Dark matter annihilation energy output and its effects on the high-z IGM

19 Pages·2014·1.75 MB·English

MNRAS445,850–868(2014) doi:10.1093/mnras/stu1780
Dark matter annihilation energy output and its effects on the high-z IGM
Ignacio J. Araya1‹ and Nelson D. Padilla2
1DepartmentofPhysicsandAstronomy,UniversityofSouthernCalifornia,920BloomWalk,LosAngeles,California,CA90089-0484,USA
2DepartamentodeAstronom´ıayAstrof´ısica,PontificiaUniversidadCato´licadeChile,Av.Vicun˜aMackenna4860,Santiago,Chile
Accepted2014August27.Received2014August24;inoriginalform2013February21
D
o
ABSTRACT w
n
Westudythecaseofdarkmatter(DM)self-annihilation,inordertoassessitsimportanceasan lo
a
energyinjectionmechanism,totheintergalacticmedium(IGM)ingeneral,andtothemedium de
d
within particular DM haloes. We consider thermal relic WIMP (weakly interacting massive fro
particle) particles with masses of 10GeV and 1TeV, and we analyse in detail the clustering m
h
properties of DM in a (cid:2) cold dark matter cosmology, on all hierarchy levels, from haloes ttp
s
and their mass function, to subhaloes and the DM density profiles within them, considering ://a
adiabatic contraction by the presence of a supermassive black hole. We then compute the ca
d
correspondingenergyoutput,concludingthatDMannihilationdoesnotconstituteanimportant em
feedback mechanism. We also calculate the effects that DM annihilation has on the IGM ic.o
u
temperatureandionizationfraction,andwefindthatassumingmaximalenergyabsorption,at p
z∼10,forthecaseofa1TeVWIMP,theionizationfractioncouldberaisedto6×10−4and .co
m
the temperature to 10K,and in the case of a 10GeV WIMP,the IGMtemperature could be /m
n
raisedto200Kandtheionizationfractionto8×10−3.WeconcludethatDMannihilations ra
s
cannotberegardedasanalternativereionizationscenario.Regardingthedetectabilityofthe /a
WIMPthroughthemodificationstothe21cmdifferentialbrightnesstemperaturesignal(δTb), rticle
we conclude that a thermal relic WIMP with mass of 1TeV is not likely to be detected -ab
s
from the global signal alone, except perhaps at the 1–3mK level in the frequency range tra
c
30 < ν < 35MHz corresponding to 40 < z < 46. However, a 10GeV mass WIMPmay be t/4
4
detectable at the 1–3mK level in the frequency range 55 < ν < 119MHz corresponding to 5
/1
11<z<25,andatthe1–10mKlevelinthefrequencyrange30<ν<40MHzcorresponding /8
5
to35<z<46. 0/1
7
4
Keywords: cosmology:theory–darkages,reionization,firststars–darkmatter. 9
1
7
0
b
y
g
u
e
1 INTRODUCTION For example, assuming that the DM consists of particles that st o
were originally in thermal equilibrium with the radiation in the n
0
The dark matter (DM) is a component of the total density of the earlyUniverse,dependingonthemassoftheparticles,thestruc- 5
A
Universewhichcanbemeasuredonlythroughgravitationalinterac- tureformationscenariowillbetopdownorhierarchical.Inthetop p
tionsand(likely)weakinteractions.Itbehavesasdust-likematterin down structure formation scenario, the first structures to collapse ril 2
0
thesensethatitiscollisionlessandthatitsenergydensitydecreases are large (on the mass scale that corresponds to present-day su- 1
9
asthethirdpowerofthescalefactor.Therearemanycompelling perclusters),becausealloverdensitiesatsmallerscalesareerased
evidencesofitsexistence,althoughallofthemareindirect.These by free streaming. This is the case for a hot dark matter (HDM)
aremainlydynamicalevidences,whichrelyonthegravitationalin- model, in which the DM particles were relativistic at the time of
fluenceofDM.Someexamplesoftheseevidencesaretherotation decoupling.Inthehierarchicalstructureformationscenario,small
curvesoflate-typegalaxies(Rubin&Ford1970),galaxyclusterdy- (andalsolarge)structuresstarttocollapseafterdecoupling,because
namics(Zwicky1933)andstronglensing(Paczyn´ski1987),among DMparticleswerenon-relativisticatthetimeofdecoupling,andso
others. The effects of DM are very important not only for deter- thefreestreamingisnegligible.Thisisthecaseforanycolddark
miningthedynamicsofgalaxiesorgalaxyclusters,butalsoforthe matter(CDM)model(Blumenthaletal.1986),includingthecur-
evolutionoftheUniverseasawhole,andforthegalaxyformation rentlyfavoured(cid:2)CDMmodel(seee.g.thePlanckresults;Planck
processesandstructureformationscenarios. CollaborationXVI2013),whichwestudyinthiswork.
ThepostulatedCDMparticlehasbeennamedaWIMP(orweakly
interactingmassiveparticle),becauseitweaklycouplestonormal
(cid:5)E-mail:[email protected] matter(forintroductionstothetopicofDMastrophysics,seeRaffelt
(cid:3)C 2014TheAuthors
PublishedbyOxfordUniversityPressonbehalfoftheRoyalAstronomicalSociety
DMannihilation effectsonthehigh-zIGM 851
2008;D’Amico,Kamionkowski&Sigurdson2009).Inthestandard Ripamonti,Mapelli&Ferrara2007a;Valde´setal.2007;Chuzhoy
scenario,thisparticleisconsideredtobeathermalrelic.Thismeans 2008;Natarajan&Schwarz2009;Cumberbatchetal.2010,among
thatitscreationandannihilationprocessesfromtheearlyUniverse others)havestudiedtheeffectsofDMannihilation(andalsodecays,
radiationfieldwereinthermalequilibriumwitheachother,andthus, indifferentDMmodels)ontheIGMtemperatureandontheglobal
thepresent-dayDMnumberdensityistherelicabundanceafterthe ionization fraction. This extra heating may be detectable through
freeze-outofthisprocesses,withannihilationbeingfavouredafter theglobal21cmdifferentialbrightnesstemperature,measurablein
theUniversehadcooledenough(Kolb&Turner1990).Thisfact theMHzfrequencyrange.Thisdifferentialbrightnesstemperature
alone allows for the computation of the interaction cross-section arisesasaresultofthecouplingoftheneutralhydrogenspintemper-
for the creation–annihilation process given only the present-day ature(thataccountsforthehyperfinetransition)totheIGMkinetic
abundanceofDM(itsdensityparameter(cid:6) ). temperature, thus causingadifference between thespintempera-
m,0
Animportantfeatureofaself-annihilatingthermalrelicWIMPis ture and the cosmic microwave background (CMB) temperature.
thattheannihilationproductsarestandardmodel(SM)particles,and Also,theenergyinputmaycontributeasanextraheatingmecha-
thus,theymayinjectenergytotheintergalacticmedium(IGM)orto nismwithinhaloesthatcanadduptothestandardSNeandAGN
D
themediumwhereaparticularDMhaloisembedded.Globally,the feedbackmechanisms(Ascasibar2006;Natarajanetal.2008),and o
w
annihilationsofWIMPsareunlikelytooccurbecausetheirabun- maychangethetemperatureofthehaloenvironment. nlo
dancehaslongagofrozen.However,theremayberegionswhere Regarding the observational evidence for the energy injection ad
e
encountersamongWIMPsaremoreprobableduetoanenhanced due to DM annihilations, no definite conclusion has been drawn d
density. These regions are precisely the regions where structures yet. However, different observations of anomalous enhancements fro
m
form. Now, it is a direct consequence of the (cid:2)CDM model that in detected signals from high-energy phenomena in the galac- h
DM is clumpy, due to the hierarchical growth of structure. Thus, tic halo and in the cosmic background suggest its presence. ttp
s
itispreciselyintheinnerregionsofDMhaloesandsubstructures Thesesignalscorrespondmainlytogamma-raydetections[Fermi ://a
thattheannihilationsaremoreprobable.Theenergyoutputdueto (Abdo et al. 2009), H.E.S.S (Aharonian et al. 2008) and EGRET ca
d
annihilationswillbeproportionaltothesquareofthedensity;thus, (Sreekumaretal.1998)experiments]andtomeasurementsofcos- e
m
theselocaldensityenhancementsareofgreatimportance(Natara- micraypositronfractions[PAMELA(Adrianietal.2009experi- ic
jan,Croton&Bertone2008). ment)].Forexample,Cline,Vincent&Xue(2010)andFinkbeiner .ou
p
ThefirsthierarchyindensityenhancementsaretheDMhaloes etal.(2010)analysethedatasetsbyPAMELA,FermiandH.E.S.S., .c
o
thatdonotbelongtoanygravitationallyboundstructureatapar- andconcludethattheobservedsignalscouldbeaccountedforbyan m
ticular redshift. Their average clustering can be characterized by ∼1TeVWIMP.Theyalsofindanenhancedinteractioncross-section /m
n
thehalomassfunction,whichcanbecomputedbyPress–Schecter (Sommerfeld enhancement) for the annihilation process (∼102 ra
s
theory (e.g. Press & Schechter 1974; Bond et al. 1991; Lacey & timesthethermalrelicWIMPvalue).Asimilarresultwasalsoob- /a
Cole1993;Mo&White1996;Sheth,Mo&Tormen2001,among tainedbyElsa¨sser&Mannheim(2004)withrespecttotheEGRET rtic
le
others). gamma-raydata(althoughthereareotherinterpretationsofit,for -a
Thesecondhierarchyindensityenhancementsarethesubstruc- exampleintermsof∼60GeVmassiveneutrinosbyBelotskyetal. bs
tures embedded in a particular halo. These subhaloes will retain 2008).However,thereissomedebateregardingthecompatibilityof trac
theiridentitiesevenaftercoalescing.Thus,insideeachDMhalo,it thepresenceofSommerfeldenhancementwiththenon-detectionof t/4
4
isexpectedthatthefullhierarchyofDMhaloesisnestedwithin, gamma-rayfluxfromknownDMstructures(likegalaxyclusters). 5/1
until the lowest mass haloes that formed. The number density of In this line, the study by Pinzke, Pfrommer & Bergstrom (2009) /8
5
thesesubhaloescanbemeasuredinMilkyWay(MW)sizedhalo determines that the attempts of explaining the Fermi, PAMELA 0/1
simulations, and by approximating their abundance by a power- and H.E.S.S. combined data with ∼1TeV WIMPs and cross- 74
law behaviour (Via Lactea II simulation, Diemand et al. 2008; sectionsenhancedbyfactorsof102–103,areincompatiblewiththe 91
7
Acquariusprojectsimulation,Springeletal.2008). EGRET upper limits on the gamma-ray emission from the Virgo 0
b
Apart from the number density of haloes and subhaloes, it is galaxycluster.RegardingtheFermigamma-rayexcess,Hooper& y g
important to quantify the form of the density distribution within Goodenough(2010)arguethatitcanbewellaccountedforbynon- u
e
s
a particular structure, in order to compute the annihilation power exoticprocesses(decayingpions,inverseCompton,pointsources, t o
output.AnexampleofsuchadistributionisgivenbytheNFWpro- etc.),exceptwithin1◦.25(≈175pc)fromthegalacticcentre,where n 0
file(Navarro,Frenk&White1997).Also,thisDMdensityprofile thegamma-rayexcessisconsistentwiththepredictionofacusped 5
A
maybefurtherenhancedbythegravitationaleffectofbaryonson (ρ ∝ r−1.34) DM density profile, and annihilating WIMPs with p
DMthroughadiabaticcontraction(see,forexample,Sellwood& a thermal relic cross-section (no Sommerfeld enhancement) and ril 2
McGaugh2005;Natarajanetal.2008).Thus,thetotalresultingDM massesintherangem =7.3–9.2GeV.Also,Hooper,Finkbeiner 01
χ 9
distributionintheUniversecanbecomputedataparticularredshift. &Dobler(2007)foundthattheexcessmicrowaveradiationwithin
Knowing the global DM distribution, one can compute the re- 20◦fromthegalacticcentre,foundbytheWMAPexperimentand
sultingenergyoutputperbaryon,whichcanbefurthercompared called the WMAP Haze, can be well explained by a cusped DM
withwhatcanbeobtainedassumingaperfectlyhomogeneousDM profilewithρ ∝r−1.2 intheinnerkiloparsecs,andaWIMPwith
distribution(seeMapelli,Ferrara&Pierpaoli2006;Cumberbatch, massinthe100GeVtomulti-TeVrangeandthermalreliccross-
Lattanzi&Silk2010).Onecanalsocomputetheenergygenerated section.However,accordingtotherecentnullresultsinDMdirect
perunittimebyaparticularhaloofagivenmassandredshift,and detectionobtainedbytheLUXCollaboration(Akeribetal.2014),
onecancalculatewhatistheenergyreceivedbyaparticularhalo aWIMPwithmassesbetween∼10and∼100GeVisincreasingly
duetotherestoftheUniverse. disfavoured. Also, for lower mass light dark matter (LDM) par-
Thisadditionalenergyinjectionmayhaveinterestingeffectson ticle candidates (with masses in the 1–100MeV range), possible
the global IGM, as well as on the environment within particu- constraints on the mass and cross-section coming from the CMB
lar DM haloes. For example, several authors (for example, Pier- have been studied (see section 5 of Mapelli et al. 2006; Zhang
paoli2004;Furlanetto,Oh&Pierpaoli2006;Mapellietal.2006; etal.2006).However,theeffectontheCMBbymassiveWIMPs,
MNRAS445,850–868(2014)
852 I.J.ArayaandN.D.Padilla
liketheonesstudiedinthiswork,wouldbecompletelynegligible. Intwowell-studiedscenarios,theWIMPparticlesaresuchthat
Itcanbeseenthatnoclearconsensusexistsneitheronthesignifi- theyaretheirownantiparticle.Forexample,inthecaseofaKaluza–
canceoftheresultsaspositiveindirectdetectionsofDM,noronthe Klein (KK) excited state, arising naturally in compactified extra-
importanceofthesedetectionsfordiscriminatingamongdifferent dimensional models (like universal extra dimensions or UED, in
WIMPscenarios.Whatcanbesaidnonethelessisthatevidenceof whichallparticlesareabletopropagateintheextradimensionpro-
possiblenewexoticphysicsisaccumulating,andthatWIMPDM videdtheyhaveenoughenergy),theWIMPisthelightestneutral-
annihilationrepresentsawell-motivatedoption. gaugeboson-excitedstate,correspondingtothefirstexcitedstateof
Themaingoalofthisworkistocomputethisadditionalenergy theBbosonorB(1),andastheSMBboson,itisitsownantiparticle
injection,bothlocallywithinhaloesandglobally,andtoassessthe (theBbosonisasuperpositionofthephotonofquantumelectrody-
importance of this injection on the different processes mentioned namicsandtheZbosonoftheweakinteraction).Also,inthecase
above.Allthecomputationswillbedoneconsideringacosmology oftheminimalsupersymetricstandardmodel(MSSM),consider-
consistentwiththePlanck2013results(PlanckCollaborationXVI ingconservedRparity(andthusthatsuperpartnersareforbiddento
2013). In particular, we will use h = 0.6711, (cid:6) = 0.3174, decaytoSMparticles),theWIMPisthelightestneutralino(i.e.the
m,0 D
(cid:6) =0.6825,f ≡ (cid:6)b,0 =0.17,n =0.9624andσ =0.8344. lightestmasseigenstateofthefermionicsuperpartnersofthegauge o
Th(cid:2)e,0paperisorgabnariyzeda(cid:6)sm,f0ollows.InsSection2,theth8ermalrelic bosonsandtheHiggses),anditisaMajoranafermion,andthusit wnlo
WIMP,andthetwoabove-mentionedwell-motivatedparticularDM isitsownantiparticle. ad
e
modelswillbeexplainedingreaterdetail.InSection3,thestatis- Inboththesescenarios,theannihilationcross-sectionisdepen- d
ticsoftheDMclusteringonallhierarchieswillbedevelopedand dentonlyonthepresent-dayabundanceofDM.Thisfacthasbeen fro
m
theNFWuniversaldensityprofile,aswellastheadiabaticcontrac- dubbedtheWIMPmiracleandisexplainedinwhatfollows. h
tionmechanismwillbeexplained.InSection4,theDMannihila- ttp
s
tion energy output, both globally (per baryon) and locally within 2.1 TheWIMPmiracle ://a
aparticularDMhalowillbecomputed.Also,theclumpinessfac- ca
d
tor,definedastheratioofoutputbetweenthesmoothedandfully TheWIMPmiraclereferstothefactthat,giventhattheWIMPis e
m
clumpedcases,willbecalculated.InSection5,theintensityofthe athermalrelic,itscreation/annihilationcross-sectioncanbecom- ic
radiationfieldwillbecomputed,andthepowerreceivedfromitby putedwithoutneedingextramodel-dependentinput(exceptforthe .ou
p
a halo will be obtained. In Section 6, the effects of the annihila- fact that the cross-section must not depend on the energy of the .c
o
tionenergyinjectionontheglobalIGMwillbefollowedindetail, WIMP,i.e.thes-wavecross-sectionmustbedominant).Thecross- m
computing the IGM temperature, the ionization fraction and the sectionthenonlydependsonthepresent-dayDMdensityparameter /m
n
resulting21cmdifferentialbrightnesstemperatureinthisscenario. (cid:6)χ,0 =(1−fbary)(cid:6)m,0,andthepresent-dayvalueoftheHubble ras
ThegeneralconclusionswillbepresentedinSection7. parameter h (that gives the current expansion rate). As expected, /a
(cid:6) isinverselyproportionaltothecross-section(cid:8)σv(cid:9). rtic
χ,0 le
Obtainingthiscross-sectionisastandardderivationanditisdone -a
b
in,forexample,section5.2ofKolb&Turner(1990).Forthevalue s
2 WIMP MODELS AND THE THERMAL RELIC ofthecross-section,weuse trac
TheWIMPisatheorizedparticlethatallowsustoaccountforthe 4×10−27 t/44
mmiasisnilnyggmraavsistaitniotnhael.UTnhivisemrseis.sTinhgemevaisdsecnacnenoofttchoimsmehisoswinegvemrafsrsomis (cid:8)σv(cid:9)= (cid:6)χ,0h2 [cm3s−1], (1) 5/1/8
5
normalbaryonicmatter,becausecurrentbigbangnucleosynthesis which for the cosmology considered in this work corresponds to 0/1
constraints,likeforexampletheprimordialHeliumabundance(sec- (cid:8)σv(cid:9)=3.37×10−26(cm3s−1),anditissimilartothevalueused 74
9
tion10.4ofRyden2003),setstringentlimitsonthenormalmatter byHooperetal.(2007). 1
7
content. Nowwewillmotivateandexplaintwoscenariosthatgiverisenat- 0
b
Foracertainparticle,motivatedbysomebeyond-standardphys- urallytoathermalrelicWIMPwiththeabove-givencross-section. y g
ical scenario, to be a viable WIMP candidate, it must have some u
e
s
basicproperties.Forexample,ithastobechargeneutralanditcan 2.2 TheKKDMparticle t o
interactonlyweaklywithordinarymatter.Also,ithastobemassive n 0
inordertobehaveasCDM.AnLDMparticlewillbehaveaswarm Inextra-dimensionalmodels(i.e.physicalmodelsthatarebuiltcon- 5
A
orhotdarkmatterdependingonifitwasrelativisticornotatthe sideringmorethanthreespatialdimensions),theadditionalspatial p
timeofdecouplingandalsodependingonitsfree-streaminglength degreesoffreedommaybeaccessibletoallorsomeoftheparticles. ril 2
0
(Angulo&White2010),andtheresultingstructureformationsce- Also,theseadditionaldimensionsmaybecompactifieddifferently 1
9
nario or minimum halo mass will be different to the CDM case. andondifferentenergyscales.IfanSMparticlecanpropagatein
Also,aWIMPmustbestable,orhaveadecaytimelongerthan(or this compactified (and thus finitely extended) dimensions, it will
atleastcomparableto)thepresentHubbletime. appear more massive in the four-dimensional effective theory. In
Insomewell-motivatedphysicalscenarios,theWIMPisather- somecases,theseexcitedstatesofSMparticlesmayhavethedesir-
malrelic.Thismeans,aswasmentionedbefore,thatitscreationand ablefeaturesoftheWIMP.Here,wewillbrieflydiscussthelightest
annihilationreactionwereinthermalequilibriumwiththeradiation KKparticleWIMPcandidateandtheextra-dimensionalscenarioin
background in the early Universe. Then, as the Universe cooled whichitarises.WelooselyfollowtheexcellentreviewbyHooper
down,theannihilationreactionwasincreasinglyfavouredoverthe &Profumo(2007).
creation reaction due to the rest-mass energy difference between The lightest KK particle in the UED framework (Appelquist,
bothstates.Finally,astheexpansionrateoftheUniversebecame Cheng&Dobrescu2001)isB(1),i.e.thefirstKK-excitedstateof
largerthantheannihilationreactionrate,theglobalabundanceof the B gauge boson (also called the KK photon). This particle is
DM(itscomovingnumberdensity)wasfrozen,andtheprobability chargeneutral,andcanannihilatewithitself,andthusisasuitable
ofoccurrenceofanannihilationeventbecamenegligible. WIMPcandidate.
MNRAS445,850–868(2014)
DMannihilation effectsonthehigh-zIGM 853
SomeofthewaysinwhichKKparticlescaninteractwithSM differentredshifts.Finally,inSection3.4,weexplaintheeffectthat
particlesarethroughdecays,annihilationsandscatterings.Forex- thegravitationalcollapseofthebaryonicmatter[inparticular,the
ample,aheavierKKparticlecandecayintoalighterKKparticle formationofasupermassiveblackhole(SMBH)]canhaveonthe
emittingSMparticles,untilithascascadedintoB(1)(alsocalledγ ; DMdensityprofilethroughadiabaticcontraction.
1
seefig.4ofHooper&Profumo2007).
Also,moreinterestinglyinviewofthepossibleeffectsofWIMP
3.1 Thehalomassfunction
annihilationontheIGM,twoB(1)canannihilatetoleptonpairsand
photons(seefig.16ofHooper&Profumo2007). We use this formalism, as presented in Mo & White (1996), but
usingthemodifiedShethetal.(2001)halomassfunctionandbias
factor,calibratedwiththeGIFsimulations(Kauffmannetal.1999).
2.3 Theneutralino Weconsidertheoverdensityδ(x)definedasδ(x)= ρ(x)−ρ,and
ρ
SupersymmetryisaparticularsymmetrypropertyoftheLagrangian theoverdensitythresholdforcollapseδ (z),definedasthedensity
c
ofaphysicaltheorysuchthatsaidLagrangianremainsinvariantifa contrast(inthelinearapproximation)requiredforanoverdensityin
D
supersymetrictransformationisperformedonthefieldsofthethe- acertainregiontoalreadyhaveformedacollapsedobject(Navarro o
w
ory.Thesupersymmetrictransformationissuchthatthechangein etal.1997,equationA14).Weconsideralsothemassfluctuation nlo
afermionicfieldisabosonicfield,andthechangeinabosonicfield σ(R), defined as the standard deviation of the matter overdensity ad
e
isafermionicfiled.Asinquantumfieldtheory,thefieldsrepresent fieldwhensmoothedonascaleofsizeR.Finally,thedimensionless d
differentparticlecontents(theparticlesarenothingmorethanthe massparameterisdefinedasν = δc(z) ,anditisameasurement fro
σ(R(M)) m
excitedstatesofthefields),asupersymmetrictheorynaturallyhas ofthemassofacollapsedregion(DMhalo)relativetothemassof h
onebosonforeachfermionpresent,andvice-versa. thestructuresthathaverecentlycollapsedatredshiftz. ttp
s
TheparticlecontentoftheMSSMisthesameasthatoftheSM, WewillusetheShethetal.(2001)probabilitydistributionf(ν). ://a
but for every SM particle, there is an associated supersymmetric The mass fluctuation σ(R) is computed following Lacey & Cole ca
d
partner(also,theHiggsfieldisdifferentfromtheoneintheSM, (1993)andMo&White(1996),startingfromthepowerspectrum e
m
being two fields instead of just one). The naming scheme for the ofmatteroverdensities,P(k),atredshiftz.Thelatteriscomputed ic
superpartnersistomaintaintherootofthenameoftheSMparticle, byapplyingthetransferfunctiontotheprimordialpowerspectrum. .ou
p
buttochangetheendofthenamewiththesuffix-inointhecase Whencalculatingthemassfunction,weadoptaprimordialpower .c
offermionicsuperpartnersofSMbosonsandtoaddtheprefixs- spectrumoftheformP(k)=Akns,whereAisnormalizationand om
inthecaseofbosonic(scalar)superpartnersofSMfermions.For n =0.9624(PlanckCollaborationXVI2013). /m
s n
example, the superpartners of the electron and the muon are the Theeffectsonthepowerspectrumduetothegravitationalcol- ra
s
selectronandthesmuon,andthesuperpartnersofthephotonand lapse of the DM on subhorizon scales after the radiation–matter /a
theZbosonarethephotinoandtheZino.TheWIMPcandidateof equalityepochandthebaryonacousticoscillations(BAOs)areen- rtic
le
supersymmetry(SUSY)isthelightestneutralino,asexplainedin codedinthetransferfunction.Weadoptthefunctionalformgiven -a
b
thereviewbyJungman,Kamionkowski&Griest(1996). byBardeenetal.(1986,theBBKStransferfunction),whichignores s
ThephenomenologiesofKKparticles(astheyariseinUED)and theBAOs. trac
ofSUSYareverysimilar,andbothofthemcouldbetested,forex- t/4
4
ampleinparticlecolliders,bystudyinginteractionsaboveacertain 3.2 Thesubstructuremassfunction 5/1
energyscale.InthecaseofKK,thisenergyscalecorrespondsto /8
5
theinverseoftheradiusofcompactificationoftheextradimension, ThesecondlevelofclusteringoftheDMcorrespondstotheself- 0/1
whereasinSUSY,itcorrespondstothemassoftheneutralinosand boundedhaloesthatretaintheiridentitywithinbiggerhaloes.The 74
9
superpartners.Currentcolliderlowerlimitsonthemassscaleofthis presenceofthesesubstructuresisanaturalconsequenceofthehier- 1
7
newphysics[forexampletheLargeHadronCollider(LHC)limits] archicalstructureformationpicture,becausehaloesatallredshifts 0
b
are of the order of 100GeV and increasing. This mass limit will areformedbytheaggregationofsmallerparenthaloes.AstheDMis y g
motivatethevaluesthatweadoptfortheWIMPmassintherestof collisionless,DMhaloesaresignificantlylessdisruptedthanbary- u
e
s
thiswork.WeusebothaWIMPmassof10GeVandof1TeV. onicmatterduringthemergingprocesses,andthussubhaloesmay t o
surviveuptothesmallestscales(duetothenegligible,butnon-zero, n 0
freestreamingoftheWIMPs).Manyauthors(forexample,Taylor 5
3 THE DM CLUSTERING A
& Babul 2004; Giocoli, Pieri & Tormen 2008b; Pieri, Bertone & p
As it was mentioned, the gravitational collapse process is hier- Branchini2008;Giocolietal.2009;Kamionkowski,Koushiappas ril 2
0
archical and self-similar, and so different levels of clustering are &Kuhlen2010)haveinvestigatedtheclusteringpropertiesofthe 1
9
expected.Inthefollowingsubsections,weexplaintheprescriptions subhaloesbymeansoffittingananalyticformtothesubhalomass
adoptedtoaccountfortheDMclusteringatalltheselevels,consid- functionobtainedinhigh-resolutionnumericalN-bodysimulations
eringalsothedistributionofDMwithinaparticularhaloorsubhalo. andresimulations.
InSection3.1,wefollowtheanalyticapproach,usingthePress– WeusethesubstructuremassfunctionproposedbyGiocolietal.
Schechter formalism (Press & Schechter 1974; Bond et al. 1991; (2008a),becauseitalsoconsiderstheevolutionofthesubstructure
Lacey&Cole1993;Mo&White1996;Shethetal.2001,among in time, due to the combined effects of gravitational heating and
others)tocomputethenumberdensityofDMhaloesperdecadein tidalstrippinginthepotentialwellofthemainhalo.Theseeffects
mass.InSection3.2,weaccountforthepresenceofsubstructure will tend to erode the subhaloes, which are the remnants of the
by considering the substructure mass function given by Giocoli, haloesaccretedbythehosthalo.Aswasfoundbytheseauthors,the
Tormen&vandenBosch(2008a).InSection3.3,wepresentthe mass-loss(tothesmoothcomponentofthemainhalo)ofsubhaloes
universalNFWdensityprofile(Navarroetal.1997),whichgives canbeapproximatedbyanexponentialdecayofthesubhalomass
the DM distribution within each gravitationally bound halo, and onacharacteristictime-scalethatisproportionaltothedynamical
explainhowtocalculateitsparametersfordifferenthalomassesat timeofthemainhalo.
MNRAS445,850–868(2014)
854 I.J.ArayaandN.D.Padilla
Explicitly,theunevolvedsubhalomassfunction(thatconsiders 3.3 TheNFWuniversaldensityprofile
allsubhaloeswiththemasstheyhadwhentheywereaccreted)is
The last hierarchy corresponds to how the DM is smoothly dis-
universal,andisgivenby
tributedwithinaparticularhaloorsubhalo.ThisDMdistribution
dN =N x−αe−6.283x3, x = mv , (2) appears to be universal, as found by Navarro et al. (1997) using
dln(m /M ) 0 αM N-bodysimulations,andisgivenbytheNFWdensityprofile:
v 0 0
with α = 0.8 and N0 = 0.21. These values were calibrated us- ρ(r)= ρ0 , (8)
ing the GIF (Kauffmann et al. 1999) and GIF2 (Gao et al. 2004) (r/rs)(1+r/rs)2
simulations,aswellasresimulationsdonebyDolagetal.(2005). where ρ is the characteristic density and r is the scale radius.
0 s
Here, mv represents the unevolved subhalo mass (the mass that Thus, the NFW profile depends on only two parameters, and for
thehalohadwhenitwasaccreted)andM0 representsthepresent- obtaining them, we first have to compute the redshift at which a
day host mass. As a caveat of this formula, we mention that the halowithmassM(atcurrentredshiftz)wasformed.Theimplicit
GIF simulation considered DM particles of 1.4 × 1010M(cid:10)h−1, definitionofthisredshift,calledtheredshiftofcollapseanddenoted D
t1h.3e×GI1F029oMf(cid:10)1.7h3−1×,an1d09thMer(cid:10)efho−re1,faonrdstmheallDerolmagasrseessim(duolwatniotnosthoef aszc⎛oll(M,z),isgivenby ⎞ own
lo
floreneg-esrtrheoamldi.ngmass),thefunctionalformofequation(2)mayno erfc⎝(cid:6) (cid:7)δc(zcoll(M,z))−δc(z) (cid:8)⎠=1/2. (9) aded
To account for time evolution, the authors find the following 2 σ2(0.01M,z)−σ2(M,z) fro
m
relationbetweenthemassofasubhaloattimet(giventhatitwas Thechoiceofthemassofthesmallsubstructure(of0.01M)is h
amccr(ett)e=damttimexept(cid:2)m−),tan−dtimts(cid:3)u,nevolvedmass: (3) mNaovtiavrartoedetbayl.c(o1m99p7a)r.isonstoN-bodysimulations,asmentionedin ttps://ac
sb v τ(z) Then, we proceed to determine ρ0 and rs using the following ad
e
relation: m
wheremsb istheevolvedmassandτ(z)isthecharacteristicmass- ic
losstimegivenby ρ0=ρcrit(z)δ0(M,z), (10) .ou
τ(z)=τ (cid:2) σ(z,M0) (cid:3)−1/2(cid:2)H(z)(cid:3)−1, (4) where ρcrit(z) is the critical density of the Universe at redshift z, p.com
0 σ(z=0,M0) H0 andδ0(M,z)isacharacteristicoverdensity(ordensitycontrast)that /m
dependsonthemassandredshiftofthehalowhoseNFWprofile n
whereτ =2.0Gyr. ra
0 wewanttoobtain. s
Inthiswork,itwasusefultoapproximatetheaccretiontimeof /a
allsubhaloesbythetimeatwhichthemainhalocollapsed,corre- Also, rtic
aspcqounidriendghtaolfaorefditsshmiftaszsco(lsl,ewehSiecchtiwone3ta.3k)e.aInsttheremresdosfhtihftealotowkh-ibcahcikt rs= rcvi(rM(M,,zz)), (11) le-abstra
timetoredshiftz(LBT(z)),equation(3)canbeapproximatedby wherer (M,z)isthevirialradiusofahaloofmassMatredshift c
msb=mvexp(cid:2)−LBT(zcolτl)(z−)LBT(z)(cid:3). (5) zivs,ireicqaoluriazrelevsdtiprosottnrhudecintdugernetsh(iietnyrtrhaedeqicuuaisrseeadtoffwothrheitchEheinhthsateleoiantvo–eDbraeegSaeistdteeelrfn-csgoirtsaymvwiotaliotthigniyng, t/445/1/85
0
Now,therequiredsubhalomassfunctionistheonethatincludes thisdensityis200timesthemeandensityoftheUniverse).c(M,z) /1
themass-lossofsubhaloesintime,soitshouldbe dN = dN dmv. istheconcentrationparameterofthehalo. 74
Expressingthesubhalomassfunctionintermsofmsdbmisbnsteaddmovfdmmsbv, ThevirialradiusisgivenbyNavarroetal.(1997)as 917
odnNeo=btaNin0sx−αe−6.283x3, x = mK−1(z,Mh), (6) rvir(M,z)=1.63×10−2(cid:11)h−1MM(cid:10)(cid:12)1/3 0 by gue
wdmheremmisthepresent-day(atredshifαtzM),hevolvedmassofasubhalo, ×(cid:2) (cid:6)m,0 (cid:3)−1/3(1+z)−1h−1kpc, (12) st on 0
Mhisthemassof(cid:2)thehosthaloandKisdefined(cid:3)by (cid:6)m(z) 5 A
K(z,Mh)=exp −LBT(zcoll(Mτ(hz)))−LBT(z) . (7) wvihriearlera(cid:6)dmiu(sz)isisgitvheenminatpterropdeernksiptyc.parameteratredshiftz,andthe pril 20
Thecharacteristicdensityisgivenby 19
Itcanbeseenthatthebehaviourofthesubhalomassfunctionis (cid:11) (cid:12)
that of a power law with an exponential cutoff, and that all mass δ =3.41×103(cid:6) (z) 1+zcoll(M,z) 3. (13)
scales are displaced as a function of time since halo formation. 0 m 1+z
The displacement is such that a present-day subhalo of mass m,
Finally,theconcentrationparameterisimplicitlygiveninterms
correspondedtoasubhaloofgreatermass(byafactorK),andsoits
ofthecharacteristicdensitycontrastbytherelation
abundanceisdecreasedbecauseitistheabundancecorresponding
tothegreatermassatthetimeofhaloformation. δ (M,z)= 200(cid:13) c3(M,z) (cid:14). (14)
Theselasttwoequationsgivethenumberofsubhaloespresentin 0 3 ln(1+c(M,z))− c(M,z)
ahaloofmassMatredshiftz,whichhaveamassbetweenmand (1+c(M,z))
m+dm,andsocorrespondtothesubhalomassfunctionthatwe Thus, the density profile of a particular halo or subhalo is, on
need.Inthenextsection,weexplorethesmoothdensityprofileof average,completelyspecifiedbyitsmassandredshift.
aparticularhalo,inordertoaccountforthelastleveloftheDM Inthenextsection,weproceedtostudytheconsequencesofthe
clusteringhierarchy. collapse of the baryons on the density profile of the DM haloes,
MNRAS445,850–868(2014)
DMannihilation effectsonthehigh-zIGM 855
andobtainamodifieddensityprofileforthelatterconsideringthe t , the time elapsed since the spike was formed, we use
spike
growthofacentralSMBH,aprocessthatappearstobeubiquitous t =LBT(z )−LBT(z),i.e.thelook-backtimetotheredshift
spike coll
ingalaxyformation. at which the halo collapsed (defined in Section 3.3) as measured
fromthecurrentredshift.
ForthegravitationalinfluenceradiusoftheBH,weuse
3.4 Adiabaticcontraction
GM
WeconsiderthemodificationintheNFWDMdensityprofilere- rBH=0.2 σ2BH, (17)
sultingfromthegravitationalcollapseofbaryons.Baryonsbehave sph
dynamicallydifferentlyfromDMintheircollapseprocessbecause whereMBH istheBHmass,andσsph isthevelocitydispersionof
they can heat and radiate away their energy, as they are not col- thespheroidcomponent(thebulgeinthecaseofhaloestypicalof
lisionlessandhavenon-zerointernalpressure.However,baryonic late-typegalaxies).
cooling will only take place in DM haloes with masses above a For σsph, we use the observational relation between spheroid
certaincriticalmassMcrit.Afirstestimateofthiscriticalmassisthe velocitydispersionandBHmass,aspresentedinMurray,Quataert D
Jeansmasssuchthatthebaryonsinhaloeswithsmallermassesare &Thompson(2005): ow
(cid:11) (cid:12) n
pcrrietsesruiareassusupmpoerstetdha(steteheBaDrMkanpae&rtuLrboaetbio2n00fr1o)m.Hwowhiecvhert,htehehaJeloaniss M =1.5×108 σsph 4 M(cid:10). (18) load
fvoirrimaleidzeidsshtaillloiens,thaenldintehaerrerefogrime,eT,ewghmicahrkisentoatl.th(e19c9as7e)fcoornaslirdeeardya BTHhemassoftheBH20d0e[pkemnds−s,1]onaverage,onlyonthemassofthe ed from
csloilglhaptlsyedpirfofecreesnst.FMincraitllcyr,itoenrieocnotuhladtwacocroryunthtsatfothrethperedseetnacielsooffDthMe hreolsattihoanlobaentwdeoennthbeoftahctmthaasstethseishogsitviesnabmyai(nLhagaloos,orCaosruab&haPloa.dTilhlea https
itselfmayalterMcrit,butasshownbyRipamonti,Mapelli&Ferrara 2008) ://a
(cid:11) (cid:12) (cid:11) (cid:12) c
(2007b),althoughtheJeansmassisaltered,M asconsideredby a
crit M M d
Tegmark changes by an O(1) factor only (and this is considering log BH =0.84log halo −2.1 (19) em
M(cid:10) M(cid:10)
LDM,becauseforthecaseofthemassiveWIMPsunderstudy,it ic
.o
wouldbeessentiallyunchanged).Therefore,inthiswork,wewill forthecaseofamainhalo,and u
considerthatonlythosehaloes(mainandsubstructure)thatattaina (cid:11) (cid:12) (cid:11) (cid:12) p.c
M M o
massgreaterthanM ,asgiveninfig.6ofTegmarketal.(1997),are log BH =0.84log halo −2.9 (20) m
crit M(cid:10) M(cid:10) /m
abletoundergoadiabaticcontraction.Forhaloesofsmallermass, n
wewillsimplyconsiderthemtobeofpureNFWform.(However, forthecaseofasubhalo. ras
Tmaantaioknao&fiLnite2rm01e4didaitsec-musasssaBprHosceastszth∼at3c0o,uwldhicahllocwouflodrgtehneerfaotre- sityPuptrtoinfiglealolfthaehpaileocheasstofoguetrhdeirf,ftehreenatdrieagbiamticeas,llayncdoimspgrivesesnebdyden- /article
adiabaticcontraction,butonlyforveryrarehaloes.) -a
ρ (r)=ρ (r),r >r , (21) b
Incomputingtheadiabaticallycompresseddensityprofile,differ- DM NFW BH stra
entauthorshaveconsidereddifferentalgorithms(e.g.Young1980; c
Btohnleulymprfeeonsrtchtrhaipelteiaotdnaila.gb1iav9tei8cn6;cboSyneNtlrlawactoatioroadnja&nduMeetctaGol.ath(u2eg0hf0o28r0)m,0aw5t)ih.oiWncheofaccoacnopsuoidninetsrt ρDM(r)=ρNFW(rBH)(cid:11)rBrH(cid:12)−γspike, rplateau <r <rBH, (22) t/445/1/85
0
mass(i.e.anSMBH)inthecentreofthepotentialwell. /1
This algorithm assumes that the density profile is of the initial ρ (r)= mχ , 3r <r <r , (23) 74
NFWformuptotheradiusofgravitationalinfluenceoftheSMBH DM tspike(cid:8)σv(cid:9) s plateau 917
0
(definedastheradiusuptowhichitisthemaincontributortothe b
y
containedmass),andthatwithinthisradius,thedensityprofileis ρDM(r)=0, r <3rs, (24) gu
describedbyapowerlawwithadifferent(steeper)exponent.Italso e
assumes(liketheBlumenthalalgorithm)thattheoriginalorbitsof whererplateau istheradiusatwhichthedensityofthespikeequals st o
DMparticlesintheNFWprofilewerecircular. themaximumattainableDMdensity,rsistheSchwarzschildradius n 0
AccordingtoNatarajanetal.(2008),thevalueoftheexponent oftheBHandtheinnerlimitof3rs correspondstothelaststable 5 A
ofthedensityprofileintheinnerregionisgivenby ocorbmitproefssaednodne-nrosittaytinpgroBfilHe.oIfnaFDigM. 1,hawloeoshfomwasthse5a×dia1b0a1t3icMal(cid:10)ly pril 2
γspike=2+ 4−1γ, (15) actleraerdlyshdifitstzin=gui1sh(atybpleic.aTlhoefoQuSteOrssylospteemcso)r.rTeshpeofnodusrtroegthimeeNsFaWre 019
whereγ istheexponentintheinnerregionoftheuncompressed γ =1andtheinnerslopecorrespondstoγspike = 73.
DMdensityprofile(γ =1inthecaseoftheNFWprofile).
The derivation of the value of the exponent can be found in
4 DM ANNIHILATION ENERGY OUTPUT
Quinlan,Hernquist&Sigurdsson(1995).
Finally,weneedthemaximumdensityandtheradiusofgravita- HavingconsideredtheclusteringpropertiesoftheDMatallscales
tionalinfluenceoftheBHtohavethecompletedensityprofile.For and hierarchy levels, and having explained two particular DM
themaximumdensity,weuse WIMPcandidates,motivatingtheirexpectedmassesandannihila-
tioncross-sections,wecanproceedtocomputetheexpectedenergy
m
ρmax= χ , (16) outputofannihilationsinlocalstructures(haloes)andasanaverage
DM t (cid:8)σv(cid:9)
spike forthewholeIGM.Inthissection,weobtaintheluminositydueto
where m is the mass of the WIMP particle, and (cid:8)σv(cid:9) is the annihilations of individual DM haloes, considering the clustering
χ
thermal relic annihilation cross-section (see Section 2.1). For analysedpreviously,andwealsocomputetheannihilationenergy
MNRAS445,850–868(2014)
856 I.J.ArayaandN.D.Padilla
wheren isthepresent-dayDMnumberdensity,andiscalculated
χ,0
as
ρ
n =(cid:6) c,0, (29)
χ,0 χ,0m
χ
where(cid:6) =(1−f )(cid:6) isthepresent-dayDMdensitypa-
χ,0 bary m,0
rameter,ρ isthepresent-daycriticaldensityoftheUniverseand
c,0
m istheWIMPmass.Also,n isthepresent-daybaryondensity,
χ b,0
andiscalculatedas
ρ
n =(cid:6) c,0 , (30)
b,0 b,0μm
H
where (cid:6) is the present-day baryonic density parameter (equal
b,0
tof (cid:6) ),m isthehydrogenmassandμisthemeanatomic
bary m,0 H D
weight of the baryon content. The mean atomic weight, consid- o
w
ering a universe with only hydrogen and helium (a good approx- nlo
Figure1. Theadiabaticallycompresseddensityprofileofahalowithmass imation, particularly before star formation), can be computed as ad
5×1013M(cid:10),atredshiftz=1. fμra=ctifoHn+of4hfHelei,uwmhberyenfHumisbethr.eAfrsascutmioinngofahvyadlruoegoefnXan=df0H.e74isatnhde ed fro
m
Y=0.26(thehydrogenandheliumfractionsbymass,respectively) h
rateinjectedtotheIGM,perbaryon,intheUniverseasawhole. givesfH=0.92andfHe=0.08.Thus,μ=1.24willbeusedinthis ttps
Wealsomentiontheobservationalevidenceofthisannihilationen- work. ://a
ergyinjectionprocessinsatellitehaloesandintheMW.Finally,we Knowing the absorption fraction of the energy fabs(z), we can ca
d
assesstheimportanceoftheDMclumpinessintheenergyoutput, readilycomputetheenergyinjectedtotheIGMperbaryonthrough em
comparingthecaseofaperfectlysmoothuniversewiththeonewith DM annihilations. Different authors use various prescriptions for ic
clusteredDM. fabs(z). Natarajan & Schwarz (2009) consider that only the pho- .ou
p
First,letusconsiderthegenericenergyoutput.Thereactionrate tonsresultingassecondaryannihilationproductscaninjectenergy .c
o
oftheannihilationprocessisgivenby to the IGM, and calculate the photon energy spectrum for differ- m
entneutralinomodelsexplicitly.Theyconsiderthatasthephotons /m
n
(cid:13)=nχ(cid:8)σv(cid:9), (25) propagate,theyloseenergy,andcomputetheprobabilitythatthey ras
scatterofftheIGMatoms.Theyfindthatf (z)isintherangeof /a
and thus, the variation of the number density of WIMPs can be 0.1–0.2forz<50(theydonotconsiderhigahbserredshifts). rtic
calculatedas le
Cumberbatchetal.(2010)computefabs(z)fortheneutralino(see -a
b
dn 1 alsoRipamontietal.2007a),calculatingfirstthenumberofphotons s
dt = 2n2χ(cid:8)σv(cid:9). (26) andelectronsproducedperneutralinoannihilationevent,andthey trac
obtain f (z) in the range of 0.01–0.1, being higher for higher z t/4
conAssidienrththeaatnanllihtihlaetiirornesptromcaesssstihseaWvaiIlMabPlesaarsepdaersttroofytehde,weneecrgayn (anOdthcoernabsasiduethrionrgs,relidksehiMftsapueplltioezt=al.15(20000).6), simply assume that 45/1/8
output (actually, only a fraction of this energy will be available f (z) = 1, and compute the maximal effects that their WIMP 50
for affecting the environment, depending on the mechanism that abs /1
candidates can have on the IGM. We will consider that f is a 7
couples the annihilation products and the IGM or halo medium). abs 49
constant that does not depend on redshift, and we will use the 1
Thenthepowerdensityoftheenergyoutputcanbewrittenas valuesf =0.01,0.1and1,inordertoobtainresultscomparable 70
abs b
(cid:14)χ = 21mχc2n2χ(cid:8)σv(cid:9). (27) toIthneFdiigf.fe2r,enwteausthhoowrs.the results obtained for the smooth energy y gue
s
Thisequationiscompletelygeneral,andsoitappliesforhaloes i1n0jeGcetiVon),apserexbpalrayionned, fionrthtwecoadpitfiofenr.eWnteWonIlMyPcomnsaisdseersth(1emTeaVximanadl t on
andalsoforthediffuseIGM.Letusnowconsiderthepoweroutput absorptionfraction(f =1),becauseforothervalues,theresulting 05
inthedifferentsituations(wewillcallthemlocalandglobal). abs A
curves should simply be rescaled by f . It can be seen that the p
effectofalowerWIMPmassis,asexpabescted,toboosttheenergy ril 2
injectionratebyafactor∝m−1. 01
χ 9
4.1 ThesmoothDMannihilationenergyinjection
rateperbaryon
4.2 TheDMannihilationluminosityofahalo
First,weconsidertheglobalcase,assumingaperfectlysmoothDM
Having considered the global case of the energy injection to the
density.Thiscaseistreated,forexample,byMapellietal.(2006),
IGM,wenowconsidertheenergyoutputperunittime(orluminos-
Ripamonti et al. (2007a) and Cumberbatch et al. (2010). As the
ity)ofaparticularhaloofmassMatredshiftz.FollowingNatarajan
Universemaybeinfiniteinspatialextent(aswouldbethecasefor
etal.(2008),fromthegeneralformula,theenergyoutputintegrated
flat,i.e.(cid:2)CDMandEinstein–DeSitter,andopencosmologies),the
onthevolumeofthehaloisgivenby
relevantquantityistheenergyinjectedtotheIGMperbaryon.In (cid:15)
termsofthepresent-dayWIMPnumberdensity,thisiscalculated L = (cid:8)σv(cid:9)c2 ρ2 (x,t)d3x, (31)
as χ 2mχ V DM
1 m whereρ isthecompleteDMdensityprofileofthehalo,account-
E˙smooth= χ c2n2 (1+z)3(cid:8)σv(cid:9)f (z), (28) DM
χ 2n χ,0 abs ing for the smooth main halo, the substructure and the adiabatic
b,0
MNRAS445,850–868(2014)
DMannihilation effectsonthehigh-zIGM 857
D
o
w
n
lo
a
d
e
d
Figure3. Theluminositytomass(Lχ/M)ratiooftheinternalDMannihila- from
tionluminosityofindividualhaloes,asafunctionofhalomass(intherange h
FWthiegIMudraPeshm2e.adsTcseuhsre.vsTemhtoeooadtohWtteeInMdecrPguyrmviaensjcseocortrfieo1sn0proGantedeVsp.teorabaWryIoMnPE˙mχa(zss),offo1rdTiefVferaenndt dstforhiofozfmre=tr-1ed1n0a0t−s,hr4etehtddoesc1htru0iifr1ptv7sle;eMs-idnc(cid:10)ooptr)–ar.rdeTtasichpsuheolendaddirf,cfttouehrrzeevne=dtsoct5tout,erztdvh=eecsud5irno0vtet–ahsndecdafosothrhureeerdslpopclnouognrtsv-ddectasoosrchrzoeers=drpecos0upn,rodvtnhetodes ttps://acad
toz=100.Thedifferentplotscorrespondtodifferentclusteringscenarios em
cdoencotrmacptoiosnedininbtoothth(esemeaSiencthioanlos3lu.2m–i3n.o4s)i.tyThainsdlutmheinsousbitsytrcuacntubree awnidthWsuIbMstPrumctausrseesa;ndinapdaiartbicautilcarc,otnhteratcotpio-lne,ftwpitlhotacoWrrIeMspPonmdassstootfh1eTceaVse; ic.ou
p
luminositysuchthat thetop-rightplotcorrespondstothecasewithsubstructureand1TeVWIMP .c
o
mass,butwithnoadiabaticcontraction;thebottom-leftplotcorrespondsto m
Lχ =Lχ,main+Lχ,subs, (32) thecasewithaWIMPmassof1TeVandneitheradiabaticcontractionnor /m
n
where substructureandthebottom-rightplotcorrespondstothecasewithadiabatic ras
contractionandsubstructure,butaWIMPmassof10GeV.Inalltheplots, /a
Lχ,main =Lχ(Mma(cid:15)in,z) cwoerraelsspooinndcilnugdelinthee-sνty=les1. massvaluesforthethreelowestredshiftsinthe rticle
(cid:8)σv(cid:9)c2 -a
= 2mχ V ρD2M(r,Mmain,z)dV, (33) Itcanbeseenthat,forthesamehalomass,theluminosityofthe bstrac
and haloincreaseswithredshift.Thisisaresultofthestructureforma- t/4
(cid:15) Mmain dN tionprocess,becauseaDMhalothatformedearlierwillbemore 45/1
Lχ,subs =Lχ,subs(Mmain,z)= dmLχ(m,z)dm. (34) concentratedandhaveahighercharacteristicdensity.Thus,itsan- /85
mfree nihilationrate,andinconsequenceitsluminosity,willbeincreased. 0/1
Note that in the above equations, Lχ(M, z) is the DM annihi- Also,itcanbeseenforthecasesthatincludeadiabaticcontraction 74
9
lation luminosity of a smooth, adiabatically compressed halo or that for each redshift there is a discontinuity in luminosity at the 1
7
subahlo,theonlydifferenceamongthemisthatforthehaloorthe masscorrespondingtoM (thecriticalmassforbaryoncooling, 0
crit b
subhalo, the mass of the SMBH depends differently on the mass asdiscussedinSection3.4).Also,itcanbeseenthatthegreatest y g
ofthehalo.Itisimportanttomentionthatinordertoaccountfor effectontheluminositycomesfromtheWIMPmass.Inaccordance u
e
s
thesubstructureluminosity,anintegrationofthesubstructuremass withwhatwasfoundforthesmoothcase,Lχ/Misalmostinversely t o
function multiplied by the luminosity of an individual subhalo in proportionaltomχ.Therearevariationstothisproportionalityin n 0
all the substructure mass range must be performed. This integra- the cases with substructure and adiabatic contraction, for masses 5
A
tionshouldbedonebetweenm ,thefree-streamingmassofthe greater than M , because the maximum attainable mass of the p
WIMP (the minimum mass thaftreea virialized DM halo can have), compressedspikcreitisalsodependentonmχ.Itcanbealsoseenthat ril 20
and M , the mass of the main halo (no substructure can be as thesecondgreatesteffectontheluminositycomesfromthepres- 1
main 9
massive as the main halo). The free-streaming mass will depend enceofadiabaticcontraction,andthatthepresenceofsubstructure
naturallyontheWIMPmass,asm ∝m−3.Basedontheanaly- contributesatmostwithanorder∼1factor(atredshifts5and10)
free χ
sisoftheeffectsofthefree-streaminglengthonthematterpower totheluminositywhenadiabaticcontractionisabsent,andthusit
spectrumbyAngulo&White(2010),weconsiderafree-streaming istheleastsignificantcontributor.
mass of m = 10−10M(cid:10) for a WIMP mass of m = 1TeV, and InordertoassesstheimportanceofDMannihilationasapossible
χ χ
a free-streaming mass of m = 10−4M(cid:10) for a WIMP mass of feedbackmechanisminhaloes,wecomparetheobtainedlight-to-
χ
m =10GeV. massratiosfortheannihilationluminositywiththetypicalmass-to-
χ
We explicitly compute the luminosity of DM haloes, in the lightratiosinhaloesatdifferentmassscales.Forhaloeswithmasses
(cid:2)CDM cosmology, considering two different WIMP masses inthe108–1015M(cid:10)range,thelogarithmofthemass-to-lightratio
(10GeV and 1TeV) and the presence or absence of substructure (log (ϒ))atredshiftzero(nearUniverse)variesbetween1.5and
10
andadiabaticcontraction.InFig.3,weshowtheresultsintheform 3,asshownbyEkeetal.(2004)andalsobyMarinoni&Hudson
ofthelight-to-massratio(L /M)fortheannihilationluminosity,as (2002).Also,ϒ hasaminimumforhalomassesaround1012M(cid:10),
χ
afunctionofhalomass. and it increases for lower masses due to SNe feedback and for
MNRAS445,850–868(2014)
858 I.J.ArayaandN.D.Padilla
highermassesduetoAGNfeedback(seeLagosetal.2008andalso
Mutch,Croton&Poole2013).Thus,wenotethatthecorresponding
light-to-massratiosforthehaloes(ϒ−1)aremuchhigherthanthe
L /M curves that we obtain for redshift zero. Furthermore, the
χ
minimuminϒ occursformasseswheretheadiabaticcontraction
isalreadypresent,andthereforetheL /Mismaximum,andthus
χ
bothmass-to-lightratioscorrelatepositivelyinsteadofnegatively
as it would be expected in the case of star-formation-quenching
feedback.Therefore,wecanconcludethattheDMannihilationis
irrelevantasafeedbackmechanism.(However,seeAscasibar2006.)
Itshouldberememberedthat,althoughweplothaloluminositiesup
tohaloesofmassof1017,themostmassivegravitationallybound
haloestoday(atz =0)havemassesoftheorderof1015M(cid:10),so
D
more massive haloes are virtually non-existent; and furthermore, o
w
themassesofthehaloesthathaveaparticularnumberdensityata nlo
certainredshift(givenbythecharacteristicdimensionlessmassν) ad
e
fallsrapidlywithredshift.Thisisthereasonwhyeventhoughthe d
computedLχ/MratioofhaloeswithM>Mcritatz=50and100is from
comparabletothenearUniverseϒ−1,saidhaloesare,forpractical h
purposes,negligible.Forclarity,wealsoincludeinFig.3thelines Figure4. Theclumpedenergyinjectionrateperbaryon(E˙χ(z))asafunc- ttps
correspondingtoν=1atdifferentredshifts. tionofredshift,fordifferentscenarios,inthicklines.Thedottedcurvecor- ://a
respondstothecasewithsubstructureandadiabaticcontractionforaWIMP ca
massof1TeV,thelong-dashedcurvecorrespondstothecasewith1TeV de
4.3 Theglobal,clumped,DMannihilationluminosity m
WIMPmassandsubstructurebutnoadiabaticcontraction,thedot–dashed
perbaryon curvecorrespondstothecaseof1TeVWIMPwithneitheradiabaticcon- ic.o
u
tractionnorsubstructureandtheshort-dashedcurvecorrespondstoaWIMP p
Having computed the DM annihilation luminosity of a particular .c
massof10GeVwithsubstructureandadiabaticcontraction.Alsoincluded o
halo of mass M at redshift z, we now return to the problem of m
computingtheglobalenergyinjectionratetotheIGMperbaryon, asthinlinesarethepreviouscurvesofFig.2,correspondingtothesmooth /m
cases,fora1TeVWIMP(dotted)anda10GeVWIMP(short-dashed). n
butconsideringtheclumpinessoftheDMonallclusteringscales. ra
s
To do this, we simply consider that, knowing the luminosity of /a
aparticularhalo(thatalreadyincludessubstructureandadiabatic duetoDMhaloeswillbedecliningrapidlywithincreasingredshift rtic
le
contraction) and knowing the halo mass function (calculated in after the truncation of low-mass haloes due to the free-streaming -a
b
Section3.1),theenergyoutputratedensity(orluminositydensity) limitbecomingimportant,weuseastheclumpedDMinjectionrate s
duetoDMannihilationsisgivensimplyby thegreatervalueamongtheonecalculatedinequation(46),andthe trac
(cid:15) ∞ dn onecalculatedinSection4.1. t/44
(cid:14)χclumped(z)=(1+z)3 Lχ(M,z)dM(M,z)dM, (35) The results obtained for the clumped energy injection rate 5/1
mfree per baryon, considering different clustering scenarios and WIMP /8
5
wheremfreeisthefree-streamingmass,Lχ isthetotalDMannihila- masses,areshowninFig.4.Weonlyconsiderthemaximalabsorp- 0/1
tionluminosity(mainhaloplussubstructure)ofaDMhaloofmass tionfraction(fabs=1),becauseforothervalues,theresultingcurves 74
Matredshiftz, ddMn isthepreviouslycalculatedhalomassfunction shouldsimplyberescaledbyfabs. 917
(incomovingcoordinates)andthefactor(1+z)3istoconvertthe It can be seen that the case considering a WIMP with mass of 0
b
comovingdensitytoaproperdensity.Weconsidermmax=1017M(cid:10) 10GeVistheonethatgivesthehigherinjectionrate,inagreement y g
sincethenumberdensityofcollapsedDMhaloeswithmassabove with what was discussed above. However, the curve for this case u
e
thislimitingmassisnegligible. hasacutoffatz∼50insteadofatz∼80likeforthecasescon- st o
Finally,theenergyinjectionratetotheIGMperbaryoncanbe sideringa1TeVWIMP.Thiscanbeunderstoodasaconsequence n 0
directlycomputedas ofthehigherfree-streamingmass(andthus,higherminimummass 5
A
thathaloescanhave),andthefactthatthemassofthehaloesthat p
E˙clumped(z)= (cid:14)χclumped(z), (36) haveacharacteristicν =1(thehaloesthathavejustrecentlycol- ril 2
χ n (z) 0
b lapsed)decreaseswithincreasingredshiftsuchthatwithahigher 1
9
wheren (z)isthepropernumberdensityofbaryonsatredshiftz free-streamingmass,lesshaloesareallowedtoformathigherred-
b
(equalto(1+z)3n intermsofthepresent-daybaryonnumber shifts.Wecanseethatthepresenceofadiabaticcontractionisalso
b,0
densitycomputedinSection4.1). veryimportantforboostingtheannihilationenergyinjection.We
ItisimportanttomentionthattheenergyinjectionratetotheIGM alsonotethatthecurvesconsideringadiabaticcontractionhavea
perbaryoncomputedinthiswayconsidersonlythecontributionof secondarybroadpeakatlowerredshift,besidesthepeakadjacent
theDMhaloesandnotofthesmoothcomponent.Inthecaseofa tothecutoff.Thissecondarypeakispresentduetotheexistenceof
zerofree-streamingmass,alltheDMwouldbeinhaloessincethere M ,becauseforhigherredshifts,thefractionofhaloesthathave
crit
would be no minimum mass scale for the formation of collapsed M>M decreasesduetotheredshiftdependenceofthehalomass
crit
structures.However,duetothenon-zerofree-streamingmass,all function,andsodoestheenergyoutputinDMannihilations.The
haloeswithmasseslessthanthatvalue,thatshouldexistaccording curve for the case considering substructure but no adiabatic con-
tothehalomassfunction,aresimplynotallowedtoform.Asthe tractionisalmostthesameasthatforthecaseconsideringonlythe
clumpedenergyinjectionshouldbealwaysgreaterthanthesmooth NFW profile of the main halo, with neither adiabatic contraction
energyinjection(computedinSection4.1),andasthecontribution norsubstructure.Theonlyslightdifferenceoccursforintermediate
MNRAS445,850–868(2014)
DMannihilation effectsonthehigh-zIGM 859
redshiftsanditamountstoanorder∼0.1factor.Thereasonforthis atredshiftzero,E˙smooth(z)ishigheraswell,andsotheclumpiness
χ
isthat,asdiscussedinSection4.2(withrespecttotheinterpretation C(z) is not. It can also be seen that in agreement with what was
ofFig.3),thecontributiontotheluminositybythesubstructure,in discussedbefore,theadiabaticcontractionmechanismisimportant
theabsenceofadiabaticcontraction,isofatmostanorder∼1factor forlowredshifts,andthepresenceofsubstructureisirrelevantfor
fortheseredshifts,whileatthesametime,lessmassivehaloeswith increasingtheclumpinessprovidedthattheadiabaticcontractionis
lesssubstructurearethemostabundant. present(forhaloesandsubhaloeswithM>M ),asdiscussedin
crit
Inthenextsection,weobtaintheclumpinessfactor,C(z),implied theprevioussections.
byourenergyinjectioncalculations,andcompareourresultswith We show also the clumpiness factors C(z) obtained by
thoseobtainedbyCumberbatchetal.(2010). Cumberbatch et al. (2010), in Fig. 5, using different line-styles.
Thetwosetsofcurvescanbereadilycomparedasthedefinitionsof
theclumpinessfactorsarethesame.Thereare,however,differences
4.4 Theclumpinessfactor
inthemethodsforcomputingthem.Inparticular,Cumberbatchetal.
The clumpiness factor is a useful tool when accounting for DM (2010)computethehalomassfunctionusingthePress-Schechter D
clusteringinglobalDMannihilationenergyinjectioncomputations, (P-S)theory,startingfromananalyticfittoσ(M,z),andthencon- ow
n
liketheonesrequiredforcalculatingtheheatingandionizationof siderthatsubstructureandsubsubstructurefollowapurepower-law lo
theIGMduetoDM(seeSections6.1and6.2).Itsimplyrelatesthe massfunctionwithexponent−2.Ingeneral,itcanbeseenthatwe ad
e
clumpedenergyinjectiontothesmoothoneby patrehdiigchtlreesdsschliufmt.pTinheessloawtelorwclruemdsphinifets,sbuptrecdoimctpeadrawbliethclruemsppeicntestos d fro
E˙clumped(z) m
C(z)= χ . (37) Cumberbatch et al. (2010) at low redshift can be understood be- h
E˙χsmooth(z) causetheyalsoincludesubsubstructure(subsubhaloeswithinsub- ttps
WeshowthevaluesobtainedfortheclumpinessfactorC(z)in haloes),whichsignificantlyenhancestheannihilationoutputofthe ://a
mostmassivehaloesoncetheyarealreadyformed.Also,atinterme- ca
Fig.5.Themeaningsofthecurvesareexplainedinthecaption. d
diateredshift,ourinclusionofadiabaticcontractionincreasesour e
In general, it can be seen that for all the cases considered, the m
clumpinesssignal,compensatingforthelackofsubsubstructure.At ic
clumpinessisadecreasingfunctionofredshift.Thisisreasonable, .o
higher redshift, the contributions from both the substructure (and u
becauseastheredshiftincreases,theuniversewilltendtobemore p
subsubstructure) and the adiabatic contraction are expected to be .c
homogeneousanduniform,becausethegravitationalcollapseand o
negligible,andtherefore,anyremainingdifferencewouldonlybe m
formationofstructureswouldhavehadlessandlesstimetooccur. /m
Also,forallthecasesconsidered,theuniversewillbepractically duetotheuseofslightlydifferentprescriptionsforcomputingthe n
completelyhomogeneousbyredshift(z∼80),exceptforthecase halomassfunction. ras
WedonotincludeSommerfeldenhancementinourcalculations, /a
wneiothus1a0tGreedVshDifMtzW∼I5M0.P,Niontewthhiacthinthtehiusnciavseer,sealbtheocuogmheEs˙hcloummpeodg(ze-) because it is highly dependent on unknown physics (the mass of rticle
χ the new force carrier and the resulting potential, as well as the -a
ishigherthanfortheothercasesbyalmostanorderofmagnitude b
pithsyesfifceacltsicsentoardioirseicntlywhmicohditfhyis(cid:8)σnevw(cid:9) bfoyrcaeccoonusltdanatrifsaec)t;ohr.oAwsevwere, strac
alreadymentionedintheIntroduction,thepresenceofSommerfeld t/4
4
5
enhancementiscontroversialandnotsupportedbyallobservational /1
evidence. /8
5
0
Having computed the DM annihilation luminosity generated /1
within a DM halo for different scenarios, in the next section we 74
9
studytheluminosityreceivedbyaDMhaloduetotherestofthe 1
7
Universe. 0 b
y
g
u
e
s
5 EXTERNAL DM ANNIHILATION ENERGY t o
n
INJECTION ON A HALO 0
5
A
Inprevioussections,wehaveconsideredparticularWIMPmodels, p
calculatedtheDMclusteringonallscalesexplicitlyandcomputed ril 2
0
theDMannihilationluminositygeneratedwithinaparticularDM 1
9
halo,andtheenergyinjectionperbaryonontheIGM.Inthissection,
wecalculatetheintensity(Js−1m−2sr−1)oftheaverageradiation
field due to DM annihilations at different redshifts, considering
ourdifferentclusteringscenarios,andfromthisweobtainanup-
Figure5. TheclumpinessfactorC(z)asafunctionofredshift.Thethick perboundonthepowerreceivedbyahaloperunitofhalomass
curves correspond to our results. The line-styles of the thick curves are
(L(cid:10)/M(cid:10)), which can be compared with the luminosity-to-mass
givenasinFig.4.Forcomparison,wealsoincludethecurvesobtainedby
ratiosduetoannihilationsinsidethehalothatwerecomputedinthe
Cumberbatchetal.(2010)forthecaseofanNFWprofile,takenfromthe
previoussection.Forthis,wewillusetheannihilationluminosity
upperpaneloftheirfig.3,asthincurves.Theline-stylesofthethincurves
correspondtodifferentvaluesof(Mmin/M(cid:10),Mcut/M(cid:10)).Thesolidline ofahalothatwasobtainedinSection4.2,togetherwiththecos-
correspondsto(106,106),thedottedlinecorrespondsto(10−4,106),thelong- mologicalradiativetransferformalismaspresentedinsection2.1
dashedlinecorrespondsto(10−4,10−4),theshort-dashedlinecorresponds ofHaardt&Madau(1996).Thepowerreceivedfromtheradiation
to(10−12,106)andthedot–dashedlinecorrespondsto(10−12,10−12).We fieldbyanyhalowilldependonitseffectivecross-section.Forthis
referthereadertothepaperoftheauthorsforfurtherdetails. cross-section,wesimplyconsiderthenumberofelectronspresent
MNRAS445,850–868(2014)