The First Billion Years Project: The escape fraction of ionizing photons in the epoch of reionization

5.9 MB·English

by  Jan-Pieter Paardekooper

Mon.Not.R.Astron.Soc.000,1–21(2014) Printed9July2015 (MNLATEXstylefilev2.2)
The First Billion Years Project: The escape fraction of ionizing
photons in the epoch of reionization
Jan-Pieter Paardekooper1,2∗, Sadegh Khochfar3 and Claudio Dalla Vecchia4,5
1Universita¨tHeidelberg,Zentrumfu¨rAstronomie,Institutfu¨rTheoretischeAstrophysik,Albert-Ueberle-Str.2,69120Heidelberg,Germany
2MaxPlanckInstituteforextraterrestrialPhysics,POBox1312,Giessenbachstr.,85741Garching,Germany
5 3InstituteforAstronomy,UniversityofEdinburgh,RoyalObservatory,BlackfordHill,EdinburghEH93HJ,UK
1 4InstitutodeAstrofs´ıcadeCanarias,C/V´ıaLa´cteas/n,38205LaLaguna,Tenerife,Spain
0 5DepartamentodeAstrofs´ıca,UniversidaddeLaLaguna,Av.delAstrof´ısicoFrancisoSa´nchezs/n,38206LaLaguna,Tenerife,Spain
2

l
u
Accepted***.Received***;inoriginalform***
J

8

ABSTRACT
]
O Proto-galaxies forming in low-mass dark matter haloes are thought to provide the majority
of ionizing photons needed to reionize the Universe, due to their high escape fractions of
C
ionizing photons. We study how the escape fraction in high-redshift galaxies relates to the
h. physicalpropertiesofthehaloinwhichthegalaxiesform,bycomputingescapefractionsin
p morethan75000haloesbetweenredshifts27and6thatwereextractedfromtheFirstBillion
- Yearsproject,high-resolutioncosmologicalhydrodynamicssimulationsofgalaxyformation.
o
Wefindthatthemainconstraintontheescapefractionisthegascolumndensityinaradiusof
r
t 10pcaroundthestellarpopulations,causingastrongmassdependenceoftheescapefraction.
s The lower potential well in haloes with M (cid:46) 108M results in low column densities
a 200 (cid:12)
[ thatcanbepenetratedbyradiationfromyoungstars(age< 5Myr).InhaloeswithM200 (cid:38)
108M(cid:12) supernova feedback is important, but only ∼ 30% of the haloes in this mass range
2 have an escape fraction higher than 1%. We find a large range of escape fractions in haloes
v
with similar properties, caused by different distributions of the dense gas in the halo. This
7
makesitveryhardtopredicttheescapefractiononthebasisofhalopropertiesandresultsin
6
a highly anisotropic escape fraction. The strong mass dependence, the large spread and the
9
largeanisotropyoftheescapefractionmaystronglyaffectthetopologyofreionizationandis
1
0 somethingcurrentmodelsofcosmicreionizationshouldstrivetotakeintoaccount.
.
1 Keywords: radiativetransfer-methods:numerical-galaxies:dwarf-galaxies:formation–
0 galaxies:high-redshift
5
1
:
v
i
X 1 INTRODUCTION ing into determining the escape fraction observationally. Except
for two objects (Leitet et al. 2011, 2013), in the local Universe
r Observationsarenowprobinggalaxiesinthemiddleofthereion-
a no ionizing radiation has been detected directly (Leitherer et al.
izationepoch,whenthegasintheinter-galacticmediumwastrans-
1995;Deharvengetal.2001),althoughsomeobjectsshowindirect
formed from its initially neutral state into a hot, ionized plasma
evidence of photon leakage (Heckman et al. 2011; Zastrow et al.
(e.g. McLure et al. 2011; Finkelstein et al. 2012; Bouwens et al.
2011).Thelackofdetectionsmaybepartlyduetoselectionbias
2014).Mostlikelystarsingalaxiesareresponsibleforthistrans-
(Bergvalletal.2013),buttheobjectsfromwhichradiationisde-
formation,althoughthisheavilydependsonthefractionofionizing
tected have very low escape fractions, f < 4%. At z ∼ 1, no
photons produced by the stars that make it into the inter-galactic esc
objects with leaking ionizing photons have been detected (Siana
medium,theso-calledescapefractionf .Theescapefractionisa
esc etal.2010;Bridgeetal.2010),butatz ∼ 3,thehighestredshift
keyparameterinstudiesofthecontributionoftheobservedgalaxy
atwhichtheopacityoftheinter-galacticmediumforionizingpho-
population to reionization (e.g. Bouwens et al. 2012; Robertson
tonsisapproximatelylessthanunity,ionizingphotonshavebeen
etal.2013),semi-analyticmodellingofreionization(e.g.Choud-
detectedin∼10%oftheobservedobjects(Nestoretal.2013).At-
huryetal.2009;Pritchardetal.2010;Santosetal.2010;Mesinger
temptstoconstraintheescapefractionwithnumericalsimulations
et al. 2011; Raskutti et al. 2012; Mitra et al. 2013; Shull et al.
find ranges between f < 10% (Razoumov & Sommer-Larsen
2012) and numerical simulations of reionization (e.g. Iliev et al. esc
2006, 2007; Gnedin et al. 2008; Paardekooper et al. 2011; Kim
2006;Trac&Cen2007;Ciardietal.2012).Alargeeffortisgo-
et al. 2013) and f > 80% (Wise & Cen 2009; Razoumov &
esc
Sommer-Larsen2010),withlikelyastrongmassandredshiftde-
pendence(Yajimaetal.2010;Wiseetal.2014).Duetotheopacity
∗ E-mail:[email protected]
(cid:13)c 2014RAS
2 J.-P.Paardekooperetal.
oftheinter-galacticmedium,weneedtomostlyrelyonnumerical algorithm(Springeletal.2001;Dolagetal.2009),andtheredshift
simulationstolearnabouttheescapefractionduringtheepochof evolutionofthehaloeswasfollowedwithamergertreealgorithm
reionization. (Neistein et al. 2012). The cosmological parameters employed in
Thedifficultyofconstrainingtheescapefractionbothobser- theFiBYsimulationare:Ω =0.265,Ω =0.735,Ω =0.0448,
M Λ b
vationallyandnumericallymakesituncertainwhethergalaxiesare H = 71kms−1Mpc−1,andσ = 0.81,consistentwiththere-
0 8
abletoreionizetheUniverse.Theobservationallydeterminedes- centWMAPresults(Komatsuetal.2009).
cape fractions at low redshifts are too low for galaxies to con-
tributesignificantlytoreionization,whichrequiresanescapefrac-
2.1.1 Starformation
tionofatleast20%iftheluminosityfunctionisintegrateddown
to MUV = −13 to account for the unobserved galaxy popula- We use a pressure law to model star formation (Schaye &
tion(Robertsonetal.2013).However,thesamplesforwhichthe Dalla Vecchia 2008), which is designed to yield results consis-
escapesfractionsaremeasuredaresmallandmaybebiased.Fur- tent with the Schmidt-Kennicutt law (Schmidt 1959; Kennicutt
thermore it is not at all clear whether the galaxies for which es- 1998). The threshold density above which stars form is set to
cape fractions are determined at low redshifts are representative n = 10cm−3.ThisensuresthatatT = 1,000KtheJeansmass
forthepopulationofgalaxiesduringtheepochofreionization.It isresolvedbyseveralhundredsofparticles.Becauseofresolution
ishighlylikelythattheprimarysourcesofreionizationaregalax- limitations,weassumeapolytropicequationofstateforgaswith
iesthatarecurrentlybelowthedetectionlimitatz > 6,aregime densitiesabovethethresholddensity.Theequationofstateisnor-
currentlyonlyprobedbysimulations.Inordertolearnmoreabout malisedtoP /k =100cm−3K,withk Boltzmann’sconstant,
0 B B
thesourcesofreionization,wethereforeneedsimulationsofalarge andtheeffectiveadiabaticindexisγ =4/3.Therearetwotypes
eff
sampleoflow-massgalaxiesatz>6tobetterconstraintheescape ofstarsforminginthesimulation,dependingonthemetallicityof
fractionduringreionization. the gas. In low-metallicity gas, the stars that form are generally
Inthisworkwestudytheescapefractioninthelargestsample hotter (Schaerer 2002) and follow a more top-heavy initial mass
ofreionization-epochgalaxiestodate.Previouslywehaveshown function(IMF)(Bromm&Larson2004).Fortheseso-calledpop-
thattheseproto-galaxiesemitenoughphotonstoreionizetheirlo- ulation(Pop)IIIstarsweassumeaSalpeterIMF(Salpeter1955)
calvolume(Paardekooperetal.2013)(hereafterPaperI)andare withanupperlimitofM = 500M andalowerlimitof
(cid:63),upper (cid:12)
thereforelikelythemaindriversofreionization.Wenowturnour M =21M .Theuppermasslimitischosentobethemax-
(cid:63),lower (cid:12)
attentiontothereasonwhythesesourcesaresoefficientreionizers, imummassthatislikelytobeattainedbyPopIIIstars(Bromm&
bystudyingthedependenceoftheescapefractiononvariousphysi- Loeb2004;Karlssonetal.2008),whilethelowerlimitischosen
calpropertiesofthehosthalo.Ourverylargesampleofhaloeswith to be consistent with data on the chemical abundances in metal-
highspatialresolutionallowsustostudythescatteroftheescape poorstars(Omukai&Palla2001,2003)andwithourassumption
fractionbetweensimilarhaloes. ofaSalpeterslope.RecentcosmologicalsimulationsofPopIIIstar
formationinwhichradiativefeedbackwastakenintoaccountshow
thatthesestarsformwithmasseswithinthisrange(Hosokawaetal.
2 MODELSANDMETHODS 2011;Stacyetal.2012).
Withtheincreaseofthemetallicityofthestar-forminggas,the
We compute the escape fraction of ionizing radiation by post- IMF switches rapidly from the top-heavy Pop III IMF to a much
processing the output of two high-resolution cosmological hy- morebottom-heavyIMF,characteristicofpresent-daystarforma-
drodynamical simulations, that are part of the First Billion Years tion(Brommetal.2001;Schneideretal.2003;Omukaietal.2005;
(FiBY) simulation suite, with detailed radiative transfer simula- Dopckeetal.2013).Forthemetal-enrichedPopIIstarsweassume
tions. aChabrierIMF(Chabrier2003),whichhasaSalpeterslopeatthe
high-massendandextendsdowntosub-solarmasses.Thecritical
metallicityatwhichthetransitionintheIMFtakesplaceisuncer-
2.1 TheFirstBillionYearssimulation
tain,butcosmologicalsimulationssuggestthatPopIIIsupernovae
TheFiBYsimulationsuite(Khochfaretalinprep.,DallaVecchia quicklyboostthelocalmetallicityabovethecriticalmetallicity,so
etal.inprep.)wasrunusingamodifiedversionofthesmoothed- theonsetofPopIIstarformationdoesnotdependsensitivelyon
particlehydrodynamics(SPH)codeGADGET(Springel2005)that thischoice(Maioetal.2010,2011).Wechooseacriticalmetallic-
was previously developed for the Overwhelmingly Large Simula- ity of Zcrit = 10−4Z(cid:12), with Z(cid:12) = 0.02, a value roughly con-
tions (OWLS) project (Schaye et al. 2010). In the OWLS-code, sistentwithboththeory(Brommetal.2001;Schneideretal.2002)
gas cooling is computed using tables for line-cooling in photo- andtheinferredmetallicitiesofmetal-poorstars(Frebeletal.2007;
ionizationequilibriumfor11elements(Wiersmaetal.2009a):H, Caffauetal.2011).
He,C,N,O,Ne,Mg,Si,S,CaandFe,whichwerecomputedusing
theCLOUDYv07.02code(Ferland2000).FortheFiBYsimula-
2.1.2 Stellarfeedback
tions,wehaveaddedafullnon-equilibriumprimordialchemistry
network(Abeletal.1997;Galli&Palla1998;Yoshidaetal.2006) Massivestarsendtheirlivesexplodingassupernova,releasingco-
includingmolecularcoolingfunctionsforH andHD.Inthiswork pious amounts of energy and enriching their surroundings with
2
wefocusontwosimulations,FIBY SandFiBY.TheFiBY Ssim- metals. We model the mechanical supernova feedback by inject-
ulation is the same as was used in Paper I; it has a volume of 4 ing thermal energy in the neighbouring particles (Dalla Vecchia
Mpc (comoving) that was simulated with 2 × 6843 dark matter & Schaye 2012). For each Pop II supernova 1051 erg is injected
and gas particles that was run until redshift 6. The FiBY simula- once a star particle has reached an age of 30Myr, correspond-
tionhasavolumeof8Mpc(comoving)with2×13683 particles ing to the maximum lifetime of stars that end their lives as core-
thatwasrununtilredshift8.5.Thegasparticlemassinbothsimu- collapse supernovae. For Pop III supernovae we differentiate be-
lationsis1253.6M .HaloeswereidentifiedusingtheSUBFIND tweentypeIIsupernovaewhichoccurforstellarmassesbetween
(cid:12)
(cid:13)c 2014RAS,MNRAS000,1–21
Theescapefractionintheepochofreionization 3
21M (cid:46) m (cid:46) 100M and pair-instability supernovae which on the escape fraction in the entire halo sample, and disentangle
(cid:12) (cid:63) (cid:12)
occurfor140M (cid:46)m (cid:46)260M .Weinject1051ergpersuper- theeffectsofexternalfeedbackbytheUV-backgroundandinternal
(cid:12) (cid:63) (cid:12)
novaforthetypeIISNeafter10Myrand3×1052ergsafter3Myr feedbackbysupernovaexplosions.
for the pair-instability supernovae (Heger & Woosley 2002). For
more details on the thermal feedback implementation and exten-
2.2 Computationoftheescapefractionofionizingphotons
sivetestswerefertheinterestedreadertoDallaVecchia&Schaye
fromproto-galaxies
(2012).
Starsenrichtheirsurroundingswithmetalsbecauseheavyele- TheFiBY SandFiBYsimulationsthatwefocusoninthiswork
mentscreatedintheirinteriorsarereleasedinstellarwindsandSN do not contain the rare high density peaks where the most mas-
explosions.Inoursimulationsstarscontinuouslyreleasehydrogen, sivegalaxiesformduetothelimitedvolumeofthesesimulations.
heliumandmetals;wetracktheenrichmentbyall11specieslisted However,highresolutionisessentialforthecomputationofthees-
aboveindividually.ForPopIIstarstheabundancesarecalculated capefraction,sincethemainconstraintontheescapefractionisthe
fromyieldsforasymptoticgiantbranchstarsandtypeIaandtype localenvironmentofthesources(Paardekooperetal.2011).Fur-
IISNe(Tornatoreetal.2007;Wiersmaetal.2009b).ForPopIII thermore,thishighresolutionisnecessarytoresolvestarformation
stars we use different yields for type II SNe (Heger & Woosley in the abundant low-mass proto-galaxies that are likely the main
2010)andPISNe(Heger&Woosley2002).Themixingofthemet- driversofreionization(PaperI,Wiseetal.2014).
alswiththesurroundingmediumismodelledbytransferringthem Fromthesesimulationswehaveextractedallthedark-matter
toneighbouringSPHparticlesinproportionsweightedbytheSPH haloesthatcontainatleast1000dark-matterparticles,100gaspar-
kernel. ticlesand1star.Wechosealowerlimitof1000dark-matterpar-
ticlestomakesurethatthereareenoughparticlestorepresentthe
halopropertiesandminimiseresolutioneffectsinthelowestmass
2.1.3 ReionizationbytheuniformUV-background
regime.Thelowerlimitonthehalomasscoincideswiththemass
With the appearance of the first luminous sources, the epoch of inwhichthefirststarsareforming,sothischoicehassmallimpact.
reionization begins, and the gas in the inter-galactic medium is Similarly,wechosealowerlimitof100gasparticlestoavoidbias
transformedintoahotandionizedstate.Whenmodellingthefor- intheradiativetransfersimulations.Theparticle-to-particletrans-
mationofthefirststarsandgalaxiesitisimportanttotaketheef- portschemeusedinSimpleXbecomesinaccurateiftherearenot
fectofreionizationintoaccount,becauseithasprofoundimpacton enough gas particles to represent the gas in the halo. This lower
whichhaloesareabletoactivelyformstarsandaccretegasfrom limitmayleadustoexcludethehaloesinwhichsupernovafeed-
theinter-galacticmedium. back cleared away all the gas, but we were careful to choose the
In the FiBY simulations we do not follow the ionizing ra- limitsinsuchawaythatthereareonlyafewhaloesexcludedatred-
diation from the star particles self-consistently due to the com- shiftshigherthan12.Atlowerredshiftstherearelow-masshaloes
putational challenges of radiation-hydrodynamics simulations at thatweexcludebecausethegashasbeenclearedawaybyfeedback
theselargescales.Insteadreionizationismodelledusingtheuni- from the UV-background. Since there is no active star formation
form UV-background model of Haardt & Madau (2001), within goingoninthesehaloes,excludingthesehaloeswillnotaffectour
the bounds of reionization set by WMAP (Komatsu et al. 2011) results.Thenumberofhaloesweobtainedforpost-processingwith
and Planck (Planck Collaboration 2013). Prior to z = 12 we radiativetransferis11364fortheFiBY Ssimulationand64319for
assume that reionization hasn’t started yet and compute the ion- theFiBYsimulation,resultinginatotalof75801haloesbetween
izationstateofthegasfromthecollisionalionizationequilibrium redshift27and6,inthemassrange2×106−6×109M ,sev-
(cid:12)
rates.Betweenz =12andz =9wegraduallychangefromcolli- eralordersofmagnitudemorethaninpreviousstudiesofthiskind
sionalionizationequilibriumratestophoto-ionizationequilibrium (Wise&Cen2009;Razoumov&Sommer-Larsen2010;Wiseetal.
rates to mimic the process of reionization. From z = 9 onwards 2014).
we assume that the inter-galactic medium is fully ionized, so the In every halo we compute the escape fraction by comparing
photo-ionization rates are used. Self-shielding of dense gas is in- thenumberofphotonsthatareproducedbythestarsN tothe
emitted
corporated at a density threshold of n (cid:62) 0.01cm−3, above numberofphotonsthatreachr ofthemainsubhalo,N (r>
shield 200 phot
whichthefluxisdecreasedwithafraction(n/n )−2(Nagamine r ):
shield 200
et al. 2010; Yajima et al. 2012). This fraction is proportional to N (r(cid:62)r )
therecombinationrate,sothereisacontinuoustransitionbetween fesc = phNot 200 , (1)
shieldedandunshieldedregimes.Theratesintheshieldedgasare emitted
computed by interpolating between the collisional and photoion- wherer200 istheradiusatwhichtheoverdensityis200timesthe
izationequilibriumtables. criticaldensity.Thenumberofphotonsthatescapefromthehalo
The disadvantage of using a uniform UV-background is that arecomputedusingdetailedradiativetransfersimulations,thede-
reionizationhappensindependentlyfromtheactualnumberofion- tailsofwhichwedescribebelow.
izingphotonsproducedbythestarsthathaveformedinthecom-
putationalvolume.IntheFiBYsimulationsthetime-scaleinwhich
2.2.1 Radiativetransfer
theUV-backgroundstartstotakeeffectisconsistentwiththeioniz-
ingphotonoutputfromproto-galaxies(PaperI,butseeSect.2.2.1). Theradiativetransfersimulationswerecarriedoutwithamodified
The disadvantage of assuming that reionization takes place uni- versionoftheSimpleXalgorithm(Paardekooperetal.2010).Sim-
formlyisthatweneglectthespatialdistributionofthereionization pleXcomputesthetransportofionizingphotonsonanunstructured
sources.Theeffectsofreionizationwillthereforehappentooearly gridcreatedfromthepositionsoftheSPHparticles.Theparticles
insomeregionsandtoolateinotherregions.Ontheotherhand, areconnectedtotheirnearestneighboursbymeansoftheDelau-
thisapproachhastheadvantagethatreionizationaffectsallhaloes naytriangulation(Delaunay1934)andgridcellsareformedbythe
atthesametime.Thiswaywecanstudytheeffectofreionization Voronoi tessellation (Dirichlet 1850;Voronoi 1908),which is the
(cid:13)c 2014RAS,MNRAS000,1–21
4 J.-P.Paardekooperetal.
f
HI
f
HeI
f
HeII
f f
HeIII HII
Figure 1. Results of a 1-dimensional simulation of the ionized region Figure2.Frequencybinspacingintheradiativetransfersimulations.Top
aroundasinglesourceincludinghydrogenandhelium.Thesimulationcon- panel: Fluxes for four different source types as a function of frequency
tains100particlesandwasrunfor30Myr.Thecomputationaldomainis normalisedtothehydrogenionizationfrequency,ν0,H =3.3×1015Hz.
6.6kpc.Asinglesourceislocatedattheedgeandisassumedtohavea ThebluelinedenotesthefluxofaPopIIIstarof1Myr,thegreenlinethe
spectrumofa100,000Kblackbody.Solidlinesdepictasimulationwith fluxofaPopIIstarof1MyrwithmetallicityZ =0.004,theredlinethe
100frequencybinsevenlyspacedinlogspace,whilethedashedlinesde- fluxofaPopIIstarof1MyrwithmetallicityZ=0.02andtheorangeline
pictasimulationwith10frequencybinsspacedtooptimallyrepresentthe thefluxofaPopIIstarof10MyrwithmetallicityZ=0.004.Theshaded
source spectrum and cross sections. Upper panel: Spherically averaged areasrepresentthefrequencybinspacing.Bottompanel:Thecrosssections
species fraction as a function of radial distance from the source. Lower ofneutralhydrogen,neutralheliumandsinglyionizedheliumasafunction
panel: Spherically averaged temperature as a function of radial distance offrequency.Theshadedareasrepresentthefrequencybinspacing.
fromthesource.
edgeofthedomain.Thesourceemits5×1048 ionizingphotons
mathematical dual of the Delaunay triangulation. Contrary to the per second with a spectral shape of a black body of temperature
SPHkernel,thisresultsinaconsistentvolumediscretisation(Heß 100,000K.Thedensityofthegasis0.001cm−3;weassumethe
&Springel2010). gastobeofprimordialcompositionwithahydrogenmassfraction
Ionisingphotonsaretransportedfromparticletoparticleus- of0.75andaheliummassfractionof0.25.Thegasisinitiallyneu-
ingtheconnectivityoftheDelaunaygrid.Ateverygridpointin- tral,withatemperatureof100K.SimpleXiscapableoftransport-
teractionofthephotonswiththegastakesplace,inwhichcasewe ingdiffuserecombinationradiation,however,foreasycomparison
computetheabsorptionofphotonsbyneutralatoms,takingintoac- totheresultsofothercodesweassumetheon-the-spotapproxima-
countthepathlengththroughthecell,andrecombinationsofion- tion1inthistest.WeshowinFig.1thetemperatureofthegasand
izedatoms.Thismethodoftransportingphotonshastheadvantage thespeciesfractionsafter30Myr.Wefindthattheseareinexcel-
thatthecomputationtimedoesnotscalelinearlywiththenumber lentagreementwiththeresultsobtainedwithotherstate-of-the-art
of sources in the simulation. This is a necessary requirement for radiativetransfercodes(Masellietal.2009;Cantalupo&Porciani
thecomputationoftheescapefraction,asthemostmassivehaloes 2011;Pawlik&Schaye2011;Friedrichetal.2012).
containmorethan140,000starparticles.Reducingthenumberof FortheFiBYprojectwehaveaddedanewprescriptionforthe
sourcesbymergingstarparticlesorusingonlytheyoungestsources sourcesinthesimulation,sothatwenolongerhavetoassumethat
thatcontributethemajorityofionizingphotonscouldleadtoincor- thesourcespectrafollowablack-bodyspectrumasinPaardekooper
rectresults(Paardekooperetal.2011). etal.(2010)andPaardekooperetal.(2011).Theionizingemissiv-
Amongthemodificationstothemethodistheself-consistent ityandspectrumofthestarparticlesarecomputedfromstellarsyn-
computation of the temperature of the gas as described in thesismodelsforbothPopIII(Raiteretal.2010)andPopII(Lei-
Paardekooper (2010), including photo-heating, recombination thereretal.1999)stars,takingintoaccounttheageandmetallicity
cooling,collisionalionizationcooling,collisionalexcitationcool- of the stellar population. There is considerable uncertainty in the
ing and cooling from free-free emission. To compute the photo- totalnumberofionizingphotonsproducedbyaPopIIstellarpopu-
heatingaccurately,thespectrumofthesourcesistakenintoaccount lationbetweendifferentstellarsynthesismodels,dependingonthe
bydividingthephotonpacketsinmultiplefrequencybins. physics included. For example, the BPASS (Eldridge & Stanway
Wehavefurthermoreincludedtheabsorptionofionizingpho- 2009)modelsincludeamoredetailedtreatmentofmassivestellar
tons by helium atoms and the corresponding heating and cooling binarieswhichleadstoafactor∼ 1.5higherratioofionizingto
processesinthetemperaturecomputation.Wehavetestedthenew
implementation in standard test problems. In Fig. 1 we show the
results of the new solver in a 1-dimensional simulation with 100 1 Intheon-the-spotapproximationitisassumedthatallionizingphotons
particlesand1sourcesampledwith10frequencybins.Thesizeof producedbyrecombinationsdirectlytothegroundstateareimmediately
thecomputationaldomainis6.6kpc,withthesourcelocatedatthe reabsorbed.
(cid:13)c 2014RAS,MNRAS000,1–21
Theescapefractionintheepochofreionization 5
non-ionizing UV flux compared to the GALAXYEV (Bruzual & 2.2.2 Creatingall-skymaps
Charlot2003)models(Mostardietal.2013).Addingtheeffectof
SimpleX computes the ionization state of the gas and the escape
rotation could even lead to a factor of ∼ 5 difference in the ion-
fractionofionizingphotonsonthebasisofaDelaunaytriangula-
izing photon production (Leitherer et al. 2014). A change in the
tion of the SPH particles. This means that the angular resolution
numberofionizingphotonsproducedbythestarshasalargeeffect
of the radiative transfer calculation at the virial radius where the
ontheescapefraction,soitisimportanttokeeptheseuncertainties
escapefractioniscomputed,isdeterminedbythenumberofSPH
inmind.
particlesthere.ThesizeoftheVoronoicellsinwhichtheradiative
transferiscomputeddependsonthepositionoftheparticles,result-
Compared to the simulations presented in Paper I we have
ingindifferentcellsizesatthevirialradius.Thisisnorestriction
made one important change. Whereas in Paper I we computed
incomputingtheescapefraction:thisdependsonthetotalnumber
the spectra for the Pop II stars using the GALAXYEV package
ofabsorptionsinsidethevirialradius,andthusonthegasdensity
(Bruzual&Charlot2003),forconsistencywithsubsequentstudies
whichiscomputedfromtheSPHparticles.However,todetermine
oftheFiBYsimulationswehereusetheStarburst99spectra(Lei-
theangulardistributionoftheescapefractionweneedtoknowthe
thereretal.1999)fortheradiativetransfersimulations.Wecom-
escapefractioninequal-areapixelsonasphereatthevirialradius.
parethetotalionizingemissivityofthehaloesbetweenStarburst99
For this reason we use a different technique to generate full-sky
andGALAXYEVinAppendixA,showingthatduringtheepochof
maps,namelypost-processingtheoutputfromtheradiativetrans-
reionizationbothmodelsprovidesimilarresultsintermsofioniz-
fersimulationswitharay-tracingroutine.
ingemissivity,andonlydeviatefromeachotheratz<7,whenold
Wedefineasphereatr thatwepixelateusingtheHEALPix
populationscontributemoreionizingphotonsinBruzual&Charlot 200
algorithm(Go´rskietal.2005)with3072pixels.Raysarecastfrom
(2003)thaninStarburst99.
everysourcetoeverypixel,computingthenumberofphotonsalong
theraythatisnotabsorbedandreachesr ,whichisgivenby
200
InPaperIwehavecomputedthenumberofionizingphotons
produced by the stars assuming continuous star formation in the Nray =Nray e−τ, (2)
r200 emitted
star particles, which was necessary for the interpolation between
whereτ istheopticaldepthalongtheray.Wethenaddupthecon-
snapshotsinthereionizationmodel,andtoaccountforionization
tributionsofallraysineverypixeltocomputetheescapefraction
ofthegasbypreviousepisodesofstarformationthatwemissdue
onthefull-skymap.Wecomputeτ byaveragingthenumberden-
tothepost-processingnatureofoursimulations(seeRazoumov&
sityandionizationstateofhydrogenandheliumineverytetrahe-
Sommer-Larsen(2006)forasimilarapproach).Wehavefoundthat
dronontheDelaunaygridoverthe4verticesandcomputingthe
duetothefastrecombinationtime-scaleofthegasinsidethehaloes,
lengthoftheraythroughthetetrahedronbycalculatingtheinter-
previousepisodesofstarformationhavenosignificantimpacton
sectionbetweentherayandthetrianglesthatconstitutethefaces
theescapefraction.Theassumptionofconstantstarformationin
ofthetetrahedron(Mo¨ller&Trumbore1997).Becausetheioniza-
PaperIisthereforeastrictupperlimitontheionizingemissivity,
tionstateofthegasisalreadygivenbytheSimpleXsimulations,
especiallyinlow-masshaloeswhereonlyasinglestellarpopulation
weonlyneedtocasteveryrayonce.Evenso,duetothescalingof
formed.Althoughtakingintoaccounttheeffectofphoto-heatingon
thecomputationtimewiththenumberofsourcesandthenumber
thegaswouldincreasetherecombinationtime(bothbyincreasing
ofpixels,forthelargesthaloesthiscomputationtakessignificantly
thetemperatureanddecreasingthedensitythroughD-typeioniza-
longerthanthefullradiativetransfersimulationswithSimpleX.
tionfronts),wehavedecidedtotakeamoreconservativeapproach
In most haloes the escape fraction computed with the ray-
inthisworkandassumeaburstmodeforstarformation.Thedif-
tracing method is lower than computed with SimpleX. This is
ferenceinescapefractionsbetweenPaperIandthisworkshowthe
causedbythewaytheopticaldepthiscomputedinbothmethods.
limitsofthepost-processingapproachforthecomputationofes-
In SimpleX ionizing photons travel along the edges of the trian-
cape fractions. In Appendix B we show the difference in escape
gulation so the optical depth along the path is determined by the
fractionbetweenthetwoapproaches.Unfortunatelyitisextremely
density and the ionization state of a single particle (see Fig. 5 in
computationallydemandingtoperformcosmologicalradiationhy-
Paardekooper et al. 2010). In the ray-tracing routine the optical
drodynamicssimulationswiththeresolutionandthelevelofdetail
depth is computed through tetrahedra that consist of 4 particles,
in the radiative transfer that we present here, so in this work we
thedensityandionizationstateareaveragedoverthese4particles.
havetorelyonpost-processing.
Thisingeneralleadstoanoverestimateoftheopticaldepthinthe
ray-tracingapproach.Forourpurposesthisisnotabigissuesince
Inthesimulationspresentedhere,thesourcespectrumissam-
therelativeescapefractionbetweenpixelsisnotlikelytobepro-
pledin10frequencybins,whichissufficienttoeliminateerrorsas-
foundlyaffectedbythis.
sociatedwithfinitefrequencyresolutioninsimpletestcasesusing
black-bodyandpower-lawsources(Mirochaetal.2012).InFig.1
thesolidlinesdepictasimulationwith100frequencybinsevenly
spacedinlog-space.Wehaveverifiedthatthesimulationhascon- 3 RESULTS
vergedforthisnumberoffrequencybins.Thedashedlinesdepict
3.1 Absorptionoftheionizingradiation
asimulationwith10frequencybinsspacedtooptimallyrepresent
the spectrum and cross sections, showing no difference with the Theescapefractionofionizingradiationinahalodependssolely
convergedsimulation.InFig.2weshowthefrequencybinspacing onthefractionofphotonsthatisabsorbed.Ifweconsiderahaloof
togetherwithexamplesofsourcespectraandthecrosssectionsfor uniform(hydrogen-only)gasdensitywithasinglemonochromatic
neutral hydrogen, neutral helium and singly ionized helium. We sourceinthecentre,whetheranyionizingphotonswillescapede-
haveverifiedthatthisfrequencybinspacingincombinationwith pendsonwhethertheStro¨mgrenspherelieswithinthevirialradius
thesourcespectrausedinthisworkgivessimilaraccurateresults ornot.Inotherwords,ifrecombinationsareabletobalancetheion-
astheblack-bodycaseshowninFig.1. izationsfromthesourcephotons,theescapefractionwillbezero.
(cid:13)c 2014RAS,MNRAS000,1–21
6 J.-P.Paardekooperetal.
Inthelimitofcompletelyionizedgastherecombinationratehasa overtimeareemittedinthefirst∼5Myr,becausethatiswhenthe
strongdependenceonthegasdensity,scalingasn2. massive,hotstarsarestillaround(seeFig.2).Althoughstellarpop-
H
Althoughthegasdensityisgenerallynotuniformlydistributed ulationsyoungerthan5Myraregenerallysurroundedbycolumn
ingalaxies,thegasdensityaroundthesitesofstarformationcanbe densitiesN(r<10pc) (cid:38) 3×1020cm−2,thesepopulationsareable
H
sohighthattherecombinationrateofthegaspreventsanyphotons to penetrate higher column densities: a Pop II stellar population
from escaping. In idealised simulations of disc galaxies this pre- youngerthan5Myrisabletofullyionizeahomogeneouscloudof
vents any ionizing photons from escaping except when feedback radius10pcifthecolumndensityislessthan1.3×1021cm−2.
clearsaway(partof)thisdensegas(Paardekooperetal.2011).In Wehaveindicatedthiscolumndensitywithaverticaldashedline
ourmuchlargersamplewecannowstudywhetherthisisalsothe inFig.3.AsN(r<10pc)getshigherthanthisvalue,f getslower,
H esc
caseforhaloesinacosmologicalsimulation. aseventheyoungstellarpopulationsarenotabletopenetratethese
InFig.3weshowthetotalescapefractioninahaloasafunc- highcolumndensities.
tion of N(r<10pc), where N(r<10pc) is the spherically-averaged Tofurtherclarifythispoint,weshowinthemiddleandbottom
H H
columndensitywithin10physicalpcaroundasource,takenasthe rowofFig.3thesubsetofhaloescontainingatleastoneyoungstel-
ionizing-emissivity-weightedaverageoverallsourcesinthehalo: larpopulationandhaloescontainingnoyoungstellarpopulations,
respectively. The two subsets show a similar relation: a decrease
NH(r<10pc) = (cid:80)iN(cid:80)H(r,ii<N10iopnc,)iNion,i, (3) yinouenscgaspteelflraarcptioopnualastiNonH(sr<ar1e0pacb)legetotspheingehterra,tealmbeuicthshlaifrtgeedrbceocluaumsne
densities.Althoughthedropinescapefractionforthehaloescon-
wherethesumsareoverallstarparticlesinahalo.Thisquantity
tainingyoungstarsismoresteepthanforthehaloescontainingonly
givesusanindicationofhowdensethegasaroundthesourcesin
oldpopulations,thescalingwithdensityislessstrongthan1−n2.
thehaloisonaverage.Theradiusof10pcwaschosentobeclose H
Thereasonforthisistwo-fold.In∼ 30%ofthehaloesthereare
totheminimumsmoothinglengthofthesimulationatallredshifts
a few young stellar populations which are unobscured by dense
wherestarformationistakingplace.Wehaveverifiedthatincreas-
gas.Shortlyafterthesepopulationshadformed,asupernovaevent
ingthesphereradiusdoesnotaffecttheseresults,indicatingthat
closebyclearedawaythegasaroundthesesources.Theionizing
inmosthaloestheionizingphotonsareindeedabsorbedintheim-
photons from these sources can therefore travel through gas with
mediate vicinity of the sources. This has to do with the density
muchlowercolumndensity,causingahighoverallescapefraction.
profileofthegasinthehaloes:thedensegasisconcentratedwhere
Inthemajorityofhaloes(∼70%)thesphericallyaveragedcolumn
the star formation is taking place. However, this picture is likely
densityaroundalltheyoungstellarpopulationsishighenoughto
oversimplified as the high-redshift Universe is a violent environ-
shieldalltheproducedionizingradiation.Inthesecasesthenon-
mentforproto-galaxiesandthefrequentminormergerswillcause
zeroescapefractioniscausedbytheinhomogeneousdistribution
moredisturbedsystems.Atredshift6thedataisfromtheFiBY S
ofthedensegasaroundthesources.Theresultingescapefraction
simulationonly,resultinginlargerPoissonerrorsthanthehigher
atthevirialradiuswillthereforebehighlyanisotropic.
redshiftsduetosmallerstatistics.
ForthelaterdiscussionitisusefultolinkN(r<10pc)toother
The top row of Fig. 3 shows the results for all haloes in the H
sample.AtallredshiftsNH(r<10pc) (cid:46)1019cm−2meansthatonav- halo properties. In Fig. 4 we show the dependence of NH(r<10pc)
eragef (cid:38)0.1.Thecolumndensityinthevicinityofthesources onthevirialmassofthehalo,definedasthetotalmassofthehalo
esc
issolowthatstarparticlesofallagesareabletokeepthegasion- (dark matter, gas and stars) within the virial radius. We estimate
ized, resulting in a high escape fraction. As N(r<10pc) increases the virial radius using r200, the radius at which the overdensity
uptoN(r<10pc) ∼1021cm−2,f decreases,aHsitisgettingmore is 200 times the critical density. A higher halo mass means that
H esc theaveragecolumndensityaroundthesourcesisalsohigher,due
andmoredifficultfortheionizingphotonstopenetratethehigher
tothedeeperpotentialwellthereismoregasinthecentreofthe
columndensity.Ifthedensegasisfullyionizedanduniformlydis-
halo,wherestarsareforming.ForhaloeswithM (cid:38) 108M ,
tributedinaspherearoundthesources,thenumberofabsorptions 200 (cid:12)
1wo−ulnd2s,cawlehiacshnm2He.aInnstthhaattcaatsNet(hre<d10ecpcr)ea(cid:38)se1o0f1f9ecscmw−o2utlhdesceascleapaes tNheH(rs<c1a0ttpecr)iasnldarMge2r0d0usehtoowthaetiegfhfitcireenlattrioednisshtirpib.uFtoiornloowfethremdaesnseses
H H gasbysupernovafeedbackinlower-masshaloes.Atredshift6the
fractionwouldbeclosetozero.However,atallredshiftswefind
effectoftheUV-backgroundonhaloesofM (cid:46)2×108isclear:
thatthescalingoftheescapefractionisamuchlessstrongfunc- 200
thecolumndensityaroundthe(primarilyold)sourcesislowdueto
tionofthecolumndensity.Thisindicatesthattheassumptionofa
photo-heatingbytheUV-background.Gasinhigher-masshaloesis
uniformdistributionofthedensegasaroundthesourcesineq.3is
abletoself-shield.Thereareonlyafewhaloesinthehighestmass
asimplificationandthatthespatialdistributionofthegasisimpor-
bins,sothedropinN(r<10pc) athighmassesatredshifts8.9and
tant. H
At N(r<10pc) (cid:38) 1021cm−2 there is an increase in the es- 6islikelycausedbystatisticsratherthanphysics.
H
cape fraction. At these column densities lies the threshold for
star formation in the simulation. If we again assume a constant
gas density in a sphere with radius of 10pc around the sources,
the star formation threshold of n = 10cm−3 translates into
N(r<10pc) (cid:39) 3 × 1020cm−2. This is around the column den-
H 3.2 Thedependenceoftheescapefractiononthehalo
sity at which we see the increase in escape fraction. Haloes with
properties
N(r<10pc) (cid:46) 3×1020cm−2 contain mostly stellar populations
H
thathaveclearedawaythedensegasthroughsupernovafeedback. Intheprevioussectionwehaveshownthattheescapeofionizing
Since supernova feedback from Pop II stars sets in after 30Myr radiationfromahaloismainlydrivenbytheabsenceofdensegas
(seeSect.2.1.2),thesepopulationsarerelativelyold.Themajority in the vicinity of the most luminous sources. Here we show how
ofionizingphotonsthatareproducedbyaPopIIstellarpopulation theescapefractionscaleswithdifferenthaloproperties.
(cid:13)c 2014RAS,MNRAS000,1–21
z=6
Theescapefractionintheepochofreionization 7
1.
0.1 f)esc
d.f.(
0.01p.
0.001
Figure3.ThetotalescapefractioninahaloasafunctionofN(r<10pc)atdifferentredshifts.Here,N(r<10pc)isthesphericallyaveragedcolumndensity
H H
within10pcofasource,computedasaionizing-emissivity-weightedaverageoverallsourcesinahalo.Colour-codedisthe2-dimensionalhistogramofall
thehaloeswithinamassbin,thesolidlinerepresentsthemeanescapefractioninthesamebin.Theerrorsarethestandarddeviationaroundthemean.For
clarity,haloeswithfesc<10−4areplottedatfesc=10−4.Thetoprowincludesallhaloes,themiddlerowonlyhaloeswithatleastonestarparticleyounger
than5Myrandthebottomrowonlyhaloescontainingnostarparticleyoungerthan5Myr.Thedottedverticallineindicatesthecolumndensityatwhichthe
Stro¨mgrenspherearoundastarparticleofage5Myroryoungerwouldbe10pc.Thehistogramonthetopshowsthenumberofhaloesineachbin(grey)
andthenumberofhaloescontainingatleastonestarparticleyoungerthan5Myr(red).Atredshift6thedataisfromtheFiBYSsimulationonly,resultingin
largerPoissonerrorsthanthehigherredshifts.
1.
)
0.1 pc)
<10
(rNH
0.01 d.f.(
p.
0.001
Figure4.Thesphericallyaveragedcolumndensitywithin10pcofasource,computedasaionizing-emissivity-weightedaverageoverallsourcesinahalo,
asafunctionofhalomassM200fordifferentredshifts.Colour-codedisthe2-dimensionalhistogramofallthehaloeswithinabin,thesolidlinerepresents
themeancolumndensityinthesamebinandthedashedlinethemeancolumndensityforhaloescontainingatleastonestellarparticlewithageequaltoor
lessthan5Myr.Theerrorsarethestandarddeviationinthemean.Thehistogramonthetopshowsthenumberofhaloesineachbin(grey)andthenumber
ofhaloescontainingatleastonestarparticleyoungerthan5Myr(red).Atredshift6thedataisfromtheFiBYSsimulationonly,resultinginlargerPoisson
errorsthanthehigherredshifts.
3.2.1 Mass havevirialmassesintherange2×106M (cid:46)M (cid:46)6×109M .
(cid:12) 200 (cid:12)
Thelowestmassissetbythesmallesthaloesinwhichstarforma-
tion takes place, while the largest mass is a result of the size of
InFig.5weshowtheescapefractionasfunctionofhalomass,us-
thesimulationvolume.Highermasshaloesthanwhatwefinddo
ingthreedifferentprobesofthemass.Thefirstisthevirialmass,
formattheseredshifts,buttheyforminrarehigh-sigmapeaksthat
definedintheprevioussection.Thehaloesweconsiderinthiswork
(cid:13)c 2014RAS,MNRAS000,1–21
8 J.-P.Paardekooperetal.
1.
0.1 f)esc
d.f.(
0.01p.
0.001
1.
0.1 f)esc
d.f.(
0.01p.
0.001
1.
0.1 f)esc
d.f.(
0.01p.
0.001
Figure5.TheescapefractionasafunctionofhalomassM200(top),stellarmassM(cid:63)(centre)andgasmassMgas(bottom)fordifferentredshifts.Colour-
codedisthe2-dimensionalhistogramofallthehaloeswithinamassbin,thesolidlinerepresentsthemeanescapefractioninthesamebinandthedashed
linethemeanescapefractionforhaloescontainingatleastonestellarparticlewithageequaltoorlessthan5Myr.Theerrorsarethestandarddeviationin
themean.Forclarity,haloeswithfesc <10−4areplottedatfesc =10−4.Thehistogramonthetopshowsthenumberofhaloesineachbin(grey)andthe
numberofhaloescontainingatleastonestarparticleyoungerthan5Myr(red).Atredshift6thedataisfromtheFiBYSsimulationonly,resultinginlarger
Poissonerrorsthanthehigherredshifts.
oursimulationvolumedoesnotcapture.However,themassrange inthismassrangewithoutyoungstarsthecolumndensityofthe
wecoveriswherethemajorityofionizingphotonsduringreioniza- gasclosetothesourcesishighenoughtoabsorballtheionizing
tionisexpectedtobeproduced(Barkana&Loeb2001).Thestellar radiationfromolderstarparticles,soforradiationtoescapesuper-
massinthehaloesliesbetween1250M (cid:46) M (cid:46) 2×108M , novafeedbackmustclearawaypartofthedensegassurrounding
(cid:12) (cid:63) (cid:12)
wheretheminimummassisgivenbytheSPHparticlemass.The thesourcesinorderfortheescapefractiontobehigherthanzero.
gas mass in the haloes lies in the range 1×105M (cid:46) M (cid:46) Aswemovetohigherhalomasses,theaverageescapefractiongets
(cid:12) gas
8×108M . lower.Duetothehighercolumndensitiesinthismassrange,most
(cid:12)
Wefindastrongdependenceoftheescapefractiononmass, of the ionizing radiation produced by young star particles is ab-
independentofwhichquantityweusetospecifythemass.Mini- sorbed.Withafewexceptions,theescapefractioninhaloeswith
haloeswithvirialmassM200 (cid:46)3×106M(cid:12)alwayshaveanescape mass M200 (cid:38) 109M(cid:12) is below 0.1%. We find the similar be-
fractionhigherthan10%,duetotheirlowgascontent.Thesehaloes haviouroftheescapefractionasafunctionofstellarmassandgas
formprimarilyPopIIIstarswhichareluminousenoughtoionized mass.Inthehighestmassbinsatredshift8.9and6theaveragees-
most of the gas in the halo, hence the high escape fraction. The capefractionishigherthan10%.Thisiscausedbyrecentfeedback
average escape fraction for haloes with M ∼ 107M is still events in some of the massive haloes that have cleared away the
200 (cid:12)
high ((cid:38) 10%), but there are many haloes with zero escape frac- densegassurroundingthesources(seeFig.4).Unfortunatelythere
tion.Haloesinthismassrangethatcontainatleastoneyoungstar arenotenoughhaloesinthismassrangetodistinguishwhetherthis
particlehavehighescapefractions,becausestarparticlesyounger isaphysicaleffectorwhetheritiscausedbythestatistics.
than5Myrareabletoionizemostofthegasinthehalo.Thehigh Intheprevioussectionwefoundacleardistinctionbetween
escapefractioniscausedbyacombinationofhighionizingpho- the escape fraction in haloes containing at least one star parti-
tonproductionandrelativelylowcolumndensity(seeFig.3)due cle with age equal to or below 5 Myr and the haloes containing
to the small gas mass. The majority of these haloes contain only only older populations. Due to the higher production of ionizing
Pop II stars, so this is not caused by supernova feedback which photons in young stellar populations, for a given column density
sets intoo late, when themassive stars thatproduce the majority aroundthesources,thetotalescapefractionishigherinthehaloes
ofionizingphotonshavealreadydisappeared.However,inhaloes that are actively forming stars. In Fig. 5 the dashed line repre-
(cid:13)c 2014RAS,MNRAS000,1–21
Theescapefractionintheepochofreionization 9
sentsthemeanescapefractionforhaloescontainingyoungstars. measuringthestarformationrateareconsistentinmassivehaloes
While at the high-mass end there is no difference (these haloes inwhichcontinuousstarformationtakesplace.
have enough mass to continuously form stars), haloes with virial InFig.6weshowtheescapefractionasafunctionofstarfor-
massM (cid:46)108M thatcontainayoungstellarpopulationhave mationrateandspecificstarformationrate.Thestarformationrate
200 (cid:12)
onaverageahigherescapefraction.Thereasonforthisisthatthere inthehaloesrangesfrom0(inhaloesthatdidn’tformastarparticle
arelesshaloeswithzeroescapefractioncomparedtothetotalsam- inthelast5Myr)to2.3M yr−1.Wefindthattheaverageescape
(cid:12)
ple,becausethelow-masshaloesdonotcontainenoughdensegas fractionisaround10%forallhaloeswithstarformationratebelow
toshieldtheradiationfromtheluminousyoungstars. 10−3M yr−1.Thisreflectsthelackofdensegasinthesehaloes.
(cid:12)
Therelationbetweentheescapefractionandthemassissim- Ateveryredshift,around5%ofallhaloeshavestarformationrates
ilarfortheredshiftrange15 (cid:38) z (cid:38) 9,forallthreemassprobes. higherthan10−3M yr−1.Thesehaloesaremassiveenoughtore-
(cid:12)
At lower redshifts there are massive haloes that were not present tainthestar-forminggasinthepresenceofsupernovafeedbackand
athigherredshifts,butinthemass-rangeprobedinallthreehigh- thushavealargeN(r<10pc).Ingeneralthisresultsinaverylowes-
H
estredshiftbinstheaverageescapefractiondoesnotchangemuch. capefraction,f <10−3,withonlyafewoutliersoff >10−2
esc esc
ThisisdifferentfromtheresultsinPaperIwherewefoundthatthe wherefeedbackfromanearbystellarpopulationhasclearedaway
escapefractionwouldbehigheratlowerredshiftinaconstantmass thedensegasaroundrecentlyformedstarparticles.Atredshift6,
bin.Thisdifferenceiscausedbythewaytheionizingemissivityis feedbackfromtheUVbackgroundsuppressesstarformationinthe
computed:thecontinuousstarformationratemodelusedinPaperI majorityofhaloes(86%).
overestimatestheionizingemissivityandthustheescapefraction, Thedecreaseinescapefractionatthehigheststarformation
andtheeffectisstrongeratlowerredshifts. ratesiscausedbythemostmassivehaloesinthesample.Inthepre-
Between redshift 9 and 6 the scaling of the escape fraction vioussubsectionwehaveseenthatthereisastrongdependenceof
withmasschangesdrastically.ThisiscausedbytheuniformUV theescapefractiononthemassofthehalo.Tocircumventthemass
backgroundthatinthesimulationcomesintofulleffectatredshift dependence,weshowtheescapefractionasafunctionofspecific
9.InhaloeswithvirialmassM200 (cid:46) 108M(cid:12) theescapefraction star formation rate (the star formation rate divided by the stellar
isveryhigh(fesc (cid:38) 0.2)becausemostofthegasinthesehaloes mass)inFig.6aswell.Thespecificstarformationscaleswiththe
isionizedandheatedbythebackground.Inthismassrangethere number of ionizing photons produced per solar mass in stars. It
areveryfewhaloeswithzeroescapefraction.Therearenoyoung rangesfromzeroto200Gyr−1 inourhalosample.Haloeswitha
stellar populations present because the star formation rate is too specificstarformationratehigherthan50Gyr−1haveescapefrac-
lowintheheatedgas.TheUVbackgroundalsoaffectsthehaloes tionsabove10%.Thesearehaloesthathaverecentlyformedmost
moremassivethan108M(cid:12),althoughtheescapefractionstilldrops of their stars and therefore have the highest ionizing photon pro-
below10%inthismassrange.Similarly,weseeanescapefraction ductionpersolarmassinstars.Duetothehighnumberofioniz-
above10%inhaloeswithlowstellarmass(M(cid:63) (cid:46) 104M(cid:12))and ing photons, the young stars can ionize the dense gas around the
lowgassmass(Mgas (cid:46) 106M(cid:12))atredshift6duetotheeffectof sourcesandescapemoreeasily.Haloeswithspecificstarformation
theUVbackground.Haloeswithhigherstellarmassandgasmass rate below 50Gyr−1 show average escape fractions below 10%,
areaffectedlessduetotheirabilitytoself-shield. becausethenumberofionizingphotonsproducedpersolarmass
isnotsufficienttoionizetheimmediatesurroundingsoftheyoung
stars.
3.2.2 Starformationrate
Therelationbetweenstarformationrateandescapefractionistwo-
fold.Ontheonehand,ahighstarformationratemeansthatthere
3.2.3 Age,baryonfractionandstarformationefficiency
aremanyionizingphotonsproducedthatcanpenetratehighcolumn
densities(seeFig.3).Italsomeansthatthereisahigherrateofsu- The dense gas that is responsible for the absorption of ionizing
pernovaeoccurringandthisfeedbackmakesiteasierforradiation photonsinthehaloismainlyredistributedbysupernovafeedback.
toescapebyredistributingthedensegasinthehalo.Ontheother Thereforetheonsetofthefirstfeedbackevents,andthustheageof
hand,ahighstarformationrateimpliesthatthereiscooldensegas thehalo,couldbeanimportantfactorfortheescapefraction.On
inthehalothatislikelytoabsorbmanyionizingphotons. theotherhand,youngstarsemitordersofmagnitudemoreioniz-
Therearetwodifferentwaysofestimatingthestarformation ingphotons,makingiteasiertopenetratehighgascolumndensities
rateinthesimulation.Wecancomputethestarformationrateinev- whichisespeciallyimportantinlow-masshaloes.Herewedefine
erygasparticlefromthedensityandtemperature,andaveragethis theageofthehaloastheaverageageofthestellarpopulationin
overallthegasparticlesinthehalo.Thisistheinstantaneousstar thehalo,whichrangesfrom0.01Myrto750Myr.
formationratethatisusedinthestarformationrecipeinoursimu- In the top panel of Fig. 7 we show the escape fraction as a
lations.Althoughthisgivesagoodmeasureoftheamountofdense function of the average age of the star particles in the halo. At
gasinthehalo,itdoesn’tgiveanyinformationonrecentepisodes z > 6, the escape fraction is the highest, 20−30%, for haloes
of star formation and therefore doesn’t necessarily scale with the with an average age below 5Myr. These are low-mass haloes
number of ionizing photons produced in the halo. We therefore (∼ 107M )thatinthelast5Myrunderwentthefirstepisodeof
(cid:12)
useadifferentwaytoestimatethestarformationrate,thatismore starformation,hencetheyoungaverageage.Inmostcasesthegas
consistentwithhowthestarformationrateisdeterminedobserva- columndensitiesintheselow-masshaloesarenothighenoughto
tionally.Wecomputetheaveragestarformationrateinahaloby shieldthelargenumberofionizingphotonsproducedbytheyoung
averagingthestellarmassformedoverthelast5Myr.Thishasthe stellar populations. At these young ages there are therefore only
advantagethatthestarformationrateisproportionaltotheionizing a few haloes with zero escape fractions. Going to higher average
photonproduction.Thedrawbackofthismethodisthatthefixed age,theescapefractiondropstobelow10%forhaloeswithanage
resolutionofthestarparticlesimposesaminimumstarformation larger than 5Myr. We do not find an upward trend of the escape
rateof2.5×10−4M yr−1.Wehaveverifiedthatthetwowaysof fractionaroundanageof30MyrwhenthePopIIsupernovafeed-
(cid:12)
(cid:13)c 2014RAS,MNRAS000,1–21
10 J.-P.Paardekooperetal.
1.
0.1 f)esc
d.f.(
0.01p.
0.001
1.
0.1 f)esc
d.f.(
0.01p.
0.001
Figure6.Theescapefractionasafunctionofstarformationrate(top)andspecificstarformationrate(bottom)fordifferentredshifts.Colour-codedisthe
2-dimensionalhistogramofallthehaloeswithinabin,thesolidlinerepresentsthemeanescapefractioninthesamebinandthedashedlinethemeanescape
fractionforhaloescontainingatleastonestellarparticlewithageequaltoorlessthan5Myr.Theerrorsarethestandarddeviationinthemean.Forclarity,
haloeswithfesc < 10−4areplottedatfesc = 10−4.Thehistogramonthetopshowsthenumberofhaloesineachbin.Atredshift6thedataisfromthe
FiBYSsimulationonly,resultinginlargerPoissonerrorsthanthehigherredshifts.
backcommences.Inoursamplethedecreaseinionizingemissivity masshaloes((cid:46) 107M )thatinthelast5Myrunderwentthefirst
(cid:12)
ofthestellarpopulationafter5Myrismoreimportantthantheon- episodeofstarformation.Supernovafeedbackhasnotyetcleared
setofsupernovafeedback.Thedashedlineshowsthatthereisno awayanygasinthehalo,hencethehighbaryonfraction.However,
cleardifferencebetweenhaloeswithandwithoutyoungstars,al- duetotheirlowmass,thegascolumndensityislowandtherefore
thoughatredshift11.9thelattersampleshowsonaverageaslightly theescapefractionishigh.
higherescapefraction.Thisisnotthecaseattheotherredshifts.At Recently,Wiseetal.(2014)reportedastrongtrendofthees-
redshift6therearenohaloeswithaverageagebelow100Myrdue cape fraction with the star formation efficiency (defined as f =
(cid:63)
to the effects of the UV background. Star formation in low-mass M(cid:63) ),withtheescapefractionrisingfromtwopercentatthelow-
Mgas
haloesissuppressedbytheionizingbackground.Theonlyhaloes eststarformationefficiencies(f ∼ 10−4)tonearlyunityatthe
(cid:63)
thatarestillabletoformstarsinareionizedUniversearemassive highest star formation efficiencies (f ∼ 10−1). The reason for
(cid:63)
haloesthathaveformedstarscontinuouslyinthepastandtherefore thisisthatahighf impliesahighionizing-photon-to-baryonra-
(cid:63)
havehighaverageages. tio,thusmakingiteasierforthephotonstoionizetheinter-stellar
The importance of the amount of dense gas in the halo for mediumandsubsequentlyescape.
shieldingoftheionizingradiationmakesthebaryonfraction,de- IntheFiBYsimulationsthereisastrongcorrelationbetween
finedasfb = MM(cid:63)+2M00gas,apotentiallyinterestingquantitytocon- stellarmassandgasmassinthehaloes(Khochfaretalinprep.),so
straintheescapefraction.Forexample,supernovafeedbackwould f isameasureofthescatteraroundthisrelation.Formosthaloes,
(cid:63)
lowerthebaryonfraction,makingiteasierforradiationtoescape. thecauseofthescatterareprocesseslikeaccretionandfeedback
Inourhalosamplethebaryonfractionrangesfrom0.001to0.3. thatcausethegasmasstodifferfromtheaverageforagivenstellar
However, there are very few haloes with a baryon fraction above mass.However,atthelow-massendthestellarmassislimitedby
thecosmicmean Ωb =0.17. the SPH particle mass. Therefore, due to the probabilistic imple-
ΩM
InthemiddlepanelofFig.7weshowtheescapefractionas mentationofstarformation,inhaloeswiththelowestgasmasses
a function of the baryon fraction. At every redshift, haloes with thatformonlyasinglestarparticle,thestellarmasscanbeabove
thelowestbaryonfractionsshowhigh(> 10%)escapefractions, the average gas mass - stellar mass relation, but never below. In
as expected. However, the exact baryon fraction that results in a thesehaloeswewillthereforeoverestimatef .Inthesimulations
(cid:63)
highescapefractionchangeswithredshift.Atz = 15.2abaryon f rangesfrom10−4 to0.6.Wefindthehighestf atthelowest
(cid:63) (cid:63)
fraction of 0.015 results in an average escape fraction of ∼ 0.3. redshifts,becauseittakestimetogrowthestellarmassneededfor
On the other hand, at z = 8.9 the same baryon fraction yields suchahighf .
(cid:63)
an average escape fraction of ∼ 0.05. In general, at all redshifts InthebottompanelofFig.7weshowtheescapefractionas
arelativelylowbaryonfractionmeansthatthestarformationrate afunctionoff .Wedon’treproducetherisingtrendoftheescape
(cid:63)
isclosetozeroandthustheescapefractionishigh:thedensegas fractionreportedbyWiseetal.(2014).Instead,theaverageescape
has been cleared by feedback and no longer shields the ionizing fractionisconstantovertherangeoff for15(cid:38)z(cid:38)9.Exceptions
(cid:63)
radiation. Because the star formation rate is so low, the ionizing areafewhaloesatredshifts15.2and11.9withveryhighf and
(cid:63)
photonproductioninthesehaloesisverysmall. highf .Inthesecasesthehighf iscausedbytheoverestimate
esc (cid:63)
At redshifts 15.2 and 11.9 there is a population of haloes off inhaloeswithlowgasmass,whichinturnalsoleadstoahigh
(cid:63)
withhighbaryonfractionandhighescapefraction.Thesearelow- escapefraction.However,thishappensonlyin4haloesateachof
(cid:13)c 2014RAS,MNRAS000,1–21