The Effect of Absorption Systems on Cosmic Reionization

6.2 MB·English

by  Marcelo A. Alvarez

DRAFTVERSIONJANUARY20,2012
PreprinttypesetusingLATEXstyleemulateapjv.03/07/07
THEEFFECTOFABSORPTIONSYSTEMSONCOSMICREIONIZATION
MARCELOA.ALVAREZ1 ANDTOMABEL2
DraftversionJanuary20,2012
ABSTRACT
Weuselarge-scalesimulationstoinvestigatethemorphologyofreionizationduringthefinal,overlapphase.
Ourmethodusesanefficientthree-dimensionalsmoothingtechniquewhichtakesintoaccountthefinitemean
freepathduetoabsorptionsystems,λ ,byonlysmoothingoverscalesR <λ . Thelargedynamicrangeof
abs s abs
ourcalculationsisnecessarytoresolvetheneutralpatchesleftattheendofreionizationwithinarepresentative
2
volume; we find that simulation volumes exceeding several hundred Mpc on a side are necessary in order to
1
properlymodelreionizationwhentheneutralfractionis(cid:39)0.01−0.3. Ourresultsindicateastrongdependence
0
ofpercolationmorphologyonalargeanduncertainregionofmodelparameterspace.Thesinglemostimportant
2
parameter is the mean free path to absorption systems, which serve as opaque barriers to ionizing radiation.

n If these absorption systems were as abundant as some realistic estimates indicate, the spatial structure of the
a overlapphaseisconsiderablymorecomplexthanpreviouslypredicted.Inviewofthelackofconstraintsonthe
J
meanfreepathatthehighestredshifts,currenttheoriesthatdonotincludeabsorptionbyLyman-limitsystems,

9 andinparticularthree-dimensionalsimulations,mayunderestimatetheabundanceofneutralcloudsattheend
1 of reionization. This affects predictions for the 21 cm signal associated with reionization, interpretation of
absorptionfeaturesinquasarspectraatz∼ 5−6,theconnectionbetweenreionizationandthelocaluniverse,
]
O and constraints on the patchiness and duration of reionization from temperature fluctuations measured in the
cosmicmicrowavebackgroundarisingfromthekineticSunyaev-Zel’dovicheffect.
C
Subjectheadings:cosmology: theory—darkages,reionization,firststars—intergalacticmedium
.
h
p
- 1. INTRODUCTION Directsimulationofalltheprocessesinvolvedinreioniza-
o tion is not currently possible. Theoretical studies focus ei-
On scales relevant to the large-scale structure of reioniza-
r
ther on the small scale hydrodynamics and radiative transfer
t tion–hundredsofMpc–numericalcalculationscannotself-
s (e.g.,Miralda-Escudéetal.2000;Gnedin2000;Haimanetal.
a consistentlytrackallthesourcesandsinksofionizingradia-
2001; Shapiro et al. 2004; Ciardi et al. 2006; Pawlik et al.
[ tion. ResolvingtheIGMonthesmallscalesrelevanttoQSO
2009;Raicevic&Theuns2011;McQuinnetal.2011),oron
absorption systems optically-thick at the Lyman edge, also
2 thelargescalemorphologyofreionization(e.g.,Abel&Wan-
known as Lyman limit systems (e.g., Weyman et al. 1981;
v delt 2002; Furlanetto et al. 2004; Iliev et al. 2006; McQuinn
Tytler1982;Mo&Miralda-Escudé1996),isparticularlydif-
2 etal.2007;Trac&Cen2007;Shinetal.2008;Friedrichetal.
ficult. Theseabsorptionsystemswerelikelytohaveimpeded
3 2011). On the largest scales, it has been somewhat surpris-
theprogressofreionization(e.g.,Miralda-Escudéetal.2000;
1
ing how well simplified “semi-numerical” approaches (e.g.,
6 Gnedin&Fan2006).
Zahnetal.2007;Mesinger&Furlanetto2007;Thomasetal.
. Whatarelikelyvaluesforthecomovingabsorptionsystem
3 2009;Choudhuryetal.2009;Alvarezetal.2009;Santosetal.
mean free path at the end of reionization, λ ? Prochaska
0 abs 2010;Zahnetal.2011a;Mesingeretal.2011a)matchmuch
et al. (2009) determined the abundance of Lyman-limit ab-
0 sorption systems, dN/dz=1.9[(1+z)/4.7]5.2, over the red- morecomputationallyexpensiveradiativetransfertechniques
1 (e.g., Abel et al. 1999; Gnedin & Abel 2001; Sokasian et al.
shift range 3.6 < z < 4.4, which translates into a comov-
:
v ing mean free path of λ = c/H(z)/(dN/dz) ∼ 415 Mpc 2001;Abel&Wandelt2002;Ilievetal.2006;McQuinnetal.
i at z=3.7. This dependeanbcse on redshift is steeper than re- 2007; Trac & Cen 2007). In order to efficiently survey the
X
wideparameterspaceofλ andfocusonthelargescalemor-
portedpreviously(e.g.,Storrie-Lombardietal.1994),andim- abs
ar plies λabs ∝(1+z)−6.7. However, Songaila & Cowie (2010) pthheosloimgyplainfidedovseemrail-lnpurmogerreicssaloafprperiooancizha.tOiounr,wimephleamveenchtaotsioenn
extended the constraints out to z∼6, finding a much shal-
lower dependence, λ ∝ (1+z)−3.44, with a measurement is similar to that of Zahn et al. (2007), with the the addition
abs ofatreatmentofphotonconsumptioninabsorptionsystems.
of λ (cid:39)34 Mpc at z∼5.7. Comparison of the absorption
abs Ourapproach,inwhichwetreatabundanceofabsorbersasan
spectraandhydrodynamicsimulationscanalsoconstrainλ ,
abs input parameter, is complementary to that taken by Crociani
with Bolton & Haehnelt (2007b) reporting 20 < λ < 50
abs et al. (2011), where the semi-numerical approach was used
Mpc at z∼6. The evolution of the mean free path at z<6
to determine the large scale distribution of absorbers, rather
leavesopenawiderangeofpossiblevaluesduringthereion-
than to model the effect of the absorbers on the progress of
izationepoch, fromseveraltohundredsofMpc. Ourgoalis
reionization,aswedohere.
todeterminethesensitivityofthehistoryandmorphologyof
Our results are parametrized by different values of for the
reionizationtothiswiderangeofpossiblemeanfreepaths.
minimum source halo mass, ionizing efficiency of collapsed
Electronicaddress:[email protected] matter, and the absorption system mean free path. We de-
1CanadianInstituteforTheoreticalAstrophysics,UniversityofToronto, scribeoursimulations in§2, followedbythemain resultsin
60St.GeorgeSt.,Toronto,ONM5S3H8,Canada §3,endingwithadiscussionin§4.Allcalculationsweredone
2KavliInstituteforParticleAstrophysicsandCosmology,StanfordUni-
using(Ω ,Ω ,h,σ ,n )=(0.27,0.73,0.72,0.8,0.96),consis-
versity,MenloPark,CA94025,USA m Λ 8 s
tentwithWMAP7-yeardata(Komatsuetal.2011). Alldis-
2 ALVAREZ&ABEL
TABLE1
SIMULATIONPARAMETERS
τes Mmin λabs tev ζ z1/2 z0.9 zov λ0
[M(cid:12)] [Mpc/h] [Myr] [Mpc]
0.06 108 8 - 20 8.1 5.9 4.3 87
0.06 108 32 - 16 8 6.3 5 181
0.06 108 256 - 14.5 8 6.8 6.2 294
0.06 109 8 - 78 8.2 6.6 5.6 98
0.06 109 32 - 60 8.1 6.8 6 210
0.06 109 256 - 40 8.1 7.24 6.9 375
0.09 108 8 - 137 11 9.2 8 94
0.09 108 32 - 110 10.9 9.5 8.5 201
0.09 108 256 - 98 10.9 9.9 9.5 345
0.09 109 8 - 1200 11.1 9.8 9.1 105
0.09 109 32 - 920 11 10 9.3 232
0.09 109 256 - 820 11 10.3 10 443
0.12 108 8 - 1050 13.5 12.5 11.1 99
0.12 108 32 - 850 13.5 12.2 11.4 216
0.12 108 256 - 750 13.4 12.6 12.2 396
0.12 109 8 - 2.2e4 13.5 12.5 11.9 109
0.12 109 32 - 1.7e4 13.5 12.6 12.1 248
0.12 109 256 - 1.5e4 13.5 12.9 12.7 500
0.09a 108 8 - 102 10.9 9.7 8.3 - FIG. 1.—Bubblemeanfreepath,λb,asafunctionofmeanneutralfrac-
0.09a 108 32 - 94 10.8 9.9 9.6 - tion,xHI. Allmodelsexhibitacharacteristicscaleof∼10Mpcatthehalf-
0.09a 108 256 - 92 10.8 10 9.8 - ionizedepoch. Duringpercolation,lowervaluesofλabs leadtolowerval-
0.06 108 - 10 15 8.2 6.9 6.4 300 ues of λb. The solid line corresponds to the evaporating minihalo model
00..0066 110088 -- 15000 1155 88..10 66..41 55..63 216740 λwbith∝txevH−2I=/3.100 Myr. At xHI <0.1 the relation approaches approximately
theedgeofabubbleisλ /λ ,andtheionizationrateissup-
aObtainedusingsharpk-spacefiltering abs b
pressed.
tancesarecomoving.
A more realistic treatment would involve allowing F, λ ,
b
andλ tovaryspatially,obtainingaspatially-dependentso-
2. SIMULATIONS abs
lutionofequation(1),whichcanthenbeaveragedtofindthe
Table 1 shows our simulation parameters: ζ, the ioniz-
globalreionizationhistory. Wehavechosenthesimpleruni-
ing efficiency, λ , the comoving absorption system mean
abs formmodelofequation(1)asastartingpoint. Althoughthis
free path, t , the photoevaporation time for the evolving
ev choice does not affect our results on the morphology of the
λ models, and M , the minimum halo mass of galaxies.
abs min ionizationfieldatfixedionizedfraction,theassumptionsun-
Alsoshownaretheredshiftswhentheglobalionizedfraction
derlyingequation(1)shouldbekeptinmindwheninterpret-
equals0.5,0.9,and1–z ,z ,andz ,respectively,andthe
0.5 0.9 ov ingthereionizationhistoryandphotonconsumptionrateswe
bubblemeanfreepathwhentheneutralfractionis0.1, λ ≡
0 find.
λ (x =0.1). TheparameterswerevariedtogiveaThomson
b HI
scatteringopticaldepthof(τ−,τ0,τ+)=(0.06,0.09,0.12),corre- 2.2. Spatialvariations
sponding to the 2-σ constraint from WMAP (Komatsu et al.
Our three-dimensional model is based on that of Furlan-
2011),withtheHeIfractiontrackingHI,andinstantaneous
HeIIreionizationatz=3. Allsimulationsused40963cellsin ettoetal.(2004),laterextendedtothreedimensionsbyZahn
etal.(2007). Itsmainassumptionisthataregionisfullyion-
a2Gpc/hbox.
ized if its collapsed fraction is greater than some threshold,
2.1. Meanreionizationhistory ζfcoll >1. As shown in Alvarez et al. (2009), by smoothing
the linear density field over a range of scales, one can effi-
Wemodelthemeanreionizationhistorywtihtwospatially
ciently determine when each point is first reionized, z . We
uniformmeanfreepaths: thatcorrespondingtoionizedbub- r
do not smooth over scales with radii larger than λ , since
bles, λ , and Lyman-limit systems, λ . Ionizing radiation abs
b abs absorptionsystemsshieldradiationfromthesedistances. We
isattenuatedbytheirsuperposition,sothatλ−1 =λ−1+λ−1.
mfp b abs assumethattheabsorptionsystemscontributeaspatiallyuni-
The spatially averaged ionizing flux is F(z) = λ (z)(cid:15)(z),
mfp formopacity,andthereforethescalebeyondwhichwedonot
where (cid:15)(z) is the ionizing photon emissivity. We set (cid:15)(z)=
smoothisthesameeverywhere.
ζnH,0f˙coll(z), whereζ isnumberofionizingphotons percol- Thisapproachisdesirablebecausetheorderinwhichpoints
lapsed hydrogen atom, corrected for recombinations outside are ionized is the same as that in which the Furlanetto et al.
ofabsorptionsystems. Themeanionizationrateis (2004)criterionisfirstmetaroundeachpoint. However, the
reionizationredshiftsdonotgenerallyresultinthecorrectav-
dx λ (z)
=ζf˙ (z) abs . (1) erage ionization fraction for a sharp smoothing filter in real
dt coll λ (z)+λ (x)
abs b space(seebelowforsolutionsusingasharpfilterink-space,
When λ (cid:29)λ , photons typically reach the edges of bub- which does conserve photons; Zahn et al. 2007). To obtain
abs b
bleswithoutbeingabsorbedbyinterveningLyman-limitsys- a self-consistent solution, we determine a correction to the
tems, and the photoionization rate is equal to the emission reionization redshift at each point, z (z ), by matching the
r,c r
rate. When λ (cid:28)λ , the probability of a photon reaching simulated reionization history to the global reionization his-
abs b
ABSORPTIONSYSTEMS&REIONIZATION 3
FIG. 2.—Globalreionizationhistories(bottom)andphotonconsumption(top). Differentcolorsindicatedifferentvaluesofλabs,whilethepanelsindicate
differentThomsonscatteringopticaldepths,τes=0.06,0.09,and0.12,fromlefttoright. Notethatinallcasestheearlypartofthereionizationhistoryis
insensitivetoλabs,whenthebubblesarerelativelysmaller.Lowervaluesofλabsdelaytheendofreionization.
tory, giventhatwewanttoexcludescalesR >λ . However,for
f abs
(cid:90) ∞dp (cid:90) ∞dxdt comparisonwehavecalculatedafewcaseswithsharpk-space
dz= dz, (2) filtering,showninTable1.
dz dt dz
zr zr,c The ionization history is found by integrating a diffusion
where dp/dz is the distribution of uncorrected simulated equationforthedistributionindensityofpointsthathavenot
reionization redshifts, and dx/dt is given by equation (1). crossedthebarrier(Bondetal.1991),
Thiscalibrationproceduredoesnotchangetheoveralltopol-
∂Π(Λ,δ˜,z) 1∂2Π ∂δ˜∂Π
ogy we find at fixed ionized fraction, but does change the = − , (3)
overallreionizationhistory. ∂Λ 2 ∂δ˜2 ∂Λ ∂δ˜
Wecalculateλb(x)byrandomraytracing. Ifaraystartsin with boundary condition Π(Λ,0,z)=0, where δ˜≡B(Λ,z)−
an ionized region, it is extended in a random direction until
δ, δ is the density contrast, B(Λ,z) = δ (z)−erf−1(1−
it reaches a neutral cell. The bubble mean free path is then c
(cid:113)
the average over the lengths of all such rays. The average ζ−1) 2(σ2 −Λ)isthebubblebarrier(Furlanettoetal.2004),
convergeswithontheorderof104rays. min
andσ2 isthevarianceofthez=0lineardensityfieldatthe
About30logarithmically-spacedsmoothingscalesaresuf- min
scaleofM ,withthegrowthofstructureencodedinthetime
ficienttoconvergeonthereionizationmorphology. Because min
dependenceofδ (z).
we determine the reionization redshifts simultaneously, the c
Wesmoothonlyoverscaleslargerthanλ bystartingin-
amountofdatastoredandtheoperationcountaregreatlyre- abs
tegrationofequation3atΛ ≡σ2(λ )–thevarianceonthe
duced compared to determining the ionization field at each i abs
scale of the mean free path is matched to the one obtained
redshift. Since the assumption that points are either highly-
usingatop-hatfilterinrealspace–withinitialcondition
ionized or neutral is made in both cases, no information is
lostbyfollowingthemoreefficientapproach. 1 (cid:34) (δ˜−B(Λ,z))2(cid:35)
Π(Λ,δ˜,z)= √ exp − i . (4)
2.3. Sharpk-spacefiltering i 2πΛi 2Λi
Using a sharp k-space filter leads to a method which does The total ionized fraction is obtained by integrating over the
conserve photons (Zahn et al. 2007) in the long mean free distributionatΛ=σ2 ,
min
pathlimit,attheexpenseofusinganunphysicalfilterwhich
(cid:90) ∞
“rings” in real space, and an ambiguity in the assignment of x(z)=1− Π(σ2 ,δ˜,z)dδ˜. (5)
a smoothing scale R to the filter k – particularly troubling min
f f 0
4 ALVAREZ&ABEL
IftheinitialconditionforΠisspecifiedatΛ=0,(i.e. starting
atascale→∞),thenthiscorrespondstosmoothingoverall
scales,and
(cid:20) (cid:113) (cid:21)
x(z)=ζerfc δ (z)/ 2σ2 =ζf (z). (6)
c min coll
2.4. Evolvingλ duetominihalos
abs
To model an evolving value of λ , we assume that the
abs
absorption systems are minihalos(e.g., Abel & Mo 1998).
Given that the halo cross section increases as M2/3, while
dn/dlnM ∝ M−1, small objects should dominate the mean
free path, subject to photoevaporation effects (e.g., Haiman
etal.2001;Shapiroetal.2004).
Consider a uniform distribution of dark matter halos with
anumberdensityn (z)≡dn/dlnM . Wewillassumethatin
h h
neutralregions, halosatmassM havegasfractionsofunity
h
and would serve as Lyman-limit absorptions systems, while
in ionized regions, such halos survive for a time t before
ev
beingevaporatedbytheionizingbackground. Thenumberof
absorptionsystemsevolvesaccordingto
ddnazbs = ddlnzx(nh−nabs)+ξev(1+z)−5/2nabs(z), (7) eFlsIGco.n3s.i—der(etdoph)eNree,uitnraaldfrdaictitoionntvost.hreedrsehfeifrtefnocrethceasteh,refeorevwohlvicinhgλλababss→mo∞d-
(right-mostdottedcurve). (bottom)Absorptionmeanfreepathvs. redshift.
whereξ ≡(H Ω1/2t )−1(cid:39)260(t /100Myr)−1. Wechoose As the last neutral patches are cleared away, so are the last reservoirs of
ev 0 m ev ev coldneutralgasinwhichtheminihaloscansurviveasabsorptionsystems,so
Mh =5.7×103M(cid:12)[(1+z)/10]3/2, corresponding to the cos- λabsincreasesattheendofreionization. Astheevaporationtimedecreases,
mological Jeans mass, and consider three models, in which thedestructionrateofabsorptionsystemsincreases,leadingtolongermean
t =10, 50, and 100 Myr. These timescales are relatively freepaths. AlsoshownaretheconstraintsonλabsfromBolton&Haehnelt
ev (2007b)atz=5and6.
long compared with the rather short photoevaporation times
simulation volumes in excess of several hundred Mpc on a
forlow-masshalosfoundbyShapiroetal.(2004). However,
sidearenecessaryinordertoproperlymodelthelasttenper
if reionization is “photon-starved”, as observations are sug-
gesting(e.g.,Bolton&Haehnelt2007b), thenthefluxcould centofthereionizationprocess. AtxHI<0.1, λb∼xH−2I/3, in
be low at the end of reionization, leading to longer evapora- agreementwiththemodelofMiralda-Escudéetal.(2000),in
tiontimes. whichaconstantnumberofneutralcloudsdecreaseinsizeat
Themeanfreepathis thesamefractionalrateattheendofreionization.
The reionization history for all of the fixed λ models is
abs
λ−1(z)=πR2 (z)n (z), (8) shown in Fig. 2. The early evolution is not sensitive to λ ,
abs vir abs abs
becausethephotonmeanfreepathisdeterminedbythesize
whereR correspondstothehalovirialradius. However,the
vir of the ionized bubbles themselves. At later times, when the
boundariesoftheabsorptionsystemsarenotwell-defineddue
bubblesizesbecomecomparabletoλ ,thereionizationhis-
to hydrodynamic effects and departures from spherical sym- abs
toryissensitivetoλ ,ascanbeseenbycomparingtheevo-
metry.Forourpurposes,theminihalomodelwedescribehere abs
lution for the bracketing values of 8 Mpc/h and 256 Mpc/h.
is sufficient to incorporate the effect of an evolving absorp-
Fortheλ =8Mpc/hcase,theredshiftatwhichtheoverlap
tionsystemmeanfreepathinourcalculations. Moreaccurate abs
occursisdelayedby∆z∼1.5, relativetoλ =256Mpc/h,
modelingofabsorptionsystemsduringreionizationwillneed abs
forτ =0.09(seealsothe“z ”entriesinTable1).
to incorporate the radiative transfer of ionizing radiation in es ov
AlsoshowninFig.2isthenumberofionizingphotonsper
cosmological hydrodynamics simulations which resolve the
ionizedatom,
Jeansmass.
poWinetssttoarctrowsisththaenbianrirtiiearlagtuseosmsefosrcaλleabsa(nz)d, roendlsyhiaftlloifwtihnagt nγ(z) = ζfcoll(z) =1+ 1 (cid:90) x(z) λb(x(cid:48)) dx(cid:48). (9)
n (z) x(z) x(z) λ (x(cid:48))
scaleisbelowthecurrentvalueofλ . Thevaluesobtained HII 0 abs
abs
forλ (x)fromthesimulatedionizationfieldarethenfedback As λ →∞, x→ζf and n /n →1. We only count
b abs coll γ HII
into equations 1, 7, and 8, and the process is reapeated until photons up to the moment of overlap (x=1), as we are con-
convergenceisreached. cerned here with the consumption of ionizing photons dur-
ing the reionization process. This integral converges, since
3. RESULTS attheendofreionizationweexpectλ (x)∝(1−x)−2/3 asthe
b
Fig. 1 shows the bubble mean free path as a function of remaining diffuse neutral hydrogen patches dissappear. For
the neutral fraction for several parameter choices. At the λ =8 Mpc/h, about 3 photons per atom are consumed in
abs
half-ionized epoch, all models exhibit a characteristic bub- theabsorptionsystemsbyz . Inthelongestmeanfreepath
ov
blescaleofλ (cid:39)10Mpc,withmodelswithrarersources(i.e. case,λ =256Mpc/h,thecorrespondingfractionisabouta
b abs
increasing efficiency of ionizing radiation) leading to some- half.
what larger bubbles. During percolation, most of the varia- ShowninFig.3istheevolutioninneutralfractionandab-
tion in λ (x ) comes from variation of λ , which is most sorption system mean free path for the three minihalo ab-
b HI abs
pronouncedatx ≤0.2. Thelargevaluesofλ indicatethat sorption system models that we simulated. Models with
HI b
ABSORPTIONSYSTEMS&REIONIZATION 5
longer evaporation times lead to a higher abundance of ab- ume 100 Mpc/h across, and found a similar trend with de-
sorption systems, and hence shorter mean free paths and a creasing λ , indicating a transition to an “outside-in” mor-
abs
relative delay in the time of overlap. The mean free path in phology near the end of reionization. We find that neutral
the t =100 Myr model evolves from about 40 Mpc when patchesmayalsoremaininvoids, whereformationofioniz-
ev
x ∼0.1, to about 80 Mpc at overlap. This model is also ingsourcesisdelayedandradiationfromthenearestsources
HI
plottedasthesolidblacklineinFig.1,whichshowsthatthe isshielded. Previousstudiesthatmodeledthephysicalorigin
instantaneous λ values for the evolving λ model roughly of absorption systems as minihalos (e.g., Ciardi et al. 2006;
b abs
interpolatebetweenthoseforthefixedλ models. Thiscan McQuinn et al. 2007) also found similar effects to those we
abs
alsobeseenbynotingthatλ ≡λ (x =0.1)fortheevolving find here, although these studies were more focused on the
0 b HI
meanfreepathmodel,forwhichλ (x =0.1)(cid:39)32Mpc/h, intermediate stages of reionization and on smaller simulated
abs HI
isthesameasthatforthecorrespondingfixedmeanfreepath volumes, where the photon mean free path was not as small
modelwithλ =32Mpc/h,λ =170Mpc. relative to the bubble sizes as we find in our low mean free
abs 0
Fig. 4 shows the ionization field in a 5-Mpc/h slice at pathcasesinthefinal,percolationphase.
x=0.75forM =108M ,withλ =256Mpc/h(top)and Asmentionedintheintroduction,ourapproachiscomple-
min (cid:12) abs
λ =8 Mpc/h (bottom). The two cases are quite different, mentarytothattakenbyCrocianietal.(2011), inwhichthe
abs
withmanymoresmallneutralpatchesforλ =8Mpc/hrel- semi-numericalapproachwasusedtodeterminethedistribu-
abs
ativetoλ =256Mpc/h.AlsoshowninFig.4isthereioniz- tion of absorbers during reionization. They found that their
abs
tionredshiftina0.5-Mpc/hslice. Lowervaluesofλ result spatialdistributionisquiteinhomogenous, owingbothtoin-
abs
inamoreextendedreionizationoverlapperiod–i.e. themax- trinsicdensityfluctuationsintheIGM,aswellasfluctuations
ima in reionization redshift are higher, and the minima are in the ionizing radition field – regions far from sources have
lower. a relatively low flux, resulting in a higher abundance of ab-
sorbers. This affect was also pointed out by McQuinn et al.
4. DISCUSSION (2011), who used high-resolution hydrodynamic simulations
Wehavecarriedoutlarge-scalesimulationsofreionization post-processed with radiative transfer to determine the mean
in a 2 Gpc/h volume, including a finite mean free path to freepathasafunctionofflux,findingastronglynonlinearde-
absorption systems. Absorption systems have a significant pendenceofthemeanfreepathontheemmissivityofionizing
effect on the characteristic scales at the end of reionization. radiation.
FortheThomsonscatteringopticaldepthreportedbyWMAP, These effects imply an important improvement to our
τ (cid:39)0.09, we find that the characteristic bubble size when model is not only a time-varying mean free path, but also a
es
the universe is 90 per cent ionized is quite sensitive to λ , spatially varying one. To accomplish this, more work will
abs
withλ ∼440Mpcforλ =256Mpc/handM =109M , needtobedoneonthe“sub-grid”physicsduringreionization,
0 abs min (cid:12)
whileλ ∼94Mpcforλ =8Mpc/handM =108M . usingtheresultsofhigh-resolutioncosmologicalsimulations
0 abs min (cid:12)
Calculations using a sharp k-space filter lead to a more with radiative transfer of a background radiation field, cou-
modestextensionofthepercolationphase(seeTable1)com- pled to the hydrodynamics of the gas. However, given that
pared to the reionization history obtained by the solution of thefinalpatchestobeionizedarethosemostdistantfromthe
equation(1),onwhichthereionizationhistoriesshowninFig- most luminous sources, the delay in the very end of reion-
ure1arebased. Forτ (cid:39)0.09,thedifferenceintheredshift ization that we find here is likely to persist when the inho-
es
when the universe is 90 per cent ionized between the λ = mogeneity of the IGM opacity due to absorption systems is
abs
8 Mpc/h and λ =256 Mpc/h cases is only ∆z ∼0.3, properlytakenintoaccount.
abs 0.9
while the corresponding case using a sharp real space filter The lingering neutral clouds we find would further com-
andequation(1)is∆z ∼0.7. Thedelayintheoverlaptime plicate interpretation of quasar absorption spectra at z ∼ 6
0.9
inbothcasesis∆z ∼1.5. (e.g., Bolton & Haehnelt 2007a; Lidz et al. 2007; Alvarez
ov
Using either sharp k-space smoothing or equation (1) to & Abel 2007; Wyithe et al. 2008; Furlanetto & Mesinger
model the effect of absorption systems on the reionization 2009).AsdiscussedbyMesinger(2009),gapsinquasarspec-
history has drawbacks. While equation (1) conserves pho- tra can come from either these neutral clouds or mostly ion-
tons in the long mean free path limit, a mean flux and opac- izedbutoptically-thickabsorptionsystems. Theline-of-sight
ity are assumed, and further work will be necessary to vali- abundanceoftheseneutralcloudsatfixedionizedfractionin-
date and/or improve upon the approach layed out in section creaseswithdecreasingabsorptionsystemmeanfreepath.
2.1. Usingasharpk-spacefilterresultsinareionizationhis- As shown by Weinmann et al. (2007) and Alvarez et al.
torywhichalsoconservesphotonsinthelongmeanfreepath (2009),theLocalGroupmayhavebeenreionizedbyexternal
limit(k −→0),butattheexpenseofusinganoscillatoryfil- sources,i.e. theprogenitorsoftheVirgoClusteratadistance
F
ter which can “leak” photons from high density regions into of ∼ 20 Mpc. If λ <20 Mpc, however, local reionization
abs
lower density ones, and for which there is no unique choice wouldhavebeendelayeduntilLocalGroupprongenitorshad
fortherelationbetweenλ andk . Whileourresultsforthe formedinsufficientnumber(e.g.,Muñozetal.2009). Since
abs F
timingandphotonconsumptionoftheendofreionizaitonare the satellite population is sensitive to the timing of reioniza-
obtained from equations (1) and (9), it is important to note tion (Busha et al. 2010), lower values of λ could imply a
abs
thatmoreworkwillbenecessarytotesttheiraccuracy. How- higherlocalsatelliteabundance.
ever,resultsforthemorphologyofreionization(Figures1and Finally, our results have important implications for the in-
4)donotdependonthemodelfortheglobalhistory,andare terpretation of temperature fluctuations in the cosmic mi-
thereforemorerobust. crowave background at multipoles l ∼ 3000 (Das et al.
Our results are consistent with those of Furlanetto & Oh 2011; Reichardt et al. 2011), through the kinetic Sunyaev-
(2005), in which consumption of ionizing photons in dense Zel’dovich(kSZ)effect(e.g.,McQuinnetal.2005;Ilievetal.
systemsextendstheendofreionizationconsiderably. Choud- 2007). Inparticular, Mesingeretal.(2011b)usednearlythe
huryetal.(2009)usedthesemi-numericalapproachinavol- samethreeparametersandapproachasweusedherefortheir
6 ALVAREZ&ABEL
FIG. 4.—left:Reionizationredshiftina0.5Mpc/h-thickslicefortheλabs=256Mpc/hsimulation(top)andtheλabs=8Mpc/hsimulation(bottom). Both
caseswereforτes=0.09andMmin=108M(cid:12).Clockwisebeginningfromtop-left,regionsshownare2Gpc/h,500Mpc/h,and125Mpc/hacross.Theradiiof
thecirclescorrespondtoλabs.right:Sameasleftpanelsbutfortheprojectedionizedfractionina5Mpc/h-thicksliceatatimewhenthemeanvolume-weighted
ionizedfractionis0.75.Notethattheionizationfieldcontainsconsiderablymoresmallscalestructureintheλabs=8Mpc/hcase.
parameter study of the dependence of the kSZ signal on the R.Thomas,G.Vasilforusefuldiscussions,andJ.Chlubafor
reionization scenario. Reducing the mean free path shifts providing the diffusion solver from CosmoRec. The simu-
powertohighermultipolesbecauseoftheaccompanyingde- lations in this paper were performed on the GPC supercom-
crease in the bubble size that we find. Although the kSZ ef- puter at the SciNet HPC Consortium. SciNet is funded by:
fectisonlyjustbeginningtoconstrainthedurationofreion- the Canada Foundation for Innovation under the auspices of
ization(Zahnetal.2011b),moredetailedfutureobservations Compute Canada; the Government of Ontario; Ontario Re-
and analysis will begin to provide constraints on the patchi- search Fund - Research Excellence; and the University of
ness in addition, and understanding the effect of absorption Toronto. ThisworkwaspartiallysupportedbyNASAATFP
systemsonreionizationwillbecrucial. grant NNX08AH26G, NSF AST-0807312, and NSF AST-
0808398.
We thank M. Haehnelt, M. McQuinn, A. Mesinger,
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