Minihalo photoevaporation during cosmic reionization: evaporation times and photon consumption rates

arX

iv:a

stro

-ph/

0408

408v

1 2

3 A

ug 2

004

Mon. Not. R. Astron. Soc. 000, 000–000 (0000) Printed 2 February 2008 (MN LATEX style file v2.2)

Minihalo photoevaporation during cosmic reionization:

evaporation times and photon consumption rates

Ilian T. Iliev1, Paul R. Shapiro,2 and Alejandro C. Raga31 Canadian Institute for Theoretical Astrophysics, University of Toronto, 60 St. George Street, Toronto, ON M5S 3H8, Canada2 Department of Astronomy, University of Texas, Austin, TX 78712-10833 Instituto de Ciencias Nucleares, Universidad Nacional Autonoma de Mexico (UNAM), Apdo. Postal 70-543, 04510 Mexico,D. F., Mexico

2 February 2008

ABSTRACT

The weak, R-type ionization fronts (I-fronts) which swept across the intergalacticmedium (IGM) during the reionization of the universe often found their paths blockedby cosmological minihaloes (haloes with virial temperatures Tvir 6 104 K). When thishappened, the neutral gas which filled each minihalo was photoevaporated; as theI-front burned its way through the halo, decelerating from R-type to D-type, all thehalo gas was eventually blown back into the IGM as an ionized, supersonic wind. Ina previous paper (Shapiro, Iliev & Raga 2004, hereafter Paper I), we described thisprocess and presented our results of the first simulations of it by numerical gas dy-namics with radiation transport in detail. For illustration we focused on the particularcase of a 107

M⊙ minihalo which is overrun at z = 9 by an intergalactic I-front causedby a distant source of ionizing radiation, for different types of source spectra (eitherstellar from massive Pop. II or III stars, or QSO-like) and a flux level typical of that ex-pected during reionization. In a Cold Dark Matter (CDM) universe, minihaloes formedin abundance before and during reionization and, thus, their photoevaporation is animportant, possibly dominant, feature of reionization, which slowed it down and cost itmany ionizing photons. In view of the importance of minihalo photoevaporation, bothas a feedback mechanism on the minihaloes and as an effect on cosmic reionization,we have now performed a larger set of high-resolution simulations to determine andquantify the dependence of minihalo photoevaporation times and photon consumptionrates on halo mass, redshift, ionizing flux level and spectrum. We use these results toderive simple expressions for the dependence of the evaporation time and photon con-sumption rate on these halo and external flux parameters which can be convenientlyapplied to estimate the effects of minihaloes on the global reionization process in bothsemi-analytical calculations and larger-scale, lower-resolution numerical simulationswhich cannot adequately resolve the minihaloes and their photoevaporation. We findthat the average number of ionizing photons each minihalo atom absorbs during itsphotoevaporation is typically in the range 2-10. For the collapsed fraction in mini-haloes expected during reionization, this can add ≈ 1 photon per total atom to therequirements for completing reionization, potentially doubling the minimum numberof photons required to reionize the universe.

Key words: hydrodynamics—radiative transfer—galaxies: halos—galaxies: high-redshift—intergalactic medium—cosmology: theory

1 INTRODUCTION

The Cold Dark Matter model, due to its hierarchical na-ture, predicts that a significant fraction of the matter in the

c© 0000 RAS

2 I. T. Iliev et al.

universe before and during reionization (up to ∼ 30 − 40%at redshift z = 6 for current ΛCDM) resided in collapsedand virialized minihaloes - small haloes with virial tempera-tures Tvir below 104 K. These minihaloes, in the mass rangebetween the Jeans mass (below which gas pressure forcesprevented baryonic matter in the IGM from collapsing intodark matter haloes as they formed; i.e. MJ <

∼ 104M⊙) andthe mass for which Tvir = 104 K [M(104 K) <

∼ 108M⊙] werefilled with neutral gas, since their temperatures were too lowfor collisional ionization to be effective. Cosmic reionizationhad a profound effect on the gas content of these minihaloesand they, in turn, interfered with cosmic reionization.

This reionization began when the first sources of ioniz-ing radiation turned on, causing global I-fronts to sweep out-ward through the surrounding IGM, creating intergalacticH II regions, and ended when these H II regions became largeenough and numerous enough to overlap (Shapiro & Giroux1987). These global I-fronts were generally weak, R-typefronts, which moved supersonically, not only with respect tothe cold, neutral gas ahead of them but also with respect tothe gas photoheated to T >

∼104K behind them. As such, theyraced ahead of the hydrodynamical response of the ionizedgas, unaffected by the disturbance they created in the gas be-hind them. That situation changed, however, when the pathof one of these global I-fronts was blocked by a minihalo.Since the speed of an I-front varies inversely with the densityof the neutral gas ahead of it, the I-front must have sloweddown when it encountered the dense, centrally-concentratedgas inside a minihalo. The latter is, on average, more than ahundred times denser than the mean IGM, while at its cen-ter it is yet another hundred times denser even than that.This denser gas slowed the I-front down enough to “trap”it, long enough for the gas-dynamical back-reaction of thedisturbed minihalo gas to catch-up to the I-front and affectits progress. Thereafter, the fate of this I-front and that ofthe minihalo gas were inextricably coupled, as ionized mini-halo gas, heated to T > 104K, above the binding energy ofminihalo gravity, blew back toward the ionizing source, intothe IGM as a supersonic, evaporative wind.

The first realistic discussion of the photoevaporationof cosmological minihaloes by global I-fronts during reion-ization, including the first radiation-hydrodynamical sim-ulations of this process, was by Shapiro, Raga & Mellema(1997, 1998), developed further in Shapiro & Raga (2000a,b,2001); Shapiro (2001) and Paper I (see Paper I for amore complete description and additional references.) Asdescribed in Paper I, the dominant sources of reionizingradiation are likely to have been haloes of virial temper-ature Tvir > 104K. Star formation in minihaloes, instead,is thought to have been inhibited early on by the pho-todissociating effect on minihalo H2 molecules of the ris-ing UV background just below the H Lyman continuumedge, created by a small fraction of the sources ultimatelyrequired to complete reionization. If so, then each sourceof reionization would typically have found its sky cov-ered by minihaloes between it and the nearest neighbor-ing source haloes, and the expansion of I-fronts in the

IGM around each source to the point of overlap wouldhave depended upon the photoevaporation of interveningminihaloes (Haiman, Abel & Madau 2001; Shapiro 2001;Barkana & Loeb 2002; Shapiro, Iliev, Raga & Martel 2003;Paper I). Since the gas inside collapsed minihaloes was muchdenser than average, the rate of recombination there wasmuch higher than average, as well. Hence, minihalo pho-toevaporation was likely to have consumed multiple ioniz-ing photons per atom, thereby increasing, and potentiallydominating, the global photon consumption during reion-ization (Haiman, Abel & Madau 2001; Shapiro 2001; PaperI). Therefore, quantifying this process is crucial for deter-mining the onset, progress and duration of the reionizationof the universe.

An early start (z>∼15) and late finish (z ≈ 6) for cosmic

reionization are suggested by recent observations of the fluc-tuating polarization of the CMB ( Kogut et al. 2003) andthe Gunn-Peterson effect in the spectra of quasars at z ≈ 6(Becker et al. 2001, Fan et al. 2003), respectively. This isconsistent with the view that the volume-filling factor of ion-ized regions of the universe grew over time from z >

∼15 untilthey overlapped finally at z ≈ 6. The presence of minihaloesfilled with neutral gas was affected by this reionization, notonly because photoevaporation stripped pre-existing mini-haloes of their baryons, but also because the subsequentcollapse of baryons into newly forming minihaloes was sup-pressed by “Jeans-mass filtering” of the linear density fluctu-ations in the baryonic component after the IGM was heatedto T >

∼ 104K when it was reionized (Shapiro, Giroux, andBabul 1994; Shapiro 1995; Gnedin 2000b). These two pro-cesses acted to reduce the presence of neutral-gas-filled mini-haloes only in the ionized regions, however. The neutral vol-ume outside these regions continued to form new minihaloeswithout interruption, at the unfiltered rate of the universewithout reionization. As a result, the I-fronts which led theexpansion of the ionized regions would have advanced intoneutral gas in which minihaloes formed with ever-increasingabundance over time. The unfiltered collapsed fraction inminihaloes for ΛCDM at z = 20, 15, 9, and 6 was 3, 10,28, and 36 percent, respectively (Paper I). The question ofwhether minihalo photoevaporation was a dominant featureof reionization will ultimately be answered only when weknow the full story of cosmic reionization, since there alsoexist (somewhat more speculative) scenarios in which theformation of some minihaloes might have been suppressedby, for example, a significant X-ray background at high-z(e.g. Machacek, Bryan & Abel 2003, Glover & Brand 2003,Madau et al. 2004, Ricotti & Ostriker 2004), double reioniza-tion (Cen 2003), entropy floor (Oh & Haiman 2003), or re-duced small-scale power in the power spectrum of primordialdensity fluctuations (Somerville, Bullock, & Livio 2003).

In Paper I, we discussed the process of photoevapora-tion of cosmological minihaloes in detail. We demonstratedthe phenomenon of I-front trapping, whereby the highly su-personic, weak, R-type front propagating in the low-densityIGM slows down upon entering the minihalo, where the gasdensity is much higher than that of the IGM, and converts to

c© 0000 RAS, MNRAS 000, 000–000

Photoevaporation of cosmological minihaloes II 3

D-critical and then subcritical D-type, preceded by a shockwhich compresses the gas ahead of the I-front. The I-frontslowly propagates subsonically through the halo, stripping itof gas, layer by layer, as this gas, once ionized, expands intothe IGM as a supersonic wind towards the ionizing source.This process eventually results in a dark matter halo com-pletely devoid of gas. We described there in detail the struc-ture and evolution of the photoevaporative flows and theirdependence on the radiation spectrum of the external ioniz-ing source, considering both massive stars (Pop II and PopIII) and QSO-like spectra. We derived the evolution of theposition and speed of the I-front, the evolution of the halo’sremaining neutral mass fraction and effective opaque geo-metric cross-section for the absorption of ionizing photons,as well as the global parameters of photoevaporation likethe photoevaporation time, tev, and the number of ionizingphotons absorbed per atom in the course of that evapora-tion, ξ, and its evolution. Finally, we also discussed someobservational diagnostics of the photoevaporation process.

In this paper, we extend the results of Paper I by in-vestigating the dependence of the photoevaporation processon the mass of the halo, on the level and spectrum of theexternal photoionizing flux responsible for the I-front whichovertakes the halo, and on the redshift at which this I-frontencounters the minihalo. As discussed in Paper I, we makethe assumption that the H2 molecule formation and coolingwhich might have led to star formation inside minihaloeswere inefficient in the bulk of the minihalo population, sothe minihaloes were largely “sterile” reservoirs of neutralatomic gas. We concentrate particularly on the dependenceof the two properties of minihalo photoevaporation whichare most important for quantifying their global effect oncosmic reionization, namely, the evaporation time tev andthe net ionizing photon consumption rate per minihalo (i.e.the number of ionizing photons absorbed per minihalo atomduring the evaporation time), ξ. In § 2, we shall define thequantities tev and ξ and how we use our simulations to eval-uate them more precisely, and recall some approximationswhich have been used previously to estimate these quan-tities, for comparison. Our numerical simulations are de-scribed briefly in § 3. In § 4.1 and § 4.2, we summarize ournumerical results for tev and ξ, respectively, for haloes ofdifferent masses encountered at different redshifts by the I-fronts driven by external ionizing sources of different fluxlevels and spectra, and compare these with the previous es-timates described in § 2. In § 4.3, we show that the time-dependence of the evolving neutral mass fraction inside aminihalo during its evaporation has a universal shape, in-dependent of halo parameters and external flux level. Theimplications of these results for the theory of cosmic reion-ization are briefly discussed in § 5. Throughout this paper weuse the current concordance model for the background uni-verse and the dark matter - flat, COBE-normalized ΛCDMwith Ω0 = 1 − λ0 = 0.3, h = 0.7 and Ωbh

2 = 0.02, withprimordial density fluctuation power spectrum P (k) ∝ knp ,with np = 1, and σ8h−1 = 0.87 (e.g. Spergel et al. 2003).

2 HALO EVAPORATION TIMES AND

IONIZING PHOTON CONSUMPTION

RATES

We define the evaporation time tev to be the time at whichonly 0.1 per cent of the gas which was initially inside theminihalo remains neutral1. As in Paper I, we then obtainthe number of ionizing photons absorbed per minihalo atomduring this evaporation time by directly counting the num-ber of recombinations experienced by each of those initialhalo atoms and its first ionization:

ξ(t) =Nion

Na+

1

Na

∫ t

0

dt

∫

dV (α(2)H nHII + α

(2)HenHeII)ne, (1)

where ne is the number density of electrons, nHII and nHeII

are the number densities of H II and He II, respectively, whileα

(2)H and α

(2)He are the Case B recombination coefficients for

H II and He II, respectively, Na is the total number of atomsinitially inside the minihalo, and Nion is the number of theseatoms initially inside the minihalo which are ionized by theevaporation process. At time t = tev, practically all atomsoriginally in the minihalo are ionized, thus Nion = Na andξ = ξ(tev) is the total number of ionized photons consumedper minihalo atom. We have neglected the recombinationsof He III to He II because these generally contribute diffuseflux which is absorbed on the spot by H and He.

Haiman, Abel & Madau (2001) estimated the evapora-tion times and photon consumption rates for evaporatingminihaloes analytically as follows. The evaporation time tevwas approximated by the sound-crossing time for the char-acteristic size of the minihalo at the sound speed of the ion-ized gas, tsc = 2rt/cs(10

4K), roughly consistent with ourearlier results for the simulation of the photoevaporation ofa 107M⊙ minihalo (Shapiro, Raga & Mellema 1997, 1998;Shapiro & Raga 2000a,b, 2001; Shapiro 2001). Haloes wereassumed to be dark-matter dominated, nonsingular isother-mal spheres according to the Truncated Isothermal Sphere(TIS) model (Shapiro, Iliev & Raga 1999; Iliev & Shapiro2001). For a minihalo of mass M at high redshift z, theTIS halo solution yields

tsc = 98Myr (M7)1/3

(

Ω0h2

0.15

)−1/3

(1 + z)−110 T

−1/24 , (2)

using the adiabatic sound speed cs(T ) = 11.7(T4/µ)1/2 =

15.2 T1/24 km s−1, where µ = 0.59 is the mean molecular

weight for fully ionized gas of H and He if the abundanceof He is A(He) = 0.08 by number relative to H. Based onthis evaporation time estimate, a naıve estimate of ξ is thenmade possible by further assuming that the minihalo haszero optical depth to ionizing photons and is fully ionizedinstantaneously but remains static at its initial density for atime tsc (the optically-thin, static approximation, hereafter

1 As discussed in Paper I, the evaporation time is not sensitiveto the precise value adopted for this remaining neutral fractionin the definition, as long as the latter is much less than unity.

c© 0000 RAS, MNRAS 000, 000–000

4 I. T. Iliev et al.

referred to as “OTS”). Ignoring the contribution of He torecombinations, equation (1) then yields

ξOTS = 1 + fCint〈nH〉α

(2)H

1 + δTIStsc (3)

(Haiman, Abel & Madau 2001) (where we have added thefirst term on the r.h.s. above to their equation to accountexplicitly for the fact that each atom must first be ionizedonce before it can recombine and be ionized again), where〈nH〉 is the mean H atom number density inside a halo,Cint ≡ 〈n2

H〉/〈nH〉2 = 4442 is the effective clumping factorfor the TIS, and 1 + δTIS = 130.6 is the average overdensityof a TIS halo with respect to the cosmic mean backgrounddensity. The quantity f is an “efficiency factor” introducedto take account of details neglected by the OTS assumptions.Together, equations (2) and (3) yield

ξOTS = 1 + 206fT−3/44 M

1/37

(

Ω0h2

0.15

)−1/3

(1 + z)210 . (4)

Here, we have defined M7 ≡ M/107M⊙ and (1 + z)10 ≡(1 + z)/10. Haiman, Abel & Madau (2001) calibrated thefactor f against an optically-thin simulation of the expan-sion of a uniformly photoheated halo (i.e. no radiative trans-fer), obtaining f ≈ 1. According to this OTS approximation,ξOTS ≫ 1 for minihaloes during cosmic reionization, thusthe presence of minihaloes during cosmic reionization wouldhave dramatically increased the number of ionizing photonsneeded to complete reionization (Haiman, Abel & Madau2001). According to equation (4), that is, each atom whichhad collapsed into a minihalo before reionization endedwould have consumed many ionizing photons during its pho-toevaporation, a disproportionately large share of the ion-izing photon background compared to that which would beavailable to ionize the uncollapsed atoms in the diffuse IGM.

As shown by the detailed simulations in Paper I, forhaloes of mass 107M⊙ exposed at z = 9 to the typical levelof ionizing radiation expected during reionization, while theOTS approximation yields a reasonable, rough estimate oftev, it grossly overestimates ξ, by factors of 30-50, depend-ing on the ionizing source spectrum. This more accuratedetermination of ξ in Paper I by numerical hydrodynami-cal simulations with radiative transfer nevertheless showedthat minihalo photoevaporation could still have consumedenough photons to increase substantially the total numberrequired per baryon to complete reionization. By contrast,earlier estimates of the requirements for reionization of theIGM had concluded that only about one photon per atomwas required, but these estimates were based on calculationswithout sufficient resolution to account for the minihaloes(Gnedin 2000a; Miralda-Escude, Haehnelt & Rees 2000). Itis therefore crucial to extend the accurate determination ofξ in Paper I to the full range of minihalo masses, ionizingflux levels and redshifts expected during reionization, in or-der to evaluate the global effect of minihaloes on the processof reionization.

3 THE CALCULATION

3.1 Numerical Method and Initial Conditions

We have performed a large number of simulations of the pho-toevaporation of individual minihaloes which occurs whena weak, R-type intergalactic I-front encounters a minihaloduring cosmic reionization. Our simulation method and ini-tial conditions were described in some detail in Paper I, sowe only briefly summarize these here. Each halo is modelledas a nonsingular TIS (Iliev & Shapiro 2001), surrounded byself-similar cosmological infall according to the solution ofBertschinger (1985). We generalized this self-similar infallsolution to apply also to a low-density ΛCDM backgrounduniverse at high redshift (Iliev & Shapiro 2001). At high-z, the halo radius for the current background cosmologyadopted here is rt = 0.754 M

1/37 (1 + zcoll)

−110 kpc, the to-

tal atomic number density at the center is n0 = 3.2(1 +zcoll)

310 cm−3, and the halo virial temperature and velocity

dispersion are Tvir = 4000 M2/37 (1 + zcoll)10 K and σV =

5.2 M1/37 (1 + zcoll)

1/210 km s−1, respectively. For haloes with

Tvir 6 104K (minihaloes), the gas atoms are neutral. Weconsider haloes in the mass range between the Jeans massof the uncollapsed IGM prior to reionization, MJ = 5.7 ×

103(

Ω0h2/0.15

)−1/2×

(

Ωbh2/0.02

)−3/5 [

(1 + z)10]3/2

M⊙,

and the mass for which Tvir = 104K according to the TISmodel, M(104K) = 3.95×107(Ω0h

2/0.15)−1/2[(1+z)10 ]−3/2.

Our simulations use the two-dimensional, axisymmet-ric Eulerian gas-dynamics code CORAL, with radiative-transfer, Adaptive Mesh Refinement (AMR), and the vanLeer Flux-Vector Splitting Algorithm, which resolves shockswell and properly tracks I-fronts ranging from fast, super-sonic R-type to sub-critical, subsonic D-type, as describedin detail in Paper I (and references therein). All our simu-lations were set up as we described in detail in Paper I. Wefixed the size of the computational box in all simulationspresented here to be the same in units of the halo radius(with the long side ∼ 10 times larger than the halo radius),

xbox ∝ rvir ∝ M1/3halo(1 + zcoll)

−1, so that the fully-refinedgrid resolution is equivalent in all cases. As in Paper I, weconsider three different cases for the spectrum of the exter-nal source of ionizing radiation: (1) starlight with a 50,000Kblack-body spectrum (hereafter, referred to as “BB5e4”),representative of massive Population II stars; (2) starlightwith a 100,000K black-body spectrum (hereafter, “BB1e5”),as expected for massive Population III stars; and (3) QSO-like, with a power-law spectrum with spectral index −1.8(hereafter, “QSO”). As in Paper I, we express the unatten-uated external flux of ionizing photons as a dimensionlessquantity, F0, the flux in units of that from a source emit-ting Nph = 1056s−1 ionizing photons per second at a properdistance d of 1 Mpc, or equivalently, emitting 1052 s−1 at adistance of 10 Kpc, so F0 ≡ F/1056 s−1/[4π(1 Mpc)2] =Nph/d2

Mpc.

c© 0000 RAS, MNRAS 000, 000–000

Photoevaporation of cosmological minihaloes II 5

Table 1. Numerical convergence test for the evaporation time tev and photon consumption rate per atom ξ as functions of grid resolutionfor minihalo of initial mass 107M⊙, exposed to initial flux of F0 = 1, starting at z = 9.

spectrum Ncell (∆x)min[pc] rt/(∆x)min tev [Myr] ξ

BB5e4 128 × 256 27.0 23.4 465 6.41BB5e4 256 × 512 13.5 56.7 300 5.73BB5e4 512 × 1024 6.7 113 195 5.39BB5e4 1024 × 2048 3.4 227 150 5.14QSO 128 × 256 27.0 23.4 112 6.20QSO 256 × 512 13.5 56.7 103 5.45QSO 512 × 1024 6.7 113 100 5.20QSO 1024 × 2048 3.4 227 97 4.95

BB1e5 128 × 256 27.0 23.4 215 5.16BB1e5 256 × 512 13.5 56.7 178 4.03BB1e5 512 × 1024 6.7 113 145 3.39BB1e5 1024 × 2048 3.4 227 128 3.33

3.2 Numerical Resolution

To begin, we have studied the numerical convergence of oursimulations with increasing spatial grid resolution, for thequantities tev and ξ, in order to establish the limits of ro-bustness and reliability of our results. We have performed aseries of four simulations, with fixed halo parameters andexternal flux level, with our fully-refined grid resolutionNcells = Nr × Nx, in 2D, axisymmetric (r, x)-coordinates,increasing by factors of 2 from 128× 256 up to 1024× 2048,for each of the three incident spectra. The results are shownin Table 1. We note that with increasing resolution both theevaporation time tev and the number ξ of ionizing photonsconsumed per atom decrease, leading ultimately to conver-gence in both quantities at our highest resolution. The con-vergence is more readily achieved for the cases with harderspectra (QSO and BB1e5) than for the softer spectrum caseBB5e4. In all cases ξ converges faster than tev.

3.3 Parameter space

In order to study how the photoevaporation of minihaloesby global I-fronts during reionization depends upon thehalo mass, the external ionizing flux level and spectrum,and the redshift at which the I-front encounters the mini-halo, we have performed the following large set of simu-lations. We considered a range of halo masses from a lowof 104M⊙, which is close to the Jeans mass at the corre-sponding epoch, to 4×107M⊙, close to M(104 K), the massat which the halo virial temperature is Tvir ∼ 104 K (e.g.M(104 K) = 4 × 107M⊙ at z=9). The intergalactic I-frontwhich encountered the minihalo in each case was assumedto reach the boundary of our simulation volume at a start-ing redshift zi (and we assumed zcoll = zi in defining theinitial parameters of the TIS model for the halo). We tookzi to range from 1 + zi = 20, consistent with the early startfor reionization implied by the recent WMAP results, to1 + zi = 7, at which epoch the data from QSO absorptionline spectra suggest that reionization was just ending. Theflux levels assumed ranged from F0 = 0.01 to F0 = 103.

We argued in Paper I that the interval F0 = 0.01 − 100corresponds approximately to the typical range of fluxes ex-pected when a global I-front encountered minihaloes dur-ing the reionization epoch. Here we extend this range up toF0 = 103, to accommodate cases of very high flux due eitherto a rare, very strong source or to the minihalo’s proximityto the source. In all our simulations the ratio of box sizeto halo radius was kept fixed at Lbox/rt ∼ 10, which provedsufficient to follow the evaporation flow until full evaporationwas achieved. Such a box size guaranteed that our transmis-sive boundary conditions were self-consistent at all times,since the simulation volume (fixed over time in proper co-ordinates) was always large enough to contain the sphereof radius equal to the turn-around radius in the cosmolog-ical infall profile centered on the minihalo as that radiusincreased over time. We show the complete set of simulationparameters and results for tev and ξ in Figure 1. In order tomake such a large set of simulations feasible in terms of bothcomputing time and storage requirements, we limited the de-gree of refinement of the mesh so that the fully-refined meshhad 512× 1024 cells in all cases. Our numerical convergencetests above showed that, at this resolution, the simulationsare already largely converged. To correct for any remainingsmall differences from the fully-resolved limit, we used thehighest-resolution simulations in Table 1 to re-normalize ourplotted results for tev

2. Furthermore, in order to verify thisapproach, we performed a limited set of additional higher-resolution (1024×2048 fully refined) simulations for differenthalo masses and external fluxes as presented in Table 2.

2 For BB5e4 and BB1e5 cases in Figure 1 which are not listed inTables 1 and 2, the plotted points for tev are the actual valuessimulated at fully-refined grid resolution 512 × 1024, adjustedslightly to correspond to the converged limit of our results for thehighest grid resolution of 1024×2048, based upon the convergencetest results in Table 1. The adjusted values are 0.77 (0.88) timesthe simulation values for resolution 512 × 1024 for cases BB5e4(BB1e5), respectively.

c© 0000 RAS, MNRAS 000, 000–000

6 I. T. Iliev et al.

Table 2. Evaporation times and ionizing photon consumption rates per atom for several simulations at highest grid resolution (1024×2048fully refined), all for zi = 9 and rt/(∆x)min = 227.

spectrum F0 M7 (∆x)min[pc] tev [Myr] tev [Myr] (fit) ξ ξ (fit)

BB5e4 1 10−3 0.34 8.5 7.5 1.75 1.71BB5e4 10 4 5.4 170 137 10.6 11.2BB5e4 0.5 1 3.4 210 193 4.7 4.8BB5e4 10 1 3.4 85 75 6.85 7.3BB5e4 102 1 3.4 49 47 9.07 8.5BB5e4 103 1 3.4 37 38 9.80 8.3QSO 102 1 3.4 25 21 10.9 11.0QSO 103 1 3.4 9 10 13.6 14.6BB1e5 102 1 3.4 41 41 5.28 5.7BB1e5 103 1 3.4 29 34 6.41 6.2

Figure 1. Photoevaporation times tev for individual minihaloes(top panels) and total ionizing photon consumption per halo atomξ (bottom panels) vs. halo mass M (left), redshift z at which I-front first encounters the minihalo (middle) and dimensionlessionizing flux F0 (right), for the three different types of sourcespectra, BB5e4 (circles), QSO (triangles) and BB1e5 (squares),labelled by (M, F0, z) to indicate the parameters which we keptfixed in each case as we varied the quantity on the x-axis. Thered (green) lines on the two left panels correspond to redshift1 + z = 15 (1 + z = 20). Also, note the different vertical scale ofthe lower right panel only.

4 RESULTS

4.1 Evaporation time

Our simulation results for the evaporation time tev are plot-ted for all cases in Figure 1 (upper panels). The evaporationtime increases significantly with increasing halo mass. For

Figure 2. Fractional errors of the fitting formulae in equations (5)and (6) compared with the simulation results for each of the casesplotted in Figure 1, for the evaporation times (top panels) andtotal ionizing photon consumption rates ξ (bottom panels), re-spectively, vs. halo mass M (left), redshift z at which I-front en-counters the minihalo (middle) and dimensionless ionizing flux F0

(right), for BB5e4 (blue circles), QSO (red triangles) and BB1e5(green squares) spectra.

the smallest minihaloes tev ∼ 10 Myr, much shorter thanthe Hubble time at that epoch, while for the larger mini-haloes tev ∼ 100 Myr, closer to, but still shorter than thecurrent Hubble time. The redshift dependence of the evapo-ration time is even stronger than the mass dependence, withtev increasing with redshift. However, due to the relativelysmall range of the relevant redshifts, the overall variation oftev with redshift is only by factor of ∼ 3 for fixed halo mass

c© 0000 RAS, MNRAS 000, 000–000

Photoevaporation of cosmological minihaloes II 7

Figure 3. Comparison of simulation results for the evaporationtimes and photon consumption rates with predictions of the OTSapproximation. Ratios of tev/tsc (top panels) and ξ/ξOTS (bot-tom panels) vs. halo mass M (left), redshift z at which I-frontencounters the minihalo (middle) and dimensionless ionizing fluxF0 (right), for each of the cases plotted in Figure 1 for BB5e4(blue circles), QSO (red triangles) and BB1e5 (green squares)spectra.

and external flux. Since the Hubble time tH ∝ (1 + z)−1.5,the evaporation times of higher-redshift haloes are a muchlarger fraction of the Hubble time, although still generallysmaller than tH . Finally, the evaporation times depend in-versely on the level of the external ionizing flux, becomingextremely long (tev ∼ few hundreds of Myr) for low flux lev-els (F0

<∼ 0.1), and quite short (tev <

∼ 30 − 40 Myr) for highflux levels (F0

>∼ 102).

The logarithmic slope of the dependence of tev on massand the linear slope of its dependence on the initial redshiftshow no significant dependences on the source spectrum forthe range of masses, fluxes and redshifts we have studied, al-though the overall normalization does vary. The dependenceof tev on the flux is more complicated, with a slope whichvaries with F0.

A simple analytical fit to these results is given by

tev = AMB7 F

C+D log10 F0

0

[

E + F(

1 + z

10

)]

Myr (5)

where A = (150, 97, 128), B = (0.434, 0.437, 0.465), C =(−0.35,−0.357,−0.358), D = (0.05, 0.01, 0.056), E =(0.1, 0.3, 0.24), and F = (0.9, 0.7, 0.76) for cases (BB5e4,QSO, BB1e5), respectively. We have fit the functional de-pendences based on all the points plotted in Figure 1. Therelative errors of these fitting formulae are plotted in Fig-

ure 2 (upper panels) for all the cases in Figure 1. Table 2shows the comparison between the simulation values and thefitting formulae for several of our highest-resolution simula-tions. In most cases the errors are <

∼10%, and in all but twocases the errors are <

∼20%.We have compared our simulation results for

tev with the OTS estimate of this quantity byHaiman, Abel & Madau (2001) in Figure 3. The OTSapproximation, recall, assumes that the sound-crossing timeof the halo once it is photoheated by ionizing radiation,tsc = 2rt/cs, where cs is the speed of sound at 104 K, is agood approximation to the evaporation time. According toequation (2), this means tev,OTS = tsc ∝ rt ∝ M1/3(1+z)−1,so the OTS evaporation time for a fixed halo mass decreaseswith increasing redshift in inverse proportion to the red-shift, rather than increasing linearly with redshift, as in ournumerical results, according to the scaling in equation (5),while the increase of tev with halo mass in the OTSprediction is somewhat less steep than our result. Finally,the OTS approximation completely ignores the significantdependence of tev on the magnitude and spectrum of theionizing flux, making tsc a significant underestimate of tev(by up to an order of magnitude) for low fluxes and asimilarly significant overestimate for high fluxes.

4.2 Ionizing photon consumption

The simulation results for the number of ionizing photonsper minihalo atom required to evaporate a minihalo, ξ, areplotted in Figure 1 (lower set of panels). The photon con-sumption per atom increases steeply with increasing halomass. For the smallest minihaloes, ξ ∼ 2, regardless of thespectrum of the ionizing source, i.e. each atom on averagerecombines just once during the evaporation of these haloes,or, equivalently, is ionized twice by the time it is expelledfrom the halo in the evaporative wind. On the other hand,for the larger minihaloes, the photon consumption rate issignificantly larger, ξ ∼ 5 − 8, so each atom in these haloesrecombines multiple times on average during the evaporationof the halo. A BB1e5 source evaporates a large minihalo afactor of ∼ 2 more efficiently in terms of photon consump-tion than a QSO or BB5e4 source, although the differencesin efficiency between different spectra almost disappear forthe smaller minihaloes. The ionizing photon consumptionrate grows approximately linearly with redshift, and henceis noticeably higher at higher redshifts. Finally, opposite tothe trend for the evaporation time, the photon consumptionrate grows with increasing flux, with only 2-3 photons peratom needed in the case of low ionizing flux (F0

<∼ 0.1), but

up to 7-16 needed in the case of high flux levels (F0>∼ 102).

Therefore, and somewhat counter-intuitively, the low fluxlevels are much more efficient than the high flux levels, al-though the evaporation process takes much longer in theformer than in the latter cases.

A good fit to these results is given by

ξ = 1+AMB+C log10 M7

7 FD+E log10 F0

0

[

F + G(

1 + z

10

)]

, (6)

c© 0000 RAS, MNRAS 000, 000–000

8 I. T. Iliev et al.

where A = (4.4, 4.0, 2.4), B = (0.334, 0.364, 0.338), C =(0.023, 0.033, 0.032),D = (0.199, 0.24, 0.219), E = (−0.042,−0.021,−0.036),F = (0, 0, 0.1), G = (1, 1, 0.9) for the BB5e4, QSO, andBB1e5 spectra, respectively. The relative errors of these scal-ing laws are plotted in Figure 2, lower panels. In most casesthe errors are <

∼10%, and in all cases but one the errors arebelow 20%.

We have compared our simulation results for ionizingphoton consumption with the predictions of the OTS ap-proximation in Figure 3. The OTS prediction for ξ is givenby equation (4) with f = 1. The number of recombina-tions per atom, ξOTS − 1, scales with mass and redshift asM1/3(1 + z)2. While the logarithmic slope of this OTS de-pendence on the halo mass is roughly similar to that of oursimulations according to equation (6), the increase of ξ withredshift of the OTS prediction is much stronger than thatof the simulation results. In addition, the dependence of ξon the ionizing flux and spectrum is quite strong in the nu-merical results, while it is completely ignored in the OTSapproximation. Unlike the evaporation time, tev, however,the overall normalization of ξ in the OTS approximation isoverestimated by at least an order of magnitude in all cases,as compared with the simulation results, and by more thantwo orders of magnitude in some cases, particularly at highredshift and/or low ionizing flux levels.

4.3 Evolution of the neutral mass fraction

As described in Paper I, the first phase of the encounter be-tween an intergalactic I-front and a minihalo is the weak,R-type phase, during which the I-front enters the halo andphotoionizes an outer layer of halo gas, predominantly onthe side facing the ionizing source, before the gas has timeto respond hydrodynamically and move. This phase endswhen the I-front slows to the R-critical speed and makes atransition to D-type, thereby trapping the I-front until itcan burn through the remaining neutral, shielded minihalogas and evaporate it. We identify the fraction of the origi-nal minihalo gas mass which was photoionized in this initialR-type phase when the front became R-critical as Mcr. Wehave plotted in Figure 4 the values of Mcr for our simulationcases in Figure 1. In Paper I, we derived an analytical ap-proximation which can be used to estimate Mcr, called theInverse Stromgren Layer (ISL) approximation, where ISL isthe ionized boundary layer of the minihalo defined by the“outside-in” Stromgren length for each impact parameter(see Paper I for details). In the static, ionization-equilibriumapproximation, the mass contained in this boundary layer,MISL, which is readily obtained for our TIS model by numer-ical integration as shown in Paper I, can be used to estimateMcr as the complement of MISL, Mcr = Mtot − MISL.

In Paper I, we compared our simulation results for theionized mass fraction at the transition from R-type to D-type for a 107M⊙ halo exposed at zi = 9 to a source of fluxF0 = 1 and found an excellent agreement with MISL. Herewe extend the comparison to determine how well that agree-

Figure 4. Neutral mass fraction of minihalo at the moment ofI-front trapping, Mcr. Points show results from simulations withBB5e4 (circles), QSO (triangles) and BB1e5 (squares) spectra, re-spectively. For comparison, we plot the analytical prediction forthis quantity derived from the Inverse Stromgren Sphere (ISL)approximation in Paper I (with argument on x-axis shifted as de-scribed in text), in which Mcr,analyt ≡ 1−MISL/Mtot (solid line),and a convenient fit to the ISL approximation and simulation re-sults, according to equation (7) (dashed line).

ment holds up when we vary the halo parameters, source fluxlevel, and redshift. We find that the dependence of the simu-lation results on these parameters follows that predicted bythe ISL approximation fairly well, where the latter predictsthat MISL should depend upon (M, F0, z) only in the com-bination F0M

−1/3(1 + z)−5. We find, however, that the fullrange of simulation results for Mcr is better fit by this ISLscaling law if we multiply the quantity F0M

−1/3(1+z)−5 inthe original ISL approximation by 10. The excellent agree-ment between this modified ISL scaling law for Mcr and thesimulation results is shown in Figure 4. We find that the fol-lowing analytical expression provides a reasonable fit bothto the ISL curve (with shifted argument) and the simulationresults,

Mcr,approx = 0.9954 exp(−0.0013x4), (7)

where x ≡ 5 + log10[F0M−1/37 (1 + z)−5

10 ].In Fig. 5 we show the evolution of the neutral mass

fraction, Mn/Mcr vs. t/tev as the minihalo evaporates, forall of the cases plotted in Figure 1. This time-variation of theneutral mass fraction of the evaporating minihaloes is ap-parently a nearly universal function, when the neutral massMn(t) is expressed in units of Mcr and t is expressed in unitsof tev for each case, independent of the halo parameters and

c© 0000 RAS, MNRAS 000, 000–000

Photoevaporation of cosmological minihaloes II 9

Figure 5. Evolution of the neutral mass fraction Mn/Mcr fromthe end of the initial R-type phase of the I-front and its conver-sion from R-type to D-type to the final evaporation of the halo,for simulations with BB5e4 (top), QSO (middle) and BB1e5 (bot-tom) spectra (solid lines). We also show the best fits in each case,as described in the text (long-dashed red lines).

external flux, but allows for some dependence on the spec-trum of the external source. The scatter in all cases is largelydue to the decrease of the flux level in time due to cosmo-logical expansion, for a source located at fixed comovingdistance from the target minihalo. Accordingly, the relationis tightest in the QSO case, since in these runs the evapo-ration times are shorter. The average shape of the neutralmass time-dependence function is well-fit by

Mn(t) = Mcr

(

1 −t

Atev

)B

, (8)

where A = (1.07, 1.03, 1.03) and B = (2.5, 2, 2) for theBB5e4, QSO and BB1e5 spectra, respectively (red dashedlines in Fig. 5).

5 IMPLICATIONS FOR REIONIZATION AND

CONCLUSIONS

The widespread presence of minihaloes filled with neutralgas during the epoch of reionization made encounters be-tween the weak, R-type intergalactic I-fronts which ionizedthe IGM and these minihaloes a common occurrence. Thisinteraction affected both the minihaloes and the I-fronts pro-foundly. The trapping of the I-fronts by minihaloes whichcover the sky as seen by the more massive haloes whichare generally believed to have been the primary sources of

cosmic reionization slowed the advance of those I-fronts inthe IGM, at large, and wasted some of the ionizing radi-ation which would otherwise have been available to ionizethat IGM. The minihaloes, in turn, were transformed by thephotoevaporation which followed their trapping of the inter-galactic I-fronts, into barren, dark-matter haloes devoid ofbaryons. In order to explore this process and its implicationsfor cosmic reionization, we performed a large set of high res-olution gas dynamical simulations which include radiativetransfer, the first in their kind. In Paper I, we discussed ourresults in detail for the illustrative case of a 107M⊙ halowhich is overrun by an intergalactic I-front at zi = 9, froma source of flux F0 = 1, for different source spectra. Here wehave summarized a much larger set of simulations designedto quantify the dependence of minihalo photoevaporation onthe halo mass, source flux level and spectrum, and redshift.In Paper I, we discussed the comparison between our simu-lation results for 107M⊙ haloes and previous related work.Here, we extend this comparison to include the broader setof cases presented in this paper.

We noted in Paper I that the approximate analysis ofBarkana & Loeb (1999) which was used to argue that pho-toevaporation of minihaloes affected a significant part ofthe baryon fraction collapsed into haloes before the end ofreionization was inconsistent with the results of our detailednumerical simulations for 107M⊙ haloes. Barkana & Loeb(1999) had modelled this process by a static approximationlike the ISL approximation in Paper I, to determine whatportion of the halo’s mass is shielded from external ionizingphotons and what portion is exposed to photoheating. Theythen assumed that only that portion which was instanta-neously heated in this way, enough to unbind it from the halogravitational potential well, would evaporate from the halo.From this static approximation, they concluded that onlyminihaloes smaller than ∼ few × 105 − 106 M⊙ (dependingon the redshift and the source spectrum) were evaporatedcompletely during reionization, while the larger minihaloeswould have retained a significant fraction of their gas (up to∼ 50% of the gas for haloes of mass ≈few×107M⊙). In PaperI we demonstrated that this model does not account prop-erly for the dynamics of photoevaporation, and, as a result,it fails to anticipate the fact that even minihaloes as large inmass as M = 107M⊙ must ultimately evaporate completely.The larger set of simulations we have presented here showfurther that this neglect of dynamics by Barkana & Loeb(1999) in estimating the dependence of photoevaporation onhalo mass is not correct; all minihaloes exposed to ionizingradiation during reionization evaporate completely. All ofthe gas initially in a minihalo would eventually have becomeunbound, outflowing with speeds of vwind = 20− 40 kms−1,leaving behind just a dark halo. The static approximationfails because it ignores the dynamical nature of photoevap-oration. As evaporation proceeds, layers of gas are continu-ously stripped away, exposing the gas layers within to theionizing radiation. Thus, although the inner halo region self-shields initially, all the gas is eventually “unshielded” and

c© 0000 RAS, MNRAS 000, 000–000

10 I. T. Iliev et al.

photoheated to T > 104 K, well above the halo virial tem-perature, and, hence, boils out of the halo.

In Paper I we pointed out that simulations of the pho-toevaporation of 107M⊙ haloes by an intergalactic I-frontduring reionization did not support the suggestion by Cen(2001) that external ionizing radiation can cause the implo-sion of the minihalo gas, leading to globular cluster forma-tion. We did not observe such an effect in our simulationsreported there. The broad range of cases considered hereallow us to extend this comparison to the full range of pa-rameter space expected for minihaloes exposed to the effectsof externally-driven I-fronts during reionization. In no casehave we yet found evidence of the implosion effect predictedby Cen (2001).

Our simulations show that the number of ioniz-ing photons per minihalo atom needed to photoevapo-rate the minihalo is typically in the range ξ ∼ 2 −10, significantly larger than the value of ξ ∼ 1 pre-viously estimated to be sufficient to reionize the IGM(Gnedin 2000a; Miralda-Escude, Haehnelt & Rees 2000).On the other hand, we have also shown that recent claims(Haiman, Abel & Madau 2001) that the photoevaporationof minihaloes can require up to hundreds of ionizing pho-tons per atom are very significantly overestimating ξ.

We have quantified here the dependence of evapora-tion times and photon consumption rates on the halo mass,source flux level and spectrum, and redshift of encounterbetween I-front and minihalo. We have provided convenientfitting formulae for our simulation results as functions ofthe parameters (M, F0, z), which should be useful in furtheranalyses of the role of this process in cosmic reionization.To go beyond this step, it is necessary to determine the dis-tribution in time and space of these parameters. The effecton reionization will depend upon the luminosity function ofsource haloes and the mass function of minihaloes as theyevolve in time and fluctuate in space, including the clus-tering of both the sources and minihaloes. These may de-pend, in turn, upon the feedback of reionization on galaxyformation, so it is likely to be necessary to solve the prob-lem of reionization in some detail before the true impact ofminihaloes as screens and photon sinks is known. That iswell beyond the scope of this paper. Here, instead, we shallprovide a simple first estimate of the average effect of mini-halo photon consumption on the photon budget required forreionization, based on the mean Press-Schechter (PS) distri-bution of minihaloes, as follows.

Let the ionizing photon consumption rate per minihaloatom be given for a single minihalo by ξ(M, z, F0). Then itsaverage over all minihaloes at redshift z is given by

ξ(z, F0) =1

ρtotfcoll,MH

∫ Mmax

Mmin

ξ(M,z, F0)Mdn(M, z)

dMdM, (9)

where dn(M, z)/dM is the well-known PS mass function ofhaloes,

fcoll,MH ≡1

ρtot

∫ Mmax

Mmin

Mdn(M, z)

dMdM (10)

Figure 6. Collapsed fraction in minihaloes fcoll,MH (top panel)for the ΛCDM universe, mean photon consumption per minihaloatom, ξ, due to photoevaporation of minihaloes (middle panel),and mean excess photon consumption per total atom (i.e. allatoms, including both minihaloes and IGM) compared to thenominal requirement of one per atom (bottom panel). Resultsare for ionizing fluxes F0 = 0.1 (long-dashed), 1 (solid), 10 (short-dashed) and 100 (dotted), for case BB5e4.

is the collapsed mass fraction in minihaloes, ρtot is themean matter density at the corresponding redshift, Mmin

is the Jeans mass in the uncollapsed IGM, MJ , and Mmax =M(104K), is the halo mass corresponding to virial tempera-ture 104 K. Assuming that ξ(M, z, F0) is given by the fittingformula in equation (6), we can calculate ξ(z, F0) for anygiven redshift z and ionizing flux F0. Results for F0 = 0.1,1, 10 and 100 and 7 6 1 + z 6 21 are shown in Figure 6,for case BB5e4. We see that ξ depends strongly on the ion-izing flux (ξ ∝ F

0.199−0.042 log10 F0

0 , in equation [6]), but isapproximately independent of the redshift. Thus, the con-tribution of minihalo photoevaporation to the mean globalphoton consumption rate is largely dictated by the ionizingflux levels and the current collapsed fraction in minihaloesand can require up to ∼ 1 additional ionizing photon peratom in the universe (both minihalo and IGM atoms) tofinish reionization, as compared with the requirement whenminihaloes are neglected, for the flux levels we considered

c© 0000 RAS, MNRAS 000, 000–000

Photoevaporation of cosmological minihaloes II 11

here. Hence, minihalo photoevaporation can potentially dou-ble the number of ionizing photons required to reionize theuniverse.

We defer the complete treatment of minihalo and sourcebias which would account for the spatially and temporally-varying fluxes during reionization to a future paper (Iliev,Scannapieco & Shapiro, in preparation). A brief sum-mary of our first results along these lines can be foundin Iliev, Scannapieco, Shapiro & Raga (2004), where it isshown that the statistical bias which causes the minihalomass function to be enhanced around source haloes relativeto the mean PS mass function boosts the relative contribu-tion of minihaloes as photon sinks and compensates for thedeclining mean collapsed fraction at high redshift.

This shows that minihalo photoevaporation will be animportant feature of reionization, causing a modest, but sig-nificant slow down of the global I-fronts and delaying theiroverlap. Only a more detailed, self-consistent treatment ofthe global process for a given reionization scenario will deter-mine if reionization was photoevaporation-dominated, sincethe influence of minihaloes is strongly dependent on both theflux and, through the evolving collapsed fraction, the red-shift at which the I-fronts encountered the minihaloes. Ad-ditionally, minihalo formation in some places and at someepochs might be partially suppressed by e.g. an early X-ray background, double reionization, or reduced small scalepower. Such different scenarios might be distinguished obser-vationally by detecting the fluctuations of the redshifted 21-cm emission or absorption lines from minihaloes (Iliev et al.2002, 2003; Cen 2003) or the kind of absorption line sig-natures at UV and optical wavelengths discussed in PaperI.

ACKNOWLEDGMENTS

We thank Andrea Ferrara and Garrelt Mellema for manyuseful discussions. This research was supported by NSFgrant INT-0003682 from the International Research Fellow-ship Program and the Office of Multidisciplinary Activitiesof the Directorate for Mathematical and Physical Sciences,the Research and Training Network ”The Physics of the In-tergalactic Medium” set up by the European Communityunder the contract HPRN-CT2000-00126 RG29185, NASAATP grants NAG5-10825 and NNG04G177G and Texas Ad-vanced Research Program grant 3658-0624-1999.

REFERENCES

Barkana R., Loeb A. 1999, ApJ, 523, 54Barkana R., Loeb A. 2002, ApJ, 578, 1Bertschinger E., 1985, ApJS, 58, 39Cen R. 2001, ApJ, 560, L592Cen R. 2003, ApJ, 591, 12Glover S. C. O., Brand P. W. J. L. 2003, MNRAS, 340, 210Gnedin N. Y. 2000, ApJ, 535, 530Gnedin N. Y. 2000, ApJ, 542, 535

Haiman Z., Abel T., Madau P., 2001, ApJ, 551, 599Iliev I. T., Shapiro P. R., Scannapieco E., Raga A. C. 2004,in Outskirts of Galaxy Clusters: Intense Life in the Sub-urbs (Proceedings of IAU Colloquium 195), ed. A. Diafe-rio, Cambridge University Press, in press

Iliev I. T., Scannapieco E., Martel H., Shapiro P. R. 2003,MNRAS, 341, 81

Iliev I. T., Shapiro P. R. 2001, MNRAS, 325, 468Iliev I. T., Shapiro P. R., Ferrara A., Martel H. 2002 ApJ,572, L123

Machacek M. E., Bryan G. L., Abel T. 2003, MNRAS, 338,273

Madau P., Rees M. J.,Volonteri M., Haardt F., Oh S. P.2004, ApJ, 604, 484

Miralda-Escude J., Haehnelt M., Rees M. J., 2000, ApJ,530, 1

Oh S. P., Haiman Z. 2003, MNRAS, 346, 456Ricotti M., Ostriker J. P. 2004, MNRAS, 352, 547Shapiro P. R. 1995, in The Physics of the InterstellarMedium and Intergalactic Medium, A. Ferrara, C.F. Mc-Kee, C. Heiles, P.R. Shapiro eds., (ASP Conference Series,Volume 80), p.55

Shapiro P. R. 2001, in Relativistic Astrophysics: 20th TexasSymposium, eds. J. C. Wheeler & H. Martel (AIP Conf.Proc. 586), pp. 219-232

Shapiro P. R., Giroux M. L. 1987, ApJ, 321, L107Shapiro P. R., Giroux M. L., Babul, A. 1994, ApJ, 427, 25Shapiro P. R., Iliev I. T., Raga A. C. 1999, MNRAS, 307,203

Shapiro P. R., Iliev I. T., Raga A. C. 2004, MNRAS, 348,753 (Paper I)

Shapiro P. R., Iliev I. T., Raga A. C., Martel H. 2003 inThe Emergence of Cosmic Structure, eds. Holt, S. S. &Reynolds, C. S. (AIP Conf. Proc. 666), pp. 89-92

Shapiro P. R., Raga A. C. 2000, in Astrophysical Plasmas:Codes, Models, and Observations, eds. S. J. Arthur, N.Brickhouse, & J. J. Franco, Rev.Mex.A.A. (SC), 9, 292

Shapiro P. R., Raga A. C. 2000, in Cosmic Evolution andGalaxy Formation: Structure, Interactions, and Feedback,eds. J. Franco, E. Terlevich, O. Lopez-Cruz, & I. Aretx-aga, (ASP Conference Series, vol. 215), pp. 1-6

Shapiro P. R., Raga A. C. 2001, in The Seventh Texas-Mexico Conference on Astrophysics: Flows, Blows, andGlows, eds. W. Lee & S. Torres-Peimbert, Rev.Mex.A.A.(SC), vol. 10, pp. 109-114

Shapiro P. R., Raga A. C., Mellema G. 1997, in Structureand Evolution of the Intergalactic Medium from QSO Ab-sorption Line Systems (Proceedings of the 13th IAP As-trophysics Colloquium), eds. P. Petitjean & S. Charlot,pp. 41 -45

Shapiro P. R., Raga A. C., Mellema G. 1998, MmSAI, 69,463

Somerville R. S., Bullock J. S., Livio M. 2003, ApJ, 593,616

Spergel D. et al. 2003, ApJS, 148, 175

c© 0000 RAS, MNRAS 000, 000–000