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 be important. 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ZaldarriagaM.,FurlanettoS.R.,HernquistL.,2004,ApJ,608,622 ZhengW.,KrissG.A.,TelferR.C.,GrimesJ.P.,DavidsenA.F.,1997, ApJ,475,469 ZygelmanB.,2005,ApJ,622,1356 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)