The Impact of Small-Scale Structure on Cosmological Ionization Fronts and Reionization

0.48 MB·English

by  Ilian T. Iliev

DRAFTVERSIONFEBRUARY2,2008
PreprinttypesetusingLATEXstyleemulateapjv.11/12/01
THEIMPACTOFSMALL-SCALESTRUCTUREONCOSMOLOGICALIONIZATIONFRONTSAND
REIONIZATION
ILIANT.ILIEV1,EVANSCANNAPIECO,2ANDPAULR.SHAPIRO3
AcceptedbyApJ
ABSTRACT
Thepropagationofcosmologicalionizationfrontsduringthereionizationoftheuniverseisstronglyinfluenced
by small-scale gas inhomogeneities due to structure formation. These inhomogeneitiesinclude both collapsed
minihalos, which are generally self-shielding, and lower-density structures, which are not. The minihalos are
denseandsufficientlyoptically-thicktotrapintergalacticionizationfronts,blockingtheirpathandrobbingthem
5 ofionizingphotonsuntiltheminihalogasisexpelledasanevaporativewind. Thelower-densitystructuresdonot
0
trap these fronts, but theycan slow them downby increasing the overallrecombinationrate in the intergalactic
0
medium(IGM).Inthispaperwestudytheeffectsofbothtypesofinhomogeneities,includingnonlinearclustering
2
effects,andwefindthatbothIGMclumpingandcollapsedminihaloshavesignificantyetqualitativelydifferent

n impacts on reionization. While the number density of minihalos on average increases strongly with time, the
a densityofminihalosinsideHIIregionsaroundionizingsourcesislargelyconstant.Thustheimpactofminihalos
J
isessentiallytodecreasethenumberofionizingphotonsavailabletotheIGMatallepochs,whichisequivalent
0 toareductionintheluminosityofeachsource.Ontheotherhand,theeffectofIGMclumpingincreasesstrongly
2
withtime,slowingdownreionizationandextendingit. Thuswhiletheimpactofminihalosislargelydegenerate

withtheunknownsourceefficiency,IGMclumpingcanhelpsignificantlyinreconcilingtherecentobservations
2
ofcosmicmicrowavebackgroundpolarizationwithquasarabsorptionspectraatz∼6,whichtogetherpointtoan
v
earlybutextendedreionizationepoch.
5
3 Subjectheadings:hydrodynamics—radiativetransfer—galaxies:halos—galaxies:high-redshift—intergalactic
0 medium—cosmology:theory
1
1
4 1. INTRODUCTION laythefinaloverlapuntilz≈6(Wyithe&Loeb2003;Ciardiet
0 al.2003). Finally,severalauthorshaveexploredthepossibility
Recent polarization observations of the cosmic microwave
/ ofearlypartialreionizationduetoadecayingparticle(Chen&
h background by the Wilkinson Microwave Anisotropy Probe
Kamionkowski2004;Hansen&Haiman2004),complemented
p (WMAP)implythatreionizationwasfairlyadvancedatz ∼15
- (Kogutetal. 2003). Thiscameasasurprise. Thepriorredetec- bylaterfullreionizationfromastrophysicalsources.
o The role of small-scale inhomogeneities as sinks of ioniz-
tionoftheGunn-Petersoneffectinthespectraofhigh-redshift
r ing photons has mostly been ignored in this context. Never-
t quasarshadsuggestedthatreionizationwasonlyjustendingat
s theless,overalargerangeofredshifts,therecombinationtime
a z∼6(Whiteetal2003;Fanetal.2004). Thatwasconsistent
t atthemeanIGMdensityisontheorderofthecorrespond-
: withpredictionsofthemostaccuratenumericalsimulationsin rec
v thecurrentΛCDMparadigm,whichhadallpredictedthistran- ingHubbletime,asillustratedinFigure1. Thustheabsorption
Xi sition at z <8- 10 (Ciardi et al. 2000; Gnedin 2000a; Ra- ofionizingphotonsduringreionizationhappenspredominantly
re ∼ in overdense regions. In hierarchical models like Cold Dark
r zoumovet al. 2002;Ciardi et al. 2003). Despite manypoorly
a understood details concerning the star formation rate, the es- Matter (CDM), the smallest structuresare the first to collapse
gravitationallyanddominatethephotonconsumptionbothdur-
cape fraction of ionizing radiation, and the differences in nu-
ing the ionizationof a regionand afterwards, while balancing
merical treatments of reionization, z ∼ 15 had seemed un-
re recombinations.
likely,andsuchanextendedperiodofreionization,impossible.
When the first sources turned on, they ionized the neutral,
Nowtheraceisontoreconciletheearlyonsetofreionization
opaqueIGMaroundthembypropagatingweakR-typeioniza-
suggestedbyWMAPwiththehigh-redshiftGunn-Petersonef-
tionfronts(I-fronts). Thistypeoffrontmovesoutwardsuper-
fect,whichimpliesneighboringionizedpatchesfinallygrewto
sonicallywithrespecttoboththeneutralgasinfrontofitand
overlapatz∼6(Haiman&Holder2003;Cen2003;Wyithe&
theionizedgasbehindit,soitracesaheadofthehydrodynam-
Loeb2003;Ciardietal.2003). Onesuggestionisthattheuni-
ical response of the IGM. This process was first described by
verse had two reionizationepochsbut recombinedin between
Shapiro(1986)andShapiro&Giroux(1987),whosolvedan-
(Cen2003),yetthisignorestheunavoidablespreadinredshifts
alyticallyforthetime-varyingradiusofasphericalI-frontsur-
intrinsic to any such IGM transition (Scannapieco, Schneider,
roundinga point source in the expanding IGM and then used
& Ferrara 2003; Barkana & Loeb 2004; Furlanetto & Loeb
thissolutiontodeterminewhenHIIregionswouldgrowtothe
2005). Othersuggestionsinvolvefine-tuningtheionizingpho-
pointofoverlap,therebycompletingreionization.Inthisstudy
tonemissivityfordifferentsourcehalomasses,theescapefrac-
theeffectofdensityinhomogeneityonthemotionoftheI-front
tion,andthe(possiblymetalicity-dependent)InitialMassFunc-
was described by a mean gas clumping factorC≡hn2i/hni2.
tion (IMF), in ways intendedto accelerate early ionization, to
A clumpy gas hasC>1, which causes the ionized gas to re-
buildupalargeenoughτ ,butslowdownlateionization,tode-
es
1CanadianInstituteforTheoreticalAstrophysics,UniversityofToronto,60St.GeorgeStreet,Toronto,ONM5S3H8,Canada
2KavliInstituteforTheoreticalPhysics,KohnHall,UCSantaBarbara,SantaBarbara,CA93106
3DepartmentofAstronomy,UniversityofTexas,Austin,TX78712-1083
1
2
FIG.1.—Timescales.HubbletimetH (dashedline)andrecombinationtimetrec=(αBnH)- 1atthemeanIGMdensity(solidline)vs.redshiftz(lowerpanel)and
theratioofthesetimescales(toppanel).
combinemorefrequently,increasingtheopacityoftheHIIre- importance of unshielded and shielded overdenseregions and
gion to ionizing photons, which reduces the flux reaching the their sizes, densities, and abundances. These IGM inhomo-
I-frontandslowsitdown. Thisapproachhasformedthebasis geneities can be divided into two major types, both of which
formanymorerecentsemi-analyticaltreatmentsofreionization have been modeled only crudely in the majority of reioniza-
(e.g. Haiman & Holder2003; Wyithe & Loeb2003; Venkate- tionstudies. Pre-virializedobjects,suchasfilamentsandstill-
san,Tumlinson,&Shull2003). Theideathatreionizationpro- collapsing halos, are usually described in terms of the mean
ceededbythepropagationofweak,R-typeI-frontswhichmove clumpingfactordescribedabove.Currentsemi-analyticalmod-
too fast to be affected by the gas dynamical disturbance they els of reionization either assume a constant clumping factor
create is also the basis for most of the numerical simulations (Cen 2003; Haiman & Holder 2003; Tumlinson, Venkatesan
of reionizationcarriedouttodate(e.g.Razoumovetal. 2002; & Shull 2004), a clumping factor derived from linear theory
Ciardi, Ferrara,& White, 2003; Sokasianetal. 2004). Inpar- (Miralda-Escudé,Haehnelt&Rees2000;Chiu,Fan&Ostriker
ticular,allnumericalstudiesthataddradiativetransfertoapre- 2003;Wyithe&Loeb2003),orignoreclumpingaltogether(i.e.
computedinhomogeneouscosmologicaldensity field (i.e. the assumeC=1)(Onken&Miralda-Escudé2004).Inpracticeall
“static”limit)areassumingthatthereisnosignificantbackre- theseapproachesareover-simplifiedsincetheclumpingfactor
actiononthegas4. oftheIGMgasisdominatedbythehighly-overdensenonlinear
Theassumptionofeitherameanclumpingfactororthestatic regionsandevolvesstronglywithredshift.
limit to modelthe effectof density inhomogeneityon cosmo- Modeling of virialized inhomogeneities in previous studies
logicalI-frontsisnotcorrectevenonaverage,however,unless has been even more approximate. An importantdividing line
theclumpsareeitheropticallythinorabsorbonlyasmallfrac- thatseparatestwodistinctpopulationsofvirializedhalosisthat
tion of the ionizingflux. If a clumpis self-shielding,then the definedbythevirialtemperature,T =104K. Inorderforstars
vir
I-frontthatencountersitwillnotremainaweakR-typefrontif toforminsidehalos,thegasmustcoolbelowthevirialtemper-
thesizeoftheclumpislargerthanitsStrömgrenlength(i.e.the ature to become self-gravitating and gravitationally unstable.
length of a columnof gaswithin which the unshieldedarrival Radiative cooling in a purely atomic gas of primordial com-
rate of ionizingphotonsjustbalancestherecombinationrate). position is ineffective below 104K, however, so “minihalos”–
In thatcase the denser gasof the clumpmust slow the I-front halosinthemassrange104M⊙∼<M∼<108M⊙,withvirialtem-
down enough that the disturbed gas inside the clump catches peraturesbelow104K–areonlyabletoformstarsbyforming
uptotheI-frontandaffectsitsprogress. ThistransformstheI- H molecules, which have the potentialto cool the gas below
2
frontfromsupersonic,R-type,tosubsonic,D-typeand“traps” thevirialtemperature,byrotational-vibrationallineexcitations.
the I-front inside the clump (Shapiro, Iliev & Raga 2004). If TheH thatformsinminihalos,though,iseasilydissociatedby
2
the clump is gravitationally bound before the arrival of the I- UVphotonsintheLyman-Wernerbandsbetween11.2and13.6
front, then the I-front will expel the gas from the clump as a eV,whichareproducedinabundancebythefirststars,longbe-
supersonicevaporativewind,aslongastheclumpcannotbind foretheionizingbackgroundfromsuchstarsisabletoreionize
photoionizedgaswithT ≥104K. a significant fraction of the universe (e.g. Haiman, Rees, &
The impact of small-scale inhomogeneitieson the global I- Loeb 1997; Haiman, Abel & Rees 2000; Ciardi et al. 2000).
fronts that reionized the universe depended upon the relative Thusa genericpredictionof currentstructureformationmod-
4AnexceptiontothisisthecodedevelopedinGnedin(2000a)andRicotti,Gnedin,&Shull(2002),whichcombinesanapproximatetreatmentofradiativetransfer
withnumericalcosmologicalgasdynamics.
3
elsisalargepopulationofminihalosthatareunabletocooland The structure of this work is as follows. In §2 we general-
formstars. izetheapproachofShapiro&Giroux(1987)toaccountforthe
Inthatcase,thedominantsourceofphotonsforreionization effectof minihaloevaporationonthe time-varyingradiusofa
wouldhavebeenthemoremassivehalos(i.e.M∼>108M⊙)with spherical I-front. This will require us to calculate the statisti-
T ≥104K,inwhichatomiclinecoolingisefficientenoughto callybiasedabundanceofminihalosatthelocationofthefront
vir
enable starsto form. Fromthe pointof view ofsuch a source and incorporate the simulation results for the ionizing photon
halo, all lines of sight will intersect a minihalo at a distance consumption rates per minihalo. In §3 we model the global
less than the mean spacing between sources (Haiman, Abel progress of reionization by summing the results from Section
& Madau 2001; Shapiro 2001; Shapiro, Iliev, Raga & Martel 2 over a statistical distribution of source halos, leading to the
2003; Shapiro, Iliev & Raga 2004). Thus, the intergalactic I- eventualoverlapof neighboringH II regions and the comple-
frontsmusthavefoundtheirpathsblockedbyminihalosinev- tion of reionization. Our results and conclusionsare given in
erydirection,whichtrappedthefrontsuntiltheminihaloswere §4.
evaporated.
In Shapiro, Iliev & Raga (2004)and Iliev, Shapiro & Raga 2. THEPROPAGATION OF A COSMOLOGICAL IONIZATION
(2004),weusedhigh-resolutionnumericalgasdynamicalsimu- FRONT ABOUT A SINGLE SOURCE
lationswithradiativetransfertostudytheencounterbetweenan 2.1. CosmologicalIonizationFrontsinaClumpyIGM
intergalacticI-frontandaminihaloindetail,forawiderangeof
Whenasourceofionizingradiationturnsonintheexpand-
conditionsexpectedduringreionization. Theseresultsyielded
ing,neutralIGM,aweak,R-typeI-frontpropagatesoutward.If
the number of ionizing photons absorbed per minihalo atom
the IGM were static, this frontwould decelerate continuously
duringthetimebetweenthearrivaloftheI-frontandtheevap-
fromthemomentofturnon,until,withinatimecomparableto
oration of the minihalo gas, ξ, as a function of the minihalo
therecombinationtime,italmostreachedthesizeoftheStröm-
mass,sourcefluxlevelandspectrum,andtheredshiftoftheen-
grensphere.ThisStrömgrensphereisjustlargeenoughthatthe
counter. Thisisafundamentalingredientwewillneedhereto
total recombination rate of ionized atoms inside it equals the
determinehowthepresenceofminihalosaffectedglobalreion-
ionizingphotonluminosityofthecentralsource. Atthispoint
ization.
theI-frontdropstotheR-criticalspeedoftwicethesoundspeed
This trapping of intergalactic I-fronts by minihalos, com-
oftheionizedgas,andthefronttransformsfromR-typetoD-
binedwiththeincreasedrecombinationrateinsidealreadyion-
type,precededbyashock. ThereaftertheI-frontisaffectedby
izedregionsduetosmall-scaleclumpingoutsidetheminihalos,
thedynamicalresponseoftheIGM.
may help to explainhow reionizationcould have started early
This is not the case, however, in the expanding, average
andendedlate. AstheglobalI-frontsadvancedintofreshneu-
IGM. Shapiro & Giroux (1987) showed that, while it is for-
tralregions,theygenerallyencounteredminihalosthatformed
mally possible to define an “instantaneous” Strömgren radius
at the unfiltered (i.e. not affected by any radiation feedback)
(whichgrowsintimeinproportiontothecosmicscalefactor),
rate of the universe without reionization. The mass fraction
theactualI-frontgenerallydoesnotreachthisradius. Instead,
collapsedintominihalosinsuchregionsgrewovertime, from
the I-front remains a weak R-type front as long as the source
8%to24%to31%fromz=15to9to6,sotheaveragenumber
continues to shine, and it would not be correct, therefore, to
ofextraphotonsconsumedperatombyphotoevaporationmust
describe the cosmological H II region as a Strömgren sphere,
alsohaveincreasedwithtime. Thismayhaveenabledminiha-
a misnomer which unfortunately appears in the literature of
los to slow the advanceof the globalI-fronts, with increasing
reionization.
effecttowardlatetimes. ReionizationsimulationsbyCiardiet
WeshallfollowtheapproachofShapiro&Giroux(1987),in
al. (2003)forexample,which neglectedminihalos,foundthat
whichtheHIIregionisboundedbyanI-frontwhosespeedis
ifoneassumesahighescapefractionofionizingphotonsfrom
determinedbytheI-frontcontinuityjumpcondition,whichbal-
thesourcehalos,thenthelargevalueofelectronscatteringop-
ancesthe outwardflux of ionizingphotonsagainstthe inward
ticaldepth,τ observedbyWMAPcanbeachievedbythefirst
es fluxofnewlycreatedions. Thefluxthatreachesthefrontwill
stars ingalaxieswithmassM∼>109M⊙ assources. However, bedeterminedbysolvingtheequationoftransferbetweenthe
inthiscase, reionizationiscompletedfartooearly. Minihalos
source and the front. We will assume that the IGM is spheri-
mayhavethepotentialtoreconcilethisdiscrepancy,increasing
callysymmetricoutsidethesource. ForsimplicitytheI-fronts
thedurationoftheepochofreionizationandallowingforasim-
aretakentobe“sharp”,i.e. thewidthofthetransitionbetween
ilarhighvalueofτ ,whilepostponingtheredshiftofoverlap.
es theionizedregioninsideandtheneutralregionoutsidethefront
Inthispaper,weconsidertheimpactofbothminihalosand
is small comparedto its radius. The actualwidth is compara-
moregeneralIGMclumpingindetail, andattempttoquantify
ble to the absorptionmean free path on the neutralside. This
their effectsonthe durationofthe reionizationepoch. Driven
assumption of small mean free path is generally a good ap-
by measurements of the cosmic microwave background, the
proximation for a “soft” Population II (Pop. II) stellar spec-
numberabundanceofgalaxyclusters,andhighredshiftsuper-
trumwheremostionizingphotonshaveenergiesnearthe ion-
novadistanceestimates(e.g. Spergeletal. 2003;Eke,Cole&
izationthresholdofhydrogen,forwhichtheabsorptioncross-
Frenk1996;Perlmutteretal. 1999)wefocusourattentionon
theΛCDMcosmologicalmodelwithparametersh=0.7,Ω = sectionduetoneutralhydrogenislarge. However,I-frontsare
0an.3d,ΩΩΛa=re0.t7h,eΩtobt=al0m.0a5tt,eσr,8v=ac0u.8u7m,,ananddnbpa=ry1o,nwichedreenΩsi0ti,eΩs0Λin, ssiovmeePwohpautlawtiiodnerIIfIor(P“ohpar.dII”Is)psetacrtsraalnidkeththeopsoeweexrp-elacwtedspfeocrtmraaos-f
b QSOs. In thesecasesa largerfractionoftheionizingphotons
units of the critical density (ρ ), σ2 is the variance of linear
fluctuationsfiltered on the 8h-c1rMit pc8scale, and n is the index areathigherenergies,correspondingtolowerionizationcross-
p sectionsofneutralhydrogenandhelium,andthusourapproxi-
oftheprimordialpowerspectrum.TheEisenstein&Hu(1999)
mationofsharpI-frontsislessaccurate.
transferfunctionisusedthroughout.
Adoptingthispicture, we consideran ionizingsourceemit-
4
˙
tingN ionizingphotonsperunittime.Wedefinethecomoving Equation(9)hasasolution
γ
radiusoftheionizedregionasr (t)anditscomovingvolumeas
VI =4πrI3/3. WeareinterestedIinHIIregionsthatatalltimes τ(z)=κ 1- Ω0(1+z)3+ΩΛ 1/2 , (10)
are much smaller than the scale of the current horizon. The
jump condition across the I-frontis given by a balance of the where κ≡ 3H02t1Ω0 annd the(cid:2)arbitrary consta(cid:3)nt oof integration is
fluxofneutralatomsandphotoionizingphotons. Intheframe chosensothatτ(z=0)=0.Athighredshift,beforeandduring
ofthefront,itcanbewrittenas reionization,wehaveΩ0>>ΩΛ/(1+z)3andthesolution(10)
n u =β- 1F, (1) becomes
H,1 1 i τ(z)=κ[1- Ω1/2(1+z)3/2]. (11)
where n is the undisturbed hydrogen number density (in 0
H,1
proper coordinates) on the neutral side of the front, u = Inthislimitequation(8)simplifiesto
1
a(drI/dt) is the I-front peculiar velocity, a = 1/(1+z) is the τ(t) eτ′
tsocaclreeafatecteoarc,hanidonβizieisdtHheantoummbtheartoefmioenrgizeisngonphthoetoionnsiazbesdosribdeed. y(t)=Ω0κ2e- τ(t)Zτ(ti) dτ′(κ- τ′)2. (12)
Intheabsenceofminihalos,β =χ ≡1+pA(He),whichcor-
i eff Equation(12)hasanexactanalyticalsolutiongivenby
rects for the presence of helium, with p = 0,1 or 2 if He is
mostly neutral, singlyionizedordoublyionizedafter the pas- y= η eηti/t tEi(2,ηti)- Ei(2,η) , (13)
sage throughthe front, and A(He)=0.08is the He abundance (1+z)3 t t
i (cid:20) i (cid:21)
bynumberwithrespecttohydrogen. Finally,F isthenumber
fluxofionizingphotonsatthecurrentpositionoftheI-front, where η≡2(1+zi)3/2/(3H0t1Ω10/2) and Ei(2,x)≡ 1∞et-2xtdt is
theExponentialintegralofsecondorder. Thesolutioninequa-
F = S(rI,t), (2) tion (13) reduces to the one in equation (10a) ofRShapiro &
4πa2rI2 Giroux(1987)foraflat,matter-dominateduniversewithΩ0=1,
where S(r,t) is the number of photons emitted by the central t=2/(3H),andthusη=(1+z)3t/t .
i i 1
source which pass througha sphere of comovingradius r per
unittime,givenatr=rI as 2.2. TheAverageEffectofMinihaloEvaporationon
S(r ,t)=N˙ - 4πr3a- 3(n 0)2Cα χ , (3) CosmologicalIonizationFrontPropagation
I γ 3 I H B eff Having outlined a formalism to describe the expansion of
i.e. thenumberofphotonsemittedbythesourceperunittime ionization fronts in a ΛCDM cosmology, we next address the
minus the number of recombinations in the current H II re- questionofabsorptionbyminihalos. Asdescribedin§1,when
gionvolume. HereC isthevolume-averagedclumpingfactor, anintergalacticI-frontencountersanindividualminihalo,itis
αB=2.6×10- 13cm3s- 1isthecaseBrecombinationcoefficient trapped until the minihalo gas is evaporated. For every mini-
for hydrogenat104K andn0H is the comovingnumberdensity haloatom,thisprocessconsumesξ ionizingphotons. Suppose
of hydrogen in present units, 1.87×10- 7(Ωbh2/0.022) cm- 3. we considerthe averageeffectof this processonthe globalI-
In all cases, it is safe to assume that the H II regionsare cos- frontwhich movesthrougha mediumcomprisedof minihalos
mologically small, and hence no ionizing photons are lost to embeddedintheIGM.Theaveragespeedthroughoutthiscom-
redshiftingbelowthehydrogenionizationthreshold. pound medium will be given by a modified I-front continuity
Combiningequations(1)-(3),theevolutionofthecomoving jump condition which takes account of the additional photon
volumeoftheionizedregionVI isgivenby consumptionduetominihalos. Inparticular,thequantityβi in
dVI ≡4πr2drI ≡ 1 N˙ - α C(1+z)3n0 V. (4) eq.(1)shouldnowbereplacedbythefollowing
dt I dt χeffn0H γ B H I βi≡(1- fcoll)χeff+[1+A(He)]fcoll,MHξ¯, (14)
Defining
where f is the total collapsed baryon fraction (i.e. over all
V = 4πrS3,i = N˙γ , (5) halo macsoslles) and fcoll,MH is the collapsed fraction of just the
S,i 3 χ α C(n0)2 minihalos. Finally, if ξ is the the numberof ionizingphotons
eff B H
consumedperminihaloatomintheencounterbetweenthein-
wecanwriteequation(4)indimensionlessform ¯
tergalacticI-frontandanindividualminihalo,thenξ istheap-
dy
=1- y(1+z)3, (6) propriate average over the distribution of minihalos at the in-
dx stantaneous location of the global I-front. Inserting eq. (14)
where y≡V/V =(r /r )3, x≡t/t and t =1/(α Cn0) is intoeq.(1)thenyields
I S,i I S,i 1 1 B H
therecombinationtimeofthemeanIGMatpresent(Shapiro& F
GirIofuwxe1d9e8fi7n)e. dτ =dx/a3=(1+z)3dx,equation(6)becomes nH,1u1= (1- fcoll)χeff+[1+A(He)]fcoll,MHξ¯, (15)
wheren referstothetotalHatomdensity,includingboththe
dy =(1+z)- 3- y, (7) IGMandH,a1llhalos.
dτ
As usualtheflux, F, in thisI-frontjumpconditionisdeter-
forwhichaformalsolutionis
mined by integratingthe equation of transfer over the ionized
y(t)=e- τ(t) τ(t)dτ′ eτ′ , (8) regionbetween the source haloand the I-front. By definition,
[1+z(τ′)]3 the minihalos originally inside this region do not affect this
Zτ(ti)
wheret is the time of source turn-onand for the flat, ΛCDM integration, however, since they will already have been evap-
i
orated by the passage of the global I-front, thereby returning
model
dτ (1+z)2 theiratomstotheIGMinsidetheHIIregion. Weassume,for
dz =- H0t1[Ω0(1+z)3+ΩΛ]1/2. (9) simplicity, that the evaporated minihalo gas shares the mean
5
clumpingfactor the IGM into which it is mixed. In thatcase, model(e.g. Peebles1980). Inthiscase,bothquantitiescanbe
thefluxattheI-frontisgivenby expressedparametrically,intermsofa “collapseparameter”θ
F = N˙γ- αBC(1+z)3(n0H)2χeff(VI- V0), (16) as ¯ 9 (θ- sinθ)2
4πa2rI2 δ= 2(1- cosθ)3, (21)
wherewehavebeencarefulnowtostartourintegrationofthe
and
thraalnosf(ei.re.eqauvaotilounmferowmhicthhe, wLhagenramnguilatniplvioedlumbyethoef mtheeasnoduercne- δ¯ = 3 3 2/3(θ- sinθ)2/3+1. (22)
L
sity,givesthemassofthesourcehalo). 5 4
(cid:18) (cid:19)
Combiningequations(15)and(16),theevolutionoftheco- These equations define the relationship between the Eulerian
movingvolumeoftheHIIregioninthepresenceofminihalos andLagrangiancomovingradiias
isthengivenby r =r δ¯(r ,r )- 1/3, (23)
dV 1 I,E I I 0
dtI = (1- f )χ +[1+A(He)]f ξ¯ sinceδ¯givenbyequation(21)aboveistheaverageoverdensity
coll eff coll,MH withinanEuleriansphereofLagrangianradiusr .
˙ I
×(cid:2) Nγ - α C(1+z)3n0χ (V - V ) (cid:3), (17) Inthepresenceofinfall,equation(17)becomes
n0 B H eff I 0 dV 1
where initially V(cid:20) =HV . We note that the solution(cid:21)s in §2.1 in dtI = (1- f )χ +[1+A(He)]f ξ¯
I 0 coll eff coll,MH
tttihhoeenpafrboessreenVncIc(eet)oofafnmmdiinnrIiihh(taa)llooinss,aeisqnwuclaeutlidlo,innifg(β1tih3ie)n,eewxqaiulclattbiaoennavl(ay1lti4idc)aahlnesdroetlhuine- ×"Nn˙0Hγ - αBC(cid:2)(1+z)3n0HχeffZV0V,EI,EdVI′,Eδ(VI′,E)2#(cid:3), (24)
clumpingfactorCareconstants,andt isredefinedas
1 wherenowδ isnotthe averageδ withinthesphere,butrather
t1≡ αBCn0Hχβeff - 1. (18) cδoamttphuetebδouansdfaorlylo,wansd. TVhI,Eec≡omVIoδ¯v- 1inigsvthoeluEmuelesraiatinsfiveosl:ume. We
(cid:18) i (cid:19) ∆V δ(V )+V δ¯(V )=[V +∆V ]δ¯(V +∆V ), (25)
Ingeneral,bothβ andCwillnotbeconstant,butnevertheless I,E I,E I,E I,E I,E I,E I,E I,E
i
thislimitprovidesausefulcheckandsomeinsight,aswellshall where∆V isasmallchangeinthesizeoftheradius.Working
I,E
seebelow. tofirstorderin∆V ,thisgives
I,E
Weadoptasimplemodeltoaccountforinfallwhencomput- ¯
ingtheflux.Theradialcoordinate,r ,adoptedinourformalism δ=δ¯+V dδ . (26)
I I,EdV
above is essentially a Lagrangian one, in which we have as- I,E
sumedthatthelocalmeandensityofthegasaroundthesource We can therefore rewrite equation (24) using the Lagrangian
is equal to the cosmic mean IGM density. In reality, all mas- volumeas
sivesourcesarefoundinoverdenseregions,duetothegravita- dV 1
tional influenceon the surroundinggas. We estimate the map dtI = (1- f )χ +[1+A(He)]f ξ¯
coll eff coll,MH
between the Lagrangian and Eulerian comoving radii using a
simple “top-hat” picture, which is a simplified version of the × (cid:2)N˙γ - α C(1+z)3n0χ VIdV′δ((cid:3)V′)- 1δ(V′)2 (.27)
modeldescribedinBarkana(2004). n0 B H eff I I I
(cid:20) H ZV0 (cid:21)
We compute the cross-correlation between a sphere of La-
Therelevantoverdensitythatappearsinequation(27)is
grangianradiusr and sphericalperturbationof mass equalto
thesourceσh2(arlo,rmI)as≡sM1s=(∞4πk/23d)krP03Ω(k0)ρWcr(itkars)W(kr ), (19) δclump(VI)¯≡VI-1V0ZV0VIdVI′δ(VI′). (28)
where P(k) i0s tIhe i2nπitiZal0 matter power s0pectrumI , linearly 1In08FMig⊙uraend2,1w01e1Msh⊙o.wNδo,tδe,thanatdδδcclulummppeaxrcoeuendds δp¯eoavkesroafrmanagses
ofradii,becauseδ isanaverageinLagrangiancoordinates
extrapolated to the present, and W(x) is the spherical top- clump
¯
hat window function, defined in Fourier space as W(x) ≡ andδ isanaverageinEuleriancoordinates. Finally,thefluxin
3 sin(x)- cos(x)x2 , where r is the Lagrangian radius of the equation(16)iscorrectedbyasimilarfactorofδclump,yielding
sourxc3e itself. Defining σ2(r0)≡σ2(r ,r ), the expected value N˙ - α C(1+z)3(n0)2χ (V - V )δ (V)
(cid:2) (cid:3) 0 0 0 F = γ B H eff I 0 clump I . (29)
of the linear overdensity of a sphere of Lagrangian radius rI 4πa2r2
aboutthesourceisthengivenby I,E
Theaveragenumberofionizingphotonsabsorbedpermini-
δ¯ (r )= 1.69D(z)σ2(r0,rI)+1. (20) haloatom,ξ¯,intheprocessofevaporatingalltheminihalosat
L I D(z ) σ2(r ) thecurrentlocationoftheI-front,r (t),mustnowbespecified.
s 0 I
¯ Iliev, Shapiro & Raga (2004) have shown that the number of
Hereweuseδ todenotetheaveragedensitywithinthesphere,
L
photonsperatomabsorbedbyaminihaloofmassM (inunits
D(z) is the linear growth factor, z is the collapse redshift for 7
the halo, andthe“+1”appearssinsce we aredefiningthe over- of 107M⊙), overtaken by an intergalactic I-front at a redshift
density as ρ/ρ¯instead of ρ/ρ¯- 1. For simplicity, we assume of z which is driven by an externalsource of flux F0, the flux
in units of that from a source emitting N =1056s- 1 ionizing
z=z , i.e. thatthegrowthofthemassafteritcollapsesdueto ph
s
photonspersecondataproperdistanced of1Mpc,i.e.
secondaryinfall,issmall.
¯
The linear overdensityδ¯L can be related to the correspond- F ≡ Nph,56 = F , (30)
ingnonlinearoverdensity,δ,bythestandardsphericalcollapse 0 d2 8.356×105s- 1cm- 2
Mpc
6
FIG. 2.—Overdensitesaroundasourceofagivenmass,aslabeled,versusLagrangiandistancefromthecenterofthesource(inunitsoftheLagrangianradius
¯
ofthehalo):themeanδissolid(coloredblueinelectronicedition),thedensityattheboundary,δ,isshort-dashed(coloredredinelectronicedition),andδclumpis
long-dashed(coloredgreeninelectronicedition).(SeetheelectroniceditionoftheJournalforthecolorversionofthisfigure.)
duringitsphotoevaporationisgivenby We begin by defining the average photon consumption rate
perminihaloatom
ξ(M,z,F )≡1+φ (M)φ (z)φ (F) (31)
0 1 2 3 0 MmaxdMdn(M,z)Mξ(M,z,F )
whereφ1(M)≡A(M7B+Clog10M7),φ2(z)≡ G+H(1+z)/10 and ξ¯nb,1(z,F0)≡ RMmin MmaxddMM dn(M,z)M 0 , (32)
φ3(F0)≡F0D+Elog10F0. Here the factors A(cid:2)-G are dependen(cid:3)t on where dn(M,z) is the PS massRMfmuinnction odfMhalos, if we assume
the spectrum of the ionizing sources: A = (4.4, 4.0, 2.4), B dM
thattheminihalosatagivenredshiftzjustformedatthatred-
=(0.334, 0.364, 0.338),C = (0.023, 0.033, 0.032), D =(0.199,
shift. If,ontheotherhand,weassumethatminihalosatzhada
0.240,0.219),E =(-0.042,-0.021,-0.036),G=(0.,0.,0.1),H=(1,
distributionofformationredshiftsz ,withz ≥z,then
1,0.9),forthecasesinwhichtheionizingspectrumistakento f f
bea5×104KblackbodyrepresentingPop.IIstars,aQSO-like MmaxdM ∞dz d2n(M,zf) Mξ(M,z ,F )
preopwreesre-lnatwingspPeocptr.uImIIswtaitrhs,srleosppeecotfiv-e1ly.8., or a 105K blackbody ξ¯nb,2(z,F0)≡ RMmin MMmminaRxzdM fz∞ddMzdfzfd2dnM(Mdz,zff) M f 0 . (33)
In order to use this result in equation (27), the quantity In both these equations, the limit M is the minimum mini-
R R min
ξ(M,z,F0) must first be averaged over the mass function of halomass(whichweassumeheretobetheJeansmassatthat
minihalos at rI(t) on the undisturbedside of the I-frontas the epoch), while Mmax =M(Tvir =104K) is the halo mass at that
H II region evolves with time. Since ξ(M,z,F0) depends on epoch for which Tvir = 104 K. In equation (33) we have ap-
F0(t), which also dependson rI(t) accordingto equation (29), proximatelyaccountedfor the distribution of minihaloforma-
equations(27)and(29)arecoupledandmustbesolvedsimulta- tiontimes,bytakingthederivativeofthemassfunction,which
neously. Weshallconsiderthreeanalyticalapproximationsfor glossesoverthefactthatthechangeinthisfunctionatagiven
the m¯inihalo mass function when averaging ξ(M,z,F0) to ob- massMincludesbothapositivecontributionfromhaloswhose
tainξ. Thefirsttwoapproaches,describedin§2.2.1,arebased masses have increased to M from lower values, as well as a
onthewell-knownPress-Schechter(PS)approximationforthe negativecontributionfromhaloswhosemasseshaveincreased
massfunctionaveragedoverallspaceatagivenredshift(Press fromM tohighervalues. Theerrorintroducedbythisapprox-
&Schechter1974).Weshallrefertothese,whichdependupon imationissmall,however(eg.Kitayama&Suto1996),andis
zandF0,butnotuponrI(t)orthesourcehaloproperties,as“un- justifiedgiventheotheruncertaintiesinvolved.
biasedminihalo”averages.Thethirdapproximation,described
in§2.2.2,isbaseduponanextensionofthePSapproachwhich 2.2.2. TheAveragePhotonConsumptionRateforBiased
takes accountof the spatial correlationbetween the minihalos Minihalos.
and the central source halos, as described by Scannapieco & To calculate the biased distribution of minihalos about a
Barkana(2002). In thislastapproximation,which we referto given source, we employ an analytical formalism that tracks
¯
as the “biased minihalo” average, ξ not only depends upon z the correlated formation of objects. Our approach, described
andF0,butalsoonrI(t)andthesourcehalomass. in detail in Scannapieco & Barkana (2002), extends the stan-
dardPSmethodusingasimpleapproximationtoconstructthe
bivariatemass functionof two perturbationsof arbitrarymass
2.2.1. TheAveragePhotonConsumptionRateforUnbiased
and collapse redshift, initially separated by a fixed comoving
Minihalos.
distance(seealsoPorcianietal. 1998). Fromthisfunctionwe
7
canconstructthe numberdensityofminihalosofmassM that Thetotalionizingphotonoutputofasource,N ,anditstime
γ
form at an initial redshiftz at a comovingdistance r from the evolutiondependonthemassofthehosthaloM ,photonpro-
s
sourcehaloofmassM andformationredshiftz : ductionper stellar baryonN, star-formationefficiency f and
s s i ∗
ionizing photon escape fraction f . We then define the total
dn (M,z,r|M ,z )= dMd2dnMs(M,z,Ms,zs,r), (34) ionizingphotonoutputpersourceeasctomthatescapesthesource
dM s s dn (M ,z ) haloas:
dMs s s fγ = f∗fescNi, (37)
where dn (M ,z ) is the usual PS mass function and
dMs s s thusthe sourceemitsa totalof Nγ = fγMΩb/(µmp) duringits
dMd2dnMs(M,z,Ms,zs,r) is the bivariate mass function that gives lifetime,whereµmpisthemeanmassperatom.
the product of the differential number densities at two points Thestar-formationefficiency f∗ andionizingphotonescape
separatedbyaninitialcomovingdistancer,atanytwomasses fraction f arehighlyuncertainingeneral,andevenmoreso
esc
and redshifts. Note that this expression interpolates smoothly forthehigh-redshiftgalaxiesresponsibleforreionization.Their
between all standard analytical limits: reducing, for example, estimatedvaluesvarybyseveralordersofmagnitudebetween
tothestandardhalobiasexpressiondescribedbyMo&White different observational and theoretical estimates (Leitherer et
(1996)inthelimitofequalmasshalosatthesameredshift,and al. 1995; Ricotti & Shull 2000; Heckman et al. 2001; Stei-
reproducingtheLacey&Cole(1993)progenitordistributionin del et al. 2001; Tan & McKee 2001). For simplicity, and
thelimitofdifferent-masshalosatthesamepositionatdifferent since the principle aim of this investigationis to show the ef-
redshifts. FurtherdetaledchecksofthemethodagainstN-body fect of small-scale structure rather than model the sources in
simulations were presented in Iliev et al. (2003) and Scanna- detail, we assume that each source produces a fixed number
pieco&Thacker(2005). Notealsothatinadoptingthisdefini- f of ionizing photons which escape from the source galaxy
γ
tionweareeffectivelyworkinginLagrangianspace,suchthat per atom in the source during the source’s lifetime. The ion-
ristheinitialcomovingdistancebetweentheperturbations.As izing photon production per atom for Pop. II low-metalicity
ashorthandwedefine ddnmMh,s(z,r)≡ ddMn(M,z,r|Ms,zs).Withthis stars with a Salpeter IMF is Ni = 3000- 10000 (Leitherer et
definitionthebiasedvaluesoftheaveragephotonconsumption al.1999). Zero-metalicity,massivePop.IIIstars, onthe other
perminihaloatomcorrespondingtothe twounbiasedcasesin hand, are estimated to produce values of Ni that rise sharply
equations(32)and(33),respectively,are withmassfrom25,000to80,000asstellarmassincreasesfrom
10M to50M ,thengraduallyreachapeakof90,000at120
⊙ ⊙
ξ¯ (z,F ,r ,M )= MMmminaxdMddnmMh,s(z,rI)Mξ(M,z,F0) (35) M⊙,andfinallydeclineslowlyto80,000by500M⊙ (Schaerer
b,1 0 I s R MMmminaxdM ddnmMh,s(z,rI)M 2u0al0P2;opT.uImIIlisntsaorsn,haVveenkhaigtehsearnv&aluSehsuolfl 2fγ00th4a)n. PWohpi.leIIisntdairvsiodf-
and R thesamemass,anon-trivialpartoftheincreasefromPop.IIto
Pop.IIIquotedabovereflectsthefactthattheassumedPop.II
MmaxdM ∞dz d2nmh,s(z ,r )Mξ(M,z ,F )
ξ¯ (z,F ,r ,M )= Mmin z f dMdzf f I f 0 IMF has many low-mass stars, which are inefficient ionizing
b,2 0 I s R MmaRxdM ∞dz d2nmh,s(z ,r )M, sources, while the Pop. III IMF is often hypothesizedto con-
Mmin z f dMdzf f I tainonlymassivestars.
(36)
withr theLagrangianraRdiusofthReI-front. Assuming, conservatively, that Ni ≥4000 for Pop. II stars
I and N ≥ 25,000 for Pop. III stars, and taking moderate
We cananticipatehowimportantthisbiaseffectislikelyto i
fiducial values for the photon escape fraction and star for-
be in determining the average minihalo consumption rate by
consideringitsimpactontheminihalocollapsedfractioninthe mation efficiency of fesc = 0.1 and f∗ = 0.1, yields fγ ≥
vicinity of a given source halo. In Figure 3, we plot the av- (40,250)(fesc/0.1)(f∗/0.1)for(Pop.II, Pop.III),respectively.
Weshallfurtherassumethatthetime-dependenceofthisioniz-
erage(unbiased)minihalocollapsedfractionversusthebiased
ingphotonoutputischaracteristicofastarburstwithaphoton
value,whichvarieswithdistancefromthesourcehalo,fortwo
luminosity
halomasses108M⊙ and1011M⊙,atredshiftsz=7andz=15.
Thesehalomassesandredshiftsillustratetherangeofbehavior N˙ = f α- 1 MΩb × 1 - α t≤ts (38)
enxopiseectienditdiaulricnognrdeiitoionnizsatfioorn.oIunrteΛrCmDsoMfthcoesGmaoulsosgiaicna-lramndoodmel-, γ γ α µmpts ( tts t>ts,
where α=4.5, i.e. we assume that(cid:16)th(cid:17)e source is steady for a
108M⊙haloscorrespondtofluctuationsthatare3.2σ(1.6σ)at
timet ,afterwhichthephotonfluxdecreasesaspoweroftime
z=15(7),respectivelywhile1011M⊙ halosare6.4σ(3.2σ)at s
(Haiman&Holder2003). Heret isthecharacteristictimefor
thesesameredshifts. s
asourcetofade,essentiallythetypicalsourcelifetime. Forin-
Thedistancebetweensourcesandminihalosismeasuredin
dividualmassivestarst ≈3Myr,butstarburstscouldinprin-
termsofthecomovingLagrangian(i.e. unperturbed)radiusof s
ciple last significantlylonger, thuswe considerboththe cases
agivenmassshellsurroundingthecentralsource,inunitsofr ,
0 t =3 Myr and t =100 Myr. We assume that the He correc-
the Lagrangianradiusof the source. Accordingto this figure, s s
tionχ is1.08forthesofterPop.IIspectrumand1.16forthe
the bias can be significant for minihalos located in the range eff
harderPop.IIIandQSOspectra.
1<r/r <10.Fortypicalsourcehalos,infact,thebiasedcol-
∼ 0∼
lapsed fraction in the neighborhood of the source hardly de-
2.4. ResultsforIndividualHIIRegions
clines with increasing redshift, in contrast with the unbiased
collapsed fraction which declines by a factor of more than 3 Wepresenttheresultsofournumericalsolutionofthespher-
betweenz=7andz=15. ical I-front evolution equations from §2.2 for two illustrative
cases. Sampleresultsfortwosourcehalos,a108M halothat
⊙
turns on at z=15 and a 1011M halo that turns on at z=7,
2.3. TheIonizingPhotonLuminosityoftheCentralSource ⊙
both with f =250,t =3 Myr, and Pop. II spectra, are given
γ s
8
FIG. 3.—Biasedcollapsedfractionofbaryonsinminihalos, fcoll,MH,asafunctionoftheLagrangiandistancefromthesourcehalo(inunitsofthesourcehalo
Lagrangianradius)forsourcesofmasses108M⊙and1011M⊙andredshiftsz=15andz=7,asindicated(solid).Forreferencewealsoshowtheunbiasedcollapsed
fractionofbaryonsinminihalosatthecorrespondingredshifts(dashed).(SeetheelectroniceditionoftheJournalforthecolorversionofthisfigure.)
FIG.4.—Bottom:TheevolutionoftheLagrangianvolumeoftheHIIregionaboutasinglesourceof(left)mass108M⊙thatturnsonatzi=15,or(right)mass
1011M⊙thatturnsonatz=7. BothsourceshavePop.IIstellarspectraandlifetimesofts=3Myr,duringwhichtheyproduceatotalof fγ=250photons/atom.
Shownarethecasesofnominihalos(solid),unbiasedminihalos(dotted),andbiasedminihalos(dashed)forIGMclumpingfactors(toptobottomineachcase)
C=0(i.e.norecombinationsinIGMgas),1(meanIGM),and10(clumpedIGM).Vmaxisthemaximumionizedvolumereachedduringthelifetimeofthesourcein
theC=0casewithnominihalos,asdefinedinthetext.Top:Ratiosoftheionizedvolumeswithunbiasedandbiasedminihalostothenominihalocase,aslabeled,
forC=1(solid)and10(dashed).(SeetheelectroniceditionoftheJournalforthecolorversionofthisfigure.)
9
FIG.5.—EvolutionofindividualHIIregionswiththesamesourceparametersasinFigure4.Top:Thecorrectionfactorβi/χeffduetominihalosforthenumber
ofionizedphotonsconsumedperatomthatcrossestheI-front,forbiased(dashed)andunbiasedminihalos(dotted)forC=0,1and10.Bottom:Comovingradiusof
theHIIregionfornominihalos(solid),unbiased(dotted)andbiasedminihalos(dashed)forC=0,1and10.(SeetheelectroniceditionoftheJournalforthecolor
versionofthisfigure.)
FIG. 6.—EvolutionofthedimensionlessionizingphotonfluxF0(t)atthecurrentpositionoftheI-front. SamenotationasinFig.4. Bottomsetof(initially-
overlapping)curvesareforC=0,middleset-forC=1andtopset-forC=10,respectively.(SeetheelectroniceditionoftheJournalforthecolorversionofthis
figure.)
FIG.7.—EvolutionoftheIGMclumpingfactorinΛCDMfromnumericalN-bodysimulations,forthegasoutsidehalos.
10
inFigures4-6. Inbothcaseswedisplayresultsforavarietyof shiftbasedonthePSformalism. We thencalculatetheevolu-
clumping factors (C=0,1,10) and successive approximations tionoftheHIIregioncreatedbyeachsourcehalo,asdiscussed
ofnominihalos,unbiasedminihalos[asgivenbyeq.(32)],and in section §2.2 and §2.3. Finally, we add the volumes of all
biased minihalos[as given by eq. (35)]. In Figure 4 we show theseHIIregions. Thisgivesthetotalionizedmassfractionat
the evolutionof the ionizedvolume. All volumesare normal- eachredshift,accordingto
mizeindihtoalVomsaaxn=dNnoγ/renc0Ho,mi.be.intahteiovnosluinmteheiognaisz.ediftherewereno f = ∞ dM ∞dz′d2ns(M,z′)V(z,z′,M,N˙ ), (39)
I,M dMdz′ I γ
The increase of the clumping factor from C = 1 to C =10 ZMs,min Zz
significantlydecreasesthemaximumionizedvolumeachieved where d2ns(M,z′) is the PS distribution of the source halos, and
by the H II regions, with and without minihalos, especially dMdz′
we assume here that source halos have masses M ≥M =
at higher redshift, as shown for the z =15 case in Figure 4. s s,min
M(104K),the massofhaloswithT =104 K.Theuniverseis
For similar reasons, H II regionsaroundsourcehalos in over- vir
fullyionizedwhentheHIIregionsoverlap,whichcorresponds
dense areas of the IGM must also be smaller than those in
to f =1.
the mean IGM. ForC=0 and no minihalos,V /V = f , so I,M
max 0 γ
r /r = f1/3. Since more realistic cases, with C > 1 and
I,max 0 γ 3.1. Time-DependentClumpingFactorofIGMOutsideHalos
minihalos,allhaveV <V ,itmustbetruethatr /r < f1/3,
I max I 0 γ As in the individual-source results above, we calculate the
in general. According to Figure 2, the overdensity δ >1
clump evolution of the ionized mass fraction for the cases of no
outsidethesourcehaloforallr/r <20,whileδ >2forall
0∼ clump minihalos, unbiased minihalos, and biased minihalos, and for
r/ro∼<4(6),forsourcehalosofmass1011(108)M⊙. Assuch, clumpingfactors ofC=0,1 and 10. In addition, we consider
theIGMrecombinationcorrectiontotheionizedvolumeatany themorerealisticcaseofaclumpingfactorthatevolvesintime
epochmustbesignificantlyenhancedbythislocaloverdensity as more and more structure forms, which was obtained from
for any fγ ∼<1000.In short, fora realistic rangeof fγ values, numerical N-body simulations by the particle-particle/particle
cosmologicalHIIregionsfromstellarsourcesaregenerallynot mesh (P3M) method, with a computational box size of 1 co-
largerthantheinfallregionsassociatedwiththeirsourcehalos. movingMpcwith1283particlesand2563cells,corresponding
Thisistruewithandwithoutminihalos. to a particle mass of 2×104M⊙ (see Shapiro 2001 and Iliev
Nextweconsidertheeffectofaddingminihalos. According et al. 2003 for details on the simulations). This box size was
toFigure4,forthe108M⊙ sourcehalo,theunbiasedminihalo chosensoastoresolvethescaleswhichcontributemostofthe
distributiondecreasestheionizedvolumeby∼20%compared clumping-onsmallerscalesthegaswouldbeJeanssmoothed,
tothenominihaloscase,whilewhenweaccountforthemini- whileonlargerscalesthedensityfluctuationsarestilllinearand
halobiasaboutthesource,theionizedvolumeisdecreasedby donotcontributemuchtotheoverallclumpingfactor. There-
afactorof2relativetothenominihaloscase. Forthe1011M⊙ sult for the IGM clumping factor is plotted in Figure 7. This
sourcehalo,thenetionizedvolumewhenminihalosarepresent clumpingfactorexcludesthematterincollapsedhalos,sinceas
(biasedornot)isabout65%ofthevolumeinthecasewithout we discussed above, these are self-shieldedand we treatthem
minihalos. separately.TheevolutionofthisIGMclumpingfactorwithred-
In the top panel of Figure 5, we plot the factor βi/χeff by shiftiswell-fitby
whichminihaloevaporationbooststhenumberofionizingpho-
tonsconsumedattheI-frontperatomthatcrossesthefront(in C(z)=17.6e- 0.10z+0.0011z2. (40)
the IGM and in minihaloscombined). In the bottom panel of
this figure, we plotthe comoving(Eulerian)radiusof the cor- 3.2. TheGlobalConsumptionofIonizingPhotonsDuringthe
respondingHIIregions. Forthe108M⊙ sourcehaloatzi=15, EpochofReionization
ignoring the minihalo bias (bottom lines) seriously underesti-
Thereionizationoftheuniversewascompletewhenthevol-
matesthephotonconsumptionbyafactorof>2ascompared
∼ ume of all the H II regions at some epoch equaled the total
tothebiasedminihalos(toplines).Furthermore,theoverallef-
volume.Wecallthattheepochofoverlap,atredshiftz .Gen-
ov
fectofaddingminihalosistoincreasethephotonconsumption
eralizing Shapiro & Giroux (1987)we define a useful dimen-
by∼100%.Forthe1011M⊙haloatzi=7,thereisasimilarin- sionless ratio of total number of ionizing photonsemitted per
creaseinthephotonconsumption,butlittledifferencebetween
hydrogenatomintheuniverseuntiloverlapatz=z givenby,
ov
thebiasedandunbiasedresults.
Finally we plotthe dimensionlessflux F0 at the currentpo- ζ = fγΩb zovdz ∞ M dM t(zov)dt′d2n0x(Ms,z′)dz′,
sition of the I-front,in Figure 6. This is importantas a check ov n0µm t s s dM dz′ dt′
H psZ∞ ZMs,min Zt(z) s
of our assumptions, which incorporate simulation results for (41)
minihalo evaporation for a range of fluxes, 10- 2 ≤F0 ≤103. where d2n0x(M,z)/dMdz is the comoving differential number
ThevalueofF0hasasignificantimpactontheionizingphoton densityofthesourcehalosofmassMformedatredshiftz.
consumption of minihalos, which is higher for higher values
of F0. In bothcases, the flux starts fairly high (F0∼>100), but 3.3. ElectronScatteringOpticalDepthThroughtheReionized
dropstoF0∼1bythetimethesourcestartstofade.Atfirstthis Universe
dropisduemainlytogeometricdilution,butlater,recombina-
For any given reionization history, the mean optical depth
tions in the ionizedvolume acceleratethis decrease according
along a line-of-sight between an observer at z=0 and a red-
toequation(29).
shiftzduetoThomsonscatteringbyfreeelectronsinthepost-
3. TOWARDS A MORE GLOBALPICTURE recombinationuniverseisgivenby
To construct models of the global reionization process, we 0 dt
τ (z)=cσ dz′n (z′) , (42)
first calculate the number density of source halosat each red- es T e dz′
Zz