Unusually Large Fluctuations in the Statistics of Galaxy Formation at High Redshift

0.2 MB·English

by  Rennan Barkana

DRAFTVERSIONFEBRUARY2,2008
PreprinttypesetusingLATEXstyleemulateapjv.9/08/03
UNUSUALLYLARGEFLUCTUATIONSINTHESTATISTICS
OFGALAXYFORMATIONATHIGHREDSHIFT
RENNANBARKANA
SchoolofPhysicsandAstronomy,TheRaymondandBeverlySacklerFacultyofExactSciences,
TelAvivUniversity,TelAviv69978,ISRAEL;[email protected]
ABRAHAMLOEB
AstronomyDepartment,HarvardUniversity,60GardenStreet,Cambridge,MA02138;[email protected]
DraftversionFebruary2,2008
4
0 ABSTRACT
0
We showthatvariousmilestonesofhigh-redshiftgalaxyformation,suchastheformationofthefirststarsor
2
the complete reionization of the intergalacticmedium, occurred at differenttimes in differentregionsof the

r universe.Thepredictedspreadinredshift,causedbylarge-scalefluctuationsinthenumberdensityofgalaxies,
a
isatleastanorderofmagnitudelargerthanpreviousexpectationsthatarguedforasharpendtoreionization.
M
Thiscosmicscatterintheabundanceofgalaxiesintroducesnewfeaturesthataffectthenatureofreionization
and the expectationsfor future probes of reionization, and may help explain the present properties of dwarf
7
galaxies in different environments. The predictions can be tested by future numerical simulations and may
1
be verified by upcomingobservations. Currentsimulations, limited to relativelysmall volumesand periodic

boundaryconditions,largelyomitcosmicscatteranditsconsequences. Inparticular,theyartificiallyproduce
2
asuddenendtoreionization,andtheyunderestimatethenumberofgalaxiesbyuptoanorderofmagnitudeat
v
8 redshift20.
3 Subjectheadings:galaxies:high-redshift,cosmology:theory,galaxies:formation
3
0
1 1. INTRODUCTION alreadystartoutfromanenhancedlevelofdensity,andsmall-
3 Recentobservationsofthe cosmic microwavebackground scalemodesneedonlysupplytheremainingperturbationnec-
0 (Spergeletal. 2003) have confirmed the notion that the essary to reach δc(z). On the other hand, large-scale voids
h/ present large-scale structure in the universe originated from shouldcontainareducednumberofhalosathighredshift. In
thisway, theanalyticmodeldescribestheclusteringofmas-
p small-amplitude density fluctuations at early cosmic times.
sivehalos.
- Due to the natural instability of gravity, regions that were
o As gas falls into a dark matter halo, it can fragment into
denser than average collapsed and formedbound halos, first
r stars only if its virial temperature is above 104K for cool-
t on small spatial scales and later on larger and larger scales.
s Ateach snapshotof this cosmicevolution, the abundanceof ingmediatedbyatomictransitions[or 500Kformolecular
a H cooling; see, e.g., Figure 12 in Bar∼kana&Loeb (2001)].
collapsed halos, whose masses are dominated by cold dark 2
:
v matter,canbecomputedfromtheinitialconditionsusingnu- The abundance of dark matter halos with a virial tempera-
i mericalsimulationsandcanbeunderstoodusingapproximate ture above this cooling threshold declines sharply with in-
X
creasing redshift due to the exponential cutoff in the abun-
analytic models(Press&Schechter 1974; Bondetal. 1991).
r danceofmassivehalosatearlycosmictimes. Consequently,
a Thecommonunderstandingof galaxyformationis basedon
a small change in the collapse threshold of these rare halos,
thenotionthattheconstituentstarsformedoutofthegasthat
due to mild inhomogeneities on much larger spatial scales,
cooled and subsequently condensed to high densities in the
can change the abundance of such halos dramatically. The
coresofsomeofthesehalos(White&Rees1978).
modulation of galaxy formation by long wavelength modes
The standard analytic model for the abundance of ha-
of density fluctuationsis therefore amplified considerablyat
los(Press&Schechter1974;Bondetal.1991)considersthe
highredshift. Inthispaperweshowthatthisresultsinmajor
small density fluctuationsat some early, initial time, and at-
newpredictionsforhigh-redshiftobservations. Theimplica-
temptstopredictthenumberofhalosthatwillformatsome
tions are particularly significant for cosmic reionization and
latertimecorrespondingtoaredshiftz. First,thefluctuations
allobservationalprobesofthisepoch.
are extrapolatedto the presenttime using the growthrate of
Thispaperisorganizedasfollows. In§2 wequantifythe
linearfluctuations,andthentheaveragedensityiscomputed
scatter in the statistics of galaxyformationproducedby this
in spheres of various sizes. Whenever the overdensity (i.e.,
amplificationeffect. Wefirstexplainin§2.1thebasicphysi-
thedensityperturbationinunitsofthecosmicmeandensity)
calideasandimplicationsusingthewell-establishedextended
in a sphere rises above a critical threshold δ (z), the corre-
c Press-Schechter model. We then present in § 2.2 a simple
sponding region is assumed to have collapsed by redshift z,
idea that yields a much more accurate model that fits an ar-
forminga halo outof all the mass thathad been includedin
ray of previoussimulationsat low redshift. We demonstrate
theinitialsphericalregion. Inanalyzingthestatisticsofsuch
the qualitative correctness of our basic assumptions as well
regions, the model separates the contribution of large-scale
asthequantitativeaccuracyofourmodelbymatchingresults
modes from that of small-scale density fluctuations. It pre-
fromrecentsimulationsat highredshift. Since high-redshift
dicts that galactic halos will form earlier in regions that are
galaxiesprovidetheUVphotonsthatleadtothereionization
overdenseonlargescales (Kaiser1984;Bardeenetal. 1986;
of the intergalactic medium (hereafter IGM), a large scatter
Cole&Kaiser1989;Mo&White1996),sincetheseregions
2
is also expected in the reionization redshift within different
regionsintheuniverse.Weconsiderthisscatterandthemod-
ifiedcharacterofreionizationin§3.1,andshowin§3.2that
existingnumericalsimulationsdonotincludefluctuationson
sufficientlylargescalesat highredshift. In § 3.3we discuss
theobservationalimplicationsofthelargecosmicscatterex-
pected at high redshift. Finally, we summarize our main re-
sultsin§4.
2. HALOMASSFUNCTIONINDIFFERENTENVIRONMENTS
2.1. BasicModel:AmplificationofDensityFluctuations
Galaxies at high redshift are believed to form in all halos
above some minimum mass M that depends on the effi-
min
ciency of atomic and molecular transitions that cool the gas
within each halo. This makes useful the standard quantity
of the collapse fraction F (M ), which is the fraction of
col min
massinagivenvolumethatiscontainedinhalosofindivid-
ualmassM orgreater. Ifwe setM tobe theminimum
min min
halomassin whichefficientcoolingprocessesare triggered,
then F (M ) is the fraction of all the baryons in the uni-
col min
versethatlieingalaxies. Inalarge-scaleregionofcomoving
¯
radius R with a mean overdensity δ , the standard result is
R
(Bondetal.1991) FIG. 1.—Biasinthehalomassdistributioninsimulations. Weshowthe
δ (z)- δ¯ aexmporuesnsteodfamsaasfsraccotniotaninoefdthientaoltlalhmaloasssoifniandgiivviednuavlolmumases.MThmiisncourmgurleaattievre,
Fcol(Mmin)=erfc(cid:20)√2[S(cRmin)- RS(R)](cid:21) , (1) fWraectcioonnsFidcoelr(Mtwmoin)caissesshoowfnreadsshaifftunanctdiosnimofultahteiomnibnoimxusmizeh,alnoammealsyszM=mi7n,.
whereS(R)isthevarianceoffluctuationsinspheresofradius lbox=6Mpc(uppercurves), andz=20, lbox=1Mpc(lower curves). At
eachredshift,wecomparethetrueaveragedistributionintheuniverse(dotted
R, and S(R ) is the variance in spheres of radius R cor- curve)tothebiaseddistribution (solidcurve)thatwouldbemeasuredina
respondingmtion the region at the initial time that contmaiinned a simulationboxwithperiodicboundaryconditions(forwhichδ¯Risartificially
settozero).
mass M . In particular, the cosmic mean value of the col-
min
lapse fraction is obtained in the limit of R by setting
¯ →∞
δ and S(R)to zero in this expression. Throughoutthis sec-
R
tionourresultsassumethisstandardmodel,knownastheex- andlatereionization(z=7).Weconsideraresolutionequalto
tendedPress-Schechtermodel,whichweapplytoauniverse thatof state-of-the-artcosmologicalsimulationsthatinclude
with cosmological parameters matching the latest observa- gravityandgashydrodynamics. Specifically,weassumethat
tions [specifically, the running index model of Spergeletal. thetotalnumberofdarkmatterparticlesin the simulationis
(2003)]. Wheneverwe considera cubic region, we estimate N =3243, and that the smallest halo that can form a galaxy
itshaloabundancebyapplyingthemodeltoasphericalregion must be resolved into 500 particles; Springel&Hernquist
ofequalvolume. Notealsothatweconsistentlyquotevalues (2003)showedthatthisresolutionisnecessaryinordertode-
ofcomovingdistance,whichequalsphysicaldistancetimesa terminethestarformationrateinanindividualhaloreliablyto
factorof(1+z). withinafactoroftwo. Therefore,ifweassumethathalosthat
Our results are based on a simple idea. At high redshift, cool via molecular hydrogen must be resolved at z=20 (so
galactic halos are rare and correspond to high peaks in the thatM =7 105M ), andonlythosethatcoolviaatomic
min ⊙
Gaussian probability distribution of initial fluctuations. A transitionsmu×stberesolvedatz=7(sothatM =108M ),
min ⊙
modest change in the overalldensity of a large region mod- thenthe maximumboxsizes thatcancurrentlybe simulated
ulates the threshold for high peaks in the Gaussian density are l =1 Mpc and l =6 Mpc at these two redshifts, re-
box box
field,sothatthenumberofgalaxiesisexponentiallysensitive spectively.
tothismodulation.Thisamplificationoflarge-scalemodesis Ateachredshiftweonlyconsidercubicboxeslargeenough
responsibleforthelargestatisticalfluctuationsthatwefind. sothattheprobabilityofformingahaloonthescaleoftheen-
¯
Innumericalsimulations,periodicboundaryconditionsare tireboxisnegligible. Inthiscase,δ isGaussiandistributed
R
usuallyassumed,andthisforcesthemeandensityofthebox with zero mean and variance S(R), since the no-halo condi-
toequalthecosmicmeandensity. Theabundanceofhalosas tion√S(R) δ (z)impliesthatatredshiftztheperturbation
c
≪
afunctionofmassisthenbiasedinsuchabox(seeFigure1), onthescaleRisstillinthelinearregime. Wecanthencalcu-
since a similar region in the real universe will have a distri- latetheprobabilitydistributionofcollapsefractionsinabox
¯
bution of different overdensities δ . At high redshift, when of a givensize (see Figure2). Thisdistribution corresponds
R
galaxies correspondto high peaks, they are mostly foundin to a real variation in the fraction of gas in galaxies within
regions with an enhanced large-scale density. In a periodic differentregionsoftheuniverseatagiventime. Inanumer-
box, therefore, the total numberof galaxiesis artificially re- ical simulation, the assumption of periodic boundary condi-
duced,andthe relativeabundanceofgalactichaloswith dif- tionseliminatesthelarge-scalemodesthatcausethiscosmic
ferentmassesisartificiallytiltedinfavoroflower-masshalos. scatter. Note that Poisson fluctuations in the number of ha-
We illustrate our resultsfor two sets of parameters,one cor- loswithintheboxwouldonlyaddtothescatter,althoughthe
respondingtothefirstgalaxiesandearlyreionization(z=20) variationswehavecalculatedaretypicallythedominantfac-
andtheothertothecurrenthorizoninobservationsofgalaxies tor. For instance, in our two standard examples, the mean
3
FIG. 2.—Probabilitydistributionwithinasmallvolumeofthetotalmass FIG. 3.—Cosmicscatterandnumericalbias,expressedasthechangein
fraction ingalactic halos. Thenormalized distribution ofthelogarithm of redshiftneededtogetthecorrectcosmicmeanofthecollapsefraction. We
thisfractionFcol(Mmin)isshownfortwocases: z=7,lbox=6Mpc,Mmin= show the1- σ scatter (about the biased value) inthe redshift ofreioniza-
d1pa0asn8hMeel)d⊙. vI(neurptepiacecarhlplcaiannseee)l,,),tahalenodnvgazl=wuei2th0in,tlhabeopxev=railo1udeMictphbcao,txMw(moδ¯uiRnld==b70e)×ies1x0psh5eoMcwte⊙dn((gbcioevntettnormaal itoeifosn(,1(bo+ortzta)o]nmyinoptpahenererilo)pd,hiceanssoiwmmeeulnllaotanisotnhthabetodxreeepdses(hnuidfpstpoebrniapsthaen[eemxl)pa.rsesTssfhereadcbtaiiosansaiisnfrsgahcaotliawoxnn-
plus1- σ(rightdashedline)oraminus1- σ(leftdashedline)fluctuationin forMmin =7×105M⊙ (solid curve), Mmin =108M⊙ (dashed curve), and
themeandensityofthebox. Alsoshownineachcaseisthemeanvalueof Mmin =3×1010M⊙ (dotted curve). The bias is always negative, and we
Fcol(Mmin)averagedoverlargecosmologicalvolumes(solidverticalline). showitsabsolutevalue. Whenexpressedasashiftinredshift,thescatteris
independentofMmin.
expected number of halos in the box is 3 at z=20 and 900
correspondingtoStoS+dS.Thehaloabundanceisthen
atz=7, resultinginPoissonfluctuationsofafactorofabout
2 and 1.03, respectively, comparedto the clustering-induced dn ρ¯ dS
scatterofafactorofabout16and2inthesetwocases. = 0 f(δ (z),S), (3)
c
dM M dM
Within the extended Press-Schechter model, both the nu- (cid:12) (cid:12)
(cid:12) (cid:12)
merical bias and the cosmic scatter can be simply described where dn is the comovin(cid:12)g nu(cid:12)mber density of halos with
in terms of a shift in the redshift (see Figure 3). In general, masses in the range M (cid:12)to M(cid:12) +dM. In the model of
¯
aregionofradiusRwithameanoverdensityδR willcontain Press&Schechter(1974),
a differentcollapsefractionthanthe cosmicmeanvalueata
givenredshiftz. However,atsomewrongredshiftz+∆zthis f (δ (z),S)= 1 νexp - ν2 , (4)
smallregionwillcontainthecosmicmeancollapsefractionat PS c √2πS 2
(cid:20) (cid:21)
z. Athighredshifts(z>3),thisshiftinredshiftcanbeeasily
derivedfromeq.1tobe whereν=δc(z)/√Sisthenumberofstandarddeviationsthat
thecriticalcollapseoverdensityrepresentsonthemassscale
δ¯ S(R) McorrespondingtothevarianceS.
∆z= R - (1+z) 1- 1- , (2) However,thePress-Schechtermassfunctionfitsnumerical
δ0 ×" s S(Rmin) # simulationsonlyroughly,andinparticularitsubstantiallyun-
whereδ δ (z)/(1+z)isapproximatelyconstantathighred- derestimatestheabundanceoftherarehalosthathostgalaxies
0≡ c athighredshift. Thehalo massfunctionofSheth&Tormen
shifts(Peebles1980),andequals1.28forourassumedcosmo-
(1999, see also Shethetal. 2001) adds two free parameters
logicalparameters. Thus, in ourtwo standard examples, the
biasis-2.6atz=20and-0.4atz=7,andtheone-sided1- σ that allow it to fit numerical simulations much more accu-
rately (Jenkinsetal. 2001). We note that these simulations
scatteris2.4atz=20and1.2atz=7.
followed very large volumes at low redshift, so that cosmic
scatter did not compromise their accuracy. The matching
2.2. ImprovedModel:MatchingNumericalSimulations
massfunctionisgivenby
In this subsection we developan improvedmodelthat fits
the results of numerical simulations more accurately. The ν a′ 1 a′ν2
model constructs the halo mass distribution (or mass func- fST(δc(z),S)=A′S 2π 1+(a′ν2)q′ exp - 2 , (5)
tion); cumulative quantities such as the collapse fraction or r (cid:20) (cid:21) (cid:20) (cid:21)
the total number of galaxies can then be determined from it withbest-fitparameters(Sheth&Tormen2002)a′=0.75and
via integration. We first define f(δ (z),S)dS to be the mass q′=0.3,andwherenormalizationtounityisensuredbytaking
c
fraction contained at z within halos with mass in the range A′=0.322.
4
In order to calculate cosmic scatter we must determine
the biased halo mass function in a given volume at a given
mean density. Within the extended Press-Schechter model
(Bondetal. 1991), the halo mass distribution in a region of
¯
comovingradiusRwithameanoverdensityδ isgivenby
R
fbias- PS(δc(z),δ¯R,R,S)= fPS(δc(z)- δ¯R,S- S(R)). (6)
The corresponding collapse fraction in this case is given
simply by eq. (1). Despite the relatively low accuracy of
the Press-Schechter mass function, the relative change is
predicted rather accurately by the extended Press-Schechter
model. In other words, the prediction for the halo mass
function in a given volume compared to the cosmic mean
mass function provides a good fit to numerical simula-
tions over a wide range of parameters (Mo&White 1996;
Casas-Mirandaetal.2002;Sheth&Tormen2002).
For our improved model we adopt a hybrid approach that
combinesvariouspreviousmodelswitheachappliedwhereit
has been foundto closely match numericalsimulations. We
obtain the halo mass function within a restricted volume by
startingwiththeSheth-Tormenformulaforthecosmicmean
massfunction,andthenadjustingitwitharelativecorrection
basedontheextendedPress-Schechtermodel.Inotherwords,
FIG.4.—Halomassfunctionathighredshiftina1Mpcboxatthecosmic
weset meandensity. Ourhybridmodelprediction (solidlines)iscompared with
¯ thenumberofhalosabovemass7×105M⊙measuredinthesimulationsof
fbias(δc(z),δR,R,S)= Yoshidaetal. (2003, data points are taken from their Figure 5). Thecos-
f (δ (z)- δ¯ ,S- S(R)) micmeanofthehalomassfunction(dottedlines)deviatessignificantlyfrom
f (δ (z),S) PS c R . (7) thesimulatedvalues,sincetheperiodicboundaryconditionswithinthefinite
ST c × f (δ (z),S) simulationboxartificiallysettheamplitudeoflarge-scalemodestozero.Our
(cid:20) PS c (cid:21) hybridmodelstartswiththeSheth-Tormenmassfunctionandappliesacor-
As noted, this model is based on fits to simulations at low rectionbasedontheextendedPress-Schechtermodel;indoingso,itprovides
redshifts, but we can check it at high redshifts as well. Fig- abetterfittonumericalsimulationsthanthepureextendedPress-Schechter
ure4showsthenumberofgalactichalosatz 15- 30intwo mcoosdmeoll(odgaischaeldpalirnaemse)teursse,dthinesthcealep-rienvviaoruiasnfitgΛuCreDs.MWmeocdoenlsoidfeYrotswhoidsaeetstaolf.
∼
numerical simulations run by Yoshidaetal. (2003), and our (2003) (upper curves), and their running scalar index (RSI) model (lower
predictionsgiventhecosmologicalinputparametersassumed curves).
byeachsimulation. Theclosefittothe simulateddata(with
noadditionalfreeparameters)suggeststhatourhybridmodel
(solidlines)improvesontheextendedPress-Schechtermodel
(dashedlines),andcanbeusedtocalculateaccuratelythecos- tests carried out by Springel&Hernquist (2003). They pre-
micscatterinthenumberofgalaxiesatbothhighandlowred- sented a large array of numerical simulations of galaxy for-
shifts. The simulated data significantly deviate from the ex- mation run in periodic boxesover a wide range of box size,
pectedcosmic mean [eq. (5), shown by the dottedline], due mass resolution, and redshift. In particular, we can identify
to the artificial suppression of large-scale modes outside the several pairs of simulations where the simulations in each
simulatedbox. WenotethatYoshidaetal.(2003)mentioned pair have the same mass resolution but different box sizes;
thatthelackoflarge-scalemodesmightproduceasystemat- this allows us to separate the effectof large-scale numerical
icallylowhaloabundance,particularlyintheRSImodel,but biasfromtheeffectofhavingpoorly-resolvedindividualha-
theydidnotquantifythiseffect. los. Specifically,theirsimulationsZ1andR4[seeTable1in
As an additional example, we consider the highest- Springel&Hernquist(2003)]usedthesameparticlemassbut
resolutionfirststarsimulation(Abeletal.2002),whichused R4 had a box length larger by a factor of 3.375 . The sim-
l =128 kpc and M =7 105M . The first star forms ulationsQ1 and D4 are similarly related, as are Q2 and D5.
box min ⊙
×
withinthesimulatedvolumewhenthefirsthaloofmassM In each case, the smaller simulation substantially underesti-
min
orlargercollapseswithinthebox.Tocomparewiththesimu- mated the star formationrate at high redshift[see Figure 10
lation,wepredicttheredshiftatwhichtheprobabilityoffind- inSpringel&Hernquist(2003)],Z1byafactorof5atz=15
ing at least one halo within the box equals 50%, accounting comparedtoR4,Q1byafactorof3atz=8comparedtoD4,
forPoissonfluctuations.Wefindthatifthesimulationformed andQ2byafactorof3atz=10comparedtoD5.
apopulationofhaloscorrespondingtothecorrectcosmicav- We notethattherehavebeenpreviousattemptstodevelop
erage [as given by eq. (5)], then the first star should have amodelforthehalomassfunctionindifferentenvironments,
formedalready at z=24.0. The first star actually formedin sothatthemodelwouldbeconsistentwiththeSheth-Tormen
the simulation box only at z=18.2 (Abeletal. 2002). Us- massfunctionofeq.(5)whichaccuratelyfitsthecosmicmean
ing eq. (7) we can accountfor the loss of large-scalemodes mass function measured in numerical simulations. In order
beyond the periodic box, and predict a first star at z=17.8, to identify the specific requirements for such a consistency,
aclosematchgiventhelargePoissonfluctuationsintroduced we first consider the analogous case of the extended Press-
byconsideringasinglegalaxywithinthebox. Schechter model and its relation to the Press-Schechter for-
The artificial bias in periodic simulation boxes can also mulaforthecosmicmeanmassfunction.TheextendedPress-
be seen in the results of extensive numerical convergence Schechter model is consistent with the mean mass function
5
in the sense that f¯bias- PS evaluated for an infinite box (i.e., sizes. The underlying idea was that the ionized hydrogen
inthe limitwhereδR andS(R)bothvanish)yieldsthePress- (HII) regions of individual sources began to overlap when
Schechtermassfunction: the typical size of each H II bubble became comparable to
thedistancebetweennearbysources. Sincethesetwolength
fbias- PS(δc(z),0,∞,S)= fPS(δc(z),S). (8) scaleswerecomparableatthecriticalmoment,thereisonlya
Thisconditiondoesnotsuffice,however,sincethereisanad- singletimescaleintheproblem–givenbythegrowthrateof
ditionalself-consistencytestthatanyviablemodelmustsat- each bubble – and it determines the transition time between
isfy. ConsideranyfixedscaleR. Supposeweconsideravery the initial overlap of two or three nearby bubbles, to the
large number N of spheres of radius R within the universe. final stage where dozens or hundreds of individual sources
¯
The mean density δ in each sphere is determined accord- overlap and produce large ionized regions. Whenever two
R ¯
ing to a probability distribution p(δ ); we assume that R is ionizedbubbleswerejoined,eachpointinsidetheircommon
R
largeenoughso thattheprobabilityofforminga halooutof boundary became exposed to ionizing photons from both
all the mass on the scale R is negligible, and so the distri- sources,reducingtheneutralhydrogenfractionandallowing
bution is a Gaussian with zero mean and variance S(R) (see ionizing photons to travel farther before being absorbed.
also§2.1). Thenumberofgalaxiesineachsphereisgivenin Thus, the ionizing intensity inside HII regions rose rapidly,
the extendedPress-Schechtermodelby eq.(6). As N , allowing those regions to expand into high-density gas that
the halo mass function averaged over all these spheres→m∞ust had previously recombined fast enough to remain neutral
approach the cosmic mean value, and it must also approach whentheionizingintensityhadbeenlow. Sinceeachbubble
theensemble-averagedmassfunction,wheretheaveragingis coalescenceaccelerates the process, it has been thoughtthat
¯
performedovertheprobabilitydistributionofδ . Thisyields theoverlapphase hasthe characterof a phase transitionand
R
the following self-consistency requirement, which is indeed occursrapidly.Indeed,thebestsimulationsofreionizationto
satisfiedbytheextendedPress-Schechtermodel: date(Gnedin2000)foundthattheaveragemeanfreepathof
ionizingphotonsinthesimulatedvolumerisesbyanorderof
fbias- PS(δc(z),δ¯R,R,S)p(δ¯R)dδ¯R= magnitudeoveraredshiftinterval∆z=0.05atz=7.
Our results substantially modify this commonly accepted
Z
fbias- PS(δc(z),0, ,S). (9) picture for the developmentof reionization. Overlap is still
∞ expectedto occurrapidly, but only in localized high-density
Nowwe againconsiderattemptsto constructanimproved
regions, where the ionizingintensity and the mean free path
model that is consistent with the Sheth-Tormen mass func-
rise rapidly even while other distant regions are still mostly
tion. Such a model must satisfy eq. (8) (exceptwith f on
ST neutral.Inotherwords,thesizeofthebubbleofanindividual
the right-handside), and it must also satisfy eq. (9) in order
sourceis aboutthe samein differentregions(sincemostha-
to be self-consistent. The latter equation must be satisfied
loshavemassesjustaboveM ),butthetypicaldistancebe-
separatelyforeveryscaleRlargeenoughtoavoidcollapsing min
tweennearbysourcesvarieswidelyacrosstheuniverse. The
[i.e.,thatsatisfies√S(R) δ (z)]. Previousproposedmodels
c strongclusteringofionizingsourcesonlengthscalesaslarge
≪
(Sheth&Tormen 2002; Gottlöberetal. 2003) satisfied sim-
as30–100Mpcintroduceslongtimescalesintothe reioniza-
ple consistencybutnotthe self-consistencytest. Our hybrid
tionphasetransition. Thesharpnessofoverlapisdetermined
model of eq. (7) satisfies both requirements(with respect to
notbythe growthrate ofbubblesaroundindividualsources,
theSheth-Tormenmassfunction),aresultthatfollowsimme-
butbytheabilityoflargegroupsofsourceswithinoverdense
diatelyfromthefactthattheextendedPress-Schechtermodel
regionsto deliverionizingphotonsinto largeunderdensere-
also satisfies both requirements (with respect to the Press-
gions. Simplyput, the commonview assumes thatreioniza-
Schechtermassfunction). Thus,ourhybridmodelisthefirst
tionoccurredinpatchesafewMpcinsize, andthatitended
self-consistent model that is also consistent with the Sheth-
nearlysimultaneouslyinallofthem.Inreality,however,these
Tormenmassfunction,atleastwhenfluctuationsareconsid-
¯ twostatementsarecontradictory.IfthepatchesareafewMpc
eredonlargescalesRforwhichδ isGaussiandistributed.As
R in size, thenthereis a verylargespreadin theirreionization
demonstratedin this section, our modelalso matchesresults
redshifts.Conversely,ifthespreadissmall,thisimplies(from
fromawidearrayofnumericalsimulations(Abeletal.2002;
Figure3)thatthepatchesmustbefarlargerthaniscommonly
Yoshidaetal.2003;Mo&White1996;Casas-Mirandaetal.
assumed.
2002;Jenkinsetal.2001).
Notethattherecombinationrateishigherinoverdensere-
gionsbecauseoftheirhighergasdensity. Theseregionsstill
3. IMPLICATIONS
reionizefirst,though,despitetheneedtoovercomethehigher
3.1. Thenatureofreionization recombination rate, since the number of ionizing sources in
The photons of the cosmic microwave background have theseregionsisincreasedevenmorestronglyasaresultofthe
traveled to us mostly undisturbed after neutral atoms first dramaticamplificationoflarge-scalemodesdiscussedearlier.
formed in the universe at the cosmic recombination epoch.
3.2. Limitationsofcurrentsimulations
Radiationfromthefirstgenerationofstarsisthoughttohave
reionizedthehydrogenthroughouttheuniverse,transforming The shortcomings of current simulations do not amount
theIGMbackintoahotandhighly-ionizedplasma. simply to a shift of 10% in redshift and the elimination
∼
The popular view developed in the literature of scatter, for several reasons. First, the effect that we have
(Arons&Wingert 1972; Fukugita&Kawasaki 1994; identifiedcanbeexpressedintermsofashiftinredshiftonly
Shapiroetal. 1994; Haiman&Loeb 1997; Gnedin 2000; withinthecontextoftheextendedPress-Schechtermodel,and
Barkana&Loeb 2001) maintains that reionization ended onlyifthetotalmassfractioningalaxiesisconsideredandnot
with a fast, simultaneous, overlap stage throughout the its distribution as a functionof galaxymass. The halo mass
universe. Thisviewhasbeenbasedonsimpleargumentsand distributionshouldstillhavethewrongshape,resultingfrom
hasbeensupportedbynumericalsimulationswithsmallbox thefactthat∆zineq.2dependsonM .Furthermore,inour
min
6
moreaccuratehybridmodel(§2.2),theeffectonthecollapse differentregions,resultinginasmoothevolutionofthelumi-
fractionis no longerexactly equivalentto a shiftin redshift. nosity function over this redshift range. In addition, such a
Inanycase,aself-containednumericalsimulationcannotrely surveymaybebiasedtogivearelativelyhighredshift,since
onapproximatemodelsandmustdirectlyevolveaverylarge only the most massive galaxies can be detected, and as we
volume1. haveshown,thesegalaxieswillbeconcentratedinoverdense
The second reason that current simulations are limited regionsthatwillalsogetreionizedrelativelyearly.
is that at high redshift, when galaxies are still rare, the Thedistributionofionizedpatchesduringreionizationwill
abundanceof galaxiesgrowsrapidly towardslower redshift. likelybeprobedbyfutureobservations,includingsmall-scale
Therefore, a 10% relative error in redshift implies that at anisotropies of the cosmic microwave background photons
∼
anygivenredshiftaroundz 10–20,thesimulationpredicts that are rescattered by the ionized patches (Aghanimetal.
∼
a halo mass function that can be off by an order of magni- 1996; Gruzinov&Hu 1998; Santosetal. 2003), and obser-
tude for halos thathost galaxies(see Figures1 and 4). This vations of 21 cm emission by the spin-flip transition of the
largeunderestimatesuggeststhatthefirstgenerationofgalax- hydrogen in neutral regions (Tozzietal. 2000; Carillietal.
iesformedsignificantlyearlierthanindicatedbyrecentsimu- 2002;Furlanettoetal.2003). Previousanalyticalandnumer-
lations. Thismakesiteasiertoexplainrecentobservationsof ical estimates of these signals have not included the collec-
the cosmic microwave background(Spergeletal. 2003) that tiveeffectsdiscussedabove,inwhichraregroupsofmassive
suggestanearlyreionizationatz 15–20. galaxiesmayreionizelargesurroundingareas. Thesephoton
∼
The third reason for the failure of simulations arises from transferswilllikelysmoothoutthesignalevenonscalessig-
the large cosmic scatter. This scatter can fundamentally nificantlylargerthanthetypicalsizeofanHIIbubbledueto
change the character of any observable process or feedback anindividualgalaxy.Therefore,eventhecharacteristicangu-
mechanism that depends on a radiation background. Simu- lar scales that are expected to show correlations in such ob-
lationsin periodicboxeseliminateanylarge-scalescatter by servationsmustbereassessed.
assumingthatthe simulatedvolumeissurroundedbyidenti- Thecosmicscatteralsoaffectsobservationsinthepresent-
cal periodic copies of itself. In the case of reionization, for day universe that depend on the history of reionization. For
instance,currentsimulationsneglectthecollectiveeffectsde- instance,photoionizationheatingsuppressestheformationof
scribed above, whereby groups of sources in overdense re- dwarf galaxies after reionization, suggesting that the small-
gionsmayinfluencelargesurroundingunderdenseregions.In estgalaxiesseentodaymayhaveformedpriortoreionization
the case of the formation of the first stars due to molecular (Bullocketal. 2001; Somerville 2002; Bensonetal. 2002).
hydrogen cooling, the effect of the soft ultraviolet radiation Underthepopularviewthatassumedasharpendtoreioniza-
fromthesestars, whichtendstodissociatethemolecularhy- tion, it was expectedthatdenser regionswouldhave formed
drogenaroundthem(Haimanetal. 1997;Ricottietal. 2002; more galaxies by the time of reionization, possibly explain-
Oh&Haiman2003),mustbereassessedwithcosmicscatter ingthe largerrelative abundanceof dwarfgalaxiesobserved
included. ingalaxyclusterscomparedtolower-densityregionssuchas
ourLocalGroupof galaxies(Tullyetal. 2002; Bensonetal.
2003a). Ourresultsundercutthebasicassumptionofthisar-
3.3. Observationalconsequences
gumentandsuggestadifferentexplanationaltogether.Reion-
Thespatialfluctuationsthatwehavecalculatedfundamen- izationoccursroughlywhenthenumberofionizingphotons
tally affect current and future observationsthat probe reion- produced starts to exceed the number of hydrogen atoms in
ization or the galaxy populationat high redshift. For exam- the surroundingIGM. If the processesof star formationand
ple,therearealargenumberofprogramssearchingforgalax- theproductionofionizingphotonsareequallyefficientwithin
iesatthehighestaccessibleredshifts(6.5andbeyond)using galaxiesthatlieindifferentregions,thenreionizationineach
theirstrongLyαemission(Huetal.2002;Rhoadsetal.2003; regionwilloccurwhenthecollapsefractionreachesthesame
Maieretal.2003;Kodairaetal.2003). Theseprogramshave criticalvalue,eventhoughthiswilloccuratdifferenttimesin
previouslybeenjustifiedasasearchforthereionizationred- differentregions. Sincethegalaxiesresponsibleforreioniza-
shift, since the intrinsic emission should be absorbed more tionhavethesamemassesaspresent-daydwarfgalaxies,this
strongly by the surrounding IGM if this medium is neutral. estimatearguesforaroughlyequalabundanceofdwarfgalax-
Foranyparticularsource,itwillbehardtoclearlyrecognize iesinallenvironmentstoday.Thissimplepictureis,however,
this enhanced absorption because of uncertainties regarding modifiedby several additionaleffects. First, the recombina-
thepropertiesofthesourceanditsradiativeandgravitational tionrateis higherinoverdenseregionsatanygiventime, as
effectsonitssurroundings(Barkana&Loeb2003a,b;Santos discussedabove.Furthermore,reionizationinsuchregionsis
2003). However, if the luminosity function of galaxies that accomplishedatanearliertimewhentherecombinationrate
emit Lyα can be observed, then faint sources, which do not washigherevenatthemeancosmicdensity;therefore,more
significantlyaffecttheirenvironment,shouldbeverystrongly ionizing photons must be produced in order to compensate
absorbedintheerabeforereionization.Reionizationcanthen fortheenhancedrecombinationrate. Thesetwoeffectscom-
bedetectedstatisticallythroughthesuddenjumpinthenum- bineto makeoverdenseregionsreionizeata highervalueof
beroffaintsources(Haiman&Spaans1999;Haiman2002). F than underdense regions. In addition, the overdense re-
col
Our results alter the expectation for such observations. In- gions,whichreionizefirst,subsequentlysendtheirextraion-
deed,nosharp“reionizationredshift”isexpected. Instead,a izingphotonsintothesurroundingunderdenseregions,caus-
Lyα luminosity function assembled from a large area of the ingthelattertoreionizeatanevenlowerF . Thus,ahigher
col
skywillaverageoverthecosmicscatterof∆z 1–2between abundanceof dwarfgalaxiestodayis indeedexpectedin the
∼
overdenseregions.
1Ciardi,Stoehr,&White(2003)simulateda30Mpcbox,largerthanpre- The same basic effect may be even more critical for un-
vioussimulations,butonlygravitywasdirectlysimulated,andthemassres-
derstanding the properties of large-scale voids, 10–30 Mpc
olutionwasthreeordersofmagnitudelowerthantheminimumnecessaryto
resolvemostgalactichalosathighredshift. regionsinthepresent-dayuniversewithanaveragemassden-
7
sity that is well below the cosmic mean. In order to pre- density peaks on small scales in the exponential tail of the
dict their properties, the first step is to consider the abun- Gaussianrandomfield of densityfluctuations, andintroduce
dance of dark matter halos within them. Numerical simu- a remarkably large scatter in the abundance of star-forming
lations show that voids contain a lower relative abundance galaxiesatearlycosmictimes.
of rare halos (Cen&Ostriker 2000; Somervilleetal. 2001; We have developed an improved method to calculate the
Mathis&White2002;Bensonetal.2003b),asexpectedfrom cosmicscatter(see§2.2). Thisyieldsthefirstself-consistent
the raising of the collapse threshold for halos within a void. analyticmodelthatmatchesthehalomassfunctionmeasured
Ontheotherhand,simulationsshowthatvoidsactuallyplace in various regions in numerical simulations that covered a
a larger fraction of their dark matter content in dwarf halos widerangeoftheparameterspaceofregionsize, meanden-
of mass below 1010M (Gottlöberetal. 2003). This can be sity,andredshift.
⊙
understood within the extended Press-Schechter model. At Sincethecharacteristicdistancebetweennearbysourcesof
thepresenttime,atypicalregionintheuniversefillshalosof ionizingradiationvarieswidelyacrosstheuniverse,theover-
mass 1012M⊙ and higher with most of the dark matter, and lap of the HII regions producedby these individualsources
very little is left over for isolated dwarf halos. Although a intheIGMoccursatsignificantlydifferenttimesindifferent
largenumberofdwarfhalosmayhaveformedatearlytimes cosmicenvironments. Quantitatively,wefindthatthespread
insucharegion,thevastmajoritylatermergedwithotherha- in the redshift of reionization should be at least an order of
los,andbythepresenttimetheysurviveonlyassubstructure magnitudelargerthanpreviousexpectationsthatarguedfora
insidemuchlargerhalos. In avoid, onthe otherhand, large sharpendtoreionization(see§3.1).
halos are rare even today, implying that most of the dwarf Current numerical simulations that treat gravity and hy-
halosthat formedearly within a void can remainas isolated drodynamics (Gnedin 2000; Abeletal. 2002; Yoshidaetal.
dwarf halos till the present. Thus, most isolated dwarf dark 2003)largelyeliminatethisrealcosmicscatter,andareartifi-
matter halos in the present universe should be found within ciallybiasedtowardlategalaxyformationsincetheyexclude
large-scalevoids(Barkana2003). large-scalemodes(seeFigures3and4). Wefindthatgalaxy
However, voids are observed to be rather deficient in formationwithinstate-of-the-artsimulationswith3243parti-
dwarf galaxies as well as in larger galaxies on the scale of clesisartificiallybiasedtooccurtoolatebyaredshiftinterval
the Milky Way (e.g., Kirshneretal. 1981; Ederetal. 1989; ∆z 0.5atz=7and∆z 2.5atz=20.Theboxlengthused
∼ ∼
Grogin&Geller 1999, 2000; El-Ad&Piran 2000; Peebles in state-of-the-art simulations of reionization (Gnedin 2000;
2001). Adeficitoflargegalaxiesisnaturallyexpected,since Yoshidaetal. 2003) is 1.5–2 orders of magnitude below the
the total mass density in the void is unusually low, and the minimum size necessary to treat the scatter reliably, and so
fractionofthisalreadylowdensitythatassemblesinlargeha- alternative computational schemes (Barkana&Loeb 2003c)
losisfurtherreducedrelativetohigher-densityregions. The must be implemented in order to quantify the implications
absence of dwarf galaxiesis harder to understand, given the of the large cosmic scatter on the reionization history. This
higher relative abundance expected for their host dark mat- scattershouldaffectthestatisticalfluctuationsinthenumber
ter halos. The standard model for galaxy formation may be andclusteringpropertiesofsourcesinsurveyswithanarrow
consistent with the observations if some of the dwarf halos fieldofview(suchastheHubbleDeepField),theluminosity
aredarkanddonothoststars. Largenumbersofdarkdwarf functionofLyα-emittinggalaxiesaroundthereionizationred-
halos may be produced by the effect of reionization in sup- shift,thefluctuationsinthe21cmfluxproducedbytheneutral
pressing the infall of gas into these halos. Indeed, exactly IGM,thepowerspectrumofthesecondaryanisotropiesinthe
thesamefactorsconsideredabove,inthediscussionofdwarf cosmicmicrowavebackground,andthepresentabundanceof
galaxies in clusters compared to those in small groups, ap- dwarf galaxiesin variousenvironments(see § 3.3). Simula-
ply also to voids. Thus, the voids should reionize last, but tionslimited to a smallbox may be able to study the scatter
sincetheyaremoststronglyaffectedbyionizingphotonsfrom in the number density of galaxiesby varying the mean den-
theirsurroundings(whichhaveahigherdensitythanthevoids sityofthebox,butsuchsimulationscannotprobetheglobal
themselves),thevoidsshouldreionizewhentheabundanceof structureofreionizationsincethiswouldinvolvetheradiative
galaxieswithinthemisrelativelylow. Aquantitativeanalysis transferofionizingphotonsoverdistanceslargerthanthebox
ofhowthereionizationredshiftvarieswithenvironmentmay size.
help establish a common framework for explaining the ob-
served propertiesof dwarf galaxiesin environmentsranging
fromclusterstovoids.
WethankPaulSteinhardtforsuggestingtoapplyourwork
4. CONCLUSIONS to galaxy formation in voids. We acknowledge support by
We have shown that the important milestones of high- NSFgrantAST-0204514andNATOgrantPST.CLG.979414.
redshift galaxy formation, such as the formation of the first R.B. is grateful for the kind hospitality of the Harvard-
stars and the completion of reionization, occurredat signifi- Smithsonian CfA and the Institute for Advanced Study, and
cantlydifferenttimesindifferentregionsoftheuniverse.This thesupportofanAlonFellowshipatTelAvivUniversityand
conclusion results from the fact that the temperature thresh- of Israel Science Foundationgrant28/02/01. A.L. acknowl-
old,abovewhichcoolingandfragmentationofgasarepossi- edges sabbatical support from the John Simon Guggenheim
ble, selects out dark matter halos that become exceptionally MemorialFellowship. Thiswork was also supportedin part
rare at high redshifts. Consequently, density fluctuations on byNSF grantAST-0071019andNASA grantNAG5-13292
large scales modulate the threshold for the collapse of high (forA.L.).
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