21-cm signatures of residual HI inside cosmic HII regions during reionization

1.7 MB·English

by  C. A. Watkinson

Mon.Not.R.Astron.Soc.000,000–000(2015) Printed28May2015 (MNLATEXstylefilev2.2)
21-cm signatures of residual HI inside cosmic HII regions during
reionization
C. A. Watkinson1(cid:63), A. Mesinger2, J. R. Pritchard1 & E. Sobacchi2
1DepartmentofPhysics,BlackettLaboratory,ImperialCollege,LondonSW72AZ,UK
5 2ScuolaNormaleSuperiore,PiazzadeiCavalieri7,I-56126Pisa,Italy
1
0
2
28May2015

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a
M
ABSTRACT
Weinvestigatetheimpactofsinksofionizingradiationonthereionization-era21-cmsignal,

7 focusingon1-pointstatistics.Weconsidersinksinboththeintergalacticmediumandinside
2
galaxies.AtafixedfillingfactorofHIIregions,sinkswillhavetwomaineffectsonthe21-cm

morphology: (i) as inhomogeneous absorbers of ionizing photons they result in smaller and
]
O more widespread cosmic HII patches; and (ii) as reservoirs of neutral gas they contribute a
non-zero21-cmsignalinotherwiseionizedregions.Botheffectsdampthecontrastbetween
C
neutralandionizedpatchesduringreionization,makingdetectionoftheepochofreionization
.
h with21-cminterferometrymorechallenging.Herewesystematicallyinvestigatetheseeffects
p usingthelatestsemi-numericalsimulations.Wefindthatsinksdramaticallysuppressthepeak
- in the redshift evolution of the variance, corresponding to the midpoint of reionization. As
o
previouslypredicted,skewnesschangessignatmidpoint,butthefluctuationsintheresidual
r
t HI suppress a late-time rise. Furthermore, large levels of residual HI dramatically alter the
s
a evolution of the variance, skewness and power spectrum from that seen at lower levels. In
[ general, the evolution of the large-scale modes provides a better, cleaner, higher signal-to-

noiseprobeofreionization.
2
v Keywords: Keywords:darkages,reionization,firststars–intergalacticmedium–methods:
0 statistical–cosmology:theory.
7
9
1
0
. 1 INTRODUCTION ationfromgalaxiesescapesintotheintergalacticmedium(IGM).
1
0 These two physical mechanisms have naturally been the focus in
Reionization, the process in which the first galaxies ionized their
5 themodellingofreionization,withbothsemi-numerical(e.g.Zahn
surroundings and ultimately the entire Universe, is poorly con-
1 etal.2007;Mesinger&Furlanetto2007;Choudhuryetal.2009;
strainedatpresent.The21-cmsignal,asproducedbyahyperfine
: Mesingeretal.2011;Santosetal.2010;Thomas&Zaroubi2011)
v transitioninneutralhydrogen(HI),providesanexcellenttracerof
andnumericalsimulations(e.g.Trac&Cen2007;Zahnetal.2007;
i the neutral IGM. It is therefore the hope that future observations
X Baeketal.2009;Trac&Gnedin2011;Ilievetal.2014)beingused
ofthe21-cmbrightnesstemperatureusingimmenseradioarrays,
r e.g.theLOwFrequencyARray1 (LOFAR),theMurchisonWide- to refine our understanding of both. These simulations describe
a howionizedregionsform,growandultimatelymergetoionizethe
field Array2 (MWA), the Precision Array for Probing the Epoch
entire IGM. However, it is not enough to understand the sources
of Reionization3 (PAPER), the Hydrogen Epoch of Reionization
alone,wemustalsoconsiderthesinksofionizingradiation,i.e.the
Array4(HERA)andtheSquareKilometreArray5(SKA),willdra-
atomsthatonceionizedwillrecombine,wastingionizingphotons
maticallyimproveourunderstandingofreionization.
thatwouldotherwisecontributetoreionization.Sinkswillhavetwo
Tomakeoptimaluseoftheseobservationsitisvitalthatwe
maineffectsonthe21-cmsignal:
build insight into the wide range of physical processes involved
inreionization.Twoofthemostimportantfactorstoconsiderare (i) asabsorbersofionizingphotonstheyimpactonboththetim-
the star formation history and the degree to which ionizing radi- ingofreionizationandthemorphologyofcosmicHIIpatches;
(ii) asreservoirsofneutralgastheycontributeanon-zero21-cm
signalinotherwiseionizedregions.
(cid:63) Email:[email protected]
Sinks can reside in both the IGM and the interstellar
1 http://www.lofar.org/
medium (ISM), i.e. inside and outside of galaxies. In the ion-
2 http://www.mwatelescope.org/
3 http://eor.berkeley.edu/ ized IGM at reionization redshifts, sinks mostly consist of
4 http://reionization.org/ diffuse structures and partially self-shielded gas clumps called
5 http://www.skatelescope.org/ Lyman-limit systems (LLSs, e.g. Furlanetto & Oh 2005,
(cid:13)c 2015RAS
2 C.A.Watkinson,A.Mesinger,J.R.Pritchard&E.Sobacchi
McQuinn,Oh,&Faucher-Gigue`re2011;Mun˜ozetal.2014). 2 THESIMULATIONS
LLSs are inhomogeneous in both space and time (Crociani et
To investigate the impact of residual HI on 21-cm statistics, we
al. 2011; Choudhury et al. 2009); as such, they can dramatically
makeuseofthepublicsimulationtool,21CMFAST6.Thedensity
impactthedurationandmorphologyofreionization(effectiabove;
andvelocityfieldsin21CMFASTaregeneratedwithstandardper-
Sobacchi & Mesinger 2014). Although they are mostly ionized
turbation theory (Zel’dovich 1970). To generate ionization fields
(e.g. McQuinn, Oh, & Faucher-Gigue`re 2011; Rahmati et al.
theexcursion-setapproachofFurlanettoetal.(2004)isappliedto
2013), LLSs can also contribute to effect (ii). Indeed Sobacchi
theevolveddensityfields:todeterminewhetheraregionisionized,
& Mesinger (2014) recently estimated that LLSs on average
thisalgorithmcomparesatdecreasingradiithenumberofionizing
contributeafewpercentneutralfractioninsidecosmicHIIpatches
photonstothenumberofbaryons(plusrecombinations).
duringreionization.
Inthiswork,westudyhowsinksofionizingphotonsimpact
the21-cmsignal,focusingon1-pointstatistics.Todothis,werun
four21CMFASTrealisationsoftheionizationfieldduringreioniza-
Sinks are also present inside galaxies where HI can self tion,varyingtheimpactofHIonboththelarge-scalereionization
shieldindenseclumps.Atlowerredshifts(e.g.Wolfe,Gawiser,& morphologyandthelevelofresidualHIinsidecosmicHIIregions
Prochaska2005),sinksinsidegalaxies[mostlyso-calledDamped (effectsiandiidiscussedintheintroduction).Wediscussthede-
Lyman-alphasystems(DLAs);Prochaskaetal.2005;Noterdaeme tails below, showing a summary of the four simulations used in
etal.2012]providealargerreservoirofHIcomparedtosinksinthe Table1andFigure1.AllsimulationsareL = 300Mpconaside
IGM.RecentobservationssuggestthatthecontributionofDLAsto with a resolution of 4003. The density and velocity fields are the
thecosmologicalmassdensityisroughlyconstantuntilitincreases sameinalloftheruns.
from z = 2.3 to z = 3.5 by almost a factor of 2 (Noterdaeme
et al. 2012). There is currently no consensus in explaining these
2.1 Homogeneousrecombinations(21CMFASTV1)
trends, making extrapolations to reionization redshifts uncertain.
Theabundanceandionizationstructureofsuchgalacticsinksare Wecallourreferencesimulation‘NoLLS’.Sinkshaveaminimal
more susceptible to the local environment, including stellar feed- impactinthissimulation.Itisgeneratedwiththecurrentpublicre-
backandbaryoniccooling.Thereforeitisdifficulttoquantifythe lease,21CMFASTV1.Theresultingionizationfieldsaresimilarto
contributionofgalacticsinkstoeffect(ii)above.Usinga‘tuning- onesobtainedwithradiativetransfersimulations(e.g.Zahnetal.
knob’ approach (similar to the one we use below), Wyithe et al. 2011).In21CMFASTV1,aregionisionizedifits(time-integrated)
(2009) predict that by reducing the contrast between cosmic ion- number of ionizing photons exceeds the number baryons plus a
izedandneutralpatches(effectii),galacticHIcouldindeeddamp globalaveragenumberofrecombinations,n¯rec:
the21-cmpowerspectrumby10-20%.GalacticHIalsocontributes
ζf (x,z,R,M )≥1+n¯ , (1)
toeffect(i)byreducingtheefficiencywithwhichionizingradiation coll min rec
escapesintotheIGM,usuallyquantifiedwiththeso-calledioniz- wheref (x,z,R,M )isthefractionofmatter,insidearegion
coll min
ingphotonescapefraction,fesc.Thisclearlyhasaneffectonthe of scale R, which resides in halos with mass greater than Mmin
timing of reionization. However, reionization studies include the (taken here to correspond to a virial temperature of 104 K). The
escapefractionandtheassociateduncertaintiesinthesourceterm. ionizing efficiency can be expanded as ζ = f f N , where
esc ∗ γ/b
f isthefractionofionizingphotonsthatescapegalaxies,f is
esc ∗
thefractionofgalacticbaryonsinsidestars,andN isthenum-
γ/b
berofionizingphotonsproducedperstellarbaryon.Thisionization
InthisworkweexaminetheeffectthatbothLLSs(IGMsinks)
criterionineq.(1)ischeckedinanexcursion-setfashion(Furlan-
and differing levels of galactic HI (galactic sinks) have on the ettoetal.2004),startingfromamaximumradiusR 7downto
statistics of the 21-cm brightness temperature, with emphasis on max
thecellsize.Atthelastsmoothingscale,theunresolved,sub-cell
1-pointstatistics.WeanalysethesimulationpresentedinSobacchi
& Mesinger (2014), here after SM2014, to isolate the impact of HIIfractionissettoζ(1+n¯rec)−1fcoll(x,z,Rcell,Mmin).
The ‘NoLLS’ run has a different reionization evolution and
IGMsinksonthe1-pointstatisticsviaeffects(i)and(ii).Wealso
morphologyfromtheotherthreeruns.8Inordertofacilitatedirect
testhowthestatisticsofthissimulationarealteredinthepresence
morphologicalcomparisons,wevaryn¯ in‘NoLLS’tomatchthe
ofgalacticsinksduetoeffect(ii). rec
redshiftevolutionofthefillingfactorofHIIregions,QHIIfromthe
otherruns.
The rest of this paper is structured as follows. In Section 2 6 http://homepage.sns.it/mesinger/Sim.html
wedescribethesimulationsthatweusetomodelreionizationin- 7 Rmax is a free parameter based on ionized photon mean free path at
cluding sinks in both the IGM and in galaxies; in Section 3 we theredshiftsofinterest,seeStorrie-Lombardietal.(1994);Miralda-Escude
present our analysis methods and examine the 21-cm maps and (2003);Choudhuryetal.(2008).Inour‘NoLLS’runitissetto30Mpc.
theirprobability-densityfunctions;wethenconsidertheevolution Howeverthereisnomaximumsmoothingradiusenforcedbythesimula-
ofthe1-pointstatisticsandourabilitytoconstrainthemwithra- tionsofSM2014sincetheionizedphotonmeanfreepathisdeterminedby
dio telescopes present and future in Sections 3.4 and 3.5, Sec- thepropertiesof‘LLS’,whicharecomputedself-consistently.
tion 4 considers how much the power spectrum might be altered 8 The mean free path, Rmax, framework does not capture the sub-grid
recombinationphysicsoftheSM2014model,discussedinthenextsection.
inthepresenceofgalacticsinksandfinallyinSection5wesum-
Nevertheless,theseauthorsfindthatachoiceofRmax ∼ 10Mpcresults
mariseourfindings.Weusethefollowingcosmologicalparameters
inareionizationmorphologyandevolutionroughlyconsistentwiththeir
(Ωm,ΩΛ,Ωb,h,σ8,n)=(0.28,0.72,0.046,0.70,0.82,0.96),con- sub-grid model. This ‘effective ionizing photon horizon’ is in fact much
sistent with recent measurements by the Planck satellite (Planck smallerthanthephysicalmeanfreepath(definedbasedontheinstantaneous
Collaborationetal.2014). recombinationrateandemissivity),andisempiricallyderived.
(cid:13)c 2015RAS,MNRAS000,000–000
21-cmsignaturesofresidualHI 3
integrating over the entire density distribution, the recombination
Table1.Modelsummary.
rateperbaryoninanionizedcellis:
dn (cid:90) +∞
(cid:104)xHI(cid:105)M in HII drtec(x,z)= PV(∆,z)∆n¯HαB[1−xHI(∆)]2d∆,
Model Properties regions at z = 0
(3)
9.20,Q ∼0.5
HII
where P (∆,z) is the full (non-linear, sub-grid) distribution of
V
21CMFASTV1; sinks are in-
overdensities: ∆ ≡ n /n¯ (Miralda-Escude et al. 2000), and
cludedasaconstantintheionizing H H
NoLLS efficiency.CosmicHIIpatchesare 0.0 xHI(∆)isequilibriumneutralfractioncomputedusingtheempir-
icalself-shieldingprescriptionofRahmatietal.(2013).Thenthe
fullyionized.
total,time-integratednumberofrecombinationsperbaryon,aver-
Self–consistent treatment of local
agedoverthelocalcosmicHIIpatchcanbewrittenas:
recombinations & UVB feedback
of SM2014. Small absorbers act (cid:28)(cid:90) z dn dt (cid:29)
n¯ (x,z)= rec dz , (4)
as sinks of ionizing radiation to rec dt dz
LLS slow the progress of reionization, 0.03 zIN HII
reducethesizeofionizedregions where z is the ionization redshift of a simulation cell. Photo-
IN
&suppressthe21-cmtemperature heating feedback from reionization (i.e. the depletion of the gas
contrastbetweenHII(over-dense) reservoir available for star-formation), is included in this simula-
andHI(under-dense)regions. tionwithasimilarapproach,inwhichtheminimumhalomassca-
‘LLS’withtheadditionofafixed pable of retaining gas is computed using the local (cell’s) photo-
percentageofremnantHIingalax- ionization history (Sobacchi & Mesinger 2013). Photo-heating
LLS + ies; its contribution to the 21-cm 0.1fcoll:0.04 feedbackhasasub-dominantimpactonreionizationmorphology,
GalHI signal cosmic inside HII regions 0.5fcoll:0.07 comparedwithinhomogeneousrecombinations;however,sincethe
furthersuppressesthecontrastbe-
sourcesandsinksarespatiallycorrelated,theeffectsareadditivein
tweenionizedandneutralpatches.
thattheybothprolongreionizationhistory.
‘LLS’withbothsterileminihalos ‘LLS’(correspondingtothe‘FULL’simulationofSM2014)
(cooledviaH2)aswellasatomic isourfiducialmodelforreionizationhistoryandmorphology.The
cooling halos 100% neutral. De-
Mini + impact of inhomogeneous, sub-grid HI on the large-scale reion-
scribes the most intense impact
100% 0.19 ization morphology can be estimated by comparing the ‘NoLLS’
thatremnantHIcouldhaveonthe
GalHI 21-cm signal by suppressing the and‘LLS’runs(seeFigure2).Our‘LLS’runalsoincludesresid-
contrast between cosmic ionized ual HI inside the cosmic HII patches, contributing on average a
andneutralpatches. mass-averagedneutralfractionofafewpercentinsideHIIregions
(SM2014;seetable1).
2.3 InhomogeneousrecombinationsintheIGM+galacticHI
Ourremainingrunsusethesamelarge-scalereionizationmorphol-
ogyandredshiftevolutionas‘LLS’(seeFigure2).Howeverhere
we explicitly increase the level of residual HI inside the ionized
regions. Since the amount of HI inside galaxies is very difficult
to predict, we adopt a ‘tuning-knob’ approach similar to Wyithe
etal.2009.Specifically,wecomputetheaverage9 collapsedfrac-
tioninsideeachcell,usingtheconditional(dependentonthecell’s
Figure1.Evolutionofmass-averagedionizedfractionforour4mainmod- density)massfraction(e.g.Bondetal.1991;Lacey&Cole1993)
els(byconstruction,thevolume-averagedevolutionisthesameinallmod- normalized to the mean predicted by Sheth-Tormen (e.g. Jenkins
els).TotrackevolutionofthemorphologyofHII regions,weplotthese et al. (2001); see also the discussion in Barkana & Loeb 2004;
fourmodelsasafunctionoftheHIIvolume-fillingfactorhereafter.
Mesinger et al. 2011). We then assign a fixed fraction, α, of the
cell’sbaryonicmasstobeneutral,sothateachcell’sgalacticHI
2.2 InhomogeneousrecombinationsintheIGM numberdensityisα∆celln¯Hfcoll(Mmin,z,∆cell,Rcell).Keeping
withthesimplicityofthe‘tuning-knob’approach,weuseauniform
As the next step, our ‘LLS’ run includes inhomogeneous recom- valueofM correspondingtoavirialtemperatureof104K.We
min
binations in the IGM using the model of SM2014 (their ‘FULL’
emphasise that this implementation considers only the additional
model).Suchrecombinations,drivenby∼kpcscalegasclumpscan
signal that galactic sinks contribute; their impact on reionization
slowthegrowthoflargecosmicHIIpatches,primarilyresultingin timingisassumedtobeaccountedforintheionizingefficiencyof
reductionoflarge-scalereionizationstructure.SM2014implement
sources.
these local recombinations by modifying the ionization criterion
We study different values of α; for the sake of brevity, we
fromeq.(1)to:
only present the simulations with 10% and 50% of the galactic
ζf [x,z,R,M¯ (x,z)]≥1+n¯ (x,z). (2) massasneutralhydrogen.Toprovideasenseoftherangeofval-
coll min rec
uesthatαmighttakewecanextrapolatelowerredshiftconstraints
InthemodelofSM2014,boththenumberofrecombinationsand
minimumhalomasshostinggalaxiesdependonthelocalproper-
tiesofeach∼1Mpc-scalereionizationcell.Takingself-shielding 9 WeignorethePoissonnoiseassociatedwithhalodiscreteness,sincethis
intoaccountviatheparametrizationofRahmatietal.(2013),and affectsscalestoosmalltobeobservablewiththeSKA.
(cid:13)c 2015RAS,MNRAS000,000–000
4 C.A.Watkinson,A.Mesinger,J.R.Pritchard&E.Sobacchi
from DLAs. The contribution of galactic neutral hydrogen to the
Table2.Instrumentalspecificationsassumedfornoisecalculations.LO-
critical mass is found to be ΩHI ∼ 10−3 for z (cid:46) 4 (Prochaska FARandSKAparametersaretakenfromMellemaetal.(2013),HERA
et al. 2005; Noterdaeme et al. 2012). This can be related to α as fromPoberetal.(2014)/privatecommunicationswithA.LiuandMWA
α = ΩHI/(Ωbfcoll),whichpredictsα ∼ 0.05towardstheendof parametersfromTingayetal.(2013).
reionization,i.ez∼6,risingtoα∼1atz>11asf decreases
coll
withincreasingredshift.HoweverattheredshiftsforwhichΩ has
HI
beenconstrained,α (cid:46) 0.03;assuch,hereafterwewillmakethe Parameter MWA LOFAR HERA SKA
distinctionbetween‘reasonablelevelsofgalacticsinks’,(α(cid:46)0.1),
Numberofstations
and‘largelevels’(α(cid:38)0.5). (Nstat) 128 48 331 450
ItisimportanttonotethatthetailoftheMiralda-Escudeetal.
Effectivearea
(2000)densitydistributionshould,inprinciple,includegasinside
(Aeff/m2) 21.5 804 8.4e3/Nstat 106/Nstat
galaxies.However,thedistributionwascalibratedtothelowerover-
densityIGM,andhasbeenshowntobeapoorfittothematterdis- Maximumbaseline
tribution inside lower-redshift galaxies, even neglecting feedback (Dmax/m) 2864 3000 360 104
(e.g.Bolton&Becker(2009);McQuinn,Oh,&Faucher-Gigue`re
Integrationtime
(2011)).Moreover,theaccuracyoftheMiralda-Escudeetal.(2000) (tint/hours) 1000 1000 1000 1000
PDFisuntestedbothatthehigh-redshifts(z∼10)ofinteresthere,
as well as in predicting the conditional (dependent on the local, Bandwidth
(B/MHz) 6 6 6 6
large-scaleoverdensity)massfraction.Henceinourapproachwe
use the conditional, excursion-set approach, which is much bet-
ter tested in predicting the collapse fraction. However, since the
excursion-set galactic HI is added on top of the SM2014 neutral 3 BRIGHTNESS-TEMPERATUREMAPS&MOMENTS
fractionfield,ourapproachisboundtodoublecountsomegalac-
3.1 Computing1Dstatisticsandnoise
ticHI,andthevaluesofαshouldonlybetreatedasapproximate
(lowerlimits).Wefindthatduetotheinside-outnatureoftheepoch Once the ionization boxes have been calculated as described, the
ofreionization(i.e.biasedHIIregions),theexcursion-setapproach brightnesstemperature(δTb)canbecomputedaccordingto:
generally predicts much higher collapsed fractions inside HII re- T −T
gions than MHR00 (see e.g. table 1); thus we do not expect the δTb = 1s +zγ(1−e−τν0)
double-countingtohavealargeimpactonourconclusions.
(cid:20) (cid:21)
T −T H(z)/(1+z)
≈27. s γ x (1+δ)
T HI dv /dr
s r (5)
(cid:18)1+z 0.15 (cid:19)1/2(cid:18)Ω h2(cid:19)
× b mK,
10 Ω h2 0.023
m
whereδT isdependentontheoverdensityδ = ρ/ρ−1,veloc-
b
itygradientdv /dr andcosmology.ThegasspintemperatureT
r s
measures the occupation levels of the two hyperfine energy lev-
elsinvolvedintheHI21-cmtransitionanddeterminesthe21-cm
optical depth τ . It is assumed in this work that T (cid:29) T , i.e.
ν0 s γ
T ismuchhigherthanthatofthecosmicmicrowavebackground
s
T .Whilstthisisexpectedtobeareasonableapproximationduring
γ
reionizationduetoX-rayheating,theinfluenceofspintemperature
fluctuationsmaynotbenegligibleintheearlystagesofreioniza-
tion,i.e.Q <0.2(Pritchard&Furlanetto2007;Mesingeretal.
2.4 InhomogeneousrecombinationsintheIGM+extreme HII
2013;Gharaetal.2014).
galacticHI
In our analysis of the simulations described above we use
Finally,weconsideranextrememodelwiththemaximalamount the same approach as Watkinson & Pritchard (2014), hereafter
ofresidualneutralhydrogen,‘Mini+100%GalHI’.Followingthe WP2014.WemeasurethevarianceS2andtheskewS3ofasimu-
sameprocedureasoutlinedintheprevioussection,weassign100% latedboxaccordingto
oefffetchteivbelayryaosnsiucmmesastshaint m10in3ih≤alosTvwirithhaalosviarisalHteI.mTpherisatumreodoefl 1 N(cid:88)pix(cid:2) (cid:3)2
S = δT −δT ,
103 ≤T <104K,i.e.thosethathavecooledbymolecularcool- 2 N i b
vir pix
i=0
ing, have been sterilized by a Lyman-Werner background so that (6)
theycannotformstars(e.g.Haiman,Abel,&Rees2000;Ricotti, 1 N(cid:88)pix(cid:2) (cid:3)3
S = δT −δT ,
Gnedin,&Shull2001;Mesinger,Bryan,&Haiman2006).More- 3 N i b
pix
i=0
over, this model assumes that the time-scale of minihalo photo-
evaporationismuchlongerthanthedurationofreionization(e.g. where δTi is the brightness temperature of the ith pixel, the
Shapiro et al. 2004; Iliev et al. 2005; Ciardi et al. 2006). Our average brightness temperature in a box is given by δTb =
‘Mini+100% Gal HI ’ run therefore serves to illustrate the most Np−ix1(cid:80)Ni=p0ixδTi and Npix is the total number of cells in a box.
extremeimpactthatresidualgalacticHIcouldhaveon21-cmob- We will consider two normalisations for the skew, the standard
servations. skewnessS /(S )3/2andthedimensionalskewnessS /S which
3 2 3 2
(cid:13)c 2015RAS,MNRAS000,000–000
21-cmsignaturesofresidualHI 5
Figure2.Brightness-temperaturemapsfor300Mpcboxesnormalisedtoz=9;mapsareresolvedtopixelsof(3Mpc)3andpresentedwiththepixeldepth
flattenedintothepage.Mapscorrespondto,fromLLS(left),LLS+10%GalHI(middle-left),LLS+50%GalHI(middle-right)andminihalos+100%GalHI
(right).Thetoprowisforz=7.0whentheHIIvolumefillingfactoris0.92;thebottomrowcorrespondstoz=9.2atwhichtimetheHIIvolumefillingfactor
is0.50.Blackregionsarethosewithzerobrightnesstemperature.
wasfoundtobeamorenaturalnormalisationfortheskewofthe obeys the properties discussed above. We then use a test statistic
neutral-fractionboxesofWP2014. forthemthmoments,N−1(cid:80)Npix(x −x )m,todetermineunbi-
pix i=0 i i
Wemakeorderofmagnitudecalculationsfortheerrorinduced asedestimatorsforthemomentsofEquation6.Wethencalculate
byinstrumentalnoise(usingtheinstrumentparametersintable2), thenoise-inducedvarianceoneachunbiasedestimator.Thesede-
butdonotconsiderforegroundresiduals.Weassumethenoiseis rived1-σerrorsarethenpropagatedontothenormalisedskewness
independentoneachpixelbutidenticallydistributedinamanner estimatorsγ =Sˆ /Sˆ andγ(cid:48) =Sˆ /(Sˆ )3/2togive
3 3 2 3 3 2
welldescribedbyGaussianrandomnoisewithzeromeanandstan-
2
darddeviationσ2 where V = (2S σ2 +σ4 ),
noise Sˆ2 N 2 noise noise
σn2oise =2.9mK(cid:18)1A05tmot2(cid:19)(cid:18)∆10θ(cid:48)(cid:19)2 Vγ3 ≈ (S12)2VSˆ3 + ((SS23))42VSˆ2 −2(SS23)3CS2S3, (8)
(cid:115)
×(cid:18)11+0.0z(cid:19)4.6 (cid:18)1M∆Hνz100thinoturs(cid:19). (7) Vγ3(cid:48) ≈ (S12)3VSˆ3 + 49((SS23))52VSˆ2 −3SS234CSˆ2Sˆ3,
where
This expression is derived with the noise on the brightness tem-
√ 3
rpaeyrafitulrliengdefsaccrtiobredisbdyefi∆neTdNas=η T=sys/Aηf /∆Dν2tintiwnhwehreichtheAar- VSˆ3 = Npix(3σn2oiseK4+12S2σn4oise+5σn6oise), (9)
f tot max tot
is the total effective area of the array, D = λ/∆θ; the sys- and
max
tpeemrattuermepaetrathtuerefreTqsuysenicsieasssoufmiendtetroesbt.eBseaftourreatemdeabsyurthinegstkhyetmemo-- CSˆ2Sˆ3 = N6pixS3σn2oise. (10)
ments we smooth and resample the boxes to correspond roughly
with the resolution expected from the various instruments; to do
3.2 Brightness-temperaturemaps
so,weuseaGaussiansmoothingkernelwithwidthR =6Mpc
pix
forLOFAR/MWAandR = 3MpcforHERA/SKA(although Itisusefultotakealookatthemapsthemselvesbeforeanalysing
pix
we note that in this work, LOFAR/ MWA results are only pre- theirstatistics.InthemapsofFigure2,‘NoLLS’,‘LLS’,‘LLS+
sented for larger smoothing scales). We also reduce the assumed 10%GalHI’, ‘LLS + 50%GalHI’ and ‘minihalos + 100%GalHI’
frequencyresolutionofeachtelescopetocorrespondtoR ,i.e. arerepresentedfromlefttoright;thetopstripcorrespondstoz=
√ pix
(cid:112)
∆ν =H ν Ω R /[c (1+z)]. 7.0andQ ∼ 0.92andthebottomstripcorrespondstoz=9.20
0 0 m pix HII
Toestimatehowthenoisewe’vesofardiscussedwillpropa- andQ ∼ 0.50.Allmapshavebeensmoothedandresampledto
HII
gateontotheskewnessandvarianceweassumethateachpixelhas producepixelsof3Mpconaside,thepixeldepthhasbeenflattened
ameasuredsignalx =δT +n ,wherethenoiseonthepixeln intothepage.
i i i i
(cid:13)c 2015RAS,MNRAS000,000–000
6 C.A.Watkinson,A.Mesinger,J.R.Pritchard&E.Sobacchi
Asmentionedabove,theimpactofinhomogeneousIGMsinks
canbeseenbycomparingthelarge-scalereionizationmorphology
oftheleftpaneltotheotherpanels.AsdiscussedinSM2014,re-
combinationsinunresolvedstructuresresultinadramaticsuppres-
sionoflarge-scaleHIIregions.ComparingatafixedQHII,thedis-
tributionofHIIpatchsizesisbothnarrowerandshiftedtosmaller
scales.Theimpactisdramaticenoughthatpartiallyionizedcells,
hosting unresolved HII bubbles, become important. The ionized
fractionofthesecellsisonly<10%,butsincetheycorrespondto
∼
the densest(of thosenot yet reionized)∼Mpc-scale patches, this
issufficienttodampthesignalfromtheneutralcosmicpatches(as
evidenced by the ‘yellow’ vs ‘orange’ IGM surrounding the HII
bubbles in the ‘NoLLS’ and ‘LLS’ panels). This damps the con-
trastbetweenthefullyionizedandfullyneutralregions.Wecaution
however that unresolved, sub-grid HII regions are taken into ac-
countonlyinasimplisticfashioninourmodel,asdescribedabove;
furtherworkisrequiredtoaccuratelymodelsuchsub-gridphysics.
The impact of an increasing level of residual HI inside the
cosmic HII patches can be seen by comparing the insides of the
‘black’regionsinthepanelsoftheFigure2fromlefttoright.The
modelofSM2014(our‘LLS’)onlyresultsinafewpercentlevel
of HI inside HII regions, and so on this colour scheme the HII
regions are still black. However the extreme galactic HI models,
‘LLS + 50%GalHI’ and ‘minihalos + 100%GalHI’, show a no-
table level of HI inside the cosmic HII regions, with small-scale
structuretracingthedistributionofgalaxies.Thisfurtherreduces
thecontrastbetweenneutralandionizedregions,complicatingan
interferometricdetectionofreionization.Inprinciplehowever,re-
solving this small-scale HI structure with high-resolution 21-cm
interferometrycouldconstrainmodelsofgalacticHI.
3.3 Probabilitydenstityfunctions Figure3.BrightnesstemperaturePDFsatQHII ≈0.3,0.5,0.7(z =10.8,
9.2,8),toptobottom.Allmapsaresmoothedtoaco-movingpixelsizeof
ThesetrendsarequantifiedinFigure3,wherewepresenttheprob- 3Mpc.
abilitydensityfunctions(PDFs)fromthesemaps,withlinescor-
respondingto:‘NoLLS’(pinkdashedw/stars),‘LLS’(redsolid),
dashedw/circles)and‘NoLLS’(pinkdottedw/stars);wewilluse
‘LLS+50%galacticHI’(blackdottedw/triangles)and‘minihalos
thiskeyfortheremainsofthepaper.
+100%galacticHI’(greendashedw/circles).Plotscorrespondto
We begin by reviewing what simulations that ignore recom-
Q ≈0.3,0.5,0.7(z=10.8,9.2,8),toptobottom.
HII
binations in unresolved systems predict for the evolution of the
The presence of residual HI inside cosmic HII regions sup-
brightness-temperature moments. Studies of statistics from such
presseszero-valuedpixelsinthePDFs.Thebi-modalnatureofthe
modelsidentifiedseveralimportantpropertiesthatwouldprovide
PDFismaintainedfor‘LLS’butthesharppeakatδT = 0mK
b
invaluableconstraintsonthetimingofreionizationifborneoutin
issmearedintoawiderpeakcentredonδT ∼ 1mKbecauseofa
b
reality(e.g.Furlanettoetal.2004;Harkeretal.2009;Watkinson&
fewpercentlevelofresidualHIintheionizedIGM(SM2014).It
Pritchard2014).Itisusefulforthediscussionthatfollowstorefer
isonlywithlargelevelsofgalacticsinksthatthebi-modalnature
tothepinkdottedlinesw/starsinFigure4.Theamplitudeofthe
isentirelysuppressed,
varianceismaximisedatthehalf-waymarkofreionizationasthe
Also evident in these PDFs is the impact of sub-cell, unre-
weightofthenon-zerodistributionofthebrightness-temperature’s
solvedHIIregions;thesepreferentiallyoccurinthehighestdensity
PDFanditsspikeatzeroarebalanced(Figure4top).Thelate-time
pixelshostingnewlyforminggalaxies,whichiswhythereisasup-
dominanceofthezero-temperaturespikeinducesarapidincrease
pressionofthehighδT tailin‘LLS’whencomparedto‘noLLS’.
b
inskewnessatlatetimes(Figure4middle).Itcouldbearguedthat
Thiseffectconspireswiththelossofasharpzeropeaktofurther
thiseffectisdrivenbythevariancebecomingverysmallasreion-
reducethevarianceinthemaps.
ization draws to a close, however the dimensional skewness still
exhibits a late-time signature but in the form of a turnover (Fig-
ure4bottom).Atthehalfwaypointbothskewnessstatisticspass
3.4 Observingthebrightness-temperaturemoments:
fromnegativetopositive.Atearlytimestheyexhibitaminimum
Instrumentalnoise
asthePDFtransitionsfrombeingdominatedbythedensityfield
Figure 4 shows evolution of the variance (top), skewness (mid- tobeingdominatedbytheneutralfraction(Lidzetal.2008);the
dle) and dimensional skewness (bottom) as a function of the HII locationofthisminimumdependsontheoverlapofX-rayheating
volume-fillingfactor.Linescorrespondto‘LLS’(redsolid),‘LLS andreionizationepochs(Mesingeretal.2013).
+ 10% galactic HI’ (blue dot-dashed), ‘LLS + 50% galactic HI’ Unresolved HI sinks can dramatically change this picture.
(blackdottedw/triangles),‘minihalos+100%galacticHI’(green Already with ‘LLS’ the magnitude of the variance is suppressed
(cid:13)c 2015RAS,MNRAS000,000–000
21-cmsignaturesofresidualHI 7
by up to a factor of two during the mid phases of reionization.
HeretheadditionalsignalinHIIregions,reducesthecontrastbe-
tweenionizedandneutralregions.Moreimportantly,smallerHII
regions mean that there is more sub-pixel, unresolved ionization
structure in ‘LLS’ compared with ‘NoLLS’. This suppression of
bi-modality(onresolvablescales)intheionizationstructureisfur-
therexacerbatedbythemoredisjointHIIregionsin‘LLS’which
get ‘smeared-out’ when the maps are smoothed; i.e. the HII re-
gionsin‘LLS’arelesslikelytoberesolvedby21-cminterferom-
eters.Theseeffectsdramaticallysuppressthesmall-scalecontrast
between ionized and neutral regions; the strength of the turnover
associatedwiththehalfwaypointofreionizationissuppressedto
the point that even SKA could have difficulty in constraining it.
Fortunately,aswewillseeinthenextsection,furthersmoothingof
the maps emphasises the turn-over feature, improving our ability
toconstrainthismid-phasesignature.Theturn-overinthevariance
occursatQ ∼0.4in‘LLS’ratherthannearlyexactlyhalfway
HII
throughreionizationaspredictedby‘NoLLS’.Thisisbecausethe
ionizationmapisnolongerwelldescribedasatwo-phasefieldin
whichthenumberofpixelsineachphaseperfectlybalanceatthe
midpointofreionization.
TheinclusionofgalacticHIreducesthevarianceevenfurther
(seealsoWyitheetal.2009).Reasonablelevelsofgalacticsinksdo
notdramaticallyaltertheevolutionofthevariance(ascouldalready
beseenfromthemapsinFigure2).However,ifthegalacticHIis
tiedtoafixedfractionoff asinoursimpleillustrativemodels,
coll
thegrowthofcollapsedstructurewillboostthe21-cmvarianceat
latetimes.Intheextrememinihalo+100%GalHImodel,structure
growthtotallydominatesthesignalthroughout.SKAcouldeasily
detectsuchafeature.
WeseeinthemiddlepanelofFigure4thatthetransitionfrom
negativetopositiveskewnessincludingsinksoccursatQ ∼0.6
HII
(insteadofQ ∼0.5aspredictedby‘NoLLS’).Thistransitionis
HII
reducedtolowerQ inthepresenceofextremelevelsofgalactic
HII
sinks.Forreasonablelevelsofgalacticsinks,thisispredominantly
duetoeffect(i),wheresmallHIIregionsareunresolved.Weeven Figure 4. Variance (top), skewness (middle) and dimensional skewness
see this trend to a lesser degree when smoothing of ‘NoLLS’ on (bottom)ofbrightnesstemperaturemeasuredfrom‘observed’mapswith
large scales smears-out the HII regions (see Figure 6). For rea- aco-movingpixelsizeof3Mpcasafunctionofvolume-fillingfactorof
sonablelevelsofgalacticsinks,thissignatureandtheturn-overin ionizedregions.BeigeshadingsdepictSKA’s1-σ instrumentalerrorsfor
thevariancebracketthemid-pointandwillconstrainthetimingof the‘LLS’simulation.Withinthehatchedregion,spin-temperaturefluctua-
reionizationmoretightlythaneitherstatisticonitsown. tionsmaynotbenegligible.
Thelate-timerapidincreasepredictedfortheskewnessisto-
tallysuppressedbythepresenceofIGMsinks;insteadweseeaturn
overintheskewnessof‘LLS’astheHIIfillingfactorreachesabout cationofthisfeatureisdeterminedbytheoverlapofX-rayheat-
90%.Thisisbecausethenormalisingfactor,i.e.thevariance,isno ing and reionization (Mesinger et al. 2013), and here we assume
longertendingtowardszeroandweinsteadseeatransitionfroma TS (cid:29)Tγ throughout.Nevertheless,extremelevelsofresidualHI
skewnessdrivenbytheionizationfieldtoonedrivenbythedistri- willalmosttotallysuppressthissignature.Atitsdefaultresolution
butionofIGMsinks.Thisturnoveriswipedoutbythepresenceof SKA could struggle to detect this feature, depending on the tim-
evensmallamountsofgalacticsinks.Theseareimportantpointsto ingsofreionization;however,itsnon-detectionwouldstillprovide
beawareofaswebeginoureffortstoconstrainreionizationwith upperlimitsonthisphase.
21-cm observations. Detecting a turnover in the skewness would Figure5showsthevarianceasafunctionofthepercentageof
implythepresenceofIGMsinks(aswouldreducedvariance)and galacticmassinsinksattheendofreionization,forthissimulation
thatreionizationisinitsfinalphases.Thenondetectionofsucha z = 5.8. To reduce noise the maps are smoothed to a radius of
late-timesignatureintheskewnesswouldnotnecessarilymeanthat R = 10 Mpc and the beige shadings correspond to, from
smooth
wearenotobservingtheendofreionization.Itsabsencecombined lighttodark,theMWA,LOFARandSKA1-σinstrumentalerrors.
withevolutioninthevariancewithredshiftwouldinsteadimplythe NotethatthesignalwhenmeasuredwithSKA-likeresolution,i.e.
presenceofgalacticsinks. 3 Mpc pixels, is greater by around a factor of ten. The blue dot-
SKAwillbesensitivetothedetailsofskewnesswehavede- dashedlinemarksthelevelofresidualhydrogenasconstrainedby
scribed above, but only in the dimensional skewness, which ex- DLAsatlowerredshift,i.e.α=0.03atz∼4.
hibitsmoredistinctfeaturesandbetterrobustnesstonoise. Therearetwowaystoviewthispost-reionizationsignal,one
Theearly-timeminimumintheskewnessstatisticsisseento isthatitisintrinsicnoisetobeovercomeinourquesttoconstrain
berobustacrossthemodels.Thisisunderstandable,sincethelo- reionization, the other is that it provides a constraint on residual
(cid:13)c 2015RAS,MNRAS000,000–000
8 C.A.Watkinson,A.Mesinger,J.R.Pritchard&E.Sobacchi
Figure5.VarianceasafunctionofpercentageofremnantgalacticHIat
z=5.80(QHII = 1)measuredin‘observed’mapssmoothedtoaradius
ofRsmooth =10Mpctoreducenoise,thesignalisaroundafactorof10
greaterwiththedefaultpixelsizeof3Mpc.Theverticallinecorrespondsto,
α = 0.03,illustratingthelevelofresidualhydrogenobservedingalaxies
at lower redshifts. MWA, LOFAR and SKA 1-σ instrumental errors are
depictedwithbeigeshadingsfromlightesttodarkestrespectively.
HI post-reionization.Clearlytheamplitudeofthevarianceatthe
endofreionizationisverysensitivetothelevelofgalacticsinks.
ThisintrinsicnoisewillbedetectablebySKA,whichwilltherefore
bewellplacedtoconstraingalacticsinkswiththisstatistic.Wedo
notexplicitlyplotHERAerrorsbuttheyareindistinguishablefrom
SKA, i.e. negligible, on this plot and so should perform equally
wellatthistask.Beforethesetelescopescomeonline,bothLOFAR
andMWAshouldbeabletosetsomeconstraintsonthisquantity,
althoughatthelowerandmoreprobablevaluesofα,MWA’ssig-
nal to noise will be too poor to do more than set upper limits on
remnantgalacticHI.Theflip-sideofthisisthattheintrinsicnoise
from galactic sinks should not be a limiting factor for these first
generationinstruments.
Figure 6. Variance (top), skewness (middle) and dimensional skewness
3.5 Observingthebrightness-temperaturemoments:
(bottom)ofbrightnesstemperatureasafunctionoftheHIIvolume-filling
Smoothingtoreducenoise
factormeasuredin‘observed’mapssmoothedtoRsmooth=10Mpc.Beige
As in Watkinson & Pritchard (2014), we investigate the effects shadingsdepict1-σinstrumentalerrorsforthe‘LLS’simulation;tonescor-
of smoothing the maps to beat down instrumental noise. Figure respondtoMWA,LOFARandSKAfromlightesttodarkest.Withinthe
hatchedregion,spin-temperaturefluctuationsmaynotbenegligible.
6 (top) shows the evolution of the variance when smoothed to
Rsmooth =10MpcasafunctionofHIIvolume-fillingfactor.LO-
FAR will be well placed to constrain the presence of IGM sinks
4 BRIGHTNESS-TEMPERATUREPOWERSPECTRUM
through the reduction in the variance, could constrain/ exclude
largelevelsofgalacticsinksandwillbeabletooffertimingcon- The brightness temperature’s 1-point statistics form the focus of
straints from the mid-point turnover in the variance (unless the thispaper,butthe21-cmpowerspectrumwillbeanincrediblyim-
model is extreme and this signature does not exist). We see that portantstatisticwhenitcomestoconstrainingreionization.Assuch
smoothingrecoversthelate-timeturnoverinbothskewnessstatis- itisworthunderstandingtheimpactofgalacticneutralhydrogenon
ticsforreasonablelevelsofgalacticsinks.Thisisfortunateasthe this statistic. We concentrate on the spherically-averaged dimen-
late-timemaximuminthedimensionalskewnesswillbedetectable sionless power spectrum which we describe using ∆2δTb(k,z) =
evenbyLOFARforawiderangeofgalacticHIlevels.Themid- k3/(2π2V)(cid:104)|δ21(k,z)|2(cid:105)kinwhichδ21 =δTb(k,z)/δTb(z)−1,
pointofreionizationwillbeboundedbySKAandHERAmeasure- δT (z) is the redshift dependent average brightness temperature
b
mentsoftheskewness,whichwillinformuswhenQ <0.7,and calculatedfromthesimulation,V isthevolumeofthesimulated
HII
constraintsfromthevarianceofQ ≈ 0.4.Theearly-timemin- box and the angle brackets denote an average over k-space. To
HII
imumshouldstillbedetectableinbothskewnessstatisticsdespite modelinstrumentalerrorsonthesphericallyaveragedpowerspec-
suppressionbythepresenceofIGMsinks.SKAistheonlyinstru- trum,wetaketheapproachoutlinedintheappendixofMcQuinn
mentthatislikelytoactuallydetectthisfeature;howeverLOFAR etal.(2006)adoptingalogarithmicbinwidthof(cid:15)=0.5.Incalcu-
andHERAshouldbeabletouseitsabsencetosetupper-limitson latingthenumberdensityofbaselinesweassumeafillednucleus
√
thisphase.Ifresidualhydrogenlevelsareextremethenthissigna- limitedbytheminimumseparationofstations, A ,followedby
eff
ture,likeeveryother,willbesuppressed. anr−2dropoffinstationdensity.Thisassumesasmoothdensityof
(cid:13)c 2015RAS,MNRAS000,000–000
21-cmsignaturesofresidualHI 9
Figure 8. Dimensional power spectrum of the brightness temperature as
a function of the HII volume-filling factor for k = 1Mpc−1 (top) and
k = 0.1Mpc−1 (bottom). Within the hatched region, spin-temperature
fluctuationsmaynotbenegligible.
spectrum at small k can cleanly pick-up large-scale structure.
As such its timing signatures are more robust. For example, the
mid-point turnover in the k = 0.1Mpc−1 dimensional power
spectrumisrobustlyatQ ∼0.5forthemorereasonablemodels,
HII
althoughitshiftstosmallerQ withlargelevelsofgalacticsinks
HII
and is completely wiped out in the extreme galactic HI model.
We see that whilst reasonable quantities of galactic sinks have
Figure7.Dimensionlesspowerspectrumofthebrightnesstemperatureas little qualitative impact, large quantities do impact on the nature
afunctionofkMpc−1forz = 10.8(top),z = 9.2(middle),andz = 8
of the power spectrum’s evolution with its evolution no longer
(bottom).
welldescribedbyasimpleinvertedparabolaonlargescales.This
results from additional small-scale power (evident in the plots of
Figure7andthetopplotofFigure8)andfromthefactthatatlate
stations,whichforLOFARisaparticularlycrudeapproximation.
timesthereisstillsubstantialresidualHIthatdrivesthepower.
Whilst the exact configuration of SKA is yet to be decided, both
LOFARandMWAhaveoptimizedstationpositioning,assuchour
powerspectrumerrorsareindicativeonly.
Thedimensionlesspowerspectraforoursimulationsarepre- 5 CONCLUDINGREMARKS
sented in Figure 7 for Q ≈ 0.3, 0.5, 0.7 (z =10.8, 9.2, 8),
HII
In this paper we have examined the impact of sinks on the 21-
fromtoptobottomrespectively.AsalreadydiscussedinSM2014,
cmstatisticsofreionization.Weconsidertwoclassesofsink,IGM
IGMsinkscansuppresslarge-scale21-cmpower,resultinginmuch
sinksandgalacticsinks,whichhavetwoeffectsonthe21-cmsig-
steeperpowerspectrathroughoutreionization.FromFigure7we
nal:
see that adding additional galactic HI does not have a large im-
pactonthepowerspectrum,forreasonablevaluesofα.Moreex-
(i) As absorbers of ionizing photons they impact on both the
treme levels of galactic HI result in a further steepening of the timingofreionizationandthemorphologyofcosmicHIIpatches.
power spectrum, since: (i) the contrast between ionized and neu-
(ii) Asreservoirsofneutralgastheycontributeanon-zero21-
tralpatchesisreduced,and(ii)thedistributionofgalacticHIcon-
cmsignalinotherwiseionizedcosmicpatches.
tributesadditionalpoweronsmall-scales.
The evolution of the dimensional power spectrum WeusethesimulationsofSobacchi&Mesinger(2014)toinves-
δTb2(k,z)∆2δTb(k,z) as a function of the HII volume fill- tigatetheimpactofIGMsinksonthe21-cm1-pointstatisticsdue
ing factor is plotted in Figure 8 for k = 1Mpc−1 (top) and toeffects(i)and(ii).Inadditionwestudyhowthecontributionof
k=0.1Mpc−1(bottom).Unlikehigher-ordermoments,thepower galacticsinksto(ii)affectsthe21-cmstatistics.Wealsoconsider
(cid:13)c 2015RAS,MNRAS000,000–000
10 C.A.Watkinson,A.Mesinger,J.R.Pritchard&E.Sobacchi
anextreme,illustrativemodelinwhichgasinsidesterileminihaloes IlievI.T.,MellemaG.,AhnK.,ShapiroP.R.,MaoY.,PenU.L.,
andatomically-cooledgalaxiesisfullyneutral. 2014,MNRAS,439,725
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ization morphology (Sobacchi & Mesinger 2014; effect i), will JenkinsA.,FrenkC.S.,WhiteS.D.M.,ColbergJ.M.,ColeS.,
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large-scalemodesforcleanerreionizationprobes.
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Similarly,therapidincreaseoftheskewnesspreviouslyasso-
S.R.,2006,ApJ653,815
ciatedwiththeendofreionizationisalsosuppressedbyinhomo-
McQuinnM.,OhS.P.,Faucher-Gigue`reC.-A.,2011,ApJ,743,
geneoussinks.Thisisbecausethevarianceisnon-zeroevenpost-
82
reionization(duetoresidualHI;effectii)andsoweinsteadseea
MellemaG.etal.,2013,Exp.Astron.,36,235
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Wefindthatmorereasonablelevelsofgalacticsinks,willnot
MesingerA.,DijkstraM.,2008,MNRAS,390,1071
change the qualitative evolution of the moments during reioniza-
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potentialtodramaticallyalterthequalitativeevolutionofmoments. NoterdaemeP.etal.,2012,A&A,547,L1
ThisprovidesanexcitingopportunityforSKA(andevenLOFAR) O’MearaJ.M.,ProchaskaJ.X.,WorseckG.,ChenH.W.,Madau
toconstrainlevelsofgalacticHIinthehigh-redshiftUniverse. P.,2013,ApJ,765,137
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ACKNOWLEDGEMENTS
RahmatiA.,PawlikA.H.,RaicevicM.,SchayeJ.,2013,MNRAS,
CWissupportedbyanSTFCstudentship.JRPacknowledgessup- 430,2427
portunderFP7-PEOPLE-2012-CIGgrant#321933-21ALPHAand RicottiM.,GnedinN.Y.,ShullJ.M.,2001,ApJ,560,580
STFCconsolidatedgrantST/K001051/1. SantosM.G.,FerramachoL.,SilvaM.B.,AmblardA.,Cooray
A.,SantosG.,2010,MNRAS,406,2421
SchayeJ.,2001,ApJ,559,507
ShapiroP.R.,IlievI.T.,RagaA.C.,2004,MNRAS,348,753
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