Descending from on high: Lyman series cascades and spin-kinetic temperature coupling in the 21 cm line

0.3 MB·English

by  Jonathan R. Pritchard

Mon.Not.R.Astron.Soc.000,000–000(0000) Printed5February2008 (MNLATEXstylefilev2.2)
Descending from on high: Lyman series cascades and spin-kinetic
temperature coupling in the 21 cm line
Jonathan R. Pritchard⋆ and Steven R. Furlanetto
DivisionofPhysics,Mathematics,&Astronomy,CaliforniaInstituteofTechnology,MailCode130-33,Pasadena,CA91125,USA
5February2008
6
0
ABSTRACT
0
We examinethe effectof Lyman continuumphotonson the 21 cm backgroundin the high-
2
redshiftuniverse.Thebrightnesstemperatureofthistransitionisdeterminedbythespintem-

n peratureTs,whichdescribestherelativepopulationsofthesingletandtriplethyperfinestates.
a Once the first luminoussources appear, Ts is set by the Wouthuysen-Fieldeffect, in which
J Lyman-series photons mix the hyperfine levels. Here we consider coupling through n > 2
5 Lymanphotons.Wefirstshowthatcoupling(andheating)fromscatteringofLynphotonsis
2
negligible,becausetheyrapidlycascadetolower-energyphotons.Thesecascadescanresult

ineitheraLyαphoton–whichwillthenaffectTs accordingtotheusualWouthuysen-Field
2
mechanism–orphotonsfromthe2s → 1scontinuum,whichescapewithoutscattering.We
v
show that a proper treatment of the cascades delays the onset of strong Wouthuysen-Field
1
8 couplingandaffectsthepowerspectrumofbrightnessfluctuationswhentheoverallcoupling
3 isstillrelativelyweak(i.e.,aroundthetimeofthefirststars).Cascadesdampfluctuationson
8 smallscalesbecauseonly∼1/3ofLynphotonscascadethroughLyα,buttheydonotaffect
0 thelarge-scalepowerbecausethatarisesfromthosephotonsthatredshiftdirectlyintotheLyα
5 transition.We also commenton the utility of Lyn transitionsin providing“standardrulers”
0 withwhichtostudythehigh-redshiftunvierse.
/
h
Keywords: cosmology:theory–galaxies:high-redshift–atomicprocesses
p
-
o
r
t
s
1 INTRODUCTION visibility measurements, has become possible (Morales&Hewitt
a
: 2004). Three such arrays (LOFAR1, MWA2, and PAST3) will
v One potentially promising probe of the cosmic dark ages is 21
soon be operational, opening a window onto this new low fre-
i cm tomography. It has long been known (Hogan&Rees 1979;
X quencyband. Beforeadetectioncanbemade, however, thereare
Scott&Rees 1990) that neutral hydrogen in the intergalactic
still major scientific and technical challenges to be met. Iono-
r medium(IGM)maybedetectableinemissionorabsorptionagainst
a spheric scattering and terrestrial interference are two serious is-
the cosmic microwave background (CMB) at the wavelength of
sues.Alsoworryingistheneedtoremoveforegrounds,whichare
theredshifted21cmline,thespin-fliptransitionbetweenthesin-
manyordersofmagnitudestrongerthanthesignal.Multifrequency
glet and triplet hyperfine levels of the hydrogen ground state.
subtractiontechniques(Zaldarriaga,Furlanetto&Hernquist2004;
The brightness of this transition will thus trace the distribution
Morales&Hewitt2004;Santos,Cooray&Knox2005),exploiting
of HI in the high-redshift universe (Field 1958, 1959a), which
thesmoothnessoftheforegroundspectra,havebeenproposed,but
gives the signal angular structure as well as structure in redshift
theireffectivenesshasyettobetested.Thechallengesaregreat,but
space.Thesefeaturesarisefrominhomogeneitiesinthegasdensity
soaretheopportunities.Itisthuscrucialtounderstandthenature
field, the hydrogen ionization fraction and the spin temperature.
ofthe21cmsignalaswecommencethesesearches.
Madau,Meiksin&Rees (1997) showed that the first stars could
Fluctuations in the 21 cm signal arise from both cosmo-
causearapidevolutioninthesignalthroughtheireffectonthespin
logical and astrophysical sources. Most previous work has fo-
temperature.Consequently,the21cmsignalcanprovideunparal-
cussed on the signal due to density perturbations (Madauetal.
leledinformationaboutthe“twilightzone”whenthefirstluminous
1997; Loeb&Zaldarriaga 2004) or from inhomogeneous ion-
sourcesformedandtheepoch ofreionizationandreheatingcom-
ization (Ciardi&Madau 2003; Furlanetto,Sokasian&Hernquist
menced.
2004; Furlanetto,Zaldarriaga&Hernquist 2004). An additional
Despite the theoretical promise of this probe, it is only
with improvements in computing power that building radio ar-
rays with sufficient sensitivity, capable of correlating billions of
1 Seehttp://www.lofar.org/.
2 Seehttp://web.haystack.mit.edu/arrays/MWA/.
⋆ Email:[email protected] 3 SeePenetal.(2005).
c 0000RAS
(cid:13)
2 Pritchardand Furlanetto
sourceoffluctuationsisthespintemperature,whichdescribesthe 2 21CMFORMALISMANDTHE
relativeoccupationofthesingletandtriplethyperfinelevels.These WOUTHUYSEN-FIELDMECHANISM
levelsmaybeexcitedbythreeprimarymechanisms:absorptionof
The21cmlineofthehydrogenatomresultsfromhyperfinesplit-
CMB photons, atomic collisions, and absorption and re-emission
tingofthe1S groundstateduetotheinteractionofthemagnetic
ofLyαphotons (theWouthuysen-Field effect;Wouthuysen1952;
momentsoftheprotonandtheelectron.TheHIspintemperature
Field1959a).Thefirsttwoprocessesrelyuponsimplephysics,but
T isdefinedviatherelativenumberdensityofhydrogenatomsin
thelastoneallowsustostudythepropertiesofluminoussources, s
the1S singletandtripletlevelsn /n = (g /g )exp(−T /T ),
whichdeterminethebackgroundradiationfield. 1 0 1 0 ⋆ s
where(g /g )=3istheratioofthespindegeneracyfactorsofthe
1 0
Barkana&Loeb(2005b,henceforthBL05)studiedthesignal twolevels,andT ≡ hc/kλ = 0.0628K.Theopticaldepth
⋆ 21cm
generated bythefirstgenerationof collapsedobjects. Thesehigh ofthistransitionissmallatallrelevantredshifts,sothebrightness
redshiftobjectsarehighlybiased,leadingtolargevariationsintheir temperatureoftheCMBis
number density.This,combinedwiththe1/r2 dependence ofthe
T −T
flux,causeslargefluctuationsintheLyαbackground,whichcanbe Tb =τ(cid:18) s1+CzMB(cid:19), (1)
probedthroughtheireffectonthe21cmtransition.Exploitingthe
anisotropyinducedbypeculiarvelocities(Bharadwaj&Ali2004; wheretheopticaldepthforresonant21cmabsorptionis
Barkana&Loeb 2005a), they showed that information about the 3cλ2hA n
Lyα radiation field could be extracted from the power spectrum τ = 32πk T (1+1z0)(HdvI /dr). (2)
B s r
of 21 cmfluctuations and separated intothosefluctuations corre-
Heren isthenumberdensityofneutralhydrogen,A =2.85×
latedanduncorrelatedwiththedensityfield.Thefeaturesofthese HI 10
10−15s−1 isthespontaneous emissioncoefficient,anddv /dr is
spectraallowextractionofastrophysicalparameterssuchasthestar r
thegradientofthephysicalvelocityalongthelineofsightwithr
formation rate and bias. However, it is not a trivial task to relate
thecomovingdistance.WhenT <T thereisanetabsorption
theemissivitytoadistributionofLyαphotons.Thebackgroundin s CMB
of CMB photons, and we observe a decrement in the brightness
theLyαlineiscomposedoftwoparts:thosephotonsthathavered-
temperature.
shifteddirectlytotheLyαfrequencyandthoseproducedbyatomic
Thespintemperatureisdeterminedbythreecouplingmecha-
cascadesfromhigherLymanseriesphotons.Tocalculatethislatter
nisms.RadiativetransitionsduetoabsorptionofCMBphotons(as
component, BL05assumed thatatomiccascadeswere100% effi-
wellasstimulatedemission)tendtodriveT → T .Spinflips
cientatconvertingphotonsabsorbedataLymanresonanceintoa s CMB
from atomic collisions drive T → T , the gas kinetic tempera-
Lyαphoton,whileinrealitymostcascadesendintwophotonde- s k
ture.Finally,theWouthuysen-Fieldeffect(Wouthuysen1952;Field
cayfromthe2Slevel.
1958),whichisthemainfocusofthispaper,alsodrivesT →T
s k
Inthispaper,wecalculatetheexactcascadeconversionprob- (seebelow).Thecombinationthatappearsin(1)canbewritten
abilitiesfrombasicatomicphysics.Inaddition,wediscussthepos-
T −T x T
sibilityoflevelmixingbyscatteringofLynphotonsviaastraight- s CMB = tot 1− CMB , (3)
T 1+x (cid:18) T (cid:19)
forwardgeneralisationoftheWouthuysen-Fieldeffect.Wethenap- s tot k
plythecascadeefficienciestocalculatetheLyαfluxprofileofan wherextot = xα+xc isthesumoftheradiativeandcollisional
isolatedsource.Theexistenceofdiscretehorizons,determinedby couplingparameters.Thelatteris
themaximumdistanceaphotoncantravelbeforeitredshiftsintoa x = 4κ1−0(Tk)nHT⋆, (4)
givenLymanresonance,imprintsaseriesofdiscontinuitiesintothe c 3A T
10 CMB
profile,whichcaninprinciplebeusedasastandardruler.Weapply
whereκ1−0 istabulatedasafunction of Tk (Allison&Dalgarno
theseresultstothepowerspectraof21cmfluctuationsduringthe
1969; Zygelman 2005). The spin temperature becomes strongly
epoch of thefirststars, showingthat thesecorrections cannot be
coupledtothegastemperaturewhenx &1.
tot
ignoredwhenextractingastrophysicalparameters.
AschematicdiagramoftheWouthuysen-Fieldeffectisshown
Thelayoutofthispaperisasfollows.In§2weintroducethe inFigure1;itmixesthehyperfinelevelsthroughabsorptionandre-
formalismfordescribing21cmfluctuationsandthedominantcou- emissionofLyαphotons.Quantumselectionrulesallowtransitions
pling mechanism, the Wouthuysen-Field effect. In §3 we discuss forwhichthetotalspinangularmomentumF changesby∆F =
the possibility of direct pumping by Lyn photons. Next, in §4, 0,±1(except0 → 0),makingonlytwoofthefourn = 2levels
we detail the atomic physics of radiative cascades in atomic hy- accessibletoboththen = 1singletandtripletstates.Transitions
drogen. The results are applied to the Lyα flux profile of an iso- toeitherofthesestatescanchangeTs.Thecouplingcoefficientis
latedsourcein§5andtothe21cmpowerspectrumfromthefirst 4P T
x = α ⋆ , (5)
galaxiesin§6.Wealsodiscusssomeofthelimitationsofthisfor- α 27A T
10 CMB
malism.Finally,wesummariseourresultsin§7.InanAppendix,
whereP istheLyαscatteringrate(Madauetal.1997).Ifresonant
α
wereview the equations needed tocalculateanalytically theEin-
scatteringofLyαphotonsoccursrapidlyenoughT willbedriven
s
steinAcoefficientsforthehydrogenatom.Throughout,weassume
toT ,thecolourtemperatureoftheradiationfieldattheLyαfre-
α
(Ω ,Ω ,Ω ,h,σ ,n ) = (0.3,0.046,0.7,0.7,0.9,1.0),consis-
m b Λ 8 s quency (Field 1958; Madauetal. 1997). In parallel, the repeated
tentwiththemostrecentmeasurements(Spergeletal.2003).
scattering of Lyα photons by the thermal distribution of atoms
During the preparation of this paper, Hirata (2005) sub- brings Tα → Tk (Field 1959b; Hirata 2005). Consequently, the
mitted a preprint covering similar material. We have confirmed Wouthuysen-Field effect provides an effective coupling between
agreement where there is overlap. The main results of this thespintemperatureandthegaskinetictemperature.
work were discussed at the “Reionizing the Universe” confer- WecanalsowritetheWouthuysen-Fieldcouplingas
ence in Groningen, The Netherlands (June 27-July 1, 2005; see 16π2T e2f
http://www.astro.rug.nl/∼cosmo05/program.html). xα= 27A T⋆ mα cSαJα, (6)
10 CMB e
c 0000RAS,MNRAS000,000–000
(cid:13)
Lyman seriescascades andthe21cm line 3
depends only on the matter power spectrum. Pµ2 contains cross-
correlations between matter fluctuations and both δ and δ ,
xα xHI
makingitanidealprobeoffluctuationsintheradiationbackground.
In particular, at sufficiently high redshifts such that x ≪ 1, it
HI
probesvariationsintheLyαbackground. Linearcombinations of
thesethreetermscanbeusedtoextractdetailedinformationabout
othertypesoffluctuations(Barkana&Loeb2005a).
3 DIRECTPUMPINGBYLYMANSERIESPHOTONS
Ofcourse,theradiationbackgroundcontainsphotonsthatredshift
intoalltheLymantransitions,notjustLyα.Themainpurposeof
thispaper is to examine how these affect T . The existing litera-
s
tureassumesthatallLynphotonsareimmediatelyconvertedinto
Lyαphotonsbyatomiccascades(e.g.BL05).Inreality,thereare
twodifferentcontributionstoconsider:oneduetoscatteringofthe
Lynphotonitselfandtheotherduetoitscascadeproducts.Inthis
section,wediscussthedirectcontributionofLynscatteringtothe
couplingofT andT ,whichoccursinamannerexactlyanalogous
s k
totheWouthuysen-Fieldeffect.Forthiseffecttobesignificanttwo
requirementsmustbefulfilled.First,thescatteringrateofLynpho-
Figure1.Hyperfinestructureofthe2P and1Slevelofthehydrogenatom.
tonsmustbesufficienttocoupleT andT ,theLyncolourtem-
LevelsarelabelledaccordingtothenotationnFLJ,wheren,L,andJare s n
theusualradial, orbital angular momentumandtotal angular momentum perature.Second,itmustbesufficienttodriveTn → Tk.Wewill
quantumnumbers.F = I+J isthequantumnumberobtainedfromthe arguethatneitherconditionissatisfiedinpractice.
nuclearspinandJ.Allowedtransitionsobey∆F = 0, 1(except0 TheIGMisopticallythick τ ≫ 1toallLymanseriestran-
± →
0).Thoserelevant fortheWouthuysen-Fieldeffect areindicated bysolid sitionswith n . 100. Consequently, a Lyα photon emitted by a
curves,whiledashedcurvesindicatetheremainingallowedtransitions. starwillscattermanytimes(∼τ ∼106 ;Gunn&Peterson1965)
beforeitfinallyescapesbyredshiftingacrossthelinewidth;each
ofthesescatteringscontributestotheWouthuysen-Fieldcoupling.
ALynphotoncanescapebyredshiftingacrossthelinewidth,but
wheref =0.4162istheoscillatorstrengthfortheLyαtransition,
α
a transition to a level other than n = 1 will also remove it. The
S is a correction factor of order unity (Chen&Miralda-Escude´
α
probability for a decay from an initial statei to a final statef is
2004; Hirata 2005) that accounts for the redistribution of pho-
givenintermsoftheEinsteinA coefficientsby
ton energies due to repeated scattering off the thermal distribu- if
tion of atoms, and Jα is the angle-averaged specific intensity of P = Aif . (9)
Lyα photons by photon number. For reference, a Lyα flux of if A
J = 1.165 × 10−10[(1+z)/20]cm−2s−1Hz−1sr−1 yields f if
α P
x =1(correspondingtoP =7.85×10−13[1+z] s−1). Appendix A summarises the expressions needed to compute
α α
Fluctuations in the brightness temperature arise fromfluctu- the Einstein Aif coefficients. For the Lyman series transitions
ations in the density, the Wouthuysen-Field coupling, the neutral PnP→1S ≈ 0.8 (see Table 1) so that a Lyn photon will scatter
fractionxHIandtheradialvelocitycomponent.Tolinearorder oforderNscat ≈ 1/(1−PnP→1S) ∼ 5timesbeforeundergoing
acascade.
x
δ =βδ+ α δ +δ −δ , (7) Becauseacascadeoccurslongbeforeescapeviaredshifting,
Tb x˜ xα xHI drvr
tot thecouplingfromdirectpumpingisnegligible.Recallthatthescat-
where δa is the fractional perturbation in a, δ is the fractional tering rate PX for the photon type X =Lyα, Lyβ, etc. may be
density perturbation, and x˜tot = xtot(1+xtot). β is a param- expressedas(Field1959a)
eter describing the thermal history of the gas, which we assume
n P =N n˙ (10)
to have cooled adiabatically, so that β ≈ 0.2. Naoz&Barkana HI X scat X
(2005)showedthatβslightlyincreasesandexhibitsmildscalede- in terms of the production rate of photons per unit volume n˙ .
X
pendence,asgastemperaturefluctuationsdonotexactlytrackthe It is then clear that, for similar production rates (i.e., for sources
densityfluctuations.Wenotethat,onscales0.01Mpc−1 < k < with a reasonably flat spectrum), P /P ∼ N /N ∼
103Mpc−1, β is approximately constant at z = 20 and choose 5×10−6.Thissimpleargumentshonwstαhatthecscoantt,rnibutsicoant,fαrom
toignorethissubtletyforeaseofcomparisonwithBL05.Thefirst directpumpingbyLynphotonswillbenegligiblecomparedtothat
threecomponentsofequation(7)areisotropic,butthevelocityfluc- oftheLyαphotons,becausex ∝P .
α α
tuationintroducesananisotropyoftheformδdrvr(k) = −µ2δin The second question, whether Tn → Tk, is still relevant
Fourier space (Bharadwaj&Ali 2004), where µ is the cosine of for heating of the gas by repeated scatterings. Given the reduced
theanglebetweenthewavenumberkoftheFouriermodeandthe number of scattering events, it seems unlikely to be the case,
lineofsight.Thisallowsustoseparatethebrightnesstemperature but a full calculation using a Monte Carlo method or follow-
powerspectrumPTb intopowersofµ2(Barkana&Loeb2005a) ingChen&Miralda-Escude´ (2004)isrequiredtorigourously an-
PTb(k)=µ4Pµ4(k)+µ2Pµ2(k)+Pµ0(k). (8) stewrienrgthaismqoureesetifofinc.ieLnatcskouorfceeqoufilhiberaitu,monwaoupledr mscaaktteerLinygnbsacsaist-.
Theanisotropyissourcedonlybydensityfluctuations,sothatPµ4 Chen&Miralda-Escude´ (2004) have shown that Lyα heating is
c 0000RAS,MNRAS000,000–000
(cid:13)
4 Pritchardand Furlanetto
much smaller than previous calculations indicated (Madauetal.
1997), because T ≈ T , whichreduces theheat transferred per
α k
collision.ThisisunlikelytobethecasefortheLyn.
FollowingMadauetal.(1997),wecanestimatethemaximum
heating from a single Lyn scattering by assuming that all of the
atomicrecoilenergyforastationaryatomisdepositedinthegas.
Momentumconservationthendemands
∆E
E˙ =− hν P , (11)
n (cid:28) E (cid:29) n n
where h∆E/Ei ∼ 10−8 is the fraction of energy lost by a Lyn
photonafterscatteringfromastationaryhydrogenatom,andhν is
n
theenergyofthephoton.AssumingtheproductionrateofLynpho-
tonsiscomparabletothatoftheLyαphotonsandtakingx = 1,
α
wethenobtainE˙ ∼ 0.002[(1+z)/10]KGyr−1.Thisismuch
n
smallerthantheLyαheatingrate,evenincludingtheT ≈T cor-
α k
rection,sowedonotexpectLynscatteringtobeasignificantheat
source.Furthermore,ifT ≈ T theratewouldbemuchsmaller
n k
thanthisestimate,asinChen&Miralda-Escude´(2004).
4 LYMANSERIESCASCADES Figure2.EnergyleveldiagramforthehydrogenatomillustratingLyβand
Lyγcascades.Markeddecaysaredistinguishedascascades(solidcurves),
Anexcitedstateofhydrogen mayreachtheground stateinthree
Lyn(dashedcurves),Lyα(dot-dashedcurve), andthetwophotondecay
ways.Firstly,itmaydecaydirectlytothegroundstatefromannP
(dottedcurve).Notethattheselectionrules(∆L= 1)decouplethe3P
state(n > 2),generatingaLynphoton.Secondly,itmaycascade and2P levels,preventingLyβfrombeingconverted±intoLyα.
tothemetastable2S level.Decayfromthe2S levelproceedsvia
aforbiddentwophotonprocess.Finally,itmaycascadetothe2P
level, from which it will produce a Lyα photon. We are primar-
ilyinterestedinthefractionofdecaysthatgenerateLyαphotons, n frecycle PnP→1S n frecycle PnP→1S
whichwillincreasetheLyαfluxpumpingthehyperfinelevels.
16 0.3550 0.7761
ThefractionofcascadesthatgenerateLyαphotonscanbede-
2 1 1 17 0.3556 0.7754
terminedstraightforwardlyfromtheselectionrulesandthedecay
3 0 0.8817 18 0.3561 0.7748
rates. As an example, consider the Lyβ system. Absorption of a 4 0.2609 0.8390 19 0.3565 0.7743
Lyβ photon excites the atom into the 3P level. As illustrated in 5 0.3078 0.8178 20 0.3569 0.7738
Figure2, the3P level can decay directlyto the ground state, re- 6 0.3259 0.8053 21 0.3572 0.7734
generatingtheLyβ photon,ortothe2S level,whereitwilldecay 7 0.3353 0.7972 22 0.3575 0.7731
by two photon emission. The selection rules forbid Lyβ photons 8 0.3410 0.7917 23 0.3578 0.7728
frombeingconvertedintoLyαphotons.Incontrast,the4P level, 9 0.3448 0.7877 24 0.3580 0.7725
10 0.3476 0.7847 25 0.3582 0.7722
excitedbyabsorptionofLyγ,cancascadeviathe3Sor3Dlevels
11 0.3496 0.7824 26 0.3584 0.7720
tothe2P levelandthengenerateLyα.
12 0.3512 0.7806 27 0.3586 0.7718
To calculate the probability f that a Lyn photon will
recycle 13 0.3524 0.7791 28 0.3587 0.7716
generate a Lyα photon, we apply an iterative algorithm. The ex-
14 0.3535 0.7780 29 0.3589 0.7715
pression 15 0.3543 0.7770 30 0.3590 0.7713
f = P f (12)
recycle,i if recycle,f
Xf Table1.Recyclingfractionsfrecycleanddecayprobabilitiestotheground
relatestheconversionprobabilityfortheinitiallevelitothecon- state,PnP→1S.
versionprobabilitiesofallpossiblelowerlevelsf.Thedecayprob-
abilitiesarecalculatedusingequation(9).Wetheniteratefromlow
agreementwiththoseofHirata(2005).Atlargen,theconversion
tohighn,calculatingeachf inturn.
recycle fractionsasymptotetof ≈0.36becausenearlyallcascades
recycle
In our particular case of an optically thick medium, we can
pass through lower levels. We emphasise again that the quantum
ignoredirecttransitionstothegroundstate.ThesegenerateaLyn
selectionrulesforbidaLyβphotonfromproducingaLyαphoton.
photon, which will rapidly be reabsorbed and regenerate the nP
Finallywebrieflycommentonthetwophotondecayfromthe
state.Therefore, suchdecays willnotaffectthenet populationof
2S level(foramoredetaileddiscussion,seeHirata2005).Selec-
photonsorofexcitedstates.Weincorporatethisintothecalculation
tionrulesforbidelectricdipoletransitionsfromthe2Sleveltothe
bysettingAnP→1S =0(Furlanettoetal.2005). ground state, but the second order twophoton decay process can
marisRedesuinltsTafbolre t1heanldowpleosttteLdyimnaFnigsuerreie3s.4trTahnesisteiornessualtrseasruemin- occur with Aγγ = 8.2s−1 ≪ A2P→1S. At z . 400 the CMB
fluxdensityissufficientlysmallthatradiativeexcitationsfromthe
2S levelarenegligible.Additionally,attherelevantdensitiescol-
4 Code for calculating the conversion factors is available at lisional excitation to the 2P level is slow compared to the two
http://www.tapir.caltech.edu/ jp/cascade/. photon process (Breit&Teller1940).Consequently, the2S level
∼
c 0000RAS,MNRAS000,000–000
(cid:13)
Lyman seriescascades andthe21cm line 5
ultimatelytruncatedatn ≈ 23toexcludelevelsforwhichthe
max
horizonlieswithintheHIIregionofatypical(isolated)galaxy,as
onlyneutralhydrogencontributesto21cmabsorption(BL05).The
averageLyαbackgroundisthus
nmax
J (z)= J(n)(z)
α α
nX=2
nmax zmax(n) (1+z)2 c
= dz′f (n) ǫ(ν′,z′),
Z recycle 4π H(z′) n
nX=2 z
(14)
whereν′ istheemissionfrequencyatz′correspondingtoabsorp-
n
tionbythelevelnatz
(1+z′)
ν′ =ν , (15)
n n(1+z)
andǫ(ν,z)isthecomovingphotonemissivity(definedasthenum-
berofphotonsemittedperunitcomovingvolume,perpropertime
andfrequency,atfrequencyν andredshiftz).Tocalculateǫ(ν,z),
wefollowthemodelofBL05
d
Figure3.RecyclingfractionsforLynphotons.Notethatthevalueslevel ǫ(ν,z)=n¯0bf∗dtFgal(z)ǫb(ν), (16)
roefsfoantafnrceecyacreleco≈nv0e.r3te5d9ianntodLthyaαtnpohnoetoonfs.thephotonsincidentontheLyβ wheren¯0b isthecosmicmeanbaryonnumber densitytoday, f∗ is
theefficiencywithwhichgasisconvertedintostarsingalacticha-
los(andwithwhichLyman-continuumphotonsescapetheirhosts),
willpreferentiallydecayviathistwophotonprocess.Thesetransi- and Fgal(z) isthefraction of gasinside galaxiesat z. Wemodel
tionsmaythemselvesaffectTs,becauseboththe2Sand1Slevels thespectraldistributionfunctionofthesourcesǫb(ν)asaseparate
havehyperfinestructureandanyimbalanceinthedecayconstants powerlawǫb(ν)∝ναs−1betweenLyαandLyβandbetweenLyβ
herewouldaffectthe1S populations. However, evenwithoutde- andtheLymanlimit.Inourcalculations,wewillassumePopula-
tailedcalculations,wecanseethattheresultantcouplingmustbe tionIIIstarswithspectralindexαs =1.29betweenLyαandLyβ,
small.CascadesthatdonotgenerateLyαmustreachthe2S level, normalisedtoproduce4800photonsperbaryonbetweenLyαand
so the fractionof Lyn photons that undergo two photon decay is theLymanlimit,ofwhich2670photonsareemittedbetweenLyα
f (n)=1−f (n)≈0.64(SeeTable1).Eachsuchdecay andLyβ.Incontrast,forPopulationIIstarsthenumbersare0.14,
γγ recycle
hasNscat,γγ =1becausetheresultingphotonsarenotreabsorbed. 9690, and 6520 respectively (BL05). In calculating Fgal, we use
ThisismuchsmallerthanN ≈106.Consequently,onlyifthe theSheth&Tormen(1999)massfunctiondn/dm,whichmatches
scat,α
couplingperscatteringweremanyordersofmagnitudelargerfor simulations better than the Press&Schechter (1974) mass func-
twophotondecaythanforLyαscatteringcouldthiseffectbesig- tion, at least at low redshifts. We assume that atomic hydrogen
nificant. coolingtoaviraltemperatureTvir ≈104Ksetstheminimumhalo
mass.Wewillnormalisef∗ sothatxα = 1atz = 20;thisyields
f∗ =0.16%whenweincludethecorrectfrecycle.
To see what fraction of photons from a given source are
5 THELyαCOUPLINGAROUNDASOURCE
converted into Lyα, we integrate ǫ (ν) with the proper weight-
b
WecanseetheeffectsoftheserecyclingfractionsontheLyαcou- ing by frecycle. We find that f¯recycle = 0.63, 0.72, and 0.69 for
pling by considering the life of a photon emitted from a given αs(Lyα−Lyβ) = 1.29, 0.14, and −1.0 respectively (roughly
source. The photon initiallypropagates freely, redshifting until it corresponding to Pop. III stars, low metallicity Pop II stars, and
entersaLynresonance.BecausetheIGMissoopticallythick,the quasars;Zhengetal.1997).Thetotalfluxissignificantlylessthan
photon will then scatter several times until a cascade converts it iffrecycle = 1,ashasbeengenerallyassumed before.ThusLyα
intoaLyαphotonortwo2S →1Sphotons.Inthelattercase,the coupling willtakeplacelateriftheproper atomicphysicsarein-
photonsescapetoinfinity;intheformercase,itscatters∼τ times cluded (typically ∆z & 1 for fixed source parameters; see also
beforeredshiftingoutoftheLyαresonance.Thisestablishesase- Hirata2005).Ofcourse,thisisonlytheaveragevalue,andaround
riesofclosely-spacedhorizons,becauseaphotonenteringtheLyn agivensourcetherewillbeadistancedependence.Agaselement
resonanceatzmusthavebeenemittedbelowaredshift thatcanonlybereachedbyphotonsredshiftedfrombelowtheLyβ
resonancewillseeaneffectivef = 1.Incontrast,agasele-
[1−(n+1)−2] recycle
1+z (n)=(1+z) . (13) mentveryclosetothesourcewillhavef ≈ 0.36.Thiswill
max (1−n−2) recycle
bereflectedinthebrightnesstemperaturepowerspectrum(see§6).
The number of Lyn transitions contributing Lyα photons is thus TheLyαfluxprofileofagalaxywithMgal = 3×1010M⊙
afunctionofthedistancefromthesource.Thesehorizonsimprint and our fiducial parameters at z = 20 is plotted in Figure 4.
well-definedatomicphysicsontothecouplingstrengthbyintroduc- In our approximation, J ∝ M . Thus obtaining x ≥ 1 at
α gal α
ingaseriesofdiscontinuitiesintotheLyαfluxprofileofasource. r = 10MpcrequiresagalaxymassofMgal = 4.2×1012M⊙,
Thus theLyαflux, J ,arisesfrom asumover theLynlev- correspondingtoa14σfluctuationinthedensityfield.Obviously,
α
els,withthemaximumndeterminedbythedistance.Thesumis individual sources donot inducestrongLymancoupling onlarge
c 0000RAS,MNRAS000,000–000
(cid:13)
6 Pritchardand Furlanetto
is just becoming important, around the time of the first galaxies
(BL05). Those authors showed that Lyn transitions enhance the
small-scalefluctuationsinT ,buttheyassumedthatf = 1.
b recycle
Wewillshowhowthescale-dependentf modifythissignal.
recycle
It is possible to exploit the separation of powers to probe sepa-
rately fluctuations that correlate with the density field and those,
likePoissonfluctuations,thatdonot.Weconsidereachinturnand
comparetotheresultsofBL05.Wesetδ = 0throughout.We
xHI
also assume that the IGM cools adiabatically, with no heat input
fromX-rays.NotethatforeaseofcomparisonwithBL05,wedo
not incorporate the low-temperature corrections of Hirata (2005)
(andinanycasetheyaresmallinourexample).
6.1 Densityfluctuations
Density perturbations source x fluctuations via three effects
α
(BL05). First, the number of galaxies traces, but is biased with
respect to, the underlying density field. As a result an overdense
regionwillcontainafactor[1+b(z)δ]moresources,whereb(z)
isthe(mass-averaged)bias,andwillhavealargerx .Next,pho-
α
tontrajectoriesnearanoverdenseregionaremodifiedbygravita-
Figure4.FluxprofileofagalaxyofmassM =3 1010M⊙atz=20as tionallensing,increasingtheeffectiveareabyafactor(1+2δ/3).
×
afunctionofcomovingdistancer.Forcomparison,wehaveplottedtheflux Finally, peculiar velocities associated withgas flowing into over-
profileassumingfrecycle=1(dashedcurve),properatomicphysics(solid dense regions establish an anisotropic redshift distortion, which
curve)andincludingonlyphotonswithνα<ν <νβ(dottedcurve).Ver- modifiesthewidthoftheregioncorrespondingtoagivenobserved
ticallinesalongtheloweraxisindicatethehorizonsfortheLynresonances.
frequency. These three effects may be represented using a linear
Thehorizontaldashedlineshowsthevalueofxcatz=20,illustratingthe
transferfunctionW(k)relatingfluctuationsinthecouplingδ to
regimewherecollisionalcouplingdominates. xα
theoverdensityδ
δ ≡W(k)δ. (17)
xα
scales. Theconversion of photons fromLyn to Lyαsteepens the
fluxprofilebeyondthesimple1/r2form.Noticethatwehavenor- WecomputeW(k)foragaselementbyaddingthecouplingdue
toLyαfluxfromeachoftheLynresonances(BL05)
malisedf∗ foreachcurveseparately, sothatxα = 1atz = 20.
Becausesettingf =1weightslargentransitionsmoreheav-
recycle 1 nmax zmax(n) dx(n)D(z′)
ily(andhencesmallscales),thatcurveliesbelowtheothersatlarge W(k)= dz′ α
r.ThediscontinuitiesoccurattheLynhorizons.Intheory,theirpo- xα nX=2 Zz dz′ D(z)
sitionsyieldstandardrulersdeterminedbysimpleatomicphysics.
2
In practice, the weakness of the discontinuities, and the overlap- ×(cid:26)[1+b(z′)]j0(kr)− 3j2(kr)(cid:27), (18)
pingcontributionsofother nearbysources, makesitunlikelythat
thesediscontinuitieswillbeobservableforanisolatedsource(see whereD(z)isthelineargrowthfunctionandthejl(x)arespherical
also§6).Finally,wenotethatsharpdiscontinuitiesonlyoccurifa Besselfunctionsoforderl.Thefirstterminbracketsaccountsfor
photonundergoesacascadeimmediatelyafterenteringaLynres- galaxybiaswhilethesecond describes velocityeffects.Theratio
onance and if the resulting Lyα photons redshift out of the Lyα D(z′)/D(z)accountsforthegrowthofperturbationsbetweenz′
resonanceimmediately.Theformeriscertainlytrue,butthelatter andz.Thefactordxα/dzconvertsfromLyαfluxtothecoupling.
willaffecttheshapesignificantly(Loeb&Rybicki1999). Eachresonancecontributesadifferentialcoupling(seeeq.6)
Additionally, photon horizons affect the radiation heating of
dx(n) dJ(n)
thegasaround asource.Thecascades, whichresultfromscatter- α ∝ α , (19)
ingofLynphotons,depositsomeoftheirradiativeenergyintothe dz′ dz′
kineticenergyof thegas,and theirtotalheatingratediffersfrom withthedifferentialcomovingfluxinLyαfromequation(14).
“continuum” Lyα photons because they are injected as line pho- Because thiscorrelates with thedensity field, it iseasiest to
tons (Chen&Miralda-Escude´ 2004). Thus the Lyα heating pro- observevia
filewilldifferfrom1/r2.However,Lyαheatingistypicallymuch
x
s(Cmhaellner&thMainraoldtha-eErsscouudre´ce2s0,04so).this is unlikely to be important Pµ2(k)=2Pδ(k)(cid:20)β+ x˜tαotW(k)(cid:21). (20)
ThefirsttermprobesfluctuationsinTk andκ1−0 (allencoded in
β).WeshowPµ2 inFigure5,5contrastingcasesthatincludeonly
6 BRIGHTNESSFLUCTUATIONSFROMTHEFIRST
GALAXIES
Intheprevioussection,wesawthattheproperfrecycle affectsthe 5ofOBuLr0r5e,suwlthsofomraqdePaµn2earrroerainfacthtoeriroPfδ√(k2)π2no≈rma4l.i4sagtiroenate(Rr.thBanarkthaonsae,
spatialdistributionofxαaroundeachsource.Themostimportant privatecommunication). Notethatalloftheirdensity-induced fluctuation
manifestationofthiswilloccurwhentheWouthuysen-Fieldeffect amplitudesshouldincreasebyasimilaramount.
c 0000RAS,MNRAS000,000–000
(cid:13)
Lyman seriescascades andthe21cm line 7
Figure5.Toppanel: Pµ2 powerspectrum, whichbest illustrates the Tb Figure 6. Ratio of W(k) calculated using proper atomic physics to
fluctuationsarisingfromdensity-sourcedxα.Frombottomtotop,thecases W(k)withfrecycle=1.Fromlefttoright,verticallinesindicatethescales
include only photons fromLyαto Lyβ,from LyαtoLyδ, andforn ≤ associatedwiththeLyαandLyβresonancesandwithrHII.W(k)displays
23. Dashedlines indicate frecycle = 1while solid lines usetheproper smallamplituderipples,whicharisefromintegratinganoscillatingkernel
conversionfactors.Thedottedlineis2βPδ.Verticallinesindicatethescales (thesphericalBesselfunctionsinequation18)overfiniteextent(theLyn
correspondingtotheLyαhorizonrαandtheHIIregionsizerHII.Bottom horizons).Changingfrecyclemodifiesthephaseoftheseripplesleadingto
panel:TransferfunctionW(k). thewigglesseeninWrecycle(k)/W(k)onintermediatescales.
photons with να < ν < νβ, να < ν < νδ, and the entire Ly- wherethebnandanaredefinedbyreferencetoequation(18)and
man continuum. For the latter two models, we show results with the integral of equation (19) respectively. The former care only
f = 1andwiththeproperatomicphysics.Notethateachis aboutthelocalflux,butthelatterareaveragedovertheentireLy-
recycle
separatelynormalisedtoxα = 1.Thedottedcurveisolates2Pδβ, mancontinuum.Iffrecycle = constant,theycanceloutofW(k)
whichclearlydominatesonsmallscales.Notethatwehaveapplied andarerelevantonlyasanoverallnormalisationofx .Inactuality,
α
twocutoffstothepowerspectruminthisregime(BL05).Thefirstis f isafunctionofn,reducingthepower.Westressthatthisis
recycle
duetobaryonicpressure,whichpreventscollapseonsmallscales. becausethef (n)areessentiallyfrequencydependentandso
recycle
Thesecondisthethermalwidthofthe21cmline.Naoz&Barkana distortthefluxprofileaboutanyisolatedgalaxy(seeFig.4).
(2005)showedthatthisthermalcutoffdisplaysacharacteristican- It might be hoped that the discontinuities in Figure4 would
gulardependence,which,theoretically,allowsfluctuationsandthe leaveaclearfeatureonthepowerspectrum,especiallyoneassoci-
cutoff to be separated. To allow easy comparison withBL05, we atedwiththelossofallphotonsenteringtheLyβresonance.Sucha
donotincludethisrefinement.Additionally,weexpectpowerfrom feature,whoseangularscalewouldbedeterminedbysimpleatomic
the HII regions surrounding the sources to become important on physics,couldsetastandardrulerthatcouldbeusedtotestvaria-
scalessmallerthanthesizeofatypicalHIIregionr .Wehave tionsinfundamentalconstantsortomeasurecosmologicalparam-
HII
markedthisscaleforanisolatedgalaxyinFigure5,butnotethat eters.Sadly,ascanbeseenfromFigure5,thereisnotrulydistinct
Furlanettoetal.(2004)predictthattheHIIregionscouldbeafac- feature.Still,thepowerdoesdeclinearoundk andmeasuringits
α
torofafewlargerattheseearlytimes.Onsufficentlylargescales shapecanconstraintheangulardiameterdistance:wefindthatthe
kr ≈ 0,thesecondterminequation(20)dominates,andW(k)is amplitudeofPµ2 changesbyafewpercentiftheangulardiameter
fixedbythesourcebias. distancechanges by thesameamount. However, suchconstraints
Figure5clearlyshowsthattheLynresonancesareimportant wouldalsorequiretheastrophysicalparameterstobeknownpre-
onintermediatescales.Onlargescalesonlytheaveragefluxmat- cisely,whichwillbedifficult.
ters,butaswemovetosmallerscalesthehigher-nlevelsbecome
important.Figure6showsthatthefractionalreductioninW(k)on
smallscalesis∼ 0.63. Althoughthisisnear f¯ for Pop.III 6.2 Poissonfluctuations
recycle
stars,thatisnottheoriginofthisscaling.Inequation(18),wecan Weturnnowtobrightnessfluctuationsuncorrelatedwiththeunder-
write lyingdensityperturbations,whichcanbeextractedfromthepower
spectrumbecauseoftheredshiftspacedistortions(BL05).Specifi-
x = a f (n), (21)
α n recycle
cally,ifthenumberdensityofgalaxiesissmall,thenPoissonfluctu-
Xn
ationscanbesignificant.Tocalculatethecorrelationfunctionfrom
and
Poissonfluctuations,weagainfollowBL05andconsidertheLyα
1 flux from sources within a volume element dV at two points A
W(k)= b f (n), (22)
xα Xn n recycle andBseparatedbyacomovingdistancel.Thecorrelationfunction
c 0000RAS,MNRAS000,000–000
(cid:13)
8 Pritchardand Furlanetto
Figure 7. Top panel: Correlation function for the Poisson fluctuations. Figure8.Powerspectrum forthePoissonfluctuations. Lineconventions
Line conventions are the same as in Figure 5. Vertical lines show 2rα, arethesameasinFigure5.Verticallinesindicatethescalescorresponding
2rβ,and2rHII.Bottompanel:Ratioofthecorrelation functions assum- to2rαand2rHII.
ingfrecycle(n)andfrecycle=1.
pointssoξ = 0.Includingf (n)reducesξ ,especiallyon
P recycle P
takestheform thesmallestscales,becauseitdecreasestheefficiencyofcoupling
2 dn(z′ ) P(z′ )P(z′ )F (z′ ) fromleveln.ThebottompanelofFigure7showsthatthesuppres-
ξ (l)= dV A dMM2 A B gal B ,
P x2α ZV ZM dM rA2 rB2 Fgal(zA′ ) siononsmallscalesis(0.63)2 ≈0.40,becauseξP dependsontwo
(23) powersoftheflux.Notethat,onlargescales,ξ increaseswiththe
P
wherezB′ =z′(rB)istheredshiftofahaloatacomovingdistance proper frecycle.Thisresults fromtheway wehave normalised to
rBfromagaselementatredshiftz.Inthisexpression,weintegrate xα=1,whichreducesthefluxofagivensourceonlargescalesas
overahalfvolumesuchthatrA<rB,withthefactorof2account- frecycle increases(seeFigure4).AswithW(k),thescaledepen-
ingforthecontributionofsourcesthatarenearertoB.Thefactors dence is weak except on large scales, where rapid changes occur
P(z′)servetonormalisethefluxfromdV suchthat attheappropriatehorizons.Propertreatmentoftherecyclingfrac-
dxα≡P(z′)r12 Z MdndM(z′)dMdV, (24) tionsTishecsleeafrelyatnuerecseshsaavryetsoimuinldaerresftfaencdtsthoensthhaeppeoowfeξrPs.pectrumof
M fluctuationsuncorrelatedwiththedensityfluctuations
which makes explicit the expected 1/r2 dependence of the flux.
BweitchaiunsedVofathtedififnfeitreenstpesteadgoesfliinghtht,epiroienvtoslAutiaonnd. FBosleloewthinegsoBuLrc0e5s, Pun−δ(k)≡Pµ0 − 4PPµ2µ24 =(cid:18)x˜xtαot(cid:19)2PP(k), (25)
weaccountforthiswiththelastfactor,whichscalesthesourceflux
asshowninFigure8.Onlargescales,takingf = 1slightly
bythefractionofmassthathascollapsedattheobservedredshift. recycle
amplifiesthepower.Onsmallscales,theysignificantlyreducethe
Thisignores apossibledependence of theformationrateonhalo
power by ≈ 60%. They also affect the shape of the power spec-
mass, but inpracticehigh redshift galaxies arehighly biased and
trum,especiallyneark = π/r ,wherethelackofLyβ → Lyα
occurwithinasmallmassrangejustabovetheminimumcooling α
imprintsakneeonthepowerspectrum.Wealsonotethatthesharp
mass, so this dependence will be weak. Equation (23) is easy to
Lynhorizonsimprintweakoscillationsonthepowerspectrum,es-
understand.ForPoissonstatistics,thevarianceoffluxfromasetof
identical galaxieswouldbe∝ m2 n V;thismust thensimply peciallyiffrecycle = 1(thoughthesewilllikelybesmoothed by
gal gal photondiffusion).
beweightedbythefluxreachingeachofthetwopoints.Notealso
that,contrarytotheclaimsofBL05,ξ ∝ 1/f ,wheref isthe
P d d
duty cycle of each galaxy, because the fluctuations are weighted
6.3 NonlinearitiesintheWouthuysen-Fieldcoupling
bytwopowersof luminosity[MP(z)]butonlyonefactor ofthe
density. To this point, we have used equation (7) to compute the bright-
ThetoppanelofFigure7showshowthecorrelationfunction ness temperature fluctuations. This assumes that all the underly-
increasestowardsmall scales.Thisisaresultof the1/r2 depen- ingperturbationsarelinear; obviously, atx = 1, thismayonly
α
dence of the flux, which weights the correlations tosmall scales. be marginally satisfied. When the radiation background is large,
Including the Lyn resonances amplifies this, because the horizon the brightness temperature becomes insensitive to the coupling
scalesskewthefluxprofiletosmallradii(seeFigure4).Onlarge strengthandP willbesmallerthanourestimate.Whenwillsuch
Tb
scalesthecorrelationfunctiondecreasesasthetwopointsAandB correctionsbecomeimportant?Oneobvioustestiswhetherthetyp-
sharefewersources.Forr>2r asinglesourcecannotaffectboth icalfluctuationT iscomparabletothemaximumbrightnesstem-
α b
c 0000RAS,MNRAS000,000–000
(cid:13)
Lyman seriescascades andthe21cm line 9
peraturedecrementbetweencoupledanduncoupledgas,δT (i.e.,if Nextweconsidered theincreased Lyαfluxthat resultsfrom
T =T ineq.1).Butnonlinearitiesmaybeimportantevenifthis atomiccascades.Followingtheselectionrulesandtransitionrates,
s k
conditionisnotsatisfied.Auniversewithdiscretestronglycoupled wecalculatedthe probabilitythat aLynphoton isconverted into
regionsseparatedbyuncoupled IGMcouldhavesmallrmsvaria- Lyαandshowedthatf → 0.36asnincreases.Thisissig-
recycle
tions,eventhoughnonlinearitiesareextremelyimportantinfixing nificantlysmallerthanthevaluef =1usuallyassumed[e.g.,
recycle
thebrightnesstemperatureofthestrongly-coupledregions. byBL05].ForatypicalPop.IIIsourcespectrum,weshowedthat
Insteadwemustlookdeeperatthenatureofthefluctuations. only63% of theemittedphotons willbeconverted intoLyα,de-
First,notefromFigure4thatindividual galaxiesmostlikelypro- laying the onset of coupling (x = 1) for a given set of source
α
vide only weak coupling: x ≪ 1 except near tothe sources, at parameters.
α
least if small galaxies (near the atomic cooling threshold) are re- Incorporating thecorrectf modifiesthefluxprofileof
recycle
sponsibleformostoftheradiationbackground.Thisisnotsurpris- anindividual source andreduces thecoupling on smallscales by
ing: because of Olber’s paradox, each logarithmic radius interval aboutafactorof3forfixedsourceparameters.Inaddition,thecas-
contributesequallytothebackgroundfluxinahomogeneousuni- cade process imprints discontinuities onto the flux profile. Using
verse. The higher Lyman-series photons, together with the finite thecorrectatomicphysicsreducestheamplitudeofthesedisconti-
speedoflight,alsodonotdramaticallyincreasetheweightingon nuitiesandremovesoneduetotheLyβresonance.Unfortunately,
nearby radii. Thus we expect a substantial fraction of the flux to theirweaknessislikelytofrustrateattemptstousethesedisconti-
come from large distances, where density fluctuations are weak. nuitiesasastandardruler.
Thisimmediatelysuggeststhatthedensity-dependentpowerspec- WethenrecalculatedthepowerspectraofBL05,incorporating
trumdescribedinSection6.1willnotrequiresubstantialnonlinear thecorrectf .Thisshowedareductioninpower of∼ 37%
recycle
corrections. (Figure5)onintermediatescalesfordensitycorrelatedfluctuations
Morequantitatively, thefluctuationsbecome nonlinear when andof∼ 64%(Figure8)onsmallscalesforfluctuationsuncorre-
δ =W(k)δ(k)&1;thuswerequire latedwiththedensity.Itispossibletomimicthislossofpowerby
xα
changingtheshapeofthestellarspectrum. Onsmallscales,are-
W(k)&1/[σ(R)D(z)], (26)
ductioninthestarformationratewillproduceasimilarreductionin
whereσ(R)isthetypical density fluctuationonscaleR ∼ 1/k. flux.Incorporatingtheproperfrecycle(n)isthuscrucialtocorrectly
Figure 5b shows, however, that W(k) is of order unity only for interpreting21cmobservationsnearthetimeoffirstlight.Onthe
k . 0.1 Mpc−1, where the density fluctuations are themselves otherhand,theeffectsthatwehavedescribedbecomeunimportant
tiny at these redshifts. Thus, we conclude that a linear treatment onceLyαcouplingsaturatessothatTs →Tk.Inthisregime,fluc-
isadequate for computing the Pµ2 power spectrum, because it is tuationsintheLyαfluxhavelittleeffectonthepowerspectrumand
primarilydrivenbylargescalefluctuations. perturbationsinthedensityandreionizationfractiondominate.
ThePoissonfluctuationsinSection6.2aremoreproblematic. Severalexperiments,includingLOFAR,MWA,PAST,andthe
Bydefinition, future SKA6, are aiming to detect 21cm fluctuations of the type
wehavediscussedhere. Hopefully, theywillbeabletostudythe
ξP(rA−rB)=hxα(rA)xα(rB)i/x2α−1. (27) firstsourcesoflightthroughtheireffectontheIGMaroundthem.
Thus, on scales at which ξ & 1, the radiation background near ThedetailsofLyαcouplingwilldeterminetheobservabilityofthis
P
epoch.
galaxy overdensities on this scale is considerably larger than its
WethankM.Kamionkowskiforhelpfuldiscussionsandalso
averagevalue,indicatingthatnonlineareffectsareimportant.Inthe
ouranonymousrefereeforseveralhelpfulcommentsduringthere-
particularmodelwehaveexamined,ξ islargeonlyoncomoving
P
visionprocess.ThisworkwassupportedinpartbyDoEDE-FG03-
scales.10kpc,sononlineareffectsareagainnegligible.However,
920-ER40701.
the amplitude of thePoisson fluctuations increases rapidly asthe
source density decreases: in models with fewer sources at x =
α
1, or which strongly weight massive galaxies, nonlinearities may
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APPENDIXA: CALCULATINGEINSTEINA
COEFFICIENTSINTHEHYDROGENATOM
Thesimplicityofthehydrogenatompermitsustocomputematrix
elementsforradiativetransitionsanalytically.Weareinterestedin
the Einstein A coefficients, which may be written as (Sobelman
1972)
A(n,l,j,n′,l′,j′)
= 64π2e2(2j′+1) l j 1/2 2l (Rn′,l′)2. (A1)
3hλ3 (cid:26) j′ l′ 1 (cid:27) > n,l
Here{...}denotestheWigner6−jsymbolandweassumespin-
′ ′
1/2particles.Inthisexpression,thematrixelementRn,l hasthe
n,l
usualquantumnumbersn,l,andn withn−l−1≡n .Weusethe
r r
sets(n,n ,l)and(n′,n′,l′)todescribetheupperandlowerlevels
r r
inthespontaneoustransition.Inaddition,weletl bethegreater
>
of l and l′. Because they are separated by ∆l = l′ −l = ±1,
2l =l+l′+1.
>
Wenextdefine,forn′−l ≥1,
>
r=n′−l −1≥0. (A2)
>
Wealsolet
u=(n−n′)/(n+n′); (A3)
v=1−1/u2 =−4nn′/(n−n′)2; (A4)
w=v/(v−1)=4nn′/(n+n′)2. (A5)
′ ′
Thus we only need Rn,l ≡ a R, where a is the Bohr radius.
n,l 0 0
Thisisgivenby(Rudnick1935)
R=2l+l′+4nl′+3n′l+3(n−n′)n−n′−1
(n+n′)n+n′+1
(n+l)! 1/2
×(cid:20)(n′+l′)!(n−l−1)!(n′−l′−1)!(cid:21) P±. (A6)
7 SeereferencedataattheNationalInstituteofStandardsandTechnology,
HereP− andP+ (thesubscriptscorrespondingtothesignof∆l) http://physics.nist.gov/.
c 0000RAS,MNRAS000,000–000
(cid:13)