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Effects of Helium Recombination in the Early Universe

Permanent Link: http://ufdc.ufl.edu/UFE0042799/00001

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Title: Effects of Helium Recombination in the Early Universe
Physical Description: 1 online resource (120 p.)
Language: english
Creator: HILL,ANDREW L
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: COSMICS -- COSMOLOGY -- DENSITY -- EARLY -- FORMATION -- GADGET -- HELIUM -- HYDROGEN -- LARGE -- PERTURBATIONS -- RECOMBINATION -- SCALE -- STRUCTURE -- UNIVERSE
Physics -- Dissertations, Academic -- UF
Genre: Physics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: This study analyzes the recombination of hydrogen and helium in the early universe. Specifically, I examined whether looking at hydrogen and helium as two different entities causes any noticeable changes in the formation of structures in the early universe. This distinction has been for the most part neglected, and yet it is an interesting area of study. By tracking these two types of baryons independently, it is possible to see some differences in the resulting structure formation. Since helium recombines before hydrogen, it speeds up faster than hydrogen. This causes it to lead hydrogen, as well as cluster together sooner than hydrogen. These are the effects I study in this paper.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by ANDREW L HILL.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Fry, James N.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2011-10-31

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Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0042799:00001

Permanent Link: http://ufdc.ufl.edu/UFE0042799/00001

Material Information

Title: Effects of Helium Recombination in the Early Universe
Physical Description: 1 online resource (120 p.)
Language: english
Creator: HILL,ANDREW L
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: COSMICS -- COSMOLOGY -- DENSITY -- EARLY -- FORMATION -- GADGET -- HELIUM -- HYDROGEN -- LARGE -- PERTURBATIONS -- RECOMBINATION -- SCALE -- STRUCTURE -- UNIVERSE
Physics -- Dissertations, Academic -- UF
Genre: Physics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: This study analyzes the recombination of hydrogen and helium in the early universe. Specifically, I examined whether looking at hydrogen and helium as two different entities causes any noticeable changes in the formation of structures in the early universe. This distinction has been for the most part neglected, and yet it is an interesting area of study. By tracking these two types of baryons independently, it is possible to see some differences in the resulting structure formation. Since helium recombines before hydrogen, it speeds up faster than hydrogen. This causes it to lead hydrogen, as well as cluster together sooner than hydrogen. These are the effects I study in this paper.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by ANDREW L HILL.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Fry, James N.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2011-10-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0042799:00001


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EFFECTSOFHELIUMRECOMBINATIONINTHE EARLYUNIVERSE By ANDREWL.HILL ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2011

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c r 2011AndrewL.Hill 2

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ThisisdedicatedtoeveryonewhohelpedmegettowhereIamto day–myfriends,my family,mybrother,mymother,andmyself.Alsotomyfather, whowouldhavelovedthe chancetoseethis.Thankseveryone! 3

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ACKNOWLEDGMENTS ThankstoDr.Fryforgivingmethehelpandassistancetogett hisfar.Icouldn't havenishedthiswithoutthemanyhelpfulconversationson physicsandtheuniverse, oralloftheusefulsuggestionsandadvice.I'dalsoliketot hankEdmundBertschinger forcreating COSMICS andVolkerSpringelforcreating GADGET-2 .Bothoftheseprograms wereinvaluabletools,andareimmenselyuseful. I'dalsoliketothankallmyfriendsandfamily,mygrandpare nts,andespeciallymy momanddad,whogavemetheirsupportandencouragementthro ughouttheyears.I couldn'thavedoneitwithoutallofyou.Thanksalot! 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ..................................4 LISTOFTABLES ......................................7 LISTOFFIGURES .....................................8 ABSTRACT .........................................10 CHAPTER 1INTRODUCTIONTORECOMBINATIONANDTHEEARLYUNIVERSE ....11 2BASICSOFRECOMBINATION ...........................19 2.1UsingtheSahaEquationtoCalculatetheRecombinationT ime ......19 2.2EvolutionoftheFractionalIonizationofHydrogen ..............22 3ROUGHCALCULATIONS ..............................30 3.1TheScaleFactor ................................30 3.2RoughAnalyticalCalculationofRecombination ...............35 3.3RoughCalculationofRecombinationStartingFrom COSMICS Code ....41 4COSMICS .......................................52 4.1ModellingtheEvolutionofDensityPerturbations ..............52 4.2AdditionofHelium ...............................57 5INTRODUCTIONTOGADGET-2 ..........................60 5.1SettingUpARun ................................61 5.2Running GADGET-2 ...............................67 5.3Output ......................................67 6ANALYSISOFHELIUMRECOMBINATIONUSINGGADGET-2 .........70 6.1CreatinganAppropriateSetofInitialConditions ..............70 6.2EvolutionStartingataRedshiftof 1250 ....................73 6.3EvolutionStartingataRedshiftof 1000 ....................89 6.4EvolutionStartingataRedshiftof 1250 WithaDifferentHeliumMass ...97 7CONCLUDINGREMARKS .............................110 7.1 COSMICS .....................................110 7.2 GADGET ......................................110 APPENDIX:SETTINGINITIALCONDITIONSFORGADGET-2 ............113 5

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REFERENCES .......................................117 BIOGRAPHICALSKETCH ................................120 6

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LISTOFTABLES Table page 6-1Dataconveyingthehydrogenandheliumcompositionof”C lump1.” ......78 6-2Dataconveyingthehydrogenandheliumcompositionof”C lump2.” ......79 6-3Dataconveyingthehydrogenandheliumcompositionof”C lump3.” ......80 6-4Dataconveyingthehydrogenandheliumcompositionof”C lump4.” ......81 6-5Dataconveyingthehydrogenandheliumcompositionof”C lump1.” ......93 6-6Dataconveyingthehydrogenandheliumcompositionof”C lump2.” ......94 6-7Dataconveyingthehydrogenandheliumcompositionof”C lump1.” ......102 6-8Dataconveyingthehydrogenandheliumcompositionof”C lump2.” ......103 6-9Dataconveyingthehydrogenandheliumcompositionof”C lump3.” ......104 7

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LISTOFFIGURES Figure page 2-1Thefractionalionizationofhydrogen(Sahaequation). ..............21 2-2Thefractionalionizationofhydrogen(caseI). ...................24 2-3Thefractionalionizationofhydrogen(caseII). ...................25 2-4Thefractionalionizationofhydrogen(caseIII). ..................27 2-5Thefractionalionizationofhydrogen(caseIV). ..................29 3-1Thelineshowstheevolutionofthescalefactorwithtime ............31 3-2Thelineshowstheevolutionofthescalefactorwithtime onalog-logplot. ...32 3-3EvolutionoftheHubbleparameterwithscalefactoronal og-logplot. ......34 3-4Theevolutionofcolddarkmatterperturbations(caseI) .............39 3-5Theevolutionofcolddarkmatterperturbations(caseII ). .............40 3-6Thecolddarkmatterperturbationsat k=0.01 ..................44 3-7Thechangeincolddarkmatterperturbationsat k=0.01 ............45 3-8Thecolddarkmatterperturbationsat k=0.1 ...................46 3-9Thechangeincolddarkmatterperturbationsat k=0.1 .............47 3-10Thecolddarkmatterperturbationsat k=1 ....................48 3-11Thechangeincolddarkmatterperturbationsat k=1 ..............49 3-12Thecolddarkmatterperturbationsat k=10 ...................50 3-13Thechangeincolddarkmatterperturbationsat k=10 .............51 4-1Thedensityperturbationsofbaryonsandcolddarkmatte r. ...........55 4-2Thedensityperturbationsduringrecombination. .................56 4-3Thedensityperturbationsofhelium. ........................58 5-1Theinitialconditionsforarunof GADGET ......................66 5-2Thedistributionofmatterat z=0 .........................69 6-1Aviewoftheuniverseat z=10 (runI). .......................75 6-2Aviewoftheuniverseat z=91 (runI). .......................76 8

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6-3Themotionofaclusterinthreedimensions(runI). ................83 6-4Athreedimensionalplotofclustermotion(runI). .................84 6-5Theseparationofthecentroids(runI). .......................85 6-6Thestandarddeviationsoftheclusters(runI). ..................86 6-7Theratiosofheliumtohydrogen(runI). ......................88 6-8Aviewoftheuniverseat z=50 (runII). ......................90 6-9Aviewoftheuniverseat z=166 (runII). ......................91 6-10Themotionofaclusterinthreedimensions(runII). ...............95 6-11Theseparationofthecentroids(runII). ......................96 6-12Thestandarddeviationsoftheclusters(runII). ..................97 6-13Theratiosofheliumtohydrogen(runII). ......................98 6-14Aviewoftheuniverseat z=24 (runIII). ......................100 6-15Aviewoftheuniverseat z=101 (runIII). .....................101 6-16Themotionofaclusterinthreedimensions(runIII). ...............105 6-17Theseparationofthecentroids(runIII). ......................106 6-18Thestandarddeviationsoftheclusters(runIII). ..................107 6-19Theratiosofheliumtohydrogen(runIII). .....................109 9

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy EFFECTSOFHELIUMRECOMBINATIONINTHE EARLYUNIVERSE By AndrewL.Hill May2011 Chair:JamesN.FryMajor:Physics Thisstudyanalyzestherecombinationofhydrogenandheliu mintheearlyuniverse. Specically,Iexaminedwhetherlookingathydrogenandhel iumastwodifferententities causesanynoticeablechangesintheformationofstructure sintheearlyuniverse. Thisdistinctionhasbeenforthemostpartneglected,andye titisaninterestingarea ofstudy.Bytrackingthesetwotypesofbaryonsindependent ly,itispossibletosee somedifferencesintheresultingstructureformation.Sin ceheliumrecombinesbefore hydrogen,itspeedsupfasterthanhydrogen.Thiscausesitt oleadhydrogen,aswellas clustertogethersoonerthanhydrogen.Thesearetheeffect sIstudyinthispaper. 10

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CHAPTER1 INTRODUCTIONTORECOMBINATIONANDTHEEARLYUNIVERSE Recombinationisanimportanteraofstudyintheevolutiono ftheuniverse,and accordingly,alargebodyofworkisdedicatedtoit.Fromtre atmentsintextbooksto papersgoingintodetailonspecicaspectsoftherecombina tionprocess,thereisa lotofinformationtosiftandsortthroughtondwhateveron emightneedtoknow. Muchofthephysicsthatgovernevolutionoftheuniversedur ingthisepochisknown, butthemorecomplicatedonemakesthemodelbeingstudied,t hemoreintractable, difcult,andtimeconsumingtheproblembecomes.Becausem anyoftheintricacies andinteractionsduringtherecombinationeracanbeignore dorneglectedwhilestill givingusefulresults,thesecomplicationsareoftenlefto utofrecombinationmodels. Oneoftheearlieststudiesofrecombinationwaspresentedb yPeebles( Peebles 1968 ).Thistreatmentwasanexcellentbeginningtothesubject, andtodayrecombination ispartofallcosmologicalmodels.Thesebasicsareinclude dinmanytextbooksonthe subject,andthetheoryisknownbyeveryonestudyingcosmol ogy( Liddle&Lyth 2000 ; Peebles 1993 ; Bertschinger 2001 ; Carroll&Ostlie 1996 ; Dodelson 2003 ). Therearemanyrenementstothemostbasicmodel,someofwhi charemorewell understoodthanothers.Currentcalculationsincludingal lofthemostrelevantaspects ofrecombinationgivegoodresults,butmoreworkstillneed stobedoneonhowthe differentrecombinationtimesofhydrogenandheliumaffec tstructureformationandthe evolutionoftheuniverse( Wongetal. 2008 ).Workhasalreadybeendoneonaspectsof thissubject,butthereisstillmoretodo( Switzer&Hirata 2008a ; Hirata&Switzer 2008 ; Switzer&Hirata 2008b ).Thiswillbethesubjectofmystudy–theevolutionofstruc ture formationwhentheearlierrecombinationtimeofheliumist akenintoaccount. Toreallyunderstandrecombination,andhelium'sroleinth eprocess,itmustbe placedintothelargercontextoftheBigBang.Thegoalofthi sstudyisnottogivean exhaustiveaccountofthecreationoftheuniverse,asentir ebookscouldbewritten 11

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onthissubject–andtheyhavebeen.Foranin-depthtreatmen toftheBigBang,orof anyonepartofit,thereadermayrefertotheexcellentmater ialthatalreadyexists ( Hawking&Ellis 1975 ; Kolb&Turner 1990 ; Padmanabhan 1993 ; Liddle&Lyth 2000 ; Carroll&Ostlie 1996 ).Thebeginningsofstructureformationhavetheirseeds intheveryearlyuniverse,duringtheinationaryepoch.In ationoccurredavery shorttimeaftertheuniversebegan–fromaround 10 42 secondstosomewherearound 10 32 seconds.Introducedin1981( Guth 1981 ),themodelhasgonethroughmany revisionstogettoitscurrentstate.Itiscurrentlythebes twaytoexplainwherethe densityperturbationsthatgiverisetostructurecomefrom .Thesearisefromvacuum uctuationsgeneratednaturallyfromtheinationprocess ,andareGaussianinnature ( Guth 1997 ; Mukhanov 2005 ; Liddle&Lyth 2000 ; Durrer 2005 ).Withoutthem,the universewouldbecompletelyfeatureless,homogeneousand isotropic(inotherwords, itwouldlookexactlythesameineverydirection,withnovar iationofanykind).Of course,today,onlargeenoughscales,theuniverseishomog eneousandisotropic,butit denitelyhasfeatures.Inationmodelstellusthatthesef eaturesstartoutasverysmall inhomogeneitiesinthebackgroundoftheuniverse,created fromquantumuctuationsof theinatoneldarounditsvacuumstate–asmentioned,thes euctuationsareGaussian andadiabatic,whichtsthemodelsfornecessarycondition sforlargescalestructure formationperfectly( Peebles 1980 ; Peacock 1999 ; Liddle&Lyth 2000 ). Afterthisinitialinationaryperiod,theuniversewillun dergoaperiodofreheating, whenthenonrelativisticparticlesleftoverafterinatio nwilldecayintorelativistic particlesand,intheprocess,createaradiation-dominate duniverse.Duringthisera,the scalefactorwilldependontimeinthefollowingway: a / t 1 = 2 (1–1) andthedensitywilldependonthescalefactoras: R / a 4 (1–2) 12

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Theseparameterswilldeterminetheevolutionoftheuniver se–andthedensity perturbationswhichareofinteresttous–untilaround 10 4 years,whentheradiationand matterstarttoequalize.Duringradiationdomination,nuc leosynthesisoccurs–another importantstepontheroadtostructureformation.Itisduri ngthisprocess–whichoccurs whentheuniverseisonlyabout 100 secondsold–thatallthebaryonicmatterwhichwill onedaycreatethestructuresthatwearefamiliarwithisrs tcreated.Thehydrogen, helium-4,andotherlightelementscreatedduringthisproc essarenothingmorethan ionsatthispoint–becauseofthehighbindingenergyofnucl ei,theseatomscanform evenattheveryhot( 0.1 MeV)temperaturesofthisprimordialsoup,butelectronswo uld beinstantlystrippedaway( Liddle&Lyth 2000 ; Peacock 1999 ; Dodelson 2003 ). Asthetimepasses,theuniverseexpandsandcools.Itisthis processthatendsthe radiation-dominatederaandallowsthetransitionintothe matter-dominatedera,during whichthedependenceofthescalefactorontimechangesto: a / t 2 = 3 (1–3) andthedensity'sdependenceonscalefactorchangesto: M / a 3 (1–4) Itisduringthisera,whenthetemperaturecoolstoaround 3000 Kandaround 10 5 yearshavepassedsincetheuniversebegan,thatrecombinat ionoccurs.Withthe averagetemperatureoftheuniversecooledtothisdegree,n eutralatomscannally startforming.Atthispoint,thephotonsnolongerhaveenou ghenergytokeepthe electronsfrombindingtothepreviouslyionizedelements, andslowlyatrst,thenfaster astheuniversecoolsmoreandmore,theelectronsallgetsna tchedupbyatomsthat cannallybindelectronstothemselves.Untilthispoint,i tisonlythecolddarkmatter perturbationsthatcangrow;sincetheyonlyinteractthrou ghgravitationalforces,the radiationbackgrounddoesnotaffectthem.Ontheotherhand ,whiletheradiationis 13

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dominatingtheuniverse,thebaryonicmatterisnotfreetom ove–itisnotuntilafter thematterdecouplesfromthephotonsandformselectricall yneutralatomsthatit isabletomovefreelyundertheinuenceofgravity.Anditis notuntilthebaryonic matterisabletomovesolelyasgravitydictatesthatbaryon icmatterperturbations areabletogrow( Liddle&Lyth 2000 ; Peacock 1999 ; Peebles 1980 ; Kolb&Turner 1990 ; Padmanabhan 1993 ).Itisthegoalofthisstudytoexamineinmoredetailthe differencesthatarisebetweenthetwomodelswhen,rathert hanassumingthatall thebaryonicmatterishydrogen,weinsteadtreatthemoreac curatecaseofaround twenty-fourpercentofthebaryonicmatterformingintohel ium.Sincehelium-4hasa higherbindingenergythanhydrogen,itisabletorecombine andbecomeelectrically neutralatanearliertime.Thismeansthatitsdensitypertu rbationswillstartgrowing beforehydrogen's,leadingtopossiblyinterestingeffect sintheuniverseduringand afterrecombination( Peebles 1968 ; Seageretal. 2000 ; Switzer&Hirata 2008a ; Dubrovich&Grachev 2005 ; Wongetal. 2006 ). Theuniversecontinuestocoolastheperturbationscontinu etogrow.Finally, structurescanstartforming,creatingtherstobjects,an deventuallyleadingtothe universeweseetoday( Kolb&Turner 1990 ; Peebles 1980 ; Yoshidaetal. 2003 ; Gaoetal. 2007 ).Onlargescales,weseethattheuniverseisstillhomogene ousand isotropic,butonsmallscales,itismostdenitelyinhomog eneous.Thereasonforthis structureistheoncetinydensityperturbations.Astimepa sses,gravityactsonthem, pullingthemcloserandcloser,andmakingthemmoreandmore tightlybound.Asthe resultingstructuresbecomemoreandmoremassive,theywil lpullinmorematter,and eventuallystarsandgalaxiesarebornfromthoseoncehumbl eperturbations.Thisis whywemustunderstandheliumrecombination–weneedtoknow whateffectithason theevolvinguniverse,andthegrowingstructurethatitcon tains. Anunderstandingofthelargescalestructureoftheunivers eisthereforequite important.Thedistributionofstars,galaxies,voids,mat ter,energy,thecosmic 14

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microwaveradiationbackground,andthelike,areallimpor tant,andtounderstand themingreaterdepth,statisticsandtechniquespertainin gtotheeld,suchasdifferent distributionfunctions,aswellascorrelationfunctions, fractals,andmeasuretheory, areimportanttoonedegreeoranother( Hartle 2003 ).Tostartwith,letusintroducean equationforthedensityofobjects: n g (r)=n(1+ (r)). (1–5) Here, n isthemeandensityofaspeciessuchasbaryonsorradiation( whateveris representedby”g”),and (r) isaperturbationbasedonposition.Thisequationwill beveryusefulwhendealingwithrecombinationandstructur eformation.Itisonlythe deviationsfromthebackgroundhomogeneousdensitythatgi verisetostructure–both voidsandclustersofmatter–andthesearerepresentedbyth e (r) term. 1 From large-scalestructure,correlationfunctions,andscalin gequations,wemoveonto relativity,cosmology,andrelatedareas.Wecanstartwith theRobertson-Walkermetric ( Hartle 2003 ; Liddle&Lyth 2000 ): ds 2 = dt 2 +a 2 (t)dx 2 (1–6) Usingthismetric,itispossibletoobtaintheone-particle VaslovEquationbyusingthe LiouvilleEquationandBoltzmanDistribution( Hartle 2003 ; Liddle&Lyth 2000 ; Peebles 1993 ): @ f @ t + @ f @ x i dx i dt + @ f @ p i dp i dt =0. (1–7) 1 Amongothersources,myconversationswithDr.Frywerevery helpfulinthisarea. 15

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Now,ifwelet: dx i dt = p i ma 2 (1–8) dp i dt = m @ @ x i (1–9) wecanget: @ f @ t + p i ma 2 @ f @ x i m @ @ x i @ f @ p i =0. (1–10) Ifweintegratethisequationwith d 3 pp i andtakethesecondmoment,wecangetthe EulerEquation( Liddle&Lyth 2000 ; Peebles 1993 ): 1 a @ @ t (av i )+ 1 a (v k @ @ x k )v i + 1 a 1 +1 @ @ x k ((1+ ) ij )+ 1 a @ @ x i =0, (1–11) wherethethirdtermcontainsthepressureforce,andthefou rthtermisthegravitational force.Thecontinuityequation–averyimportantequationf orstudyingevolutionofthe universeandrecombination–is( Liddle&Lyth 2000 ; Peebles 1993 ): @ @ t + 1 a @ @ x k ((1+ )v k )=0. (1–12) Poisson'sequationforcomovingcoordinates,anothervery importantequationfor modelingrecombination,canbewrittenas( Liddle&Lyth 2000 ; Peebles 1993 ): 1 a 2 r 2 =4 G (1–13) Usingthesethreeequationsandmakingafewassumptions;na mely,that =0 and thattheperturbations,signiedby ,areverysmall, 1 (sothatalltheequationsare linear),itispossibletocombinethem.Withtheseassumpti ons,thecontinuityequation canberewrittenas(foramoredetailedtreatment,unnecess aryforourpurposesbut interestingnonetheless,see( Limaetal. 1997 )): @ @ t + 1 a r v=0. (1–14) 16

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Ifthisequationismultipliedby a 2 andthepartialderivativewithrespecttotimeistaken, itcanbeturnedinto: @ @ t (a 2 @ @ t )= @ @ t (a r v). (1–15) Withtheaboveassumptions,theEulerEquationbecomes: @ @ t (av)= r (1–16) Ifwetakethedivergenceofbothsides( r ),wecanturnthisequationinto: r ( @ @ t av)= r 2 (1–17) @ @ t (a ~ r ~ v)= r 2 (1–18) Thismakestherightsideofthecontinuityequation,Equati on( 1–15 ),equaltotheleft sideoftheEulerEquation,so: 1 a 2 @ @ t (a 2 @ @ t ) 1 a 2 r 2 =0. (1–19) And,combiningthiswiththelastequationwehaven'tusedye t,Equation( 1–13 ),we obtainanequationforthegrowthoflinearizeductuations : @ 2 @ t 2 + 2_a a @ @ t 4 G =0. (1–20) ThegrowthoflinearizeductuationsissomethingwhichIwa nttokeeptrackofwith respecttoheliumrecombination,andsoitmakessensetouse thisequationinsome simplecases.Itcanbedirectlysolvedinacoupleofcases,i ncludingonewithno curvatureandnodarkenergy,thecasewherethereisjustmat ter,andnopressure,and thecasewheretherecanbesomecurvature,butthereisnorad iation.Theseexercises helpedfamiliarizemewiththeideaoflinearizeductuatio ns,andhowtheycouldbe 17

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foundwiththehelpoftheFriedmannEquation( Tamvakis 2005 ): ( _a a ) 2 = 8 3 G k a 2 + 3 (1–21) 18

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CHAPTER2 BASICSOFRECOMBINATION 2.1UsingtheSahaEquationtoCalculatetheRecombinationT ime Inordertounderstandhowitwasabletoobtaindensitypertu rbations,itisimportant togetbetteracquaintedwiththephysicsembodiedin COSMICS .Thismeansdoingsome reviewofrecombinationeraphysics.Tostartwith,wecando aroughcalculationof whenrecombinationoccurred.Using( Liddle&Lyth 2000 ; Dodelson 2003 ): ( n c n d n a n b ) eq =exp E = T (2–1) wherethe n 'sarethenumberdensitiesofeachtypeofparticleatequili brium, E isthe changeinenergy(takentobetheionizationenergybelow,in Equation( 2–3 )),and T is thetemperature.Forthereaction: p + +e H+ r (2–2) weseethatEquation( 2–1 )gives: n H n r n p n e =exp I 0 = T (2–3) Keepinginmindthat: n b n r = =6 10 10 (2–4) where” b ”standsforbaryonsand” r ”standsforphotonsduringtheearlyuniverse,and thatwecansay: n p n b = n e n b =X, (2–5) (sothatthefractionofprotonsisthesameasthefractionof electrons),wecanshow that: 1 X X 2 =exp I 0 = T (2–6) 19

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Withthisequation,itispossibletocreateagraphshowingh owtheSahaEquation, withoutanyothermodicationsorrenements,predictsthe fractionalionizationwill evolve.First,asrecombinationisthefocushere, I 0 canbereplacedwith 0 ,the ionizationenergyforhydrogen.Thismeansthattheionizat ionfraction, X ,isactually theionizationfractionofhydrogen, X H .Further,asIwouldliketoseehowtheionization fractionofhydrogendependson 0 = T ,Iwillsubstitute x for 0 = T ,sotheequationIam graphingis: 1 X H X 2H =exp x (2–7) Solvingfor X H usingthequadraticformulagives: X H = 1 p 1+4 exp x 2 exp x (2–8) and,sincethefractionalionizationcanneverbenegative, theonlyphysicalsolutionis: X H = p 1+4 exp x 1 2 exp x (2–9) GraphingthisfunctiongivesFigure 2-1 ,whichshowshowthefractionalionizationof hydrogen, X H ,evolvesasafunctionof 0 = T .Comparingthisgraphwiththegraphs createdwithmorecomplicatedmodels,suchastheoneusedto getFigure 2-2 ,it seemsthatFigure 2-1 hasthesamegeneralshapeatearlytimes,althoughitdoes startdroppingalittlesooner.Theothermajordifferencei sthatitneverlevelsout–the graphjustkeepsgettingsmaller.Ofcourse,thatiswhymore complicatedmodelsare desirable–bytakingintoaccountmorevariables,behavior closertowhatshouldactually occurcanbeseen. Atthecrossoverregime,where I 0 and T arenearingeachother,Equation( 2–6 )can beusedtondoutapproximatelyhowmuchtimehaselapsedsin cethebeginningofthe universe.Ifwesaythat X=0.5 ,sothatabouthalfofthehydrogenhasrecombined,we 20

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Figure2-1.Thelineshowstheevolutionofthefractionalio nizationofhydrogenasa functionof x .Thefractionstartsdroppingabitbefore x=20 ,which correspondstoatemperatureofabout 0.68 eV,andkeepsdropping,getting smallerandsmaller. 21

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ndthat(since I 0 =13.6 eVforhydrogen)thistemperatureisaround: 1 0.5 0.5 2 =exp I 0 = T (2–10) T=0.620eV. (2–11) Thistranslatestoatemperatureofabout 7000 K.IfIthenuseEquation(18–42)from Hartle'sbook( Hartle 2003 ),Icanndthetimeatwhichthistemperatureoccurred: t=1.3( 1MeV T ) 2 s (2–12) t 0.5 =3.4 10 12 s (2–13) =107000years. (2–14) Thisisagoodapproximation,butmuchmoredetailedcalcula tionsarepossible.Iwilldo amorein-depthcalculationusingcomputermodellinginthe nextsection;otherstudies ofionizationarealsousefultoconsult( Jones&Wyse 1985 ; Dubrovich&Stolyarov 1997 ). 2.2EvolutionoftheFractionalIonizationofHydrogen Anotherimportantaspectoftherecombinationerainvolves theamountofhydrogen thathasformedatanypointintime.Modellingthisfraction alionizationisalsoa niceintroductiontotheuseofcomputerprogrammingasatoo lforsolvingotherwise intractableproblemsincosmology,astrophysics,andbeyo nd.Thereisaproblem (exercise8onpage82)inDodelson'sbook( Dodelson 2003 )thatprovidesuswithan equationtodescribebasicrecombination.Beingaratherco mplicatedanduglyequation, itwouldbenighimpossibletointegrateusingtraditionalm eans: dX e dx =Axlnx(B exp x x 1.5 X e (C X e x 3 +B exp x x 1.5 )) ( (Dx 1.5 ) = (1 X e )+8.227 (Dx 1.5 ) = (1 X e )+8.227+E(lnx = x)exp x = 4 ), (2–15) 22

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where A=0.0451cm 3 B=6.10 10 20 cm 3 C=3.25 10 7 cm 3 D=4.714 10 5 E=5.618 10 8 ,and: x= 0 T (2–16) and X e isthefractionalionizationofthehydrogen. Althoughtoointractabletosolveanalytically,computers givetheabilitytointegrate suchanunpleasantequation.Allthatisnecessaryistousea suitableintegrationroutine tosolvethisforthefractionalionizationofhydrogen,whi chiswhatIaminterestedin. ProgramsforintegratingdifferentialequationsbytheRun ge-Kuttamethodoforder4are ratherwell-knownandcanbefoundontheinternetorinbasic programmingtextbooks, anditwasthisroutinethatImodiedtousewithEquation( 2–15 )( Pressetal. 1992 ). Afterironingoutthekinks,Iwasabletoobtainmanygraphso ftheevolutionofthe fractionalionizationofhydrogenasafunctionof x (which,ofcourse,reliedinversely ontemperature,sincethe x valueinEquation( 2–15 )isgiveninEquation( 2–16 ),as wasmentionedearlier,andsoitisineffectameasureoftime ).Figure 2-2 shows arepresentativegraph.Inadditiontomyvalues,Iincluded thevaluesthatPeebles calculatedfortheionizationhistory( Peebles 1968 ). AfterIhadthismuchworking,Iwentbackandaltereditabit. Dr.Frysuggested thatIreplacemy lnx with lnx+0.86 ,asitwouldgiveslightlybetterresults.WhenIreran theprogramnumeroustimes,Igotnewgraphs,oneofwhichcan beseeninFigure 2-3 .ThisgurehasthevaluescalculatedbyPeeblesfortheioni zationhistorydisplayed forcomparisonaswell.Theresultsareslightlybetter,but itishardtoseemuchofa difference.Itseemsthatthehydrogenfractionmightstart droppingalittlesooner,but myestimateontheothergraphwassoroughthatthedifferenc eisnotreallydetectable. Thesamealsoholdstrueforwhenitstartslevellingout–itm ighthappenabitsooner inthisversion,butnotenoughtoreallymakeanoteof.Altho ughthesedifferences aresmall,theydoseemtobeforthebetter–theycertainlydo notmakethingsany worse–andsoIleftthisminorchangeinallfurtherrevision softhecode. 23

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Figure2-2.Thelineshowstheevolutionofthefractionalio nizationofhydrogenasa functionof x .Thefractionstartsdroppingaround x=30 ,whichcorresponds toatemperatureofabout 0.45 eV,andstartstoleveloutatabout x=150 whichisatemperatureofabout 0.09 eV.Thesquaresshowthevaluesthat Peeblescalculatedfortheionizationhistory. 24

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Figure2-3.Thelineshowstheevolutionofthefractionalio nizationofhydrogenasa functionof x ,thistimewiththe lnx+0.86 term.Thefractionstartsdropping around x=30 ,whichcorrespondstoatemperatureofabout 0.45 eV,and startstoleveloutatabout x=150 ,whichisatemperatureofabout 0.09 eV. Onceagain,thesquaresshowthevaluesforionizationhisto ryascalculated byPeebles. 25

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ThenextthingIdidwiththisprogramwascompareitsoutputt othatofPeebles inhis1968paperonrecombination( Peebles 1968 );ofcourse,thiscomparisoncan alsobeseeninFigures 2-2 and 2-3 ,wherehisdataisplottedonmygraphs.This comparisonwasimportanttomakesuretheprogram'soutputw asfairlycloseto othersimplemodelsofrecombination,whichwouldimplytha tmymodelwasatleast reasonablysuccessful.Peebleshadatableofthetemperatu revariationofthematteron page10,whichshowedhowhotthematterwasatdifferentfrac tionalionizationsofthe hydrogenforthreedifferentpresentmeanmassdensitiesof theuniverse.Tocompare myresultswithhis,Ihadtomakeafewalterationstomyequat ionsothatitwouldmatch upwitheachofhispresentmeanmassdensities.Theequation Iusedassumedavalue of n m =1 ,butPeeblesdidnotdothis.ThismeantIhadtond n m foreachofthe 0 valuesheused,andthenplugthemintomyprogram.Iknowthat ( Peebles 1993 ; Liddle&Lyth 2000 ): m cr =n m a 3 (2–17) andthat a=1 atthepresenttime,sothismeansthat: 0 cr =n m (2–18) Since 0 isthevaluePeeblegivesinhistable,and cr =1.879h 2 10 29 gcm 3 (with h=0.5 ),itispossibletogureout n m foreachofhisgivenpresentmeanmass densities.OnceIobtainedthesevalues,Iusedtheminmypro gramandcomparedmy resultswithhis.Thegraphfor 0 =1.8 10 29 gcm 3 ,therstcaseheuses,canbe seeninFigure 2-4 .Unfortunately,myvaluesforthefractionofhydrogenwere around oneandahalftimeswhatPeeblesgot.Presumablythisisbeca useheworkedthe problemoutfortyyearsago,andhiscodewasprobablyquitea bitdifferentfrommine. AtthispointIwentbacktomyoriginalequationanddidsomee venlongerruns withit,inanefforttogainmoreinformation.Tofurtherthi send,Iaddedtheequilibrium 26

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Figure2-4.Thelineshowstheevolutionofthefractionalio nizationofhydrogenasa functionof x .Thereisnotmuchchangefromthelasttwographs;the fractionstartsdroppingaround x=30 ,whichcorrespondstoatemperature ofabout 0.45 eV,andstartstoleveloutatabout x=150 ,whichisa temperatureofabout 0.09 eV. 27

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valuestothegraph,sothatIcouldseehowrecombinationact uallydifferedfromthe equilibriumprocess.Todothis,Ilookedforthevaluesof X e suchthatthedifferential onthelefthandsideisalwaysequalto 0 .Inotherwords,IsolvedEquation( 2–15 )for dX e = dx=0 : 0=(A exp x x 1.5 X e (B X e x 3 +A exp x x 1.5 )), (2–19) where A=6.10 10 20 cm 3 and B=3.25 10 7 cm 3 .Whenthisissolvedfor X e ,the resultingequilibriumequationis: X e =9.38 10 12 x 1.5 ( exp x + r exp 2x + 2.13 10 13 exp x x 1.5 ). (2–20) Bycreatingasimpleprogramtogivenumericalvaluesforthi sequilibriumequation,I wasabletoputagraphofitonthesamegraphastherecombinat ionequationwhich Ihadbeenworkingwith;thisgraphcanbeseeninFigure 2-5 .Itisclearthatthenal valuesof X e determinedfromtheseequationsarequitedifferent,as X e quicklygoesto zerointheequilibriumcase,whereasitlevelsoutatafract ionalionizationofabout 10 5 inactualrecombination. 28

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Figure2-5.Thesolidlineshowstheevolutionofthefractio nalionizationofhydrogenas afunctionof x .Thedottedline,whichdropsquicklyatabout x=50 ,or T=0.272 eV,istheequilibriumequation. 29

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CHAPTER3 ROUGHCALCULATIONS 3.1TheScaleFactor OnceIhadgottenallthatIcouldoutofmybasicrecombinatio nequation,Iwas readytomoveon.Thenextstepwastodosomeroughcalculatio nsoftheevolutionof thedensityperturbations,usingjustthebasicanalytical equationsandwithoutworrying abouttheperturbationsoftheradiation.Insteadofworryi ngaboutallthemyriadways thatdifferentkindsofmatterandradiationcouldinteract ,Istartedwithsomesimple couplingandrateequations,andguredouttheevolutionof thedensityperturbationsof darkmatter,hydrogen,andheliumundertheseconditions. TherewasachancethatIwouldwantorneedtoutilizetimeasm yindependent variable,butalltheequationsIwouldbeusingforthesecal culationsweredependent onthescalefactor, a .So,toprepareforthiseventuality,andsimultaneouslych eckto makesuretheintegrationroutineIwasusingwasworkingpro perly,Istartedwitha simplegraphof a vs. t .Thisturnedouttobeagoodidea,astherewereanumberof bugsinmycode,andittooksometimetotrackthemalldownand exterminatethem.To determine a ,IusedoneofFriedmann'sequations( Liddle&Lyth 2000 ; Peebles 1993 ; Dodelson 2003 ): H 2 =( _a a ) 2 = 8 G 3 X i k a 2 + 3 (3–1) ( _a a ) 2 = 8 G 3 critical,0 ( n m a 3 + n k a 2 +n ). (3–2) TheconstantsinfrontoftherighthandsideofEquation( 3–2 )canbeturnedintoone constant,thecommonlyusedHubbleConstant, (H 0 ) 2 =(8 G = 3) crit,0 ( Liddle&Lyth 2000 ).Somyequationfor _a is: _a=H 0 a r n m a 3 + n k a 2 +n (3–3) 30

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Figure3-1.Thelineshowstheevolutionofthescalefactorw ithtime. Usingthisequation,Iwasabletomakesomegraphsusingmypr ogram.Byrunning itfromatimeof 10 12 secondstoatimeofabout 3 10 18 seconds,Iwasabletocreate anicelookingplotthatshowedmethateverythingwasworkin gproperly.Thisplot,with n m =0.267 n k =0 ,and n =0.733 andaHubbleconstantof h=0.704 canbeseen inFigure 3-1 .Thisgraphshowsthat a 0 =1 ataroundthecurrenttimeof 4.32 10 17 31

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Figure3-2.Thelineshowstheevolutionofthescalefactorw ithtimeonalog-logplot. seconds,orabout 13.7 billionyears.Thisisasitshouldbe,soIcheckedthebehavi or of a atearlyandlatetimestoseewhatitlookedlikethere.Byplo ttingmydataona log-logplot,whichcanbeseeninFigure 3-2 ,Iwasabletoexaminetheevolutionof theuniversefromaveryearlytimetoatimealittlelatertha nthecurrentdate.The slopeatearlytimesonthislog-logplot,fromaboutoneseco ndtoatimeofaround 10 13 32

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seconds,was 1 = 2 .Thismakessense,as a / t 1 = 2 forearlytimes,whentheuniverseis radiation-dominated.Afterthispoint,untilaround 20 17 seconds(whichisgettingvery closetothepresentday),theslopewas 2 = 3 ,whichmeantthat a / t 2 = 3 –asitshouldbe whentheuniverseismatter-dominated.Allthatwasleftwas tomakesurethebehavior atlatetimeswascorrect,butthatwasslightlymorecomplic ated.Iknowthatthegraph shouldgrowfasterandfasteras a increasespastone,andthatbehaviourcanbeseen inmygraph.Togainalittlemoreinformationonwhathappens to a asitgrowslarger thanone,Isolvedfor a asafunctionof t atlatertimes: _a=H 0 a r n m a 3 +n (3–4) Ofcourse,thersttermdisappearswhen a getslarge,sothisequationbecomes: @ a @ t =H 0 a p n (3–5) Z @ a a = Z H 0 p n @ t (3–6) loga=H 0 t p n (3–7) Thismeansthatifthedataisgraphedonalog-linearscale,t hereshouldbeaconstant slopeof H 0 p n =1.95 10 18 s 1 .OnceImadethisgraph,Isawthattheslopedid appeartobeconstant,whichwasheartening.Uponmakingaro ughestimateofthe slope,Ideterminedittobearound 1.5 10 18 s 1 ,whichwascloseenoughtoconvince methatmyequationandprogramwerebothworking. TherewasonemoregraphIwantedtomakeatthispoint.Agraph oftheHubble parameter, H ,asafunctionofthescalefactor, a ,seemedtobeagoodidea,sinceI alreadyhadthenecessaryequations.Tomakethisgraph,Ist artedwithEquation( 3–3 ) andrearrangeditabit,thenaddedatermfortheradiationde nsityparameter,toendup withEquation( 3–26 ).The n k dropsout,asIassumedtheuniversewasat.Usingthis equation,IcreatedFigure 3-3 ,whichshowsalog-logplotof H asafunctionof a .To createthisgraph,Iusedvaluesof n r =0.00002 n m =0.2792 n =0.721 ,and H 0 =70.4 33

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Figure3-3.EvolutionoftheHubbleparameterwithscalefac toronalog-logplot. 34

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(km/s)/Mpc.Sincethisisalog-loggraph,itappearstohave oneslopeatearlytimes, whentheuniverseisradiation-dominated(of 2 ,sincetheradiationtermdominatesthe others),anotheratmiddletimeswhentheuniverseismatter -dominated(of 3 = 2 ,since themattertermdominates),andtheHubbleparameterseemst olevelout,decreasing slowly,aroundthepresenttime.Ascanbeseenonthegraph,t hecrossoverfrom radiationtomatterdominationoccursaround a=0.0001 .OnceIwasconvincedthatmy integrationroutinewasworking,Iwasabletomoveonanduti lizeitonsomethingmore interesting. 3.2RoughAnalyticalCalculationofRecombination Beforegettingintolong,detailedcalculationsofrecombi nationthattakeinto accounteverythingthatcanpossiblyhappen,itisimportan ttounderstandandbe abletoproducesimpleresultsfromaroughcalculationofre combination,takinginto accountonlythemostimportantfactors.Theshapesofthese graphsarewellknown, andcanbefoundinseveralplaces( Liddle&Lyth 2000 ; Peebles 1968 ; Seageretal. 2000 ; Ma&Bertschinger 1995 ).Thesegraphsshouldstartwithverysmalldensity perturbationsatveryearlytimes,andasthescalefactorin creases,theperturbations shouldgrowlinearly.Thecolddarkmatterstartsincreasin gveryearly,whilethe hydrogenandheliumperturbationscan'tstarttogrowuntil theyrecombine,leading todifferentstartingtimesforboth.Therststeptowardsd oingthisiscalculatingthe densityperturbations, ,ofcolddarkmatteralone. Assumingthat 1 sothattheequationsarelinear,itisafairlysimplematter to ndtheequationsthatneedtobenumericallyintegrated.St artingwiththecontinuity equation,Equation( 1–14 ): + 1 a r v=0, (3–8) 35

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andwith r v= ,wehave: = 1 a (3–9) Thisequationdescribestheevolutionof withtime;nowIneedasimilaronefor .To getit,Icanstartwith F=ma ,( Liddle&Lyth 2000 ; Peebles 1993 ): 1 a @ @ t (av)= 1 a r (3–10) _av+a @ v @ t = r (3–11) IfItakethedivergenceofbothsides,Ind: _a( r v)+a @ @ t ( r v)= r 2 (3–12) _a +a @ @ t = r 2 (3–13) Poisson'sEquation,Equation( 1–13 ),tellsusthat r 2 =4 Ga 2 ,so: = _a a 4 Ga (3–14) andtheequationfortheevolutionof withtimeis: = H 4 Ga (3–15) NowIhaveequationsfor @=@ t and @=@ t ,butitismoreconvenienttouse a asthe independentvariable,soIneedequationsfor @=@ a and @=@ a .Thisshouldbeeasy enough;allIneedtodoisusethechainrule: @ f @ t = @ f @ a @ a @ t (3–16) Iknowthat H=_a = a ,so: f= @ f @ a (Ha) (3–17) f 0 = f Ha (3–18) 36

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wheretheprimedenotesaderivativewithrespectto a .Withthis,Inallyhavethetwo equationsIneedtonumericallyintegrate : @ @ a = Ha 2 (3–19) @ @ a = H +4 Ga Ha (3–20) UsingtheseequationsandmyintegrationprogramfromSecti on 3.1 ,Ididanumber ofrunsofrecombination.Tostartwith,Ionlyuseddarkmatt er.Beforeaddinginthe morecomplicatedsubstances,Iwantedtomakesurethebasic programwasrunning correctly.Todothis,Iassumedthatthe and valuesforeverythingelse–hydrogen, helium,radiation,andanythingelseImightwanttoadd,lik eneutrinos–were 0 ,so thatIcouldreplace withjust c .Withtheseassumptions,Iranmyprogrammultiple times,butkeptgettinginaccurateresults.Ispentalotoft imetrackingdownbugsand annihilatingthem,eachtimemakingmyresultsabitmoreacc urate,butneverquite achievingthecorrectgraphsIwaslookingfor.Inaneffortt oimprovemyresults,I decidedtoadd n radiation totheHubblefactor,hopingtheaddedaccuracywouldxsome oftheproblemsIwashaving(untilthispoint,Iwasonlytrea tingthedensityparameters, n ,ofmatteranddarkenergyinmyHubblefactor).Iknowthat( Liddle&Lyth 2000 ): r =g 2 30 (kT) 3 ( ~ c) 3 kT c 2 (3–21) SinceIamusing a asanindependentvariable,ratherthan T ,Ineededtochangethis equationslightly.IcandothisbecauseIknowthatat a=1 a=T 0 = T ,where T 0 isthe currentbackgroundtemperatureofthecosmicmicrowavebac kground.So: r =g 2 30 (kT 0 ) 3 ( ~ ca) 3 kT 0 ac 2 (3–22) 37

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Finally,Icanusethisequationtogureoutwhat n rad willbe.Iknowthecriticaldensity attoday'stime,andIcangureoutwhat r isatthepresenttimebysetting a=1 ,so: n r = r,0 cr,0 (3–23) n r =(g 2 30 (kT 0 ) 3 ( ~ ca) 3 kT 0 ac 2 ) = ( 3H 20 8 G ) (3–24) n r = 8 3 Gg(kT 0 ) 4 90H 20 ~ 3 c 5 (3–25) where g=3.3626439 ,sinceitincludesbothphotonsandneutrinosinitscalcula tion.With thisnew n ,mynewHubblefactoris: H=H 0 r n r a 4 + n m a 3 +n (3–26) Ofcourse,assooftenhappenswiththesethings,thelackof n r wasnotthe problem.Itmadethingsabitmoreaccurate,andwascertainl yagoodthingtoaddto myequations,butitdidnotxtheproblemsIwashaving.SoIw entovermycodeagain, evenmorecarefully,andeventuallyguredouttheproblem– whichwasinthecoding. Ofcourse,ithadnothingatalltodowiththephysics,andwas purelyaprogramming error.Aftertreatingthisproblem,Igotbeautifulgraphst hatlookedjustlikeIwanted themto,whichwasquitesatisfying.Arepresentativegraph canbeseeninFigure 3-4 Thisgraphhas n H =0.0334 n He =0.01056 n c =0.223 ,and n =0.733 ,andhasan initial c =1 andaninitial c =0 ,aswellas H 0 =70.4 km/Mpc/sand T 0 =2.725 K. Tobesurethatmyprogramwasgivingmecorrectvalues,Ireal lyhadtoseetheslope ofthegraphmyprogramproduced.Tothatend,Imadeanotherg raphoftheslopeof theperturbationsofcolddarkmatterasafunctionofthelog ofthescalefactor,which canbeseeninFigure 3-5 .Asexpected,theslopestartsoffatzero,thenrapidlyrise s untilitgetstoalmostone(about 0.9 inmygraph)duringthematter-dominatedphase oftheuniverse'sevolution,andthennallydropsrapidlya sitapproachesthepresent time,duetotheincreasingdominationofdarkenergy.Itapp earsthatthisundertaking 38

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Figure3-4.Thisgraphshowstheevolutionofcolddarkmatte rdensityperturbationsas thescalefactorincreases. 39

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Figure3-5.Thisgraphshowstheslopeoftheevolutionofthe colddarkmatterdensity perturbationsgraphasthescalefactorincreases. 40

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wasasuccess;thenextstepwastoaddmattertomyequations, andtodothat,Igured thebestplacetostartwouldbewiththecodefrom COSMICS .ItwasmyhopethatIcould extracttheequationspertainingtotheessentialelements ofrecombinationwhileleaving behindthepartsofthecodethatwouldleadtosmallchanges. 3.3RoughCalculationofRecombinationStartingFrom COSMICS Code The COSMICS codegivesmeequationsthatareabitmoreaccuratethantheo nesI wasusinginthelastsection.Thecreatorsofthe COSMICS codestartedfromscratch(i.e., theEinsteinEquations)andworkedoutexactlywhattheresu ltingperturbationevolution equationswouldbe( Bertschinger 1995 ; Ma&Bertschinger 1995 ; Bertschinger 2001 ). Theequationstheyfoundfortheevolutionofthedensityper turbationsforcolddark matterare: = +3 (3–27) = _a a +k 2 (3–28) Ofcourse,sincemyprogramuses a astheindependentvariable,ratherthanconformal time ,astheirprogramuses,Iwillhavetoonceagaindividebotht heseequationsby (_a = a)a togettheproperresults. Clearly,theseequationsareabitmorecomplicatedthanthe oneswewereworking with.Whatarethedifferences?Howaretheseequationscalc ulated?Attheriskof goingintotoomuchdetail,Iwilldescribeexactlyhowtheyw ork.Firstofall,thereisthe constant, grhom ,whichhasavalueof 3.3379 10 11 H 20 Mpc 2 (assumingthattheunits of H 20 havealreadybeenaddedtotheendofthenumber;thisiswhere theunitscome from).Thiscomesfrom: grhom = 8 G 0 c 2 (3–29) = 8 G c 2 3H 20 8 Gn (3–30) = 3H 20 c 2 (3–31) 41

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Ifwetaketheunitsoutofthe H 20 andthespeedoflight,weget: grhom = 3H 20 c 2 km 2 s 2 Mpc 2 s 2 m 2 (3–32) Bycancellingthesecondsandaddingaconversionfactorfor themeters,wecanleave justtheunitswewant,so: grhom = 3H 20 1000 2 c 2 Mpc 2 (3–33) =3.3379 10 11 H 20 Mpc 2 (3–34) ThisconstantistheoneusedwhenndingtheHubblevariable inconformaltime: grho = ( grhom )(n c +n H +n He ) a +( grhom )n a 2 (3–35) andso, _a a = r grho 3 (3–36) Thisisnotenoughtond ,however,sinceitdependsonthedensityperturbationsand howtheyarechanging.Sotogettherestofthevariableswewi llneed,wehavetohave valuesfor(thesearevariablesfromthecode): dgrho =( grhom ) n c c a (3–37) dgtheta =( grhom ) n c c a (3–38) Withthesevalues,wecannd ,whichis(withthewavenumber” k ”inunitsofMpc 1 ): = 1 k 2 ( dgrho + 3(_a = a)( dgtheta ) k 2 ). (3–39) Finally,wehave ,whichis: = _a a + dgtheta 2k 2 (3–40) 42

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Theseequationsallseemright,asfarastheygo(theycouldd enitelybemore accurate,sinceIamcurrentlyonlyincluding n c in dgrho and dgtheta ),andsoImade graphswiththem,at k=0.01 0.1 1 ,and 10 Mpc 1 .Ascanbeeasilyascertained, somethingisoff,atleastwiththesmallestvaluesof k .Thisisparticularlyfrustrating asthesmallestvaluesof k shouldbetheeasiestthingtocompute.Althoughthegraph looksokayforlatertimes,theearlytimes–whenperturbati onsincreasebyafactorof 10 26 –areclearlywrong.Figures 3-6 to 3-13 showeachgraph,followedbyitsslope.They allhave n H =0.0334 n He =0.01056 n c =0.223 ,and n =0.733 ,withaninitial c =1 andaninitial c =0 ,aswellas H 0 =70.4 km/Mpc/sand T 0 =2.725 K. Ascanbeseen,theperturbationgraphsfor k=10 1 ,andeven 0.1 areprettyclose towhatwouldbeexpected,butthegraphfor k=0.01 ,whichcanbeseeninFigure 3-6 ,iswayofffromwhatitshouldbe.Ispentalotoftimetryingt ogureoutwheremy equationswentwrong,orwheresomesmall,stupiderrorinmy codewasampliedinto ahorrendouserrorcausingtherapidgrowthofperturbation satsmalltimesfor k=0.01 butdespitealotoftestingandrewriting,Icouldnotmaketh eproblemgoaway.Idid ndseveralsmallerrorsthatcausedalmostimperceptiblep roblems,buttheydidnot helpmuchwithmy k=0.01 graph.Intheend,Igaveupontryingtosolvethisproblem. Therearemuchbettercodesoutthere– COSMICS ,forinstance–whichIknowwork,and whichmodelthegrowthofdensityperturbations,whichiswh atIwasinterestedin studying.Spendingmoretimeonthiscode,whichattheveryl easthassomeserious problems,didnotseemlikeaproductiveuseofmytime.Still ,itwasworththeeffortto try–ithelpedmeunderstandrecombinationeraphysicsbett er,andallowedmetoget familiarizedwithhowthe COSMICS codeworkedinmoredetail. 43

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Figure3-6.Thisgraphshowstheevolutionofthecolddarkma tterdensityperturbations asthescalefactorincreases,with k=0.01 44

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Figure3-7.Thisgraphshowstheslopeoftheevolutionofthe colddarkmatterdensity perturbationsasthescalefactorincreases,with k=0.01 45

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Figure3-8.Thisgraphshowstheevolutionofthecolddarkma tterdensityperturbations asthescalefactorincreases,with k=0.1 46

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Figure3-9.Thisgraphshowstheslopeoftheevolutionofthe colddarkmatterdensity perturbationsasthescalefactorincreases,with k=0.1 47

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Figure3-10.Thisgraphshowstheevolutionofthecolddarkm atterdensity perturbationsasthescalefactorincreases,with k=1 48

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Figure3-11.Thisgraphshowstheslopeoftheevolutionofth ecolddarkmatterdensity perturbationsasthescalefactorincreases,with k=1 49

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Figure3-12.Thisgraphshowstheevolutionofthecolddarkm atterdensity perturbationsasthescalefactorincreases,with k=10 50

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Figure3-13.Thisgraphshowstheslopeoftheevolutionofth ecolddarkmatterdensity perturbationsasthescalefactorincreases,with k=10 51

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CHAPTER4 COSMICS 4.1ModellingtheEvolutionofDensityPerturbations OnceIhadasolidunderstandingofthephysicsunderlyingth erecombination era,Iobtainedacopyof COSMICS ,whichwaswrittenbyMaandBertschingerto modelthegrowthofdensityperturbationsintheearlyunive rse( Bertschinger 1995 ; Ma&Bertschinger 1995 ; Bertschinger 2001 ).Itwasaverylong,complicatedcode writtenin FORTRAN77 ,andittookintoaccountmanydifferentfactors,suchasmas sless andmassiveneutrinos,tightlycoupledanduncoupledbaryo nandphotonmotion,the temperatureofthebaryons,ionizationfractions,andthed ensityandpressureofthe neutrinos.Itdidallthis,andwaswritteninanoldlanguage thatIwasunfamiliarwith. Nevertheless,Iwaseagertondoutwhatkindofresultsitwo uldgiveme,soIread throughtheinformationthatcamewithitandstartedexamin ingthecode. Iknewthatbeforedecoupling,theelectronsscatteroffoft hephotons,duetothe intenseheatanddensity,andthereforetheelectronsareun abletogotoofarbefore interactingwithsomething( Peebles 1993 ; Liddle&Lyth 2000 ; Dodelson 2003 ).At thepointofdecoupling,itislikelythattherewasabout 74 percenthydrogenionsand 24 percentheliumions–thisisanappreciableamountofhelium .Thisenrichmentof heliumwilllikelyeffectthedecouplinganddeionizationp rocess,sinceheliumionizesat ahigherenergythanhydrogen,whichallowstheheliumionst orecombinesoonerthan thehydrogen.ItwasthisproblemthatIwashoping COSMICS mighteventuallyshedsome lightonforme.Usingtheequationfordensityperturbation s,Iwouldbeabletondout whateffectneutralheliumwouldhaveontheearlyuniverse( Liddle&Lyth 2000 ): @ j @ t +2 _a a j 4 G X i i i =0, (4–1) 52

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where P i i comesfromthedensitiesofallthedifferentkindsofmatter andenergyin theuniverse: = DM + H + He + rad + (4–2) Thisphysicswasallincludedin COSMICS ,soIstartedplugginginvaluesandexamining myoutputs. TherstproblemInoticedwasthat COSMICS tooktheinitialvaluesandevolved theuniversetoatimechosenbytheuserstartingfromthosei nitialconditions.The outputitgavewasthecurrentstructureoftheuniverseatth echosen a value.Thatwas ne,butIwasnotinterestedinanyoneparticulartime–Iwan tedtoknowwhatwas happeningduringthewholerecombinationera.Iknewthatin formationhadtobein theresomewhere,Ijusthadtogointotheinnerworkingsofth eprogramandgureout howtomakeitdowhatIneededitto.Ofcourse,Icouldjustrun theprogramrepeatedly withslightlydifferentendingtimes,butthatwouldbeincr ediblyinefcientandwould takemuch,muchlongertogetusabledatafrom.Instead,Icho setoundertakethemore difcult,butultimatelymorerewardingtask,ofmakingthe programgivemeusabledata. Aftersometrialanderror,Iunderstoodhow COSMICS workedwellenoughtochange ittomyspecications.Byinsertingmyowncodeandaltering theexistingprogram judiciously,Iwasabletomaketheprogramprintoutasmanyt imestepsasIrequired, betweenanystartingandanyendingtime.Needlesstosay,th ispleasedmeagood deal,andIexperimentedwithsomeshort,simplerunstoiron outtheremainingkinks. NosoonerhadIgotthismodiedprogramrunning,though,tha nIrealizedIhad anotherproblem.TheoutputIwasgettingcouldnotbeeasily graphed,becauseit showedtheevolutionofmanydifferentwavenumbers,alljum bledtogether.Iwasable tographtherstfeweasytestrunsbecausetheyonlyhadveo rtenpoints,soIjust cutandpastedthewavenumbersIneededintoaseparatele,b utIknewthiswould neverworkforreallylongruns.Lackinganyotheroptions,I wentbackintothecodeand 53

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addedmorechangesinanefforttoxthisnewestproblem.Int heend,Isucceeded, andpersuadedtheprogramtodowhatIneededitto.Mymodied versionof COSMICS nowpickedoutwhateverwavenumberIchoseandfollowedthee volutionoftheuniverse atthatwavenumberfromanystartingtimeIchosetoanyendin gtimeIchose.After somemoretesting,thingsseemedtobeworkingne,andsoImo vedontosomereal runs. Istartedoffwithlargetimeperiodsandonlyafewtimesteps ,untilIwassure thatIunderstoodwhatIwasgettingandthatitwasaccurate. Thebaryonswereall evaluatedtogether,sotheyjustgavemeoneline,butIwasab letondtheevolutionof thedarkmatterdensityperturbationsinadditiontothebar yonicdensityperturbations. AsImadebiggerandbiggerruns,withmoreandmoredatapoint s,variousproblems arose.However,theywereallprogrammingproblemswith FORTRAN77 ,notproblems withthephysicsmodelin COSMICS ,soIwasabletoironoutallthesedifcultieswithout majorchangestotheprogramitself.Oneofthelaterrunsoft heprogramgaveme thefollowinggraphs,whicharequiteinteresting.Figure 4-1 showstheevolutionof bothparticlespeciesoveralongperiodoftime.Figure 4-2 takesalookatasmaller periodoftimeintheinterestoftakingmoredatapointsinth einterestingareas.In bothgures,thedensitiesofbaryonicmatterare 0.044 ,colddarkmatteris 0.223 ,and darkenergyis 0.733 .TheHubbleconstantis 70.4 (km/s)/Mpc,thetemperatureofthe cosmicmicrowavebackground(CMB)todayis 2.725 K,theinitialfractionofheliumis 0.24 ,andthewavenumberfortheseplotsis 0.1 Mpc 1 .Inbothgraphs,weseethatthe baryonicmatteroscillatesaroundtherecombinationtime– thesolidlinesanddotsare theactualvalues,whereastheopenpointsanddottedlinesa reactuallynegativevalues ofdensityperturbationswhichhadtheirsignsippedinord ertographthemona log log plotlikethis.Theserepresentativegraphsshowthemainfe aturesoftheevolutionof densityperturbationsofdifferentspecieswithtime.Once Igot COSMICS workingproperly, Imovedontothenextstage.Seeinghowbaryonsevolveintime isquiteinteresting,but 54

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0.00010.0010.010.1 10 100 1000 a Figure4-1.Thesmooth,solidlineshowstheevolutionofthe colddarkmatterwhilethe pointsshowtheevolutionofthebaryonicmatter.Thedensit yperturbations ofeachspeciesareplottedversus a 55

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0.00010.001 10 100 a Figure4-2.Thesmooth,solidlineshowstheevolutionofthe colddarkmatterwhilethe pointsshowtheevolutionofthebaryonicmatter.Thedensit yperturbations ofeachspeciesareplottedversus a 56

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myfocusisonheliumrecombination.Somehow,Ihadtobeable totrackitseparately fromhydrogen. 4.2AdditionofHelium Delvingintothedepthsof COSMICS ,IfoundcertainthingsthatmademethinkImight beabletosplitthebaryonsintohydrogenandhelium,giving memoredataonwhat washappeningduringrecombination.Theway COSMICS wasinitiallysetuptreatedall baryonsthesame.Thesebaryonswereaffectedbythephysics ofrecombinationand treatedasonesubstance.Therewereacouplepartsofthecod ethatmadeitclearthat COSMICS shouldbeabletodistinguishbetweenhydrogenandhelium,a ndtreateach differentlyduringrecombination,buttheway COSMICS wascoded,itdidnotallowthe usertodoanythingspecicwiththedifferentbaryons,orke eptrackofthemseparately ( Siegel&Fry 2006 ). Whilerunning, COSMICS takeseverythingthatneedstobetrackedandputsitinto oneverylargearray, y .TherstthingItriedtodowasaddanotherarraytokeeptrac k oftheheliumresults, yHel ,andIgotagoodwaythroughthechangesIwantedtomake beforeIrealizedthatitwouldultimatelybefutile.Thethi ngsIwastryingtokeeptrackof in yHel (suchastheperturbationdensityandvelocityatwhichthep erturbationdensity waschangingateachpointateachtime,aswellastheothersu bstances,likecolddark matterandneutrinos)dependedontheelementsofanotherar ray,andtheonlywayto takeatimestepanditeratemyarrayswastosendmyarraytoan otherprogram, dverk whichevaluatedthederivatives.Since dverk couldonlyacceptonearrayatatime,this methodwasadeadend. ThenextthingItriedwastoaddtwoelementstothearraythat containedallthe othervaluestheprogramsolvedfor, y .Thismethodwasalotmorecomplicated,since Ihadtochangeeverypartoftheprogramthatreliedonthisar ray(andbasicallythe wholeprogramreliedonthisarray),butafteragooddealofw ork,Ihadmadeallthe changesIthoughtwerenecessary.Afterallofmychanges,th eprogramnowhad 57

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0.00010.0010.010.11 1 10 100 1000 This is lg(a) Figure4-3.Thedottedlineshowstheevolutionofthecoldda rkmatterwhilethesquare pointsandsolidlineshowtheevolutionofthehydrogenandt hexpointsand theirdottedlineshowtheevolutionofthehelium.The log ofthedensity perturbationsofeachspeciesareplottedversus log(a) twomorethingsitkepttrackof: and forhelium.OnelongrunImadewiththis newadditiontotheprogramcanbeseeninFigure 4-3 .Inthisgure,thedensityof hydrogenis 0.03344 ,heliumis 0.01056 ,colddarkmatteris 0.223 ,anddarkenergyis 0.733 .TheHubbleconstantis 70.4 (km/s)/Mpc,thetemperatureoftheCMBtodayis 58

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2.725 K,theinitialfractionofheliumis 0.24 ,andthewavenumberforthisplotis 0.1 Mpc 1 .Oncemore,weseethatthebaryonicmatteroscillatesaroun dtherecombination time,andalthoughtheheliumperturbationsseemtostartou tlowerthanthehydrogen perturbations,theycrossoverandbecomelargeraftertheo scillationsend,eventually growingfasterthanthedarkmatterandhydrogendensityper turbations.Thisdoesn't seemright,asheliumshouldrecombinebeforehydrogen,whi chwouldleadtoithaving densityperturbationsthatwouldgrowfasterthanhydrogen .There'stheaddedproblem thatallthreetypesofmattershouldendupatthesamevalue, sincetheyareallmatter. Thefactthatsomehowtheheliumperturbationsincreaseabo vethehydrogenandcold darkmatterperturbationsisaclearindicationthatintryi ngtoexpandthecapabilities of COSMICS ,Iinsteadmanagedtobreakit.Iwasdissatisedwiththeser esults,and thiswasoneofthebettergraphsIgot.Icametotheconclusio nthatthiswasanother deadend,andlackinganyotheroptions,decidedtomoveonto anotherprogramwhich promisedmenew,betterresults– GADGET-2 59

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CHAPTER5 INTRODUCTIONTOGADGET-2 GADGET-2 (whichstandsfor GA laxieswith D arkmatterand G asint E rac T )isan extremelypowerfulpieceofsoftwarewrittenbyVolkerSpri ngel,anddesignedto simulateparticleinteractionsoncosmologicalscales( Springel 2005a ).Manydifferent programshavebeencreatedtoperformthesetypesofsimulat ions,andtheyallhave theirstrongandweakpoints( Heitmannetal. 2008 ; Taskeretal. 2008 ; O'Sheaetal. 2004 ).Dr.FryrecommendedItry GADGET ,andsoIspentsometimelearningit.It seemedtodoeverythingIneededitto,andintheend,Idecide d GADGET wouldwork quitewellformypurposes. GADGET canmodeltheuniverseusingNewtoniandynamics orcosmologicalintegrationswithcomovingcoordinates,w ithorwithoutperiodic boundaryconditions.Bychangingthemanyavailableoption s,theusercansetupa varietyofuniverses,withtheirownparametersgoverningp articlemotion.Thesettings Ipersonallyfoundmostusefulwerecosmologicalcomovingc oordinatesinaperiodic boxusingtheadaptivetreeparticlemeshgrid.Thisallowed metosimulatestructure formationonacosmologicalscale,whichisexactlywhatIwa stryingtodo. Thereisnoquestionthat GADGET isacomplicatedprogram,butitisnothardto describewhatitdoes.Generalparameters(liketheperiodi cboundaryconditionswhich Iemployed)aresetinthe makefile ,afterwhichtheprogramiscompiled.(Thismeans thatitmustberecompiledwheneveryouwanttochangethepar ameters.)Oncethe programforthegalaxythatisrequirediscreated,itneedst wothingstorunproperly: Aparameterle,whichpreparestheprogramtoaccepttheini tialconditions. Aninitialconditionsle,whichdistinguishesonerunfrom another. Theparameterletells GADGET wheretoreadtheinputlefrom,wheretowritethe outputto,whentostartthesimulation,howoftentotakesna pshots(whicharetheonly wayofrecordingtheprogressoftheprogram),thecosmologi calparameters(likethe HubbleConstantandthedensityvalues,suchas n CDM ),andvariousotheroptions thatcontrolhowtheprogramruns.Theinitialconditionsl eisinabinaryformatand 60

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isrichwithinformationaboutthestateofthegalaxyatthet imethesimulationstarts. Itincludesthingssuchasthetypesofparticlesinvolved,t heirmasses,positions,and velocities,andotherfactorsthatmightcomeinhandywhent rackingtheevolutionof manyparticles. 5.1SettingUpARun Inordertomakearunof GADGET ,aswaspointedoutabove,threethingsneed tobechanged.Theprogramitselfisneveraltered–byputtin gallthechangesin separateles,usabilityandportabilityaregreatlyincre ased. GADGET ismadeupof manyles–aroundthirty–anditwouldbeverydifculttomak etheproperalterationsin multipleles.Therstlethatneedstobechangedisthe makefile .Therearemany featuresthatcanbechangedinhere,butmostareforspecial casesandarenotuseful formypurposes.Afewareworthsetting,though.Thisiswher etheperiodicboundary conditionsareturnedon,whichIalwaysused,aswellasthez oomandhighresolution settings,whichIdidnotneedtoturnonformypurposes. The makefile isalsotheplacewhereyouturnonthe TreePM option,whichwas usedforcosmologicalintegrations.Thisisahybridforcec alculationthatusestwo differentmethodstodeterminethestrengthoftheforceact ingonanindividualparticle, dependingonwhethersaidparticleisatshortrangeorlongr ange.Shortrangeforces arefoundusingatreemethod.Findingtheforcesbetweeneve rypairofparticlesgets prohibitivelycomputationallyexpensiveratherquickly– luckily,itisalsounnecessary. Onlyverysmallerrorsareintroducedbytreatingclusterso fparticlesthatare(relatively) farawayasonelargeparticle–thetreeroutineishowthesed istancesandreplacement particlesarechosen.Whentryingtondtheforceonapartic ularparticle,theprogram walksupthetree,ndingthenearestneighborsandtreating themindividuallyforthe closestparticles,andasincreasinglylargerclustersofp articlesthefurtherawayfrom thetargetparticletheprogramgoesinthetree( Hernquist 1987 ; Hockney&Eastwood 1988 ; Kravtsovetal. 1997 ). 61

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Forlongerrangeforces, GADGET reliesmoreheavilyuponanadaptiveparticlemesh routine.Thisiseasierandfasterthanwalkingthetree,asi tislesscomputationally expensive.Spaceisbrokenupintoagrid,andtheparticlesc ontributedensitytothe verticesnearestthemselves.Oncethisdensitygridiscrea ted,thegravitationalpotential canbefoundatanypointfromthisdensitydistributionusin gthePoissonEquation: r 2 =4 G (5–1) BygoingtothefrequencydomainbyusingaFouriertransform ,Poisson'sEquation becomesquiteeasytosolve: ^ =4 G ^ k 2 (5–2) Inthisnotation,thehatsmeanthattheFouriertransformha sbeentakenofthe gravitationalpotential andtheparticlemeshdensity .The k representsthecomoving wavenumber.TondthegravitationaleldfromEquation( 5–2 ),oneonlyneedsto multiplyby k andthentaketheinverseFouriertransform.Thisprocessis muchquicker thanotheralternatives,likewalkingthetreeorcalculati ngtheforcebetweeneach particlepairdirectly( Hockney&Eastwood 1988 ; Villumsen 1989 ). Thesearethemostimportantoptionsavailableinthe makefile .Ascanbeseen, essentially,itcontainsthesettingsthattelltheprogram whatkindofuniversetouse. Theparameterleisutilizedtotelltheprogramspecicinf ormationabouttherunthat istakingplaceintheuniverseestablishedbythe makefile .Itcontainsmanyoptions, butmostofthemdonotneedtobeddledwith,asthedefaultsh andlearegular cosmologicaluniversequitewell.Tomodelanexpandinguni versewithcosmological parameters,comovingintegrationhastobeturnedon.Thist ellstheprogramtouse comovingcoordinates,andratherthanregulartime(whichi swhattheprogramuses forplainNewtonianruns),theexpansionfactor a isused.Specicvaluesmustalso bechosenforthingslikethestarttime(or,inmycase,theex pansionfactor a ,since 62

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comovingintegrationwasusedformyruns),theexpansionfa ctorcorrespondingto thetimeatwhichthesimulationends,andtheamountoftimet hatpassesbetween eachsnapshotlethattheprogramwrites.Sincethesesnaps hotsaretheoutputofthe programthattheuserexaminestoseehowtheuniverseisevol ving,itisimportantto makesureenoughsnapshotsarebeingoutputtogiveanaccura teviewofthesimulation overtime.Otherparameterssethereincludethesizeoftheb oxthesimulationtakes placein(in GADGET 'sunits,aninternallengthunitof 1.0 equals 1.0 kiloparsecs/ h ,andI setmyboxto 50000 ,or 50 megaparsecs/ h ),theHubbleConstant(whichItooktobe 70.4 (km/s)/Mpc),thematterdensity n 0 ,includingallmatter(baryonicandcolddarkmatter), andwhichIsetto 0.3 ,andthevacuumenergydensity, n ,whichIsetto 0.7 Thenalpieceofthispuzzleinvolvestheinitialcondition sle.Thisleiswrittenin aspecicbinaryformat,andifallthedataitcontainsisnot inaveryspecicorder,the programwillhavenoideawhattheuniverseissupposedtoloo klikeatthebeginning ofthesimulation.Infact,iftheinitialconditionsleisw rong, GADGET willprobablyjust crash,anditcanbeverydifculttondtheproblemwhenitis burieddeepinsidea binarylethatcannotbeopenedandexaminedforproblemsin Notepad or VI .Onthe brightside,theinitialconditionslethat GADGET usesisrichwithinformation,andkeeps trackofprettymucheverythingtheusercouldwanttoknowab out.Thecorresponding drawbacktothiswealthofvariablesisthatitmakestheinit ialconditionslerather unintuitiveandlearninghowitisstructuredandwhereitpl acesalltheinformation neededtoconductananalysis,or,evenworse,howtochangei ttocreatenewinitial conditionscanbeadauntingtaskindeed.Fortunately, GADGET 'sauthoranticipatedthis problem,andtriedtoexplainthestructureoftheinitialco nditionsletotheuserasbest hecould. Thisexplanationwasdoneintwomajorways.First,thereisa briefchapterinthe user'sguidefor GADGET thatgoesovertheinitialconditionsle,whatitcontains, howit isordered,andhowtoaccessspecicvaluesofinterest( Springel 2005b ).Thisisne 63

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asfarasitgoes,butfranklyitisabitTOObrief,andcouldst andtobeextendedseveral morepages,allowingroomforamorein-depthexplanationof howtheinitialconditions leswork.Theauthoroftheprogramrealizedthatifhewasto explaineverythingthat neededtobeknownabouttheinitialconditionsleinwords, itwouldbehardtofollow andbeaninefcientuseofspace.Ratherthanwriteaverylon g,detaileddescription intheuser'sguide,heincludeda C programdesignedtoreadtheinitialconditionsle. Bystudyingthisprogram,itispossibletogureoutallthat isneededtoaccessanypart oftheinitialconditionsleonemightneedto–andonceyouc anaccesssomething,it isusuallypossibletogureoutawaytowriteoveranypartyo umayneedto,oreven createyourownsetofinitialconditions,assumingthatist hewayyouwanttogo.With thesetwolesworkingintandem,itispossibletogureouth owtheinitialconditions leswork,whichiscertainlynecessarytouse GADGET effectively. Theauthorof GADGET doessupplyonesampleinitialconditionsleforthecase inwhichIaminterestedin–acosmological,expandingunive rse.Thislebecamemy templateforconstructingfutureinitialconditionsles, soIspentalotoftimegettingto knowandunderstandit.Onceyougetusedtoit,thestructure oftheselesdoeskindof makesense.Thedataisbrokenupintoblocksofsimilarinfor mation,soonceyound theblockyouarelookingfor,thewholethingcanbereadinto whateverprogramyou areusingtoanalyzethedataandworkedwithinwhateverways eemsbest.Therst blockistheheaderinformation,containinginformationth atistrueforalltheparticles inthesimulation.Thisblocktellstheuserthingslikehowm anyparticlesofeachtype (baryons,colddarkmatter,etc.–itcankeeptrackofupto 6 differenttypesofparticles) areinthesimulation,thetimethatthesnapshotbelongingt othatheaderblockwas taken,andconstantsliketheHubbleConstant,theboxsize, orthematterandenergy densities.Thesampleinitialconditionsleincluded 32768 darkmatterparticlesand 32768 gasparticles,usedaboxsizeof 50 megaparsecs/ h ,andaninitialtimeof z=10 64

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Thenextblockofinformationintheinitialconditionslel iststhepositionsofallthe particles.Thisandthefutureblocksofinformationsplitt heparticlesintogroups,one foreachparticletypebeingused,sothattheorderof,say,t hegasparticlescanchange inthepositionblock,butallthegasparticlesstaytogethe r.Ofcourse,sincethisisa threedimensionalsimulation,the x y ,and z coordinatesofeachparticlearerecorded inthepositionblocktogether.Thenextblockcontainsthev elocityinformationforeach particle,onceagaininthreedimensions,andonceagainall particlesofthesametype staytogether.Whenusingcosmologicalintegration,these arecalculatedtobeinunits ofkm/sec,andtogetpeculiarvelocitiesfromthese,theyju stneedtobemultipliedby p a .However,ashasbeenpointedout,theseparticlesarefreet omovearoundintheir datablockinordertoallowparallelprocessingbythecompu terandmakeforquicker leinputandoutput.Thismeansthattheonlywaytotrackapa rticleistodetermineits uniqueparticleidenticationnumber–whichallparticles have–andusethisidentication numbertoseewhereaspecicparticleisinanygivensimulat ion.Theidentication numbersforalltheparticlesaregiveninthenextblockofda ta.Thesearethemost importantblocks,andtheymustbeincludedineveryinitial conditionsle.Thereare severalmoreoptionalblocks,however,whichmaybeusefuld ependingonthetypeof simulationbeingrun.Forinstance,thenextblockcontains themassesofeachparticle, andallowstheusertoinputdifferentmassesforeachpartic leifitbecomesnecessary. Figure 5-1 showsan x y overheadviewofthepositionsoftheparticlesinthis sampleinitialconditionsle.Theredparticlesarethegas particlesandtheblack particlesarethecolddarkmatterparticles.Ascanbeseen, theinitialconditionsare basedonagrid,withtheclumpsofgasatthecentersofthesqu aresoutlinedbythe colddarkmatterclumps.Clearly,theparticlesarenotallc lusteredexactlyontopofthe gridpoints,butareratherspreadoutaroundeachgridpoint .Thesegridpointsaretaken moreasapproximatecenters,aroundwhichtheaccompanying particlesareperturbed. 65

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0 0 Figure5-1.Initialconditionssuppliedwith GADGET 50 Mpc/ h by 50 Mpc/ h gridwithblack particlesdarkmatterandredparticlesgas. 66

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Onceallofthesethings(the makefile ,theparameterle,andtheinitialconditionsle) areproperlyadjustedandreadytogo, GADGET cannallyberun. 5.2Running GADGET-2 Because GADGET issuchacomplexandcomplicatedprogram,itisdifculttom ess aboutwithitsuccessfully.Unlessyouhavealotofexperien cewithitalready,tryingto alterthecodeitselfisalosingproposition. GADGET isamassivelyparallelcode,written usingthe MessagePassingInterface ( MPI ),sotomakealterationstothecode,one mustalsohaveasoundknowledgeof MPI .Andalthoughtheuseof MPI doesmakethe codemorecomplicated,thetrade-offincomplexityisdeni telyworthit,astheability torun GADGET inparallelcansavetheuseragooddealoftimewhenmakinges pecially longruns. Ihavealreadymentionedthelargenumberofsourcelesfrom which GADGET is compiled.Itwouldprobablybeapaperinandofitselftotryt oexplainexactlyhow GADGET takestheinitialconditionsthattheuserprovidesanduses thisdatatosimulate theentireuniverse,fromanearlytimetoalatertime.Thege neraltermsmentioned sofar–thatis,that GADGET usesa TreePM algorithmtomodelgravitationalforcesonthe differenttypesofparticlesinanexpandingcosmologicalu niverse–shouldbesufcientto explainthebasicsofthephysicsthat GADGET usestomakeitssimulations.AsIdidnot alteranythinginthesourceles,ifmoreinformationisdes ired,Irecommendlookingup thecodepaperon GADGET-2 andtheuserguidethatexplainshowtooperateit( Springel 2005a b ).Moreinformationcanbefoundintheoriginalcodepaperfo r GADGET-1 ,the originalincarnationof GADGET ( Springeletal. 2001 ).Theseshouldmorethansatisfy anycuriosityonecouldhaveabout GADGET-2 anditsusage. 5.3Output Instead,Iwouldliketojumpaheadtotheoutputof GADGET-2 .Imentionedbriey that GADGET outputssnapshotlescontaininginformationabouttheuni verseattimes speciedbytheuserintheparameterle.Theselesarehowt heuserexaminesthe 67

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universehehascreatedfromnothing.Theauthorof GADGET suppliesashortprogramto demonstrateanexampleofhowtotakeasnapshotleanddispl ayitgraphically,which candenitelyhelpwithanalysis.Usingthisprogramisnots trictlynecessary,however, forthesimplereasonthatthesnapshotlesareformattedin thesamewaythatthe initialconditionslesarestructured.Thismeansthatifo nehasspenttimelearning theinitialconditionsles–whichisfairlylikely,sincet omakeyourownrunsof GADGET youhavetobeabletocreatenewinitialconditions–theneve rythingthatisneededto manipulatetheinformationinthesnapshotlesisalreadyk nown.Iftheuserwantsto useadifferentgraphingprogramthantheoneinwhichtheexa mpleprogramiswritten, itshouldnotbeaproblemwithknowledgeofhowthesnapshot lesarestructured. Similarly,theuserisnotlimitedtothetypeofgraphthesup pliedexampleprogramcan produceaslongastheyknowwheretondtheinformationthey wanttoanalyzeinthe snapshotles. Still,despitethewealthofinformationavailableinthesn apshotles,oneofthe mostusefulthingsisperhapsthe x y ,and z coordinatesofeachparticleateachtime asnapshotleistaken.InFigure 5-2 ,Icreatedanothergraphwiththe z dimension attenedintoaplane.Thisgureshowstheuniverseatthepr esenttime(witharedshift of 0 ,oranexpansionfactorof a=1 ),andshowssomeimportantfeatures.Thereis agreatdealofdifferencebetweenthisgureandthegurede monstratingtheinitial conditionsdistributionofparticlesataredshiftof 10 .Ratherthanafairlyhomogeneous distribution,largeclumpsofgasparticlesarescatteredh ereandthereaboutthe grid,surroundedbylargevoidsemptyofmatter.Thisiswhat wouldbeexpected,as observationsoftheuniversetodayobviouslyshowgalaxies separatedbylargevoids,so Icanbefairlycondentthat GADGET doeswhatitisadvertisedtodo–andnowthatIknow howitworks,Icanuseittorunsimulationsofmyown,withini tialconditionsofinterest tome. 68

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Figure5-2.Distributionofgasparticlesataredshiftof 0 inan x y plane.Onceagain,the unitsareMpc/ h 69

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CHAPTER6 ANALYSISOFHELIUMRECOMBINATIONUSINGGADGET-2 6.1CreatinganAppropriateSetofInitialConditions Toexaminetheeffectsofheliumrecombination,Iobviously neededtocreateinitial conditionsthathadheliuminthem.AsmentionedinChapter 5 ,theauthorsuppliedaset ofinitialconditionswiththe GADGET-2 code,andsoitseemedtheeasiestwaytocreate myowninitialconditionswouldbetomodifythisexistingl e. Therstchangethathadtobemadewaschangingthestartingt imeofmy initialconditions.Ineededtobeabletochoosedifferents tartingtimes;evenifIhad decidedtomakeallmyrunsfromthesamestartingtime,thesu ppliedinitialconditions correspondedtoastartingredshiftof z=10 ,whichwasfartoolateinthestructure formationprocesstobeofanyuseformypurposes.Iwasablet omakethisalteration totheinitialconditionsbyscalingthemwiththeexpansion factor.Amoredetailed treatmentisgivenintheappendix,buttosummarize,theuc tuationsdependina relativelysimplewayontheexpansionfactoras: / a (6–1) v / p a. (6–2) Withtheseequations,itispossibletotakethegivensetofi nitialconditionsandscale thembacktowhateverexpansionfactoroneisinterestedin. Thenextstepistoseparateouttheheliumatomsfromthehydr ogenatoms, ratherthantreatallthebaryonicmatterasonecategory.Of course,oneofthemain differencesbetweenhydrogenandheliumduringtherecombi nationeraistheir relativevelocities.Sinceheliumwillstarttorecombineb eforehydrogen,ataredshift ofabout z=2500 ,heliumatomswillbemovingfasterthanhydrogenatomsduri ngthe recombinationera,whichdonotstartrecombininguntilclo serto z=1300 ( Seageretal. 2000 ; Switzer&Hirata 2008a ; Rubi no-Martnetal. 2008 ; Dubrovich&Grachev 2005 ; 70

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Wong&Scott 2007 ).Beforeeachelementstartsrecombining,itsuctuations are stationary–onlyaftereachelementstartstorecombinecan theseuctuationsstartto change.Sowhenitcomestochangingtheinitialconditions le,thevelocitiesofthe colddarkmatterparticlescanbeleftalone,astheyarefree tochangefromamuch earliertimethanthebaryonuctuations,andthehydrogena ndheliumvelocitiescanbe scaledasfractionsofthecolddarkmatterparticlevelocit ies.Startingfromthecontinuity equation( Liddle&Lyth 2000 ) 1 : d dt = 3H (6–3) H= 1 3a r v (6–4) = 3 1 3a r v (6–5) + 1 a r v=0, (6–6) andmakingthesubstitution: = 1 _a r v, (6–7) Iamleftwith: + _a a ( 1 _a r v)=0 (6–8) + _a a =0 (6–9) = a _a (6–10) Inordertoget asafunctionoftheexpansionfactor,Ineedtondouthow depends ontheexpansionfactor.Fortunately,Equation( 6–1 )tellsmethatthedensityuctuations areproportionaltotheexpansionfactor,soifItaketheder ivativeofthisequationwith 1 ConversationswithDr.Frywerealsoveryhelpfulinmakingt hesecalculations. 71

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respecttotime,Ind: / _a. (6–11) CombiningEquations( 6–10 )and( 6–11 ),Iget: / _aa _a (6–12) / a. (6–13) ThismeansthatIhavethreeequationsfor ,oneforeachspeciesIamkeepingtrack of.Ashasalreadybeenmentioned,hydrogenandheliumatoms donotstartmoving untilaftertheystarttorecombine,sotheirvelocitiesdon otstarttogrowaccordingto thisequationuntilaftertheirdecouplingtimes.Keepingt hisinmind,herearethethree equationsfor thatIhavebeenlookingfor: CDM = Ca (6–14) He =0,a < a HeDec (6–15) He = C(a a HeDec ),a a HeDec (6–16) H =0,a < a HDec (6–17) H = C(a a HDec ),a a HDec (6–18) BytakingtheratiosIaminterestedin,Icaneliminatetheco nstant C andndhow thevelocitiesofhydrogenandheliumrelatetothevelociti esofthecolddarkmatter particles.Takingthedecouplingredshiftstobe z=2500 forheliumand z=1300 for hydrogengivesvaluesfortheexpansionfactorof a HeDec =0.00040 and a HDec =0.00077 so: He CDM =1 0.00040 a (6–19) H CDM =1 0.00077 a (6–20) 72

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Bysolvingthesetwoequationsforwhateverexpansionfacto rIaminterestedinusingas astartingpoint,Icanchangethespeedoftheheliumandhydr ogenparticlesinorderto distinguishthemfromoneanother. Obviously,itisimportanttokeeptrackofwhichparticlesa reheliumandwhich arehydrogen.Becauseoftheway GADGET works,thespecicorderoftheparticlesin theinitialconditionslesgetsmixedupastheprogramruns ,soIamnotabletojust calltherstquarterofthegasparticlesheliumandthentre attherstquarterofthe gasparticlesasheliumintheoutputles.WhatIdidinstead wasrandomlychange twenty-fourpercentofthebaryonicmattertoheliumandlea vetheresthydrogen.Ikept trackofwhichparticlesImadeheliumthroughtheuseofalar gearray,whichIwroteto alewhenIwasdoneassigningheliumparticles.Sincethe ID numbersoftheparticles in GADGET neverchange,thisleuniquelyidentiedwhichparticlesw erehelium.By utilizingthislealongwiththeoutputsnapshotsfrom GADGET ,Iwasabletokeeptrackof whichparticleswerehydrogenandwhichwereheliumwhenana lyzingmydata.So,by manipulatingtheinitialconditionslefor GADGET inalloftheseways,Iwasabletomake runsof GADGET thatkepttrackofheliumseparatelyfromhydrogenandtheni nterpretmy results. 6.2EvolutionStartingataRedshiftof 1250 Forthisrun,Iusedthesuppliedinitialconditionslewith thealterationsoutlined above.Thematterdensityparameter, n 0 ,wassetto 0.3 ,thedensityparameterforthe cosmologicalconstantwassetto n =0.7 ,andthedimensionlessHubbleparameter was h=0.704 .Isetthestarttimeto z=1250 ,madehalfoftheparticlescolddark matterandtheotherhalfbaryonicmatter,andofthebaryoni cmatter,Imadetwenty-four percentheliumandlefttherestashydrogen.Tomakethisdif ferentiation,Ifollowedthe procedureoutlinedabove,andusingEquations( 6–19 )and( 6–20 ),Igotvelocityratios 73

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for z=1250 ,or a=0.0007994 ,of: He CDM =0.50 (6–21) H CDM =0.037. (6–22) Usingtheseinitialconditions,Iobtainedseveraloutputs napshotsatvarious times.Ataredshiftof 10 ,theuniversecanbeseeninFigure 6-1 .Clustersofmatter canclearlybeseenbythistime.Theyareevidentatearliert imesaswell–Figure 6-2 showsjustthehydrogenandhelium–whichareofprimaryinte resttome–ataredshift of 91 .Lookingatdifferentsnapshotsofmyresultsisnotthebest waytoanalyzethis data,though.Fromthesegraphics,itisclearthatthematte risclusteringproperly, asitshouldintherealworld,butthedifferencesbetweenth ehydrogenandhelium arenotparticularlynoticeable.Todeterminetheeffectst hattheearlierrecombination timeofheliumwillhaveontheresultingstructures,Ihavet oexaminetheclustering moremathematically.Thesnapshotlestellmewhereeveryt hingisatanypointin time,sobytrackingaclump'scompositionandstructureast heredshiftgetssmaller (or,equivalently,astheexpansionfactorincreases),Ica nseehowtheheliumevolves relativetothehydrogen.Tocollectthisstatisticalandma thematicaldata,Ihadto writeanewprogramwhichdeterminedallthequantitiesIwan tedtond.Tousethis program,Irstselectedaclumpofinterest.Ifedmyprogram theapproximate x y ,and z coordinatesofmyclump,aswellastheapproximateradiusof saidclump,andthenlet itloose. TheprogramthatIdesignedtofulllthisfunctionwasfairl ycomplicated.Ittook theinitialcoordinatesIsupplieditwithandfoundthecent roidofthesphereofmass dictatedbytheradiusIalsosupplied.Ofcourse, GADGET usedaperiodicboxtomodel theuniverse,soifItookaclumptooclosetoanedge,theprog ramhadtocorrectly accountfortheradiusextendingtotheothersideoftheboxw ithoutcreatinganyerrors orgivingwildlyinaccuratecentroidvalues.Sincethecent roidsareproportionaltothe 74

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0 0 Y Figure6-1.A y z viewoftheuniverseat z=10 .Theboxhassidesof 50 Mpc/ h ,andthe blackparticlesaredarkmatter,theredparticlesarehydro gen,andtheblue particlesarehelium. 75

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0 0 Y Figure6-2.A y z viewoftheuniverseat z=91 .Theboxhassidesof 50 Mpc/ h ,thered particlesarehydrogen,andtheblueparticlesarehelium. 76

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summationofallthepositionvalues,ifyouhavesomeveryla rgevaluesandsomevery smallvalues,your”centroid”willbesomewhereinthemiddl eofthebox,ratherthanat theactualcentroid.Itisthesekindsofunfortunatemistak esthattookthemosteffortto correctforandpreventinmyprogram. Oncetheprogramfoundthecentroidusingtheinitialvalues Isupplieditwith, itcheckedtoseehowfarawayfromtheestimatedpositionthe calculatedcentroid waslocated.Ifitwastoofaraway(and”toofaraway”isdicta tedinmyprogramasa certainfractionoftheradiusoftheclumpthatIsuppliedit with),theprogrampickeda pointfairlyclosetothecalculatedcentroidasthenew”ini tialposition”andcalculated thecentroidagain.Itrepeatedthisprocessuntilthecalcu latedcentroidwasclose enoughtothe”initialposition,”atwhichpointithadaclum ptoevaluate.Myprogram alsofoundthestandarddeviationsofthehydrogenandheliu mparticlesineachclump thatIwastracking,whichwasnottoohardonceIhadtheoverh eadinplacetondthe centroids–thehardestpartofndingthesevariousvaluesf oreachclumpwasclearly delineatingtheclump,regardlessofsuchthingsaswhether itwassplitacrossthebox ornot.AfterIknewwhatparticlesbelongedtotheclump,itw asnottoodifculttoget thecentroidofthehydrogenandheliumparticles,aswellas theirstandarddeviations andthenumberofeachtypeofparticleintheclump.Furtherm ore,sinceIknewhow massiveeachparticletypewas,onceIknewhowmanyparticle sofeachtypewasin theselectedclump,Icouldeasilygureoutthetotalmasses foralltheparticlesofa particularspeciesinaclump. Inordertogetagoodoverviewofmydata,Itrackedtheevolut ionoffourdifferent clumps.Thisway,Icouldmakesurenoanomalousdataorstati sticaloutliersinterfered withmyresults.ThedataforeachclumpiscollectedinTable s 6-1 6-2 6-3 ,and 6-4 Thereisalotofdatacontainedinthesetables,andseverald ifferentwaystoanalyze it.Firstofall,themotionofthecentroidsisobviousbyjus tlookingatthelocations foreachclusterastimepasses.Obviously,theyallmovearo und,somesubstantially. 77

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Table6-1.Dataconveyingthehydrogenandheliumcompositi onof”Clump1.” ParticleNum.HCentroidHeCentroidStand.Dev. zHHex(kpc)y(kpc)z(kpc)x(kpc)y(kpc)z(kpc)H(kpc)He(kp c) 380190482476220994108192494021115109943232315630018546248582031610837246662025010792269225922363018925096191611088225132188721099730433128186186502516119734110392515619707109871343140714726274256241894211101257251885511220123312941163561012623018132111262619718176111389358969139811226858180321143926885179941149910431047724761432753818055116842753018007117629861037565601782834218234117952831018133118931015108444612189293401838311778293741835111869100010303587526230438187211179230447186511181491495427943273313601891312222313431886412249115311822110263033172719125129913175219080129901076110817108131033019198751368133005198501370595094513110432634253205181465834252205431464788090510975281366612257715315366992257715330697692 78

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Table6-2.Dataconveyingthehydrogenandheliumcompositi onof”Clump2.” ParticleNum.HCentroidHeCentroidStand.Dev. zHHex(kpc)y(kpc)z(kpc)x(kpc)y(kpc)z(kpc)H(kpc)He(kp c) 3801846144651311541294453853105311944614454030025884459603307718994637933178144650964905236182624398833774180444062343036904533442218613549440763302329884405033403248239554176147592844098341221482445203462710232667245411657284383833827189444272344341606243222829150264394033562275244221341632530212820177248224325733103321143613335373150157913785649224327432603429343619328524449128911224453224348232009566243574320655903112685535602543904312447215438763120876031041756276925449713051590054495930429924496669321902318477153065589514771430571892711761244171221402467103189411356467573191811357101998813268685445447322551342845479322641342810631049103300106444854318271365444798318401368416431663 79

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Table6-3.Dataconveyingthehydrogenandheliumcompositi onof”Clump3.” ParticleNum.HCentroidHeCentroidStand.Dev. zHHex(kpc)y(kpc)z(kpc)x(kpc)y(kpc)z(kpc)H(kpc)He(kp c) 380147542571456902167124774590222042344635263003081161595458692173215424678222392477246972361314226034592320262288646031208702542254118613141248046183200262948465062033122132201147145471968461631992222954624420361200821271161665617134660219462184946809195642000224991190708924642519672850466371983718212105721967140546621192743554667619399141416115626292496194643618756496414651718843138214064428797489634647618045490384659118031101410363533010848398466381711248338467061716411201060274671464807347176161324803347266161521338129221603213477044690015183476914690215209109111191771024247268458861437647306459021440410881076138112754674744151139974680844175139911192117610111438246481421071418246492421211419610641052 80

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Table6-4.Dataconveyingthehydrogenandheliumcompositi onof”Clump4.” ParticleNum.HCentroidHeCentroidStand.Dev. zHHex(kpc)y(kpc)z(kpc)x(kpc)y(kpc)z(kpc)H(kpc)He(kp c) 3802005940062328992819139904327672812841893658300394107396613416727412398403413227480512048812363138439265342602711639380339662740243014045186300763928834189261833921233776261613740338914721560392673431326506393993383326620260222881161955839965337312601239960336022611018701627914161104045134028249104052034123250732565244272385108408823429924615407863437124754143615465645712341215340902389041202342132405315761542445171414166034311233664166534394234701267118435673188420073460122346419933458722390110211252769618642363343352116742346343452117211821087217392104298533690195004297933678194961100107317844249437043285317325436573288217347109610961325938264544132252134624548632255134491007997103300106444854318271365444798318401368416431663 81

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Sincethismotionisinthreedimensions,agraphoftheirpat hsisalittlecomplicated toproduce.ThebestwayIfoundofdoingthiswastogivethree graphs,oneforeach dimension.Ialsodecidedtojustgraphonecluster;whenIgr aphedtheentire 50 Mpc box,theindividualclustersdidnotmovetoomuch,andthein dividualmotionsofthe hydrogenandheliumwerealmostimpossibletosee.However, becauseClustersTwo andFourdoendupjoiningatverylatetimes(around z=13 orso),Ididmanagetoget thetailendofClusterFour'smotioninthesegraphs.Themai nfocusofFigure 6-3 is themotionofClusterTwo,however.Thestartingpointsofth ecentroidscanbefoundby lookingatTable 6-2 ;forthe x y graph,theystartinthemiddle,towardsthebottom.In the y z graph,theystartinthebottomleft,andinthe z x graph,theystartattheleft.The solidlinesmarkedbyplussignsindicatethemotionofthehe lium,andthedottedlines markedbytheasterisksindicatethemotionofthehydrogen. WhenClusterFourenters thegraphatlatetimes,thelinetypesarethesame,butthehy drogenisdistinguishedby diamondsandtheheliumisdistinguishedbysquares. Fromthesegraphs,onethingisclear–astimepasses,thetwo centroidsdoget closerandclosertogether,aswouldbeexpected.Strongerc onclusionsareharder todraw.Lookingatmygraphs,aswellasthedataformyotherc lusters,itappears thatformosttimes–excludingtheveryearlytimes,whereit ishardtoseeexactlywhat ishappening,andlatetimes,wherethecentroidshavebasic allyjoined–thehelium centroidmayleadthehydrogencentroidabit.IwishIcouldb eabitmorecertain,but itishardtosaywhatisgoingonforsure.Iwouldsaythattheh eliumlinesareabit smootherandthatthehydrogenmotionisfollowingtheirmot iontoasmalldegree,but theredoappeartobetimeswherethehydrogencentroidseems tobeontheinsideof thecurvesthatthemotionismaking,whichseemstoimplytha ttheheliumisfollowing thehydrogenatthesetimes.Theproblemmaylieinthefactth atitisjusttoohardto puttogetherthesethreegraphsintoareasonableapproxima tionofthreedimensional motion.Ididtrytomakeathreedimensionalgraphofthemoti onofClusterOne,andit 82

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Figure6-3.MotionofClusterTwointhreedimensions.Thedo ttedlineswithasterisksindicatethehydrogenandthesoli d lineswithplussignsindicatethehelium.83

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appearsthattheheliumisontheinsideofthecurvetheirmot ionmakesformostoftheir evolutiontime,whichwouldimplythatthehydrogenisfollo wingthehelium.Thisgraph canbeseeninFigure 6-4 ,butitiskindofhardtotellexactlywhatishappening. x axis y axis 9000 10000 11000 12000 13000 14000 15000 16000 z axis H Cluster He Cluster 22000 24000 26000 28000 30000 32000 34000 36000 38000 17500 18000 18500 19000 19500 20000 20500 21000 21500 22000 22500 23000 z axis Figure6-4.Thisisathreedimensionalplotofthemotionoft hecentroidsofClusterOne. Thesolidlineishydrogenandthedashedlineishelium.Them otionstarts inthelowerleft,andmovestotherightastimepasses. SomethingIcanbeabitmoredenitiveonistheseparationbe tweenthetwo centroids.Thereisnoquestionthatastimepasses,thedist ancebetweenthehydrogen andheliumcentroidsgetssmaller.Thismakessense–astime passes,thevelocitiesof allthebaryonicmattertendtogotoaboutthesameamount.Si ncethevelocitiesofthe clustersthemselveswillalsotendtoslowastheygetlarger ,thecentroidsofboththe heliumandhydrogenwillgotothecenteroftheseclustersan dthenslowthemselves. InFigure 6-5 ,theseparationbetweentheheliumandhydrogencentroidsi sshownfor 84

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eachcluster.Forsomethetrendismoredramaticthanforoth ers,buttheoverallmotion Figure6-5.Thisgraphshowstheseparationoftheheliumand hydrogencentroids,in kiloparsecs,astimepasses.Thesolidlinewithdiamondsis ClusterOne,the dottedlinewithcirclesisClusterTwo,thedashedlinewith asterisksis ClusterThree,andthedashdottedlinewithplussignsisClu sterFour. clearlyshowsthetwocentroidsgettingclosertogetherast imepasses.Despitestarting outatavarietyofdifferentdisplacements,allthecluster seventuallydecreasetheir separationstoaroundftykiloparsecs,whichdenitelysh owsthatastimegetscloser tothepresent,thehydrogenandheliumparticleswillinter mixandhaveoverlapping centersofmass. Anotherimportantstatisticaltoolforanalyzingthemotio noftheseclustersisthe variance,anditssquareroot,thestandarddeviation.Thes tandarddeviationcanbe thoughtofasameasureofhowcloselyclumpedtogetherthema tterparticlesare,asit 85

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isbasicallyameasureofhowfastthedensityfallsoff.Thes tandarddeviationsforeach ofmyclustersisplottedinFigure 6-6 .Thisgraphiskindofbusy,butthelatertimes, Figure6-6.Thisgraphshowsthestandarddeviationsoftheh ydrogenandhelium clustersastimepassesinkiloparsecs.Eachclusterisrepr esentedbya differentlinetype–ClusterOneisthesolidline,ClusterT woisthedottedline, ClusterThreeisthedashedline,andClusterFouristhedash dottedline. Thehydrogenisfurtherdemarcatedbyplussigns,whiletheh eliumis representedbyasterisks. wheretheyalltendtobeprettyclosetogether(whichmakest hemhardtotellapart), arenotasimportant.Clearly,thestandarddeviationsallg etmuchsmalleratthoselate times,whichmakessense.Asthetimeapproachesthepresent ,theclustersshouldbe wellformed,andthesecertainlyappeartobethat.Bythetim etheredshiftapproaches ten,allofthestandarddeviationsarearoundthe 100 kpcrange.Infact,theystayfairly 86

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constantaroundthatrangefromaroundaredshiftofthirtyo rfortyonwards.Itisthe earlierbehaviorwithwhichIamconcerned.Atearlytimes,t hestandarddeviationstend tobefairlylarge,butstarttotrenddownwardrathersharpl y.Astheheliumparticles begintomovebeforethehydrogenparticles,Iwouldexpectt hemtostarttoform clusterssooner,whichwouldthereforestarttocollapseso oneraswell.ForClusters TwoandFour,thiscanbeseeninapronouncedmanner–foralar geamountofthe time,theheliumisconsistentlymoretightlyboundtogethe rthanthehydrogencluster thatisgatheringaroundit.However,thereislittlediffer encebetweenthestandard deviationsforthehydrogenandheliumforClusterOne,andt hehydrogenstandard deviationisactuallysmallerthantheheliumstandarddevi ationformuchofCluster Three'sevolution.Sinceclustersarehardtoclearlydelin eate,acertainamountoferror isintroducedwhentryingtodothistypeofanalysisonthem– thismayaccountforthese discrepancies.Idonditencouragingthattwoofthecluste rsIwastrackingseemto givetheexpectedresults. Analmeasureoftheevolutionoftheseclustersliesinther atioofheliumto hydrogenasafunctionofthecluster'stotalmass.Inthetab lesabove,Ilistedthe numberofeachtypeofparticle,hydrogenandhelium,astime passed.Eachparticle hasamassof 4.235 10 10 M / h ,soboththeratioofheliumtohydrogenandthetotal massforeachclusteriseasytocalculate: m total =(n He +n H ) 4.235. (6–23) Theunitsare,asstatedabove, 10 10 M / h .Thisgraph,foreachoftheclustersItracked, appearsinFigure 6-7 .Itisknownthatabouttwenty-fourpercentofthebaryonicm atter formedintheBigBangishelium,withtherestbeinghydrogen ( Peebles 1993 ).Ifthe particleswerealldistributedcompletelyhomogeneouslyt hroughouttheuniverse,they wouldgivearatioof: 0.24 0.76 =0.32. (6–24) 87

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Figure6-7.Thisgraphshowstheratioofheliumparticlesto hydrogenparticlesasa functionofthecluster'smassinunitsof 10 10 M / h .Thesolidlinewith diamondsisClusterOne,thedottedlinewithcirclesisClus terTwo,the dashedlinewithasterisksisClusterThree,andthedashdot tedlinewith plussignsisClusterFour. Signicantdeviationsfromthismassratioareinteresting ,andarelikelytheresultof theearlierrecombinationtimeofhelium.Figure 6-7 clearlyshowslargedeviationsfrom thismeanvalue.Becausethisgraphisafunctionoftotalmas sratherthanredshift,it isabithardertoseehowtheseclustersevolveintime.Howev er,thedatashowsthat clustersgetbiggerastimepasses(which,asdescribedabov e,iswhatwasexpected), sotherightendsofthelinesarethelatesttimes,whilethel eftendsofthelinesareearly times.Lookingatthesegraphs,itisclearthatastimepasse s,themassratiosforallthe clustersheadtowardstheexpectedvalueof 0.32 .Infact,thebiggerclustersreachthese 88

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valuesatlatetimesandremainrelativelyconstant,wherea sthesmallerclustersarenot abletoreachthesecompletelyhomogeneousratios.Cluster sTwoandThreearehighly enrichedinhelium,whereasClustersOneandFouraredecie ntinhelium. 6.3EvolutionStartingataRedshiftof 1000 Theonlydifferencebetweenthisrunandthepreviousrunwas theinitialstarting time.Thematterdensityparameter, n 0 ,wasstillsetto 0.3 ,thedensityparameterforthe cosmologicalconstantremainedat n =0.7 ,andthedimensionlessHubbleparameter was h=0.704 .However,Ididchangethestarttimeto z=1000 .Ilefthalfofthe particlesascolddarkmatterandtheotherhalfasbaryonicm atter,andofthebaryonic matter,Imadetwenty-fourpercentheliumandlefttheresta shydrogen.Tomakethis differentiation,Ifollowedtheprocedureoutlinedabove, andusingEquations( 6–19 )and ( 6–20 ),Igotvelocityratiosfor z=1000 ,or a=0.0009990 ,of: He CDM =0.60 (6–25) H CDM =0.23. (6–26) Usingtheseinitialconditions,Iobtainedseveraloutputs napshotsatvarioustimes. Ataredshiftof 50 ,theuniversecanbeseeninFigure 6-8 .Ididnotmakethisrunlast aslong,becauseasImentionedafewtimesabove,thelatetim esarenotasinteresting tome.Itismainlytheearlyandintermediatetimesduringwh icheffectsduetothe earlierrecombinationtimeofheliumwillbemostnoticeabl e.Anearlierviewofjustthe hydrogenandheliumforthisruncanbeseeninFigure 6-9 ,whichistakenataredshift of 166 .Onceagain,Icanseethatthematterisclusteringproperly ,butthedifferences betweenthehydrogenandheliumarenotparticularlynotice able.Togetabetteridea ofhowthestructuresareevolving,Imustonceagainturntot heprogramIdesignedfor thatpurpose. Inthiscase,Idecidedtotracktwoclusters,sinceIwasnote xpectinganyreally drasticchangesfromthe z=1250 runs.Thedatafortheseclustersiscollectedin 89

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0 0 X Figure6-8.An x y viewoftheuniverseat z=50 .Theboxhassidesof 50 Mpc/ h ,and theblackparticlesaredarkmatter,theredparticlesarehy drogen,andthe blueparticlesarehelium. 90

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0 0 X Figure6-9.An x y viewoftheuniverseat z=166 .Theboxhassidesof 50 Mpc/ h ,the redparticlesarehydrogen,andtheblueparticlesareheliu m. 91

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Tables 6-5 and 6-6 .Usingthedatacontainedinthesetables,Icanfollowthesa me analysisthatIdescribedabove.First,Iwillcheckthecent roidmotions.Onceagain,I amjustgoingtolookatthemotionofonecluster–ClusterTwo ,thistime.Thismotion canbeseeninFigure 6-10 .Thestartingpointsofthecentroidscanbefoundbylooking atTable 6-6 ;inallthreecases,themotionstartsintheupperrightofth egraphs.The solidlinesmarkedbyplussignsindicatethemotionofthehe lium,andthedashedlines markedbytheasterisksindicatethemotionofthehydrogen. Inthe z x graph,itappearsthatthetwocentroidsremainfairlyclose together throughouttherangeofmotion–notalotcanbegleanedfromt hisgraph.However, lookingattheothertwographs,itdoesappearthattheheliu mleadsthehydrogena littlebit.Inthe x y graph,especiallytowardstheend,itappearsthattheheliu mdoes notchangetoomuch,andthehydrogencentroidisslowingapp roachingthehelium centroid.Thistrendseemsabitmoreclearinthe y z graph–thefeaturesinthehelium graphseemtoleadthesamefeaturesinthehydrogengraph.On eprominentfeature isthespiketothelowerrightat y=17000 kpc–thehydrogenspikeisclearlylagging byabout 500 kpc,andasimilartrendofheliumleadingcanbeseenlaterin thegraph, althoughitisnowherenearthispronounced. Thistakesmetothecentroidseparation.Onceagain,therei snoquestionthatthe twocentroidsaregettingclosertogetherwithtime,aswoul dbeexpected.Thisgraph canbeseeninFigure 6-11 .SinceIonlycarriedouttheanalysistoaboutaredshiftof 50 ,theclustersenduparound 100 kiloparsecsapart,buttheyaredenitelytrending downward,andwouldpresumablygetevencloserastimeappro achesthepresent.It istheearlyandintermediatetimesthatareofinterestform e,however,sothislatetime behaviordoesnotparticularlyconcernme. Onceagain,lookingatthestandarddeviationsshouldprovi desomeinteresting data.Thestandarddeviationsforbothoftheseclustersapp earsinFigure 6-12 .My resultsforthesetwoclustersseemtoagreewithmyexpectat ions.Formostofthe 92

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Table6-5.Dataconveyingthehydrogenandheliumcompositi onof”Clump1.” ParticleNum.HCentroidHeCentroidStand.Dev. zHHex(kpc)y(kpc)z(kpc)x(kpc)y(kpc)z(kpc)H(kpc)He(kp c) 8112416210998203413390611057199843354550034946666301791038158603032613821558130319501849185462026543381370827587448513494278544148423744824676396115069281503772143272848943044332367154454300145322837240831415528688337333693011675328961458828241266914281286793321331824712439592214247274826539141842762426772658203198574910150022735149741490627513305330141661474841891510027555408315121277222293244413617352267715605270942643155842721821092048111212621927164972707018421636627043157015239124773213516913267252189169802666811791093753461012631171912617625721725426110119511156136310125341744125783246917513257438657645042012021581751225022212417623250221006911 93

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Table6-6.Dataconveyingthehydrogenandheliumcompositi onof”Clump2.” ParticleNum.HCentroidHeCentroidStand.Dev. zHHex(kpc)y(kpc)z(kpc)x(kpc)y(kpc)z(kpc)H(kpc)He(kp c) 8113471023507521188190043482620793192005707573166644912926506197121084426760200421168056075829546426115245351817792042422017686929853825642448303792496117286924625104167789208454646593672235624792182128489249181751884963835401030119046247921750491982514216933931233343322247125372499516109919025024155899194250224732031724725103162929739252011568197962630241916612639255651561410093254281555110125180717121361664325635167251015425586163081022516781487111318892583818188109942582318075109231369146491371104262771811611156262201817411115984902753759926746180511141026611181391138910019696142613127376180401172327301181881173891393350493155280601809011735279741817411739935946 94

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Figure6-10.MotionofClusterTwointhreedimensions.Thed ashedlineswithasterisksindicatethehydrogenandthe solidlineswithplussignsindicatethehelium.95

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Figure6-11.Thisgraphshowstheseparationoftheheliuman dhydrogencentroids,in kiloparsecs,astimepasses.Thesolidlinewithasterisksi sClusterOne andthedottedlinewithplussignsisClusterTwo. intermediatetimes,theheliuminbothclustersismoretigh tlyboundthanthehydrogen inthosesameclusters.Infact,evenatthelatertimes,theh eliumstillseemstobemore tightlybound,althoughthestandarddeviationsforthetwo elementsaregettingcloser togetherby z=50 Thelastthingtocheckistheratioofheliumtohydrogenasaf unctionofthe cluster'stotalmass.Inthetablesabove,Ilistedthenumbe rofeachtypeofparticle, hydrogenandhelium,astimepassed.Onceagain,eachpartic lehasamassof 4.235 10 10 M / h ,soboththeratioofheliumtohydrogenandthetotalmassfor eachclusteris easytocalculateinthesamemanneroutlinedinSection 6.2 .Thisgraph,forbothofmy 96

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Figure6-12.Thisgraphshowsthestandarddeviationsofthe hydrogenandhelium clustersastimepassesinkiloparsecs.Eachclusterisrepr esentedbya differentlinetype–ClusterOneisthesolidlineandCluste rTwoisthe dottedline.Thehydrogenisfurtherdemarcatedbyplussign s,whilethe heliumisrepresentedbyasterisks. clusters,appearsinFigure 6-13 .Thisgraphshowsthatbothoftheselectedclusters seemtoberatherdecientinhelium.Infact,theyarerarely abovethemeanexpected valueof 0.32 throughouttheirevolution.Bylookingatthedata,itiscle arthatthelate timesforbothclustersareintheupperright,soitdoesseem thatbothclustersincrease insizethroughouttheirlifetimes,butitwouldappearthat theyarequiterichinhydrogen. 6.4EvolutionStartingataRedshiftof 1250 WithaDifferentHeliumMass Forthisrun,IusedthesamesetofinitialconditionIusedin Section 6.2 .The matterdensityparameter, n 0 ,wassetto 0.3 ,thedensityparameterforthecosmological 97

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Figure6-13.Thisgraphshowstheratioofheliumparticlest ohydrogenparticlesasa functionofthecluster'smassinunitsof 10 10 M / h .Thesolidlinewith asterisksisClusterOneandthedottedlinewithplussignsi sClusterTwo. constantwassetto n =0.7 ,andthedimensionlessHubbleparameterwas h=0.704 Isetthestarttimeto z=1250 ,madehalfoftheparticlescolddarkmatterandtheother halfbaryonicmatter,andofthebaryonicmatter,Imadetwen ty-fourpercentheliumand lefttherestashydrogen.Tomakethisdifferentiation,Ifo llowedtheprocedureoutlined above,andusingEquations( 6–19 )and( 6–20 ),Igotvelocityratiosfor z=1250 ,or a=0.0007994 ,of: He CDM =0.50 (6–27) H CDM =0.037. (6–28) 98

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TheonlychangeImadewastothemassoftheheliumparticles. GADGET allowsforeach particle'smasstobechangedseparately,sowithabitoftin keringaroundwithmyinitial conditionsle,Iwasabletogureouthowtochangethemasse sfortheparticlesIwas designatinghelium.Imadethemfourtimesasmassiveastheh ydrogenparticles,and thenranmysimulationagain. Usingtheseinitialconditions,Iobtainedseveraloutputs napshotsatvarioustimes. Ataredshiftof 24 ,theuniversecanbeseeninFigure 6-14 .Agraphshowingonly hydrogenandheliumataredshiftof 101 canbeseeninFigure 6-15 .Lookingatthese twographs,thereisplentyofclustering,asthereshouldbe .However,ataglance, itappearsthattheheliumisactuallyclusteringmoretight lythanthehydrogen,and formingtheseedsaroundwhichthehydrogenclustersform.T hisiswhatIwouldexpect, butthiseffectwasnotvisiblebyjustlookingatthegraphsi nmyrsttworuns.Itwould alreadyappearthatincreasingthemassoftheheliumpartic lesishavingadeniteeffect ontheevolutionoftheseclusters.Ineedtoseeifthisismor ethananillusion,though, andsothenextstepistoperformthesameanalysisonthisdat athatIdidonthelast tworuns. Forthisrun,Itrackedtheevolutionofthreedifferentclus ters,inordertogetanice sampling.ThedataforeachclumpiscollectedinTables 6-7 6-8 ,and 6-9 .Tostart with,IhaveincludedagraphofcentroidmotioninFigure 6-16 .ThistimetheclusterI decidedtoshowisClusterOne.Bylookingatitsdata,Icanse ethatinallthreegraphs, thecentroidsstartedinthelowerleftandevolvedfromther e.Thedashedlineswiththe asterisksshowthemotionofthehydrogen,andthesolidline swithplussignsshowthe motionofthehelium.Asexpected,thecentroidsarefurther apartandmovefasterat earliertimes,slowingdownandgettingclosetogetheratla tertimes.Inboththe x y and y z graphs,theheliumseemstobeleadingthehydrogen–itseems toalmostalways beontheinsideofthecurves,andthesharpbendsandkinksin thehydrogencurves seemtobelaggingthesamefeaturesintheheliumcurves.The hydrogenandhelium 99

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0 0 Z Figure6-14.A z x viewoftheuniverseat z=24 .Theboxhassidesof 50 Mpc/ h ,andthe blackparticlesaredarkmatter,theredparticlesarehydro gen,andtheblue particlesarehelium. 100

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0 0 Z Figure6-15.A z x viewoftheuniverseat z=101 .Theboxhassidesof 50 Mpc/ h ,the redparticlesarehydrogen,andtheblueparticlesareheliu m. 101

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Table6-7.Dataconveyingthehydrogenandheliumcompositi onof”Clump1.” ParticleNum.HCentroidHeCentroidStand.Dev. zHHex(kpc)y(kpc)z(kpc)x(kpc)y(kpc)z(kpc)H(kpc)He(kp c) 12491344033141120652817433389123792799838873826106811333334531248329188337301271529042360234689131223833933121462985134460121143046836793526781115383397612307300173447812150304813574349666710338344371214731448346051231431938336532335719836346141220632373345001268832473319029484889434349461194532625346561251032766299027804171284335244113233535934991114253492928792870356207663597412033353393561811941351553479350430578293573112293339313549112364339482081181726010435359611243934363357571240034209211918282221675435475122973713835227123303667822502268190188563790712938399553807913339397972498259316216351359881302438573358481303738612186319941391363936405137373940536163135963950014921501118433149369751488138219369941482038244174117301014681563684415544382413689815513382071382107286463167367091630738180367601636938221130692174649239361461822538212362611803538315186815926311634453598519319387053601619277387081494941548254603624619373385673616919405387361441121146936499363391961438454363921950838552147197139733504365861976338267365781965938319131189033118259036724198343771636751197883784117091082289966003691319913376203697519992375441383899248816033714420112373583720420200374221201745 102

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Table6-8.Dataconveyingthehydrogenandheliumcompositi onof”Clump2.” ParticleNum.HCentroidHeCentroidStand.Dev. zHHex(kpc)y(kpc)z(kpc)x(kpc)y(kpc)z(kpc)H(kpc)He(kp c) 12491023732004445720274314444054195663411342710681023731944445920270314044055195653401341891398363182444022022631094413219649332533667819333313744376202642979441021975032023220667102353164450742018831404505819689318932095711003431644515020122322445031197333052307948810337311945102201663179449981971329683092417902926994524920326223445115197032614266635611040299645437206203114457872022627192920305943130504541420098321145204198432210227626014346276245930201852601459542014129042733222142482260458962021922334597120008267825981901074226464591719926234746022198581751202016213547210445900199932041461351989220802019139108422176459761970920314615719668125613621181224417864606019618180446114195621234105910113145141846138195551542461721950611587888615148105646455193411154463371928412657557417452612466691906160546606191051339949632036078464201893711146324189471469130654212644971446766185264962246845185671240103446260844907446736181894888946696181971378128339297105484194628217818483104622017731126311253333912248047465201719948081464691712411178062839316147801470911649447910470741647012511070245292554776947462158894784747328158681107705 103

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Table6-9.Dataconveyingthehydrogenandheliumcompositi onof”Clump3.” ParticleNum.HCentroidHeCentroidStand.Dev. zHHex(kpc)y(kpc)z(kpc)x(kpc)y(kpc)z(kpc)H(kpc)He(kp c) 12493822230601969064952336019738674625542242106847262312620219724023498198837164271024199135131236972018273492393119865723327802603781432623406202827413235511985772692519232866764302395720207827524124196847792273423685715831238182015182302393219744778325192315488753524010198598240241461975679642619237141789372457720030925424424197898694248923243561334524666197629877244771955491752713246130590362464219589919724563195018846200318262601835324871198081071024789195671010824152290222170522497719503105212496219277102181957186219020460251491942410901250791920910570163216101622867925441185841083825396183681067717381655139236632571618744110472570418716111161064776118379104260991807310975261251797411099133910261013881152644418017112302654417916111681340980864171402695117916116302697317935117291332828744641542744217922117722747617947117771397846634861762792517859117682795417948117801370829544781872836318056119342843918103119601250720465012012890617952119082890818002118441287739396082372984018164120312981918179121151371947337202833036018380121083037118399121361105682285752993105818538124623114018534125211124767246583173135618480129233150218493129271128707 104

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Figure6-16.MotionofClusterOneinthreedimensions.Thed ashedlineswithasterisksindicatethehydrogenandthe solidlineswithplussignsindicatethehelium.105

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curvesseemtocrossabitmoreofteninthe z x graph,butthiscouldjustbeamatterof perspective. ThecentroidseparationcanbeseeninFigure 6-17 .Clearly,asintheotherruns, Figure6-17.Thisgraphshowstheseparationoftheheliuman dhydrogencentroids,in kiloparsecs,astimepasses.ThesolidlinewithcirclesisC lusterOne,the dottedlinewithasterisksisClusterTwo,andthedashedlin ewithplus signsisClusterThree. thecentroidseparationgetssmallerastimepasses.Inthis case,itlevelsoutarounda redshiftof 75 orsoatabout 100 kpc. Thenextthingthatneedstobeexaminedisthestandarddevia tion.Thiswilltellme iftheheliumisclusteringmoretightlythanthehydrogen,a sitshouldbe.Thestandard deviationsforthisdataaregraphedinFigure 6-18 .Puttingallthreeclustersonone graphmakesforagurethatconveysalotofinformation,all ontopofeachother,but 106

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Figure6-18.Thisgraphshowsthestandarddeviationsofthe hydrogenandhelium clustersastimepassesinkiloparsecs.Eachclusterisrepr esentedbya differentlinetype–ClusterOneisthesolidline,ClusterT woisthedotted line,andClusterThreeisthedashedline.Thehydrogenisfu rther demarcatedbyplussigns,whiletheheliumisrepresentedby asterisks. Ithinkthethreeclustersaredifferentenoughtoseeeveryt hingthatneedstobeseen. Thisgraphshowsamuchlargerseparationbetweenhydrogena ndheliumthaneitherof thegraphsformyrsttworuns.Atearlytimes,thestandardd eviationsforthehydrogen andheliumremainprettycloseforClusterTwo,butinCluste rsOneandThree,there isaclearandnoticeableseparationbetweenthestandardde viationsforhydrogenand helium.Sometimestheygetclosetooneanother,butatalmos talltimes,theheliumis moretightlyboundthanthehydrogen.Especiallyatlatetim es,allthreeclustersshow alargedifferenceinthestandarddeviationsforhydrogena ndhelium,whichseems 107

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toremainfairlyconstantafteraredshiftofaround 75 orsoonwards.Thesearepretty interestingresultsthatseemtoshowthattheheliumisden itelyformingclustersat earliertimesthanthehydrogen,aroundwhichthehydrogenc lustersform. Finally,thisbringsmetotheratiosofheliumtohydrogenas afunctionofeach cluster'stotalmass.Thistime,asImentioned,myheliumwa sfourtimesasmassiveas myhydrogen,givingitamassof 16.94 10 10 M / h .ThischangestheequationIneedto ndthetotalmassofaclusterto: m total =(4n He +n H ) 4.235. (6–29) Theunitsarestill 10 10 M / h .Thisgraph,foreachoftheclustersItracked,appearsin Figure 6-19 .Thisisaprettyinterestinggure.Theratiosforallthree clustersseemto haveafairlysimilarshape,sothebehaviourthatisbeingdi splayedisprobablycommon tomostclusters.Ofcourse,timestartsontheleftsideofth egure,whentheclusters areallsmallandthereforecontainlessmass.Itappearstha tatearlytimes,theclusters arehighlyenrichedinhelium,aswouldbeexpected.Astimep asses,thisratiofalls towardsthemeanvalueof 0.32 ,andatlatertimes–aroundaredshiftof 100 orsofor ClustersOneandThreeand 50 orsoforClusterTwo–theratioevenfallsbelowthis value.Noneoftheclustersstaybelowthisvalueforlong,th ough,astheyallincrease backoverthisvalueshortlyafterwards–insomecasesbyqui teabit.Presumablythisis afunctionoftheclustersIwastrackingmergingwithotherc lusters,alsohighlyenriched inhelium.Itisnotthelatertimesthatareofinteresttome, though–itistheearlyand intermediatetimes.Attheseredshifts,theratiosforallt hreeclustersshowsubstantial heliumenrichment,whichmakessense.Theheliumshouldsta rtformingclustersbefore thehydrogen,andasitdoes,thehydrogenwillgraduallyfal lintowardstheseclusters. Sincemostoftheheliumhasalreadyformedintoclusters,as timepasses,theratios willallfall(asshownhere)asthehydrogenstartsforminga roundthehelium.Themain wayofchangingtheamountofheliumintheseclustersaftert heyformwillbemerging 108

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Figure6-19.Thisgraphshowstheratioofheliumparticlest ohydrogenparticlesasa functionofthecluster'smassinunitsof 10 10 M / h .Thesolidlinewith circlesisClusterOne,thedottedlinewithasterisksisClu sterTwo,andthe dashedlinewithplussignsisClusterThree. withanothercluster,whichwillnothappenuntilafterthec lustershavealreadyhadsome timetodevelop. 109

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CHAPTER7 CONCLUDINGREMARKS 7.1 COSMICS Understandingthecreationandgrowthofperturbationsint heearlyuniverseis essentialtobeingabletomodellargescalestructureforma tionintheuniverse.Before thecreationoftheseperturbations,theuniversewascompl etelyfeaturelessand homogeneous.Oncetheystartgrowing,theyformtheseedsfo rstructureformation onlongtimescales.Inadditiontomyownrelativelysimplec alculationsandprograms modelingtheevolutionoftheearlyuniverse,Iworkedwith COSMICS ,acodedeveloped byBertschingerandexplainedindetailinhispaperonitsus e( Bertschinger 1995 ; Ma&Bertschinger 1995 ). COSMICS wasoriginallydesignedtotreatallbaryonicmatterashydr ogen,butwith somealterations,Iwasabletotrackthehydrogenperturbat ionsseparatelyfromthe heliumperturbations.Unfortunately, COSMICS isacomplicatedprogram,andbreaking thebaryonicmatterintotwodifferentgroupslikethisgave risetosomeproblemsinthe codewhichIwasunabletoresolve.Atthispoint,Iamunsurew hethertheproblemwas aprogrammingissueonmybehalf,orwhether COSMICS issetupinsuchawayasto makeseparatingtheheliumfromthehydrogenimpossiblewit houtmajorlyreworking thecode.ThisisanissuethatImaycomebacktointhefuture, ashavingaprogram like COSMICS thatevolvesthedifferentkindsofbaryonicmatterperturb ationsseparately withoutneglectinganyoftherelevantphysicswouldbeaver yusefultoolthatmightgive interestinginsightsintotherecombinationera,theevolu tionoftheseperturbations,and thegrowthoflargescalestructures. 7.2 GADGET Using GADGET ,Iwasabletomodeltheformationofstructureswhenheliumi s allowedtostartrecombiningbeforehydrogen.Throughoutt hethreerunsIperformed, severalthingsseemcertain.Thecentroidsoftheheliumpar ticlesandhydrogen 110

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particlesdobeginseparatefromeachother,implyingthatt heheliummustbemoving toformstructuresbeforethehydrogen.Ifitwasnot,thecen troidswouldbeinthe sameplace,sincethehydrogenandheliumisinitiallydistr ibutedrandomlyand homogeneously.Therefore,thisseparationincentroidsmu starisefromthefactthat theyaredifferentparticles,withdifferentproperties.O fcourse,astimepasses,the velocitiesofthehydrogenandheliumparticlestendtoappr oachthesamevalue,and thecentroidsofthetwodifferenttypesofparticlesapproa cheachotheraseverything reachesthesamevelocityandgravitypullseverythinginto towardseachcluster'scenter ofmass. Thestandarddeviationsofeachtypeofparticlearealsoint erestingtolookat.For thersttworuns,itappearsthatforthemajorityoftime,an dformostofthecluster's tracked,thestandarddeviationsoftheheliumparticlesar esmallerthanthehydrogen particles,aswouldbeexpected.Forthelastrun,whichhada nincreasedmassfor theheliumparticlesinadditiontothefactthatitstartedm ovingatanearliertime,this effectwasevenmorepronounced.Fromtheseresults,Iwould drawtheconclusionthat theheliumparticlesdotendtobemoretightlyclusteredtha nthehydrogenparticles. Sincetheheliumrecombinesrst,andcanthereforestartfo rmingstructuresearlier,this certainlymakessense. Finally,theratiosofheliumparticlestohydrogenparticl esasafunctionofeach cluster'stotalmassgivesausefulmeasureofwhethereachs tructureformedisricher inheliumthanwouldbeexpectedfromtheaveragevalue.Ofco urse,thereismuch morehydrogenthanheliumintheuniverse–aboutthreetimes asmuch.Thiswould leadmetoexpectthatstructuresformedatearlytimes,when heliumrststartsto recombine,butbeforehydrogenisable,shouldbeenrichedi nhelium,whichwillform theseedsforthesestructuresoncehydrogenisabletostart forming.Conversely,since muchoftheheliumwillbelockedupintheseearlystructures ,anystructuresthatform atintermediatetimes,afterhydrogenreallystartsrecomb ining,wouldbedecientin 111

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helium.Ofcourse,muchofthehydrogenwouldbeattractedto thegravitationalpullfrom theclustersalreadyformedfromthehelium,butwithsomuch hydrogenintheuniverse, itisentirelypossiblethathydrogenrichstructuresmight startformingintheareas relativelyfarawayfromtheheliumstructures.Thismayexp lainwhyacoupleofthe clustersItrackedinrunsoneandtwoaredecientinhelium, ratherthanenriched,which iswhatIwouldhaveexpected.Ofcourse,maybethiseffectis weakerthanIexpect–all threeoftheclustersItrackedinRunThreeweregreatlyenri chedinhelium,soperhaps byneglectingtheincreasedmassoftheheliumparticles,Iw asoversimplifyingthe problem.IdohavemorecondenceinmyresultsfromRunThree ,astheyincludemore oftherelevantphysicsthanRunsOneandTwobyincludingthe heliummass. EverythingIdidinthisstudyleadsmetobelievethatinclud ingtheearlierrecombination timeofhydrogeninthemodelingoflargescalestructurefor mationisworthdoinginany detailedcalculations.Althoughitseemstointroduceonly smalleffectsthatprobablywill notcauseanyproblemsinmostcalculations,theeffectsitd oesproducearenoticeable, andshouldbestudiedmore.Thisstudyprovidesagoodstarti ngpoint,butmorework inthisareaiscertainlyjustiedtogureouttojustwhatex tenttheseeffectsshouldbe includedinmodelsofrecombination,largescalestructure formation,andtheevolution oftheuniverse. 112

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APPENDIX:SETTINGINITIALCONDITIONSFORGADGET-2 Creatingaproperinitialconditionslefortheheliumreco mbinationrunsthatIwish todoisanimportantpartofmyproject.Usingtheinitialcon ditionsprovidedforthe Lambda-ColdDarkMatter(or CDM)modelsuppliedwith GADGET isagoodstarting point,buttheseinitialconditionswerecreatedforaspeci ctime–asmentionedin thepaper,thistimeis z=10 .Iknow,asmentionedinSection 6.1 ,thatthedensity uctationsscaleasfunctionsoftheexpansionfactordurin gtheeraofmatterdomination (whichisthetimeduringwhichIwilluse GADGET ),andsoitiseasytoscalethegiven initialconditionsletowhatevertimeisrequired. Therearetwothingsthatneedtobechangedintheinitialcon ditionsletochange thecorrespondingtime.Eachparticlehasapositionandvel ocityassociatedwithit;the positionsoftheparticlesaredeterminedbyanequationlik e: x=na+ (A–1) The na determinesthepointsofthegrid.The n variabletellswhichgridpointtobase thatparticularparticleon( n canrangefrom 1 tothenumberofpointsononesideof thegrid),andthe a variabledeterminesthedistancebetweengridpoints.Itsv alueis thelengthoftheboxdividedbythenumberofgridpoints.The isthevariablethat determinestheuctuationsizes,anditisonlythispartoft hepositionvariableforeach particlethatIneedtochange.So,toscalethesepositionsw ithtime,Ineedtogure outhowthedensityuctuations,representedinthe term,scalewiththeexpansion factor.Thevelocitytermsareabiteasier–theydonothavea constantterm,andare justfunctionsofthevelocitiesoftheuctuations,soitis agooddealeasiertoscalethe velocitiesonceIknowhowtheydependontheexpansionfacto r. 113

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Tondhowthedensityuctuationsdependontheexpansionfa ctor,Iwillstartwith Poisson'sEquation( Liddle&Lyth 2000 ; Peebles 1993 ) 1 : r 2 =4 G (A–2) ThiscalculationiseasierinFourierspace,soIwillmaketh atswitchwiththerelation: r 2 = ( k a ) 2 (A–3) where k isthewavenumberofaparticulardensityuctuation(allof whichevolve independentlyofeachother), a istheexpansionfactor,and k = a isthephysical wavenumberincomovingcoordinates. k ,thegravitationalpotentialinFourierspace, isconstantintime,aswell.Technically,allofthedensity uctuationsshouldcarrya k subscriptaswell,sincethesearedensityuctuationsinFo urierspace,butIwilltake themtobeunderstoodtosavesometime.Withtheserelations ,Ind: ( k a ) 2 k =4 G (A–4) Itisthedensitycontrastthatismostoftenusedincosmolog icalequations,anditisthe densitycontrastusedintheinitialconditionsequationde scribedin( A–1 ),sothatisthe nextsubstitutionIwillmake.Thedensitycontrastisdene dby: = (A–5) whichgives: ( k a ) 2 k =4 G (A–6) Duringmatter-domination,Iknowthat: / a 3 (A–7) 1 ConversationswithDr.Frywerealsoveryedifyinghere. 114

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So,makingthissubstitutionandsimplifying,Ind: ( k a ) 2 k / 4 G a 3 (A–8) 4 G / k 2 a k (A–9) / k 2 a k 4 G (A–10) / 2 3 k 2 a 2 H 2 k (A–11) / a. (A–12) AndsoIndthatthedensityuctuationsareproportionalto theexpansionfactor. Thisleavesthevelocitiesoftheparticlestobedetermined .Onceagain,Iwilljust lookatthematter-dominationera,where / a 3 andusingtheFriedmannEquationwith nocurvatureandacosmologicalconstantof 0 H 2 =((8 G) = 3) ( Liddle&Lyth 2000 ). First,Iwanttondhowtimedependson H : _a a =H (A–13) da dt =aH (A–14) Z t 0 dt= Z a 0 da aH (A–15) t= Z a 0 a 3 = 2 da a p (8 G) = 3 (A–16) = 1 p 8 G3 Z a 0 a 1 = 2 da (A–17) = 2 3 1 p 8 G3 a 3 = 2 (A–18) = 2 3 1 H (A–19) t= 2 3 1 H (A–20) OnceIhavethisequation,Icancombineitwithanotherequat ionfrom( Liddle&Lyth 2000 ),givenby: ~ v= t ~ r (A–21) 115

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ConvertingthistoFourierspaceusingtherelation ~ r =i ~ k = a ,Ind: ~ v= 2 3 1 H i ~ k a k (A–22) Iwanttogetridofthevectors,soIwilltakethedotproducto fbothsideswiththeir complexconjugatesandthentaketheirsquarerootstoget: v= 2 3 k aH k (A–23) and,with( A–11 ),thisbecomes: v= aH k (A–24) Since H / p H / a 3 = 2 ,andso: v / a k a 3 = 2 (A–25) v / p a (A–26) And,alongwithEquation( A–12 ),thisallowsmetostatethat: v / p a. (A–27) Sothevelocitiesoftheparticlesareproportionaltothesq uarerootoftheexpansion factor. 116

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REFERENCES Bertschinger,E.1995,arXiv:astro-ph/9506070—.2001,arXiv:astro-ph/0101009Carroll,B.W.,&Ostlie,D.A.1996,AnIntroductiontoModer nAstrophysics(Benjamin Cummings) Dodelson,S.2003,Moderncosmology(Amsterdam(Netherlan ds):AcademicPress) Dubrovich,V.K.,&Grachev,S.I.2005,arXiv:astro-ph/050 1672 Dubrovich,V.K.,&Stolyarov,V.A.1997,AstronomyLetters ,23,565 Durrer,R.2005,inLectureNotesinPhysics,BerlinSpringe rVerlag,Vol.653,The PhysicsoftheEarlyUniverse,ed.K.Tamvakis,31–+ Gao,L.,Yoshida,N.,Abel,T.,Frenk,C.S.,Jenkins,A.,&Sp ringel,V.2007,MNRAS, 378,449 Guth,A.H.1981,Phys.Rev.D,23,347Guth,A.H.,ed.1997,Theinationaryuniverse.Thequestfo ranewtheoryofcosmic origins Hartle,J.B.2003,Gravity:anintroductiontoEinstein'sg eneralrelativity(San Francisco,CA,USA:AddisonWesley) Hawking,S.W.,&Ellis,G.F.R.1975,TheLargeScaleStructu reofSpace-Time (CambridgeUniversityPress) Heitmann,K.,etal.2008,ComputationalScienceandDiscov ery,1,015003 Hernquist,L.1987,ApJS,64,715Hirata,C.M.,&Switzer,E.R.2008,Phys.Rev.D,77,083007Hockney,R.W.,&Eastwood,J.1988,ComputerSimulationUsi ngParticles(Bristol: TaylorandFrancis) Jones,B.J.T.,&Wyse,R.F.G.1985,A&A,149,144Kolb,E.W.,&Turner,M.S.1990,Theearlyuniverse(Reading ,MA:Addison-Wesley) Kravtsov,A.V.,Klypin,A.A.,&Khokhlov,A.M.1997,ApJS,1 11,73 Liddle,A.R.,&Lyth,D.H.2000,CosmologicalInationandL arge-ScaleStructure (Cambridge:CambridgeUniversityPress) Lima,J.A.S.,Zanchin,V.,&Brandenberger,R.1997,MNRAS, 291,L1 117

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Ma,C.,&Bertschinger,E.1995,ApJ,455,7Mukhanov,V.2005,PhysicalFoundationsofCosmology(Camb ridgeUniversityPress) O'Shea,B.W.,Bryan,G.,Bordner,J.,Norman,M.L.,Abel,T. ,Harkness,R.,&Kritsuk, A.2004,arXiv:astro-ph/0403044 Padmanabhan,T.1993,StructureFormationintheUniverse( Cambridge,UK: CambridgeUniversityPress) Peacock,J.A.1999,CosmologicalPhysics(Cambridge,UK:C ambridgeUniversity Press) Peebles,P.1993,PrinciplesofPhysicalCosmology(Prince ton:PrincetonUniversity Press) Peebles,P.J.E.1968,ApJ,153,1—.1980,Thelarge-scalestructureoftheuniverse(Princet on,N.J.:PrincetonUniversity Press) Press,W.H.,Teukolsky,S.A.,Vetterling,W.T.,&Flannery ,B.P.1992,Numerical recipesinC.Theartofscienticcomputing(Cambridge:Uni versityPress) Rubi no-Martn,J.A.,Chluba,J.,&Sunyaev,R.A.2008,A&A,485 ,377 Seager,S.,Sasselov,D.D.,&Scott,D.2000,ApJS,128,407Siegel,E.R.,&Fry,J.N.2006,ApJ,651,627Springel,V.2005a,MNRAS,364,1105—.2005b,UserGuideforGADGET-2,Max-Plank-Institutefor Astrophysics,Garching, Germany Springel,V.,Yoshida,N.,&White,S.D.M.2001,NewA,6,79Switzer,E.R.,&Hirata,C.M.2008a,Phys.Rev.D,77,083006—.2008b,Phys.Rev.D,77,083008Tamvakis,K.2005,inLectureNotesinPhysics,BerlinSprin gerVerlag,Vol.653,The PhysicsoftheEarlyUniverse,ed.K.Tamvakis,3–+ Tasker,E.J.,Brunino,R.,Mitchell,N.L.,Michielsen,D., Hopton,S.,Pearce,F.R., Bryan,G.L.,&Theuns,T.2008,MNRAS,390,1267 Villumsen,J.V.1989,ApJS,71,407Wong,W.Y.,Moss,A.,&Scott,D.2008,MNRAS,386,1023Wong,W.Y.,&Scott,D.2007,MNRAS,375,1441 118

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Wong,W.Y.,Seager,S.,&Scott,D.2006,MNRAS,367,1666Yoshida,N.,Abel,T.,Hernquist,L.,&Sugiyama,N.2003,Ap J,592,645 119

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BIOGRAPHICALSKETCH AndrewL.HillwasborninCaliforniaintheyear1980.Hespen tmostofhislife there,eventuallysettlingdowninNorthernCalifornia.He attendedDelOroHigh SchoolinLoomis,graduatingin1999.Interestedinspacefr omanearlyage,he decidedtoattendtheCaliforniaPolytechnicStateUnivers ityinSanLuisObispo, mostlyfortheaerospaceengineeringprogram,butpartlyfo rtheproximitytotheocean. Whenhediscoveredhewasmoreinterestedintheorythanthep racticaldisciplineof engineering,heswitchedtophysicsandgraduatedwithabac helor'sdegree.Unsureof theusefulnessofabachelor'sdegreeinphysicsonthejobma rket,Andrewthendecided hehadlittlechoicebuttogotograduateschoolandgetadoct orate.Stillinterested inspace,hedecidedastrophysicswastheeldforhim,andst artedhisresearchinto cosmologyandthelargescalestructureoftheuniverse.Fut ureplans:takeanice, longvacation,andthenhopefullyndajobwithatleastsome smallconnectionto astrophysics. 120