How does radiative feedback from a UV background impact reionization?

2.1 MB·English

by  Emanuele Sobacchi

Mon.Not.R.Astron.Soc.000,000–000(0000) Printed30January2013 (MNLATEXstylefilev2.2)
How does radiative feedback from a UV background impact
reionization?
Emanuele Sobacchi1(cid:63), Andrei Mesinger1
1ScuolaNormaleSuperiore,PiazzadeiCavalieri7,56126Pisa,Italy
3 30January2013
1
0
2 ABSTRACT
AnionizingUVbackground(UVB)inhibitsgasaccretionandphoto-evaporatesgasfromthe
n
a shallowpotentialwellsofsmall,dwarfgalaxies.Duringcosmologicalreionization,thiseffect
J can result in negative feedback: suppressing star-formation inside HII regions, thus imped-

8 ingtheircontinuedgrowth.Itisdifficulttomodelthisprocess,giventheenormousrangeof
2 scales involved. We tackle this problem using a tiered approach: combining parameterized
results from single-halo collapse simulations with large-scale models of reionization. In the

] resultingreionizationmodels,theionizingemissivityofgalaxiesdependsonthelocalvalues
O
of the reionization redshift and the UVB intensity. We present a physically-motivated ana-
C lyticexpressionfortheaverageminimummassofstar-forminggalaxies,M¯ ,whichcanbe
min
. readily used in modeling galaxy formation. We find that UVB feedback: (i) delays the end
h
stagesofreionizationby∆z <0.5;(ii)resultsinamoreuniformdistributionofHIIregions,
p ∼
peaked on smaller-scales (with large-scale ionization power suppressed by tens of percent);
-
o and(iii)suppressestheglobalphotoionizationrateperbaryonbyafactorof<2towardsthe
∼
r endofreionization.However,theimpactismodest,sincethehydrodynamicresponseofthe
t
s gastotheUVBoccursonatime-scalecomparabletoreionization.Inparticular,thepopular
a
approachofmodelingUVBfeedbackwithaninstantaneoustransitioninM ,dramatically
[ min
overestimatesitsimportance.UVBfeedbackdoesnotsignificantlyaffectreionizationunless:

1 (i) molecularly-cooled galaxies contribute significantly to reionization; or (ii) internal feed-
v back processes strongly couple with UVB feedback in the early Universe. Since both are
1 consideredunlikely,weconcludethatthereisnosignificantself-regulationofreionizationby
8 UVBfeedback.
7
6 Key words: cosmology: theory – early Universe – galaxies: formation – high-redshift –
. evolution
1
0
3
1
: 1 INTRODUCTION UVBradiativefeedbackiscrucialindevelopingmodelsofreion-
v
ization,aswellasinterpretingobservations.
Xi Early generations of luminous objects reionized the Universe
within the first ∼billion years following the Big Bang. This pro- Themaindifficultyinself-consistentlymodelingUVBfeedback
r
a cessisexpectedtobefairlyextendedandinhomogeneous.Asion- liesintheenormousdynamicalrangerequired.Ideally,wewould
izingradiationspreadoutfromearlygalaxies,itheatedtheinter- needtofullyresolvetheinternalstructureofthedominantpopu-
galacticmedium(IGM),affectingitscoolingproperties,andphoto- lationofatomically-cooledhalos(withvirialradii∼1properkpc;
evaporatedgasfromshallowpotentialwells.Inthispicture,galax- Barkana&Loeb2001)incosmologicalsimulationswhichresolve
iesforminginsideanalready-ionizedIGMwouldhaveadepleted ionizingstructureon>100comovingMpcscales(e.g.Furlanetto
∼
gasreservoir(andbyextension,fewerstars)comparedwiththose et al. 2004). Even state-of-the-art single galaxy simulations must
forminginsidetheneutralIGM.Therefore,theionizingultraviolet employ some analytic prescriptions in estimating internal feed-
background(UVB)duringreionizationresultsinanegativefeed- back (e.g. Hopkins et al. 2011); therefore compromises must be
backmechanism,hinderingthefurthergrowthofionizedregions. made.Generallyoneoftwoapproachesarefollowed.Thefirstap-
Thisnegativefeedbackcoulddelayreionization,aswellasresultin proach uses relatively small simulation boxes (∼ few cMpc on a
amoreuniformreionizationmorphology.Thereforeunderstanding side),attemptingtomodelthesub-gridinterstellarmedium(ISM)
physics,throughanalyticalprescriptions(e.g.Petkova&Springel
2011;Finlator etal. 2011;Wiseet al.2012;Johnson etal. 2012;
Hasegawa&Semelin2012).Hereitisimportanttoexplorereason-
(cid:63) email:[email protected] able extremes of all sub-grid recipes, and ensure resolution con-
(cid:13)c 0000RAS
2 Sobacchi&Mesinger
vergence.Evenso,itiscurrentlyimpossibletomodeltherelevant redshift-dependenthalomass(e.g.Barkana&Loeb2001):
scalesofreionization.Thesecondapproachsimulatesreionization
(cid:16) µ (cid:17)−3/2(cid:18)Ω ∆ (cid:19)−1/2
onlarge-scales,butdoesnotmodeltheinternalstructureofgalax- M =108h−1 m c ×
cool 0.6 Ωz 18π2
ies; instead, simple analytic (and relatively ad-hoc) prescriptions m
areusedtoaccountforUVBfeedback(e.g.Ilievetal.2007;Mc- (cid:18) T (cid:19)3/2(cid:18)1+z(cid:19)−3/2
× vir M (cid:39)
Quinnetal.2007). 1.98×104K 10 (cid:12)
Inthispaperwefollowthelaterapproach.However,ratherthan
(cid:18)1+z(cid:19)−3/2
adoptingad-hocprescriptions,webaseourtreatmentofUVBfeed- (cid:39)108 M (2)
10 (cid:12)
backonresultsfromparameterstudiesof1Dcollapsesimulations
(Sobacchi&Mesinger2013,hereafterPaperI).InPaperI,weob- where µ is the mean molecular weight, Ωz =
m
tained expressions for the baryon content of galaxies, residing in Ω (1+z)3/(cid:2)Ω (1+z)3+Ω (cid:3)and∆ =18π2+82d−39d2
m m Λ c
regionswhichwerereionizedatredshiftzIN.Inthiswork,wein- withd=Ωmz −1.
cludetheseexpressionsinsemi-numericalsimulationsofinhomo- Fittingtoabroadparameterspaceof1Dcollapsesimulations,
geneous reionization. We present large-scale reionization simula- inPaperIwepresentedthefollowingexpressionforM (techni-
crit
tions which include a physically-motivated prescription for UVB callydefinedasthetotalhalomassatwhichthebaryonfractionis
feedback.Withthesesimulations,westudytheimportanceofUVB 1/2oftheglobalvalue,Ω /Ω ):
b m
feedbackinregulatingreionization.
(cid:18)1+z(cid:19)b(cid:20) (cid:18) 1+z (cid:19)c(cid:21)d
Thispaperisorganizedasfollows.In§2wedescribeourUVB M =M J a 1− (3)
feedbackprescription,andoursemi-numericalreionizationsimula- crit 0 21 10 1+zIN
tions.In§3wepresentourresults,comparingdifferentinhomoge- withbest-fitparameters:
neousreionizationmodels.In§4wediscussourresultsandpresent
(M ,a,b,c,d)=(cid:0)2.8×109M ,0.17,−2.1,2.0,2.5(cid:1) (4)
ourconclusions.ThroughoutweassumeaflatΛCDMcosmology 0 (cid:12)
withparameters(Ωm,ΩΛ,Ωb,h,σ8,n)=(0.27,0.73,0.046,0.7, The critical mass depends on (i) redshift z; (ii) the redshift zIN
0.82,0.96),consistentwithWMAPresults(Komatsuetal.2011). whenthehaloisexposedtoaUVB;(iii)theUVBintensityJ :2
21
Unlessstatedotherwise,wequoteallquantitiesincomovingunits.
J(ν)=J (ν/ν )−α×10−21ergs−1Hz−1cm−2sr−1 (5)
21 H
whereν istheLymanlimitfrequencyandα = 5correspondsto
H
astellar-drivenUVspectrum(e.g.Thoul&Weinberg1996).
2 MODELINGREIONIZATIONWITHUVBFEEDBACK
2.2 Semi-numericalreionizationsimulations
2.1 Minimumhalomasstohostgalaxies Tomodelcosmologicalreionization,weuseaparallelizedversion
InthesimplestUVB-regulatedscenario1,therearetwofundamen- ofthepubliclyavailablesemi-numericalsimulation,21CMFAST3.
We generate the IGM density and source fields by: (i) creating a
talhalomassscalesregulatingthegasreservoiravailableforstar
3DMonteCarlorealizationofthelineardensityfieldinaboxwith
formation:(i)M ,thecharacteristicorcriticalmassbelowwhich
crit sidesL = 300MpcandN = 16003 gridcells;(ii)evolvingthe
baryons are photo-evaporated or cannot efficiently accrete onto
densityfieldusingtheZel’dovichapproximation(Zeldovich1970),
theirhosthalos;(ii)M ,thecoolingthresholdrequiredforgasto
cool andsmoothingontoalower-resolutionN = 4003 grid;(iii)using
efficientlycool,collapseandformstarsinsidethehalo.Therefore,
excursion-set theory (Press & Schechter 1974; Bond et al. 1991;
wecandefinetheminimummassofstar-forminghalosas:
Lacey & Cole 1993; Sheth & Tormen 1999) on the evolved den-
M =max[M ,M ] . (1) sityfieldtocomputethefractionofmattercollapsedinhalosbig-
min cool crit
gerthanM ,thuscontributingtoreionization(seeMesinger&
min
Here we mainly consider the advanced stages of reionization, Furlanetto2007;Mesingeretal.2011foramoredetaileddescrip-
where cooling through molecular hydrogen is expected to be rel- tionofthecode).
ativelyinefficientduetoadisassociatingbackground(e.g.Haiman The ionization field is computed by comparing the integrated
et al. 1997). This corresponds to an atomic cooling threshold of number of ionizing photons to the number of baryons, in spheri-
a constant virial temperature, T ∼ 104 K, or analogously a calregionsofdecreasingradiusR(i.e.followingtheexcursion-set
vir
approach of Furlanetto et al. 2004). Specifically, a cell located at
spatialpositionandredshift,(x,z),isflaggedasionizedif:
ξf (x,R,z,M )≥1 (6)
coll min
1 Wecautionthatwedonotdirectlyincludeinternalfeedbackprocesses, wheref (x,R,z,M )isthefractionofcollapsedmatterin-
coll min
which,dependingontheISMtreatment,couldbeimportantindetermin- sideasphereofradiusRresidinginhaloslargerthanM ,and
min
ingthestarformationrate(SFR)ofdwarfgalaxiesathigh-redshifts(e.g.
Pawlik & Schaye 2009; Finlator et al. 2011; Wyithe & Loeb 2012). Al-
though winds can push gas to the outskirts of galaxies, where it can be 2 InPaperI,weignoredself-shieldingwhencomputingthecriticalmass.
moreeasilyphoto-evaporatedbyaUVB,thisamplificationeffectismod- Theimportanceofself-shieldingonsuchdwarfgalaxiesathigh-redshiftsis
estathigh-redshifts(atredshiftsnotfarbelowzIN;e.g.Pawlik&Schaye notwellknown,withsomeauthorssuggestingthatitdoesnothavealarge
2009;Finlatoretal.2011).Hence,duringmostofreionization,itisprobably impact on the relevant photo-evaporation time-scales (e.g. Shapiro et al.
reasonabletomodelinternalfeedbackindependentlyfromUVBfeedback 2004; Iliev et al. 2005). Nevertheless, it should be noted that neglecting
(e.g.bydecreasingtheionizingefficiencyparameter,ξ,below).Sincewe self-shieldingoverestimatestheimpactofUVBfeedbackonreionization,
areprimarilyconcernedwiththerelativeimpactofUVBfeedback,wedo makingourmainconclusionsconservative.
notexpectinternalfeedbacktoalterourmainconclusions. 3 http://homepage.sns.it/mesinger/Sim
(cid:13)c 0000RAS,MNRAS000,000–000
TheimpactofUVBfeedbackonreionization 3
no feedback
instantaneous feedback
J=0.01
self-consistent
J=1
Figure1.Slicesthroughourionizationboxes,withHIIregionsinblackandHIregionsinwhite.Fromtoptobottomweshowtheruns:NF,IF,J=0.01,SC,
J=1.
ξ isanionizingefficiency,definedbelow.StartingfromR = theIGM,N isthenumberofionizingphotonsperstellarbaryon,
max γ
R = 30 Mpc (roughly corresponding to the ionizing photon f isthefractionofgalacticgasinstars,andn¯ isthemeannum-
mfp ∗ rec
mean free path in the ionized IGM, R , at z ∼ 6 Songaila & berofrecombinationsperbaryonintheIGM.Ourfiducialchoice
mfp
Cowie2010;McQuinnetal.2011)thesmoothingscale,R,isde- of ξ = 30 results in an ionization history in agreement with the
creased,andthecriterionineq.(6)isre-evaluated.Atthecellsize, WMAPobservedvalueofτ (Komatsuetal.2011);howeverwe
e
thepartialionizationsfromsub-gridsourcesareevaluated,includ- alsoexploreothervaluesbelow.Althoughourmodelsdependonly
ing Poisson noise around the mean value of the collapse fraction on the product in eq. (7), we show on the RHS some reasonable
(e.g. Mesinger et al. 2011). This algorithm results in ionization valuesforthecomponentterms.N ≈4000isexpectedforPopII
γ
fields which are in good agreement with cosmological radiative stars(e.g.Barkana&Loeb2005),andstudiesofthehigh-redshift
transferalgorithms(Zahnetal.2011). Lymanalphaforestsuggestn¯ ∼0inthediffuseIGM(Miralda-
rec
Theionizingefficiencycanbewrittenoutas: Escude´2003;Bolton&Haehnelt2007;McQuinnetal.2011;note
(cid:18) (cid:19)(cid:18) (cid:19)(cid:18) (cid:19)(cid:18) (cid:19) that we instead incorporate photon sinks in the form of a mean
N f f 1
ξ=30 γ esc ∗ . (7) free path through the ionized IGM, governed by the separation
4000 0.15 0.05 1+n¯
rec of Lyman limit systems, LLSs). On the other hand, the parame-
wheref isthefractionofUVionizingphotonsthatescapeinto
esc
(cid:13)c 0000RAS,MNRAS000,000–000
4 Sobacchi&Mesinger
ters f and f are extremely uncertain in high-redshift galaxies
esc ∗
(e.g.Gnedinetal.2008;Wise&Cen2009;Ferrara&Loeb2012), 1
though our fiducial choices are in agreement with high-redshift
galaxyluminosityfunctions(e.g.Robertsonetal.2013).
Themainimprovementinthisworkisthatweusethespatially- SC
dependentvalueofM ,computedaccordingtoeq.1(ratherthan
min
ahomogeneousvalueMmin = Mcool asiscommonlydone).We HI 0.5 IF
x
keeptrackoftheredshiftwheneachcellwasfirstionized,z (x),
IN NF
updatingthelocalvalueofM (z,z ,J )asreionizationpro-
crit IN 21
J(cid:61)0.01
gresses (for details on how we compute the inhomogeneous J ,
21
see the following section). Neutral cells are assigned the value J(cid:61)1
M (z).Wethenusetheaveragevalue,(cid:104)M (cid:105) intheioniza-
cool min R 0
tion criterium of eq. (6). The drawback of this addition is that it 7 8 9 10 11 12 13 14 15
requires“running-down”thesimulationfromhighredshifts,rather
z
than being able to independently generate the reionization mor-
phologyatagivenredshift.
Figure2.Evolutionoftheglobalneutralfractionx¯HIwithdifferentmod-
els of radiative feedback. Dot-Dashed: no feedback (NF). Short-dashed:
J21 = 0.01 inside HII regions. Dotted: J21 = 1 inside HII regions.
2.2.1 Incorporatinganinhomogeneous,self-consistentUVB Dashed:instantaneousfeedback(IF).Long-dashed:self-consistentmodel
(SC).
WealsocomputetheaverageintensityinsideeachHIIregion,for
use in eq. (1). The mean emissivity (number of ionizing photons neglectsUVBfeedback,asiscommonlydoneinreionizationliter-
emitted into the IGM per second per comoving volume) can be ature.
writtenas: • Instantaneous feedback (IF): in this extreme model, feed-
(cid:15)≈f f N n¯ fcoll ≈ξn¯ fcoll , (8) back leads to an instantaneous transition in Mmin as soon as the
∗ esc γ b t b t halo is exposed to a UVB, thus ignoring the hydrodynamical re-
∗ ∗
sponse time-scale of the gas. This is the most common prescrip-
wheren¯ isthemeanbaryonnumberdensityinsidetheHIIregion,
b tion4 for UVB feedback (e.g. Wyithe & Loeb 2003; Onken &
andt isthestar-formationtime-scale(whichwetaketobet =
∗ ∗ Miralda-Escude´ 2004; Iliev et al. 2007; Wyithe & Morales 2007;
0.3×t ).Theintensityisproportionaltothemeanfreepathinside
H Wyithe&Cen2007;Kulkarni&Choudhury2011;Alvarezetal.
theHIIregion(takentobe≈R)multipliedbytheemissivity.Ifthe
2012; Ahn et al. 2012), and we take M = M in HI re-
emissivityisspectrallydistributedasν−α(eq.5),wecanwritethe min cool
gions and M = 109M in HII regions, consistent with pre-
averageUVBintensityinagivenHIIregionofcharacteristicsize min (cid:12)
vious works. The IF and NF runs therefore bracket the expected
R(inergs−1Hz−1sr−1propercm−2)as
impactofUVBfeedback.
(cid:104)J (cid:105) = (1+z)2Rhαn¯ ξfcoll . (9) • J=1: we calculate Mmin with eq. 1, using a constant UVB
21 HII 4π b t∗ intensityJ21 = 1inHIIregions.Thisintensityislargerthanesti-
matesfromtheLymanalphaforestatz ∼ 6,J ∼ 0.1(Bolton
Duetotheclusteringofdarkmatterhalos,therelevantintensity 21
&Haehnelt2007;Wyithe&Bolton2011;Calverleyetal.2011).
atgalaxylocationswillbehigherthantheaverageUVBintensityin
ThereforethismodelplacesanupperlimittotheimpactofUVB
eq.(9).ThereforewhencalculatingM ,weuse(cid:104)J (cid:105) =
min 21 halo,HII feedback (which is more physically relevant than the upper limit
f ×(cid:104)J (cid:105) , with f = 2, consistent with results from
Jbias 21 HII Jbias providedbytheIFmodel).
Mesinger&Dijkstra(2008).Wealsonotethatthesimplificationof
• J=0.01:likethepreviousmodelbutwithJ =0.01.
auniformUVBinsideeachHIIregionseemstobereasonable(see 21
• Self-consistent(SC):wecalculateM usingeq.1,andwe
Fig.7andtheassociateddiscussion). min
useeq.9tocalculatetheinhomogeneousUVBinHIIregions.
Furthermore, this formalism is not fully self-consistent, since
our expression for M was computed for a time-independent
crit
UVB.Nevertheless,wefindthatthemeanUVBisroughlyconstant
inHIIregions,somewhatvalidatingthisapproximation.Moreim- 3 RESULTS
portantly,feedbackislargelyinsensitivetotheexactvalueofJ
21 3.1 ReionizationHistory
sincethegasisalwaysheatedtoT ∼104K(specifically,inPaper
IwefindM ∝J0.17). InFig.1,weshowslicesthroughtheionizationboxesofourfive
crit
fiducial models: NF, IF, J=0.01, SC, J=1 (top to bottom). The
correspondingelectron-scatteringopticaldepthsare:τ = 0.088,
e
2.3 Runs τ = 0.073,τ = 0.087,τ = 0.085,τ = 0.0845 consistentat
e e e e
1σwithWMAPresults(Komatsuetal.2011).
The efficient, semi-numerical approach outlined above allows us
to explore several scenarios of UVB feedback during reioniza-
tion. As already mentioned, our fiducial model assumes ξ = 30, 4 Insimulations,thisgenerallyamountstousingthesimulationtime-step
Tcool = 104K and Rmfp = 30cMpc, though we explore other (generally≈10Myr)asthefeedbacktime-scale.
valuesthroughout.Wehavefivemainprescriptionsforimplement- 5 UV-drivenreionizationis“inside-out”onlarge-scales,withoverdensities
ingUVBfeedback: hostingthebiasedgalaxiesbeingthefirsttoionize.Mesingeretal.(2012)
notedthatthiscorrelationbetweenthedensityandionizationfieldsresults
• Nofeedback(NF):weassumeMmin = Mcool,correspond- ina≈4%higherτethanwhatwouldbeexpectedassumingthetwofields
ingtoaconstantvirialtemperatureTcool.Thismodelcompletely areuncorrelated:(cid:104)xi×nb(cid:105)(cid:54)=(cid:104)xi(cid:105)×(cid:104)nb(cid:105).Forexampleourself-consistent
(cid:13)c 0000RAS,MNRAS000,000–000
TheimpactofUVBfeedbackonreionization 5
As expected, UVB feedback delays reionization and results in (McQuinn et al. 2007). In other words, reionization morphology
a more uniform distribution of HII regions (e.g. McQuinn et al. ismuchmoresensitivetox¯ thanitistoUVBfeedbackeffects.
HI
2007;Ilievetal.2007).However,ourself-consistentmodelisnot Thisimpliesthatpreviouspredictionsforreionizationmorphology
remarkablydifferentfromtheonewhichignoredUVBfeedbackal- (atfixedx¯ )arenotparticularlysensitivetothissourceofastro-
HI
together(NF).Moststrikingistheeffectofanunphysicalinstanta- physicaluncertainty.
neoustransitioninM :theIFmodeldramaticallyover-predicts
min
theimportanceofUVBfeedbackevenwithrespecttotheextreme
3.3 EvolutionofM
J=1 model. We can therefore immediately surmise that the delay min
inthehydrodynamicresponseofthebaryonstotheUVBisfunda- Theresultsoftheprevioussectionscanbemorereadilyunderstood
mentalinproperlymodelingUVBfeedback. bylookingattheredshiftevolutionoftheaveragevalueofthemin-
This qualitative result is confirmed by the evolution of the imum halo mass hosting star-forming galaxies, M¯ , shown in
min
globalneutralfractionx¯HI,showninFig.2.Asexpected,theself- Fig. 5, using the same line styles as above. This quantity is also
consistentmodelshowsintermediatefeaturesbetweentheJ=0.01 fundamentalinsemi-analyticalmodelsofgalaxyformation.
andJ=1models.SincethetransitionofMminisnotinstantaneous, By construction, the M¯min curve in the NF model simply fol-
onlythelatestagesofreionizationaredelayed.ComparingtheSC lows the T = 104 K isotherm. The curve corresponding to
vir
andNFcurves,weseethatUVBfeedbackdelaysthelaterstages the SC model is closer to the J=1 model than the J=0.01 one
ofreionizationby∆z <∼0.5.Ontheotherhand,thisdelayissig- (seeFig.7andassociateddiscussion).Itbegins“pealingoff”the
nificantlyoverestimatedbyamodelwithinstantaneousfeedback: T = 104 K isotherm at z ∼ 10, when reionization is already
vir
∆z≈1.5. well underway (x¯ ∼ 0.6). In contrast, M¯ in the IF model
HI min
immediately starts increasing as reionization progresses, since in
that model the average minimum halo mass is simply M¯ =
min
3.2 ReionizationMorphology x¯ M +(1−x¯ )109M .Becauseitignoresthedelayasso-
HI cool HI (cid:12)
We now proceed to quantify the impact of UVB feedback on ciatedwithUVBfeedback,theIFmodelover-predictsM¯minbya
the morphology of reionization. In Fig. 3 we show the ioniza- factorof∼2–3,withrespecttotheSCmodel.
tion power spectrum (∆2xx ≡ k3/(2π2V) (cid:104)|δxx|2(cid:105)k, with δxx = ForcomparisonwealsoshowtheevolutionofM¯min assuming
xHI/x¯HI−1),whileinFig.4weshowthesizedistributionsofthe a homogeneous reionization at zIN = 9.3 = zre, corresponding
HIIregions6.Panelscorrespondtodifferentstagesofreionization: totheredshiftwhenx¯HI = 0.5intheself-consistentmodel(ma-
x¯HI = 0.75,0.50,0.25, (left to right). We compare our fiducial gentasolidline).Thisformulaunder-predictsM¯minbyafactorof
models of UVB feedback: no feedback (yellow dot-dashed line); ∼ 2 around the mid-point of reionization, highlighting that the
J = 0.01 (magenta short-dashed line); J = 1 (blue dotted scatter in the reionization redshifts of galaxies can be important
21 21
line); instantaneous feedback (green dashed line); self-consistent inregulatingstarformation(e.g.Alvarezetal.2009).Thishomo-
(bluelong-dashedline). geneousreionizationmodelfurtherhighlightsthedelayinthehy-
As expected from Fig. 1, we see that ignoring feedback over- drodynamicresponseofthegastotheUVB:Mcritonlysurpasses
predicts the large-scale ionization power by tens of percent. The Mcoolatz≈8,eventhoughtheionizingbackgroundiseffectively
sametrendcanbeseenintheHIIsizedistributions,withthepeak turnedonatzre =9.3(seealsoOkamotoetal.2008).
intheNFmodelbeingshiftedby≈ 50%withrespecttotheSC It is natural to approximate this delay with the sound-crossing
model.TheIFmodelontheotherhandover-estimatesthesefeed- time-scale, tsc = 2Rvir/cs(104K) (which is comparable to the
backeffectsbyacomparableamount.Wethereforeconfirmearlier photo-evaporation time-scale, e.g. Haiman et al. 2001; Shapiro
predictions that UVB feedback results in smaller, more uniform etal.2004).ForahaloofmassM,thesound-crossingtime-scale
ionizedregions(McQuinnetal.2007).Thisiseasytounderstand: canbeapproximatedas(c.f.Shapiroetal.2004):
thelargestHIIregionscorrespondtothemostbiasedlocationsof (cid:18) M (cid:19)1/3(cid:18)1+z(cid:19)−1(cid:18)Ω h2(cid:19)−1/3
the density field, and are therefore ionized earliest. It is in these tsc ≈200Myr 108M 10 0m.15 .
regionsthatUVBfeedbackhashadthemosttimetoquenchstar- (cid:12)
(10)
formation.Smaller,late-formingHIIregionsarelessaffected,and ItwouldthereforebereasonabletoexpectM¯ toevolvefrom
min
thereforemoreabundantatfixedx¯ .
HI M athighredshiftstosomeM (T ≈const)atlowredshifts,
cool 0 vir
Nevertheless, the impact of UVB feedback is relatively minor,
withthetransitionoccurringatz∼z −∆z [where∆z isthe
re sc sc
whencomparedatfixedx¯ :thepowerspectraofallmodelsagree
HI redshift interval corresponding to eq. (10)], roughly over a time-
to within a factor of < 2. This is due to the fact that the halo
∼ scale scaling with the duration of reionization, ∆z . Indeed, we
re
bias evolves only weakly over the mass scales relevant to UVB findthatourresultsforM¯ arewellfittedby:
min
feedback; hence morphology at a given x¯ is relatively robust
HI (cid:18) M (cid:19)g
M¯ (z)=M × 0 , (11)
min cool M
cool
modelhasτe =0.085insteadofτe =0.083aswouldbeexpectedignor- withthetransitionfunction
ingthiscorrelation.
6 WecalculatethesizedistributionsfollowingtheprescriptioninMesinger g(z)= (cid:104) 1 (cid:105) . (12)
&Furlanetto(2007).Namely,werandomlychooseanionizedcellandthen 1+exp z−(zre−∆zsc)
recordthecomovingdistancetotheedgeoftheHIIregioninarandom ∆zre
direction.ThenwedefinethesizedistributionasRdp/dR,wherepisthe WetakeM (z)tocorrespondtoafixedvirialtemperatureT =
0 0
probabilitythatthisdistanceisbetweenRandR+dR.Technicallywe 5×104K,7anddefine∆z tocorrespondtotheredshiftinterval
re
define the edge of an HII region as the first cell with xHI ≥ 0.5. This betweenx¯HI =0.6andx¯HI =0.4.
choiceofthresholdcanaffectthesizedistributionssomewhat;however,we
areinterestedintherelativedifferencebetweenthemodels,whichismuch
morerobusttothischoice(Friedrichetal.2011). 7 We caution that our simple 1D simulations from Paper I are not well
(cid:13)c 0000RAS,MNRAS000,000–000
6 Sobacchi&Mesinger
1 1 1
x (cid:61)0.75
HI
SC
0.1 0.1 0.1
2(cid:68)xx 2(cid:68)xx 2(cid:68)xx NIFF
10(cid:45)2 10(cid:45)2 10(cid:45)2
xHI (cid:61)0.50 x (cid:61)0.25 J(cid:61)0.01
HI
J(cid:61)1
10(cid:45)3 10(cid:45)3 10(cid:45)3
0.1 1 0.1 1 0.1 1
k Mpc(cid:45)1 k Mpc(cid:45)1 k Mpc(cid:45)1
(cid:72) (cid:76) (cid:72) (cid:76) (cid:72) (cid:76)
Figure3.Powerspectrum∆2xxoftheionizationfieldatx¯HI = 0.75(left),x¯HI = 0.50(middle)andx¯HI = 0.25(right)withdifferentmodelsofUVB
feedback.Dot-Dashed:NF.Short-dashed:J=0.01.Dotted:J=1.Dashed:IF.Long-dashed:SC.
0.3 0.3 0.3
xHI (cid:61)0.75 xHI (cid:61)0.50 xHI (cid:61)0.25
R R R
d 0.2 d 0.2 d 0.2
(cid:144) (cid:144) (cid:144)
p p p
d d d
R R R
0.1 0.1 0.1
0 0 0
1 10 102 1 10 102 1 10 102
R Mpc R Mpc R Mpc
(cid:72) (cid:76) (cid:72) (cid:76) (cid:72) (cid:76)
Figure4.SizedistributionoftheHIIregionsatx¯HI =0.75(left),x¯HI =0.50(middle)andx¯HI =0.25(right)withdifferentmodelsofUVBfeedback.
Dot-Dashed:NF.Short-dashed:J=0.01.Dotted:J=1.Dashed:IF.Long-dashed:SC.
InFig.6wecomparethisanalyticapproximation(dottedlines)
with the results from our simulation (dashed lines). We show the 109
self-consistent model with the standard choice of the parameters HR
(inthiscasetheanalyticformulaandtheresultofsimulationare SC
indistinguishable).Totesttherobustnessoftheaboveapproxima- IF
(cid:159)
tion, we also include two ”extreme” models: (i) with molecular M NF
hydrogen cooled halos significantly contributing to reionization: n (cid:144) J(cid:61)0.01
Tcool =5×103K(magentacurves);(ii)withasignificantlyhigher Mmi J(cid:61)1
emissivityξ =100(greencurves).Theserunscorrespondtozre= 108
(9.3,9.9,12.0),∆z =(1.0,1.3,0.8)and∆z =(2.0,0.9,2.5),for
re sc
thefiducial,T =5×103K,andξ=100models,respectively.
cool
Even in the extreme cases, our formula from eq. (11) provides a
good fit. This general, physically-motivated expression facilitates 7 8 9 10 11 12 13 14 15
implementingUVBfeedbackfromaninhomogeneousreionization
z
intomodelsofgalaxyformation.
Figure5.AveragevalueofMmin versusredshift.Dot-dashed:nofeed-
back(NF).Short-dashed:ourmodelwithJ21 =0.01.Dotted:ourmodel
3.4 UVBEvolution withJ21 = 1.Dashed:instantaneousfeedback(IF).Long-dashed self-
InadditiontotheevolutionofHIIregions,UVBfeedbackaffects consistentmodel(SC).Solid:forcomparisonweshowMmin (eq.1)as-
theUVBitself.Asstar-formationissuppressedinvulnerablehalos, sumingahomogeneousreionization(HR)atzIN =9.3(correspondingto
themid-pointofreionizationintheSCmodel).
theUVBdecreases(withrespecttothecasewithoutfeedback).Our
formalismexplicitlycomputestheinhomogeneousUVBintensity,
asdescribedin§2.2.1.Itthereforeprovidesausefulpredictionof
alphaforestinhigh-redshiftquasarspectra.Theionizationrateper
theevolutionofJ duringreionization.
21 baryon is found to be remarkably constant from z ≈ 2 → 5,
TheUVBcanbedeterminedfrommeasurementsoftheLyman Γ ≈ 10−12 s−1 (Bolton & Haehnelt 2007; Faucher-Gigue`re
bkg
etal.2008).Thenfromz≈5→6,thereisevidenceofadropbya
factoroffew,thoughtheuncertaintyinthemeasurementislargeat
suited for low-redshift structure formation, where mergers and other 3D
thesehigh-redshiftswhentheLyαforestbeginstosaturate(Bolton
effects can be important. Hence, here we use a lower value of M0 than
impliedbyextrapolatingeq.(3)toz = 0;ourchoiceisroughlyconsis- &Haehnelt2007;Wyithe&Bolton2011;Calverleyetal.2011).
tentwithcosmologicalsimulationsofhomogeneousreionization(Okamoto Suchadropcouldbecausedbyeitheranincompletereionization
etal.2008). (Mesinger2010;McGreeretal.2011),and/orbyarapidevolution
(cid:13)c 0000RAS,MNRAS000,000–000
TheimpactofUVBfeedbackonreionization 7
isgenerallynotmadeinLymanforeststudies,whichtherebyim-
plicitlyassumethatreionizationisover(thoughseeGalleranietal.
2006andMcGreeretal.2011),orthatitisdrivenbyX-rayswith
109
longmeanfreepaths(e.g.Oh2001).Tohighlightthispoint,inFig.
Ξ(cid:61)100
7wealsoshowtheevolutionof(cid:104)J (cid:105) intheSCmodel(dashed
21 V
(cid:159)
M curve).
min 108(cid:144) SC regFioronms iFnigth.e7,SwCemseoedethlaitstqhueiateveflraatg,ewUitVhBaivnatelunesitoyfw(cid:104)Jit(cid:105)hinH(cid:39)II
M HII
0.2,consistentwithobservationalestimatesatz ∼6.9Comparing
Tcool(cid:61)5(cid:180)103K the solid and dot-dashed curves, we see that feedback in our SC
107 modelsuppressestheUVBbyafactorof<∼2towardstheendof
reionization. This factor is modest, compared to the uncertainties
7 8 9 10 11 12 13 14 15 inthemodelparametersandobservationalmeasurements.
The shaded region in Fig. 7 corresponds to the 1σ scatter in
z
(cid:104)J (cid:105) . This spread is roughly a factor of ∼ 2 during most of
21 HII
Figure6.AveragevalueofMmin versusredshift.Wecomparetheself- reionization,decreasingto<∼50%inthefinaloverlapstages.Such
consistentmodelwiththestandardparameterchoice(SC),withTcool = aspreadintheHII-regionaveragedintensityisroughlycompara-
5×103Kandwithξ = 100.Dashed:resultsofthesimulation.Dotted: bletotheoverallspreadintheintensity,asestimatedbyMesinger
analyticexpressionineq.11. &Dijkstra(2008)andMesinger&Furlanetto(2009)whosummed
contributions from individual galaxies. This supports our formal-
ism in §2.2.1, and the corresponding simplification of using a
roughlyuniformionizingbackgroundinsideeachHIIregion.
Calverley (cid:43) 2011
1 Bolton &Haenelt 2007 10(cid:45)12
(cid:76) 3.5 WasReionization“Self-Regulated”?
(cid:72) (cid:76) (cid:45)1
J21 0.1 (cid:72) (cid:76) 10(cid:45)13 s (cid:72) UsingtheassumptionofaninstantaneouschangeinMmin inside
g ionized regions (i.e. ourIF model), many works claim thatUVB
k
b
(cid:71) feedbackhasadramaticimpactonreionizationhistoryandthecor-
10(cid:45)2 10(cid:45)14 respondingstar-formationrate(e.g.Wyithe&Loeb2003;Onken
&Miralda-Escude´2004;Ilievetal.2007;Wyithe&Morales2007;
Wyithe&Cen2007;Kulkarni&Choudhury2011;Alvarezetal.
6 7 8 9 10 11 12 13 14 15
2012; Ahn et al. 2012). Some works dub this effect to be “self-
z regulation”,sincewhenHIIregionsform,theirsubsequentgrowth
isimpaired.However,weshowabovethatundermorerealisticpre-
Figure 7. Evolution of the UVB intensity J21 (left-hand scale) and of scriptionsofUVBfeedback,thiseffectisrelativelyminor(seealso
thecorrespondingphotoionizationrateperbaryonΓbkg(right-handscale). Mesinger&Dijkstra2008).Thisqualitativeclaimiseasytounder-
Solid:averageintensity(cid:104)J(cid:105)HII withinHIIregionsinourSCmodel;the stand:thetime-scalesassociatedwithUVBfeedbackarecompara-
shadedregioncorrespondstothespreadamongHIIregions.Dashed:aver- bletothetime-scaleofreionization.Thusintheregimeswestudy,
ageintensity(cid:104)J(cid:105)V intheentiresimulationbox.Dot-dashed:evolutionof there is not enough time for UVB feedback to efficiently delay
(cid:104)J(cid:105)HIIwhenfeedbackisneglected(NF).ForcomparisonweshowΓbkg reionization. We now ask the question, “what are the physically-
atz ≈ 6(offsetforclarity)asestimatedfromtheLyαforest(Bolton&
motivatedregimesinwhichUVBfeedbackcaninfactself-regulate
Haehnelt2007;Calverleyetal.2011).
reionization?”NaivelyonemightexpectthatthestrengthofUVB
feedback can be increased by (i) a higher UVB intensity; (ii) a
intheabundanceofLLSs(Furlanetto&Oh2005;McQuinnetal.
2011).8 moreextendedreionization;(iii)alargercontributionofsmallha-
los,moresusceptibletoUVBfeedback.Weexploreeachofthese
InFig.7weshowtheredshiftevolutionoftheUVBinourSC
inturn.
(solidcurve)andNF(dot-dashedcurve)models(J /Γ onthe
21 bkg
In Fig. 8 we show the reionization history in models with dif-
left/right vertical scales), together with z ≈ 6 observational esti-
ferent ionizing photon efficiencies: ξ =30, 50, 100 (left to right
matesbyBolton&Haehnelt(2007)andCalverleyetal.(2011).It
pairs of curves). We compare the evolution when UVB feedback
is important to note that during UV-driven reionization, the UVB
is neglected (NF; dot-dashed line) with our self-consistent ap-
is expected to be bi-modal, with (cid:104)J (cid:105) inside HII regions and
21 HII
proach (SC; long-dashed line). In fact, increasing the UVB in-
J ∼ 0 inside neutral regions, such that the volume-averaged
21
tensity (J ∝ ξ), results in smaller feedback effects. This is
UVB intensity is (cid:104)J (cid:105) ∼ (1−x¯ )(cid:104)J (cid:105) . This distinction 21
21 V HI 21 HII
because the dependence of M on J is weak in our model
min 21
(M ∝J0.17),whileforhighvaluesofξ,reionizationproceeds
crit 21
8 Notethatalargedropintheionizingbackgroundmustinvolvethephoto- faster. Therefore as the ionizing emissivity is increased, the de-
creaseinthedurationofreionizationismorerelevantthanthedelay
evaporationofLLSs(McQuinnetal.2011),sincethesesystemsregulatethe
progressofreionizationduringitsfinalstages(e.g.Furlanetto&Mesinger
2009;Crocianietal.2011;Alvarez&Abel2012).Wecautionthatourmod-
elsdonotincorporateanevolvingmeanfreepathinionizedregions.There- 9 Choosingalower/highervalueoftheionizingefficiency,ξ,wouldde-
fore,theylikelyunderestimatetheriseintheUVBinthelatestagesand lay/advancereionizationanddecrease/increasetheresultingphotoioniza-
followingreionization.However,therelativeimpactoffeedbackismore tionrate.Theobservationaldatastillallowforarelativelybroadrangeof
robusttothisuncertainty. ξ.
(cid:13)c 0000RAS,MNRAS000,000–000
8 Sobacchi&Mesinger
resultingfromtheincreaseinM ,andtheevolutionofreioniza-
crit
tionisdrivenbytheemissivity. 1
Wenextattempttoextendthedurationofreionization,inorder
togivetheUVBtimetosignificantlysuppressthegascontentof
galaxies.Themostefficientwayofsignificantlyextendingreion- Ξ(cid:61)50
ization10istodecreasetheionizingphotonmeanfreepaththrough
theionizedIGM,Rmfp(e.g.Alvarez&Abel2012;Mesingeretal. HI 0.5
2012). Our fiducial model assumes R = 30 Mpc, consistent x Ξ(cid:61)30
mfp
withobservationalandtheoreticalestimatesatz≈6(e.g.Songaila
&Cowie2010;McQuinnetal.2011).InFig.9weshowthereion- Ξ(cid:61)100
izationhistoryofourfiducialmodel,butdecreasingR to5Mpc.
mfp
Although the reionization history is significantly extended, UVB
0
feedbackhasanegligibleeffect,resultinginadelayof∆z <∼0.1. 7 8 9 10 11 12 13 14 15
This is because the effective horizon imposed by R restricts
mfp z
feedbacktoaverylimitedvolumesurroundingeachgalaxy.Reion-
izationproceedsinadisjointmanner:withtiny,isolatedHIIregions
Figure8.Evolutionoftheglobalneutralfractionx¯HIwithdifferentioniz-
emerging around newly-forming galaxies (Alvarez & Abel 2012; ingefficienciesξ =30,50,100.Dot-Dashed:nofeedback.Long-dashed:
Mesingeretal.2012).Suchareionizationscenarioisnotaffected self-consistentmodel.
byUVBfeedback.
Finally,weturntoourpoint(iii):alargercontributionofsmall
halos,moresusceptibletoUVBfeedback.Sofar,wehaveconsid- 1
ered halos down to T = 104K: this is conservatively small, R (cid:61)5cMpc
cool mfp
sincewindsandinternalfeedbackcansuppressstar-formationin-
T (cid:61)5(cid:180)103K
sidesuchhalos(e.g.Springel&Hernquist2003;Pawlik&Schaye cool
2009;Finlatoretal.2011).However,somemodelsargueforasig-
nificantcontributionofgalaxieshostedbyevensmallerhalos(so- HI 0.5
x
calledminihalos),inwhichgascollapsesthroughmolecularcool-
ing channels. In Fig. 9, we show the reionization history of such
a toy model which allows halos down to T = 5×103K to
cool
emit ionizing photons with the same fiducial efficiency ξ = 30.
In this extreme case, UVB feedback has a dramatic impact on 0
reionization history: delaying the advanced stages of reionization 7 8 9 10 11 12 13 14 15
by ∆z ∼ 1.5–2, and resulting in a slow evolution of x¯ during
HI z
thefirsthalfofreionization.Thisresultisqualitativelyconsistent
withtherecentresultsbyAhnetal.(2012)(whoassumedinstan- Figure 9. Evolution of the global neutral fraction x¯HI with Rmfp =
taneousUVBfeedback,aswellasinstantaneousfeedbackfroman 5cMpc and with Tvir = 5×103K. Dot-Dashed: no feedback. Long-
H -disassociative background; the later results in an even flatter, dashed:self-consistentmodel.
2
self-regulatedregimeduringtheearlystages).11
4 CONCLUSIONS
Wethereforeconcludethatreionizationisself-regulatingonlyif
star-formationisefficientinsidemolecularly-cooledgalaxies.We Cosmic reionization inhomogeneously heated the IGM, affecting
caution however that our implementation of UVB feedback from itscoolingpropertiesandphoto-evaporatinggasfromtheoutskirts
PaperIwascalibratedonlyforatomiccooledhalos.Furthermore, of galaxies. As a result, small dwarf galaxies inside the ionized
our models assume that the star-formation rate is proportional to patcheswouldhaveareducedgasreservoirforstar-formation.This
thetotalamountofgasinsidehalos.Wedonotmodeltheabilityof processinvolvesahugedynamicrange;thereforelarge-scalereion-
gastocoolandformstars,whichcanbedramaticallysuppressed izationsimulationsgenerallyeitherignorethiseffectorincludeit
inminihalosbyanH2-disassociativebackgroundbeforethebulkof withsimple,fairlyad-hocprescriptions.
reionization (assuming modest values for their ionizing emissivi- In this work, we use a tiered approach to study the impact of
ties;e.g.Haimanetal.1997). UVB feedback on reionization. Combining parameterized results
fortheminimumhalomasscapableofhostingstar-forminggalax-
ies(M ,seePaperI)withsemi-numericalsimulationsofreion-
min
10 Notethatwealreadyincludestarformationinhalosdowntotheatomic- ization,wepresentlarge-scale(300Mpc)reionizationsimulations
coolingthreshold.Thefractionalabundanceofthesehalosevolvesmore which include UVB feedback in a physically-motivated manner.
slowly than that of more massive halos, expected to have more efficient In our models, the ionizing emissivity of galaxies is assumed to
star-formation(e.g.Lidzetal.2007).Thereforeincreasingthedurationof be proportional to their baryonic content, and depends on the lo-
reionizationbychangingthepopulationofhosthalosisonlypossibleifwe calvaluesofthereionizationredshiftandtheUVBintensity.These
includemolecularly-cooledgalaxies(seebelow). additionswillbeincludedinanupcomingversionupdateof21CM-
11 Wecautionthatsuchsimplemodelsverylikelyoverestimatetheion-
FAST.
izingefficienciesofminihalos,byignoringthechemicalandmechanical
UVBfeedbackdelaystheendstagesofreionizationby∆z <
feedbackfromthefirststars(e.g.Tornatoreetal.2007;Whalenetal.2008), ∼
0.5,suppressesthelarge-scaleionizationpowerbytensofpercent
X-ray pre-heating of the IGM (e.g. Ricotti & Ostriker 2004), as well as
thebaryon-DMvelocityoffset(e.g.Tseliakhovich&Hirata2010).Simu- (at fixed x¯HI), and decreases the mean photoionization rate by a
lationspredictthatthesemechanismsareveryefficientinsuppressingstar- factor of <∼ 2 towards the end of reionization. These effects are
formationinminihalos. quitemodest,whencomparedtocurrentastrophysicalandobser-
(cid:13)c 0000RAS,MNRAS000,000–000
TheimpactofUVBfeedbackonreionization 9
vational uncertainties. By contrast, the popular assumption of an MesingerA.,DijkstraM.,2008,MNRAS,390,1071
instantaneouschangeinM resultsinamuchlargereffect,with MesingerA.,FerraraA.,SpiegelD.S.,2012,preprint(arXiv:1210.7319)
min
a∆z ≈1.5delayinreionization,andafactoroffewhigheraver- MesingerA.,FurlanettoS.,2009,MNRAS,400,1461
ageM . MesingerA.,FurlanettoS.R.,2007,ApJ,669,663
min
MesingerA.,FurlanettoS.R.,CenR.,2011,MNRAS,411,955
AssumingthatUVBfeedbackactsonatime-scaleroughlycor-
MesingerA.,McQuinnM.,SpergelD.N.,2012,MNRAS,422,1403
respondingtothehalosound-crossingtime,wepresentananalytic
formulaforM¯ .Thisgeneralexpressionfitsoursimulationre- Miralda-Escude´J.,2003,ApJ,597,66
min OhS.P.,2001,ApJ,553,499
sults very well, and can be easily included in models of galaxy
OkamotoT.,GaoL.,TheunsT.,2008,MNRAS,390,920
formation. OnkenC.A.,Miralda-Escude´J.,2004,ApJ,610,1
In our models, UVB feedback significantly delays reioniza- PawlikA.H.,SchayeJ.,2009,MNRAS,396,46
tion only if molecularly-cooled galaxies contribute significantly PetkovaM.,SpringelV.,2011,MNRAS,412,935
to reionization. Star-formation inside such galaxies is easily sup- PressW.H.,SchechterP.,1974,ApJ,187,425
pressedbymechanicalfeedback,X-raypre-heatingandthebaryon- RicottiM.,OstrikerJ.P.,2004,MNRAS,352,547
DMvelocityoffset.Hence,weconcludethatUVBfeedbackisun- RobertsonB.E.,etal.,2013,ArXive-prints
ShapiroP.R.,IlievI.T.,RagaA.C.,2004,MNRAS,348,753
likelytoself-regulatereionization.
ShethR.K.,TormenG.,1999,MNRAS,308,119
SobacchiE.,MesingerA.,2013,preprint(PaperI)
WethankAndreaFerraraforcommentsonadraftversionofthis SongailaA.,CowieL.L.,2010,ApJ,721,1448
work. SpringelV.,HernquistL.,2003,MNRAS,339,312
ThoulA.A.,WeinbergD.H.,1996,ApJ,465,608
TornatoreL.,FerraraA.,SchneiderR.,2007,MNRAS,382,945
TseliakhovichD.,HirataC.,2010,PRD,82,083520
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