that mini-haloes can fully account for the large optical depth τ= 0.17 measured by WMAP. Note the thick curve in the upper panel of Fig. 16 has a total τ= 0.2 (τ= 0.07 attributable to mini-haloes, and τ= 0.13 to larger haloes), so that it is consistent with the WMAP measurement. Note that raising the efficiencies in mini-haloes would not increase the optical depth attributed to mini-haloes, since feedback then would set in earlier (we have explicitly verified that τ is approximately independent of efficiencies over a range of multiplicative factors 0.1–10 for ε). Finally, note that the suppression considered here and shown in Fig. 16 provides a negative feedback in addition to the negative feedback expected from H₂ photodissociation (Haiman et al. 1997, 2000). In fact, as discussed above, pre-heating amplifies the effect of the H₂-photodissociative negative feedback.

Because an entropy floor essentially eliminates gas clumping due to mini-haloes (as shown in Section 4.1), the reionization history is likely to more closely approximate the C= 1 case than the C= 10 case until low redshifts z < 10 (when gas clumping due to larger structures predominates). An entropy floor therefore has two counter-veiling effects on reionization: by suppressing star formation in mini-haloes, it reduces the comoving emissivity. However, by reducing gas clumping, it also reduces the photon budget required for reionization. It is interesting to note that the evolution of the filling factor is non-monotonic for the C= 1 case: feedback due to early reionization naturally produces a bump in the comoving emissivity, and a pause ensues before larger haloes (which can resist feedback) collapse. This is similar to the reionization histories derived by Cen (2003) and Wyithe & Loeb (2003), but is regulated by feedback from the entropy floor rather than a Population III to II transition due to a universal metallicity increase.

6 Conclusions

In this paper, motivated by the WMAP results, we have considered the feedback effect of early reionization/pre-heating on structure formation. This feedback effect is inevitable in any reionization scenario in which star formation throughout the Universe is not completely synchronized. Our principal conclusions are as follows.

• (i)

  Fossil H ii regions have a residual entropy floor after recombination and Compton cooling that is higher than the shock entropy for mini-haloes (T_vir < 10⁴ K); thus, such haloes accrete gas isentropically. The IGM entropy depends primarily on the redshift and only weakly on the overdensity δ; it is thus largely independent of the details of structure formation. For this reason, and also because it is conserved during adiabatic accretion or Hubble expansion, the gas entropy is a more fundamental variable to track than the temperature. We provide a simple analytic formula for the temperature (and hence entropy K=T/n^2/3), in equations (33) and (35). An early X-ray background would also heat the entire IGM to similarly high adiabats. The entropy floor due to the latter would be much more spatially uniform.

• (ii)

  We apply the entropy formalism used to calculate the effect of pre-heating on low-redshift galaxy clusters to the high-redshift mini-haloes. We obtain detailed gas density and pressure profiles, which we use to calculate the impact of pre-heating on the central density, accreted gas fraction, gas clumping factor and mini-halo baryonic mass function.

• (iii)

  These quantities can then be used to calculate global effects of pre-heating. The collapsed gas fraction in mini-haloes falls by approximately two orders of magnitude, while the gas clumping factor falls to C→ 1, as for a uniform IGM. An entropy floor reduces the photon budget required for full reionization by approximately an order of magnitude, by reducing gas clumping and eliminating the need for mini-haloes to be photoevaporated before reionization can be completed.

• (iv)

  However, an entropy floor does not necessarily promote early reionization: it also sharply reduces the comoving emissivity. By reducing the central gas densities in mini-haloes, pre-heating impedes H₂ formation and cooling, and reduces the critical UV background required for H₂ suppression by two to four orders of magnitude. Thus, once a comoving patch of the IGM is reionized, no subsequent star formation in mini-haloes can take place in that volume. The patch can only be reionized by more massive haloes T > 10⁴ K, which can undergo atomic cooling. By furnishing an entropy floor, X-rays also suppress H₂ formation. Thus, contrary to conventional wisdom, X-rays provide negative rather than positive feedback for early star formation. We note, however, the results of Ricotti et al. (2002b), who showed that gas in dense parts of filaments and in the interiors of galactic haloes can experience an enhancement of their H₂ abundance, if they are engulfed by an H ii front and the flux then turns off.

• (v)

  Mini-haloes will not be observable in 21-cm emission/absorption in fossil H ii regions. Thus, 21-cm observations provide an unusual probe of the topology of reionization: mini-haloes trace out regions of the IGM that have never been ionized. If mini-haloes are seen in large numbers, this places an upper limit on the filling factor of fossil H ii regions and the X-ray background at that redshift.

• (vi)

  We have computed the reionization histories as in HH03, but taking the feedback effect of an entropy floor (and reduction of gas clumping) into account. This is different from previous models of the reionization history that take into account the feedback only in actively ionized regions of the IGM. The strong additional feedback in fossil H ii regions imply that H ii fronts at high redshift never overlap, and global reionization at high redshift does not occur. This limits the contribution of mini-haloes to the reionization optical depth τ < 0.07, almost independent of star formation efficiency in mini-haloes (if star formation is more efficient, feedback sets in earlier). Thus, the bulk of the optical depth observed by WMAP must come from more massive objects.

• (vii)

  Strikingly, we obtain a double-peaked reionization history: an early peak in which the Universe is filled with fossil H ii regions, followed by a pause before more massive haloes collapse which finally fully reionize the Universe. This is similar to ‘double reionization’ scenarios computed by other authors (e.g. Cen 2003; Wyithe & Loeb 2003), but one in which the comoving emissivity is regulated by gas entropy, rather than a Population III to II transition due to a universal metallicity increase. This last point deserves additional comment. A metallicity-regulated evolution of the emissivity requires that metal pollution is fairly spatially uniform. However, different parts of the IGM likely undergo the Population III to II transition at different epochs. Furthermore, an increase in metallicity does not necessarily result in a drop in the overall emissivity: metal line cooling probably results in greater star formation efficiency in haloes, since metals are not subject to internal UV photodissociation, unlike H₂. The factor of ∼10 drop in the H i ionizing emissivity per stellar baryon could be outweighed by the increase in the total mass of stars formed. In comparison, we argue that an entropy-regulated transition is inevitable, and therefore more robust. This is an important conclusion, given the possibility that future CMB polarization studies will be able to distinguish among different reionization histories (HH03; Holder et al. 2003).

In this semi-analytic study, we essentially assumed a single value of the entropy floor at each redshift, for all haloes. In reality, as we discussed above, there should be large fluctuations in entropy, depending on the topology/history of reionization and the accretion/merger history of haloes. It would be interesting to study these effects in detail in three-dimensional numerical simulations. While it would be challenging to resolve the dynamics of gas in the cores of mini-haloes in large cosmological simulations, the problem appears well-suited to high-resolution simulations that zoom in to study the behaviour of a single collapsing halo (Bromm et al. 1999; Abel et al. 2000). It would be interesting to re-run existing simulations of the collapse of gas in mini-haloes, but adding an entropy floor in the initial conditions. Such studies could quantify more precisely the effects of an entropy floor in suppressing H₂ formation and cooling in mini-haloes. In fact, since densities are lower and gas cooling is reduced, it should be a computationally more tractable problem. With the aid of larger volumes, several global issues mentioned above could be addressed: for instance, a global, self-consistent study of feedback as in Machacek et al. (2001), but including the effects of both the entropy floor and UV feedback. Because of its strong effect on the evolution of the comoving emissivity, gas entropy acts as a self-regulating mechanism which probably has a strong influence in controlling the progress of reionization.

¹

For the purposes of this paper, a mini-halo is defined as any halo with T_vir < 10⁴ K that cannot cool via atomic line cooling.

²

See http://www.nfra.nl/skai

³

See http://www.astron.nl/lofar

⁴

This formal solution predicts a shrinking of the ionization front. In reality, once the source turns off, recombinations throughout the ionized region will decrease the ionized fraction uniformly by a factor of (R_S/R_S,max)³, rather than reduce the size of a highly ionized volume. However, for our purposes of computing the global ionized fraction, the two interpretations are equivalent.

Appendix

Appendix: Analytic Expression for the Final Temperature

It is useful to have an approximate analytic expression for the final temperature a parcel of gas cools down to, given the initial temperature T_i, redshift z, overdensity δ and length of time spent cooling t. This allows one to quickly estimate the effect of early reionization in different situations without evolving the full chemistry code. We develop such an expression in this Appendix.

At the redshifts and overdensities of interest, Compton cooling dominates by far. The most relevant physics is Compton cooling and hydrogen recombination:  

(A1)

Note that (T−T_γ) ≈T since T≫T_γ and α∝T^−0.7 in the temperature range of interest. Assuming the gas is fully ionized x_e= 1 at some initial temperature T_i, we obtain the analytic solution:  

(A2)

where  

(A3)

This gives the temperature as a function of the ionization fraction x_e. We therefore need to know the final ionization fraction. How can we estimate it? If the gas recombines isothermally at temperature T′, the ionization fraction is given by  

(A4)

where x₀= 1 is the initial ionization fraction. If the gas cools as it recombines, substituting the instantaneous temperature T′ into the expression would overestimate the speed of recombination and underestimate x_e. We find that if we substitute T′= 2T and substitute this into equation (33), the solution of this non-linear equation for T_f is remarkably close to the full non-equilibrium solution (see points in Fig. 1). We have also verified that we obtain fairly accurate results for x_e(t). Equations (33) and (35) thus give T_f(z, δ, T_i, t).

Our neglect of recombination line cooling fails in high-density regions. This leads to at most a factor of ∼2 error in the final temperature, since recombination line cooling can cool gas down to at most ∼5000 K. The entire discussion assumes that H₂ formation and cooling is not competitive with Compton cooling, which is generally true in low-density regions.