A map of the non-thermal WIMP

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

A

map

of

the

non-thermal

WIMP

Hyungjin Kim

a

,

b

,

Jeong-Pyong Hong

c

,

d

,

Chang

Sub Shin

e

,

f

,

∗

a_{DepartmentofPhysics,KAIST,Daejeon34141,RepublicofKorea}

b_{CenterforTheoreticalPhysicsoftheUniverse,InstituteforBasicScience(IBS),Daejeon34051,RepublicofKorea} c_{InstituteforCosmicRayResearch,TheUniversityofTokyo,5-1-5Kashiwanoha,Kashiwa,Chiba277-8582,Japan} d_{KavliIPMU(WPI),UTIAS,TheUniversityofTokyo,5-1-5Kashiwanoha,Kashiwa,Chiba277-8583,Japan} e_{AsiaPacificCenterforTheoreticalPhysics,Pohang37673,RepublicofKorea}

f_{DepartmentofPhysics,Postech,Pohang37673,RepublicofKorea}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received10December2016

Receivedinrevisedform17February2017 Accepted4March2017

Availableonline8March2017 Editor:G.F.Giudice

We studythe effect ofthe elasticscatteringon thenon-thermal WIMP,which isproduced bydirect decayofheavyparticlesattheendofreheating.Thenon-thermalWIMPbecomesimportantwhenthe reheating temperatureiswellbelowthefreeze-outtemperature.Usually,twolimitingcaseshavebeen considered.Oneisthattheproducedhighenergeticdarkmatterparticlesare quicklythermalized due totheelasticscatteringwithbackgroundradiations.Thecorrespondingrelicabundanceisdeterminedby thethermallyaveragedannihilationcross-sectionatthereheatingtemperature.Theotheroneisthatthe initialabundanceistoosmallforthedarkmattertoannihilatesothatthefinalrelicisdeterminedby theinitialamountitself.Westudytheregionsbetweenthesetwolimits,andshowthattherelicdensity depends not onlyonthe annihilation rate,butalsoonthe elasticscatteringrate.Especially, therelic abundanceofthe p-waveannihilatingdarkmattercruciallyreliesontheelasticscatteringratebecause theannihilationcross-sectionissensitivetothedarkmattervelocity.Wecategorizetheparameterspace intoseveralregionswhereeachregionhasdistinctivemechanismfordeterminingtherelicabundanceof thedarkmatteratthepresentUniverse.Theconsequenceonthe(in)directdetectionisalsostudied.

©2017TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3_.

1. Introduction

The weakly interacting massiveparticle (WIMP) is one ofthe promisingdarkmatter(DM)candidatesbecauseitcanbenaturally incorporated in new physics beyond the Standard Model, and it givesinterestingobservableconsequences.

In the standard thermal history, the most important quantity todeterminetherelicdensityisathermalaveragedpair annihila-tioncross-section,



σ

annvrel



T.Taking“thermalaverage”isjustified

becausetheelasticscatteringratebetweentheWIMPdarkmatter andthebackgroundradiationismuchbiggerthantheannihilation rate, so the kinetic decoupling happenswell afterthe dark mat-ter freeze-out [1,2]. Usually, the elastic scatteringdoes not have thespecialrole todeterminetherelicdensity,butitiscrucialfor smallscalestructuressincetheinteraction suppressesthegrowth ofthe dark matterdensity perturbation [3–6]. When the reheat-ingtemperatureislow,itseffectismoreinterestingdependingon

*

Correspondingauthor.

E-mailaddresses:hjkim06@kaist.ac.kr(H. Kim),hjp0731@icrr.u-tokyo.ac.jp (J.-P. Hong),changsub.shin@apctp.org(C.S. Shin).

whetherthekineticdecouplinghappensbeforeoraftertheendof reheating[7–11].

Inthisletter,we studythepossibilitythattherelicabundance of WIMP explicitly dependson the elastic scattering rate. In the context ofself-interacting darkmatter,such possibilityisrealized bynotingthatthedominantannihilationratefrom3

→

2 scatter-ing and theelastic scatteringbetweenthe DMandthermal bath can be independent so that we can take suitable parameters to show such behavior[12].This assumption isnot usually valid in the caseof WIMPbecause the annihilationand elasticscattering cannotbetreatedindependently.

However, when the reheating temperature is well below the dark matter mass, a new possibility emerges. At the end of re-heating,thedarkmatterisproducedbythedirectdecayofheavy particles.Suchnon-thermallyproduceddarkmatterparticleshave veryhighenergies.Evolutionofthedarkmattermomentumgives a strongeffect onthe annihilationcross-section, andsuch evolu-tionisdeterminedbytheelasticscatteringrate.Consequently,the relativesizeoftheannihilationrate,theelasticscatteringrate,and the Hubblerateat theend ofreheatingcan give various mecha-nismstodetermine thefinal relicdensityofthedarkmatter. We

http://dx.doi.org/10.1016/j.physletb.2017.03.005

0370-2693/©2017TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3_.

classify the parameter space into the regions whereeach region has distinctive mechanism to determine the relic density of the darkmatter. We also provide analytic expressions andnumerical resultsforeach ofthose mechanisms.Especially,we findthatthe p-waveannihilatingdarkmatterhasmoreinterestingproperty be-causethecross-sectionhighlydependsontheexpectationvalueof thedarkmattermomentum.

Insection2,wepresentourbasicset-up.Insection3,we com-putethemomentumevolutionofthedarkmatterafterits produc-tionattheendofreheating.Theeffectofthemomentumevolution ontheannihilationcross-sectionandthecorrespondingfinal abun-danceofthedarkmatterarediscussedinsection4.Wediscussthe constraintsfrom(in)directdetectionexperimentsinsection5,and concludeinsection6.

2. Thermalhistoryofthenon-thermalWIMP

Inourset-up,thereistheearlystage ofthematterdominated Universemaintainedbya longlivedheavyparticle,

φ.

Aftermost of

φ

decay,theUniverse is“reheated”andradiation(

γ

) starts to dominatetheenergydensityoftheUniversewithareheating tem-perature,Treh

∼





φMPl.Ononehand,thedarkmatter(

χ

)canbe

producedeitherfromthescatteringoftheradiationbackground,or fromthedirectdecayof

φ

withabranchingfractionBrχ .

Ignoringthesub-leadingcontributions,thecorresponding Boltz-mannequationsofeachcomponentsaregivenas

˙

ρ

φ

= −

3H

ρ

φ

− 

φ

ρ

φ

,

˙

ρ

γ

= −

4H

ρ

γ

+

Brγ



φ

ρ

φ

,

˙

nχ

= −

3Hnχ

+

Brχ



φ

ρ

φ mφ

− σ

ann vrel



χn2χ

,

+ σ

annvrel



T

(

neqχ

)

2

,

H

=



ρ

φ

+

ρ

γ

+

ρ

χ 3M2_Pl

,

(1)

where MPl is the reduced Planck mass. Here we consider a

sit-uation that the reheatingtemperature islower than thethermal freeze-outtemperatureofthedarkmatter(Tfr).Beforetheendof

thereheating,



φ

≤

H ,thereareseveralsourcesforthedark mat-terdensity. First of all, a usual freeze-out mechanism can work withnχ

|

Tfr

∼

H

(

Tfr

)/



σ

annvrel



Tfr,whiletheresultingabundanceis subsequentlydilutedbycontinuousentropyinjection.IfBrχ isbig, quasi-static equilibriumstate can persistuntil theendof reheat-ing(



σ

annvrel



Tn2χ

∼

Brχ



φ

ρ

φ

/

mφ)[13].Alsoifmφ islargeenough tosatisfymφ



m2χ

/

T ,productionfromaninelasticscattering be-tweenthermalbathandaboostedradiationproducedby

φ

decays becomesimportant[14,15].Herewe takearathermoderate hier-archy betweenthe mass of

φ

and

χ

asmφ

/

mχ

=

O(

10

−

100), andasubGeVreheatingtemperaturesothat theinelastic scatter-ing is subdominant. For Treh



mχ anda sizable Brχ , the most

importantsource ofthe late time darkmatter abundance is the directdecay of

φ

atthe endof reheating. It is known that such non-thermalproductionoftheDMcanbesimplifiedby assuming thatthe darkmatter isinstantaneouslyproduced fromthe heavy particledecayatT

=

TrehwithaninitialamountoftheDMgiven

asnreh_χ

=

Brχ

ρ

φ

/

mφ[16–19].Insummary,weareinterestedinthe followingrangeofparameters:

Treh

≤

Tfr



mχ



mφ

∼

O

(

10

−

100

)

mχ

.

(2) Becauseofthehierarchybetweenmasses,theinitialenergyofthe DMis muchgreater than mχ .So we first considerthe evolution ofthedarkmattermomentumandthenconsideritseffectonthe annihilationrate.

3. EvolutionoftheDMmomentum

Afterthedarkmatterisproduced,it experiencestwo typesof interactions.Oneistheelasticscatteringbythebackground radia-tions (

χ γ

→

χ γ

). The other one ispair annihilationofthe dark matter into the radiations (

χ χ

→

γ γ

). The effect of pair pro-duction from thermal bath

(

γ γ

→

χ χ

)

is negligible if the DM abundance at Treh ismuch larger than the equilibriumvalue. In

principle, both of the elastic scatteringand annihilation are rel-evant forthe dark mattermomentum evolution. However, aswe discussinAppendix A,the contributionfromannihilation canbe safelyignored when themomentum distributionof

χ

hasa nar-rowwidthcompared toitsmeanvalue.Initially, darkmattersare produced fromthe decaysof

φ,

so thewidth

σ

sd,χ

∼ φ

is natu-rallysmallerthanthemeanvalueofmomentum,



p



χ

∼

mφ.Until the relaxation time when the momentum distribution arrives at its equilibriumone up to the normalization factor, the width of the distribution is still small and our simplification is justified. Thenthemomentumevolutionisgovernedbythefollowing equa-tion[20], dpχ dt

+

Hpχ

= −

nγ

 

d

(

σ

elvrel

)

p



χ,T

≡ −

nγ



σ

elvrel



pχ



χ,T

,

(3)

where pχ

≡ 

p



χ ,

· · · 

χ,T isthe averageoverthe distributionof

χ

and

γ

intherestframeofthethermalplasma,

σ

el istheelastic

scatteringcross-section,



p isthechangeofthedarkmatter mo-mentumfromsingle eventoftheelasticscattering, andnγ isthe numberdensityofthebackgroundradiation,nγ

∼

g_∗T3_.

Therighthandsideof(3)canbesimplifiedaspχ nγ times



σ

elvrel



pχ



χ,T p_χ

⎧

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎩



σ

elvrel



χ,T

(

I

),



σ

elvrel



χ,Tpχ T_m2 χ

(

II

),

σ

elvrel



χ,T_mχT

1

−

3mχ T p2 χ



(

III

),

(4)

fordifferentrangesofthedarkmattermomentum,

(

I

)

m2_χ



p_χT

,

(

II

)

pχT



m2χ



p2χ

,

(

III

)

pχT



p2χ



m2χ

.

(5) In(I), the DMisrelativisticinthe plasma restframe,andin the centerofmass(cm)frames.In(II),theDMisnon-relativisticinthe cmframe,whereasstillrelativisticintheplasmarestframe.In(III), theDMisnon-relativisticinbothframes.Inthelastcase,the addi-tionalfactorintheelasticscatteringratedrives pχ tothe equilib-riumvalue, peqχ

=



3mχ T .Fortheorderofmagnitudeestimation, we obtain



σ

elvrel



pχ

/

pχ

∼ 

σ

elvrel



p



χ

/

pχ . Thus the addi-tionalfactors ofthe scatteringrate, 1

(

I

),

pχ T

/

m2

χ

(

II

),

T

/

mχ

(

III

)

areeasily understoodfromthefact thattheallowed phasespace,



p



χ

/

pχ ,becomeswiderasthecollisionenergiesbecomehigher. Sincethecommonfactornγ dependsonthetemperature,theDM can quickly arrive at kinetic equilibrium or it can justdecouple relativisticallydependingonTreh.

Whentheinitialmomentumofthedarkmatterismuchgreater thanmχ ,Fig. 1showspossibleevolutionofthemomentumfor dif-ferentreheatingtemperatures.Atarelatively highreheating tem-perature, the elastic scattering rate is large enough to make the dark matter in kinetic equilibrium instantaneously after its pro-duction. As the temperature goes down, the momentum follows theequilibriumvalue(peqχ

∝

1/

√

a)untilthekineticdecoupling.If the reheatingtemperature isrelatively low,afterthedarkmatter

Fig. 1. Momentumevolutionofthenon-thermallyproduceddarkmatterfor differ-entreheatingtemperatures,Treh=0.2GeV (blue),30 MeV(red),5 MeV(magenta). Inthisplot,wesetmχ=300GeV.TheinitialmomentumoftheDMisgivenas preh

χ =20mχ.Thedashedlinesarethekineticequilibriumvalues,peqχ =3mχT . (Forinterpretationofthereferencestocolorinthisfigurelegend,thereaderis re-ferredtothewebversionofthisarticle.)

momentumexperiencesasmallsharpsuppressionaround Treh,it

slowlydecreasesaspχ

∝

1/a,anditcouldbecomenon-relativistic wellafterreheating(magentaline).

There isa natural connectionbetween themomentum evolu-tionandthedarkmatterpairannihilationrate.Forexample,ifthey are highly relativistic, the cross-section becomes



σ

annvrel



χ

∝

1/p2_{χ ,}anditwillincreaseastheenergyoftheparticledecreases. Therefore, ifthe darkmatter is not instantaneously thermalized, theannihilation ofthedark mattercould happen later whenthe annihilationcross-sectionbecomeslargeenoughtostartthe anni-hilation.

Sincethe corresponding darkmatter abundanceis affectedby the evolution of the annihilation cross-section, we can find the connection between the final yield of the dark matter, and the elasticscatteringrate.

4. Evolutionofthepairannihilationcross-section

When theinitialabundance(nreh

χ )ismuchgreaterthann

eq

χ ,the productionof

χ

fromthermalbathcanbeignored,andthe corre-spondingBoltzmannequationforthedarkmatternumberdensity issimplifiedas

˙

n_χ

+

3Hn_χ

= −σ

annvrel



χn2χ

.

(6)

Solvingtheaboveequation,wefindtheyieldofthedarkmatterat thepresenttime,t0

treh,as

Y_χ

(

t0

)

=

Yχ

(

treh

)

⎛

⎝

1

+

n reh χ Hreh 1



0 du

σ

annvrel



χ

⎞

⎠

−1

,

(7)

whereu

≡

√

treh

/

t,HrehistheHubblerate,andsrehistheentropy

ofthe Universe at T

=

Treh. The yields are denoted by Yχ

(

t0

)

=

(

nχ

/

s

)

t0,Yχ

(

treh

)

=

n reh

χ

/

sreh.Thetime dependenceof



σ

annvrel



χ isdeterminedbythatofpχ

(

u

)

governedbyEq.(3).Moreprecisely, wehavetoevaluatetheannihilationcross-sectionthatisaveraged overthe full time dependent momentumdistribution function of darkmatter.However, whendarkmattersare non-thermally pro-ducedbytwo-bodydecays,thewidthofthedistributionwouldbe small, andfor



t

<

1/ el,



pn



χ

= 

p



nχ

(1

+

O(

σ

sd2,χ

/

p2χ

))

pnχ . Thusitisagoodapproximationtotake



σ

annvrel



χ asthefunction of pχ

(

u

)

until the relaxationtime,



t

∼

1/ el. After relaxation,

the momentum distribution will be proportional to the equilib-riumvalue, in whichthe standard deviationandmean value are

in the same order. This leads to

O(

1) difference between



pn

_

χ and



p



n_{χ ,}butthisdoes notchangeourresultqualitatively.Solving thefullBoltzmannequationswillbediscussedinfuturework.

Two limitingcases are familiar.One is that pχ

(

u

)

quickly ar-rives at its equilibrium value within the period much shorter than the Hubble time as given in Fig. 1 with blue color. The darkmatter annihilationhappensafterits thermalizationbutstill much fasterthan the Hubbleexpansion rate. Therefore, Yχ

(

t0

)

=

Hreh

/(



σ

annvrel



Trehsreh

).

The other limit is that the initial abun-dance is too small so that



σ

annvrel



nrehχ



Hreh. Annihilation

barely happens, and theyield is preserved; Yχ

(

t0

)

=

Yχ

(

treh

).

In

bothcases,thefinalyieldsdonotexplicitlydependontheelastic scatteringcross-section.

There is an intermediate domain between these two limiting cases.Includingtheaboveexamples,weidentifythreemechanisms for the relic density ofthe DM. After the productionof the DM fromthedirectdecayoftheheavyparticles,therelicabundanceis determinedbyoneofthefollowingmechanisms:

•

(N.A.) No Annihilation: the annihilation rate,



ann

(

T

,

Eχ

),

is

always smallerthan H

(

T

)

forT

≤

Treh,regardlessofthedark

matter momentum.Thereforethedarkmatterdoesnot anni-hilateafterthereheating,andtheyieldispreserved.

•

(I.A.) Instantaneous Annihilation: the elastic scattering rate,



el

(

Treh

,

Eχ

),

isalwaysgreaterthan Hreh,sothatthe

momen-tumofthedarkmatterquicklyapproachestotheequilibrium value, andmostof theDM pairannihilationalso happensat T

Treh.Especially forthe p-wave annihilatingdark matter,

the finalabundance dependson therelativesize of



ann and



el.

•

(C.A.) ContinuousAnnihilation: theelastic scatteringrate be-comes smaller than Hreh at Eχ

mχ , so the dark matter

decouples with a relativistic energy, and travels freely af-terits production.In thiscase,



σ

annvrel



∝

1/p2χ

∝

a2,while H

∝

T2

∝

1/a2.Thereforethe annihilationcould happen con-tinuouslyuntiltheDMbecomesnon-relativistic.

Theratesaregivenby



ann

(

T

,

Eχ

)

=

nχ

σ

annvrel



χ

,



el

(

T

,

Eχ

)

=

nγ



σ

elvrel



pχ



_χ,T pχ

,

(8) where Eχ

= (

m2_χ

+

p2_χ

)

1/2.

Fig. 2 showsthe correspondingdomains, heuristically.Thered colordenotestheregionwheretheelasticscatteringrateisgreater than the Hubble parameter for given preh_χ and Treh. In this

re-gion, thedark matter momentumevolves nearly along the verti-cal direction. It quickly approaches to pdec

χ , which is defined as



el

(

Treh

,

Edecχ

)

=

Hreh.Ifpdecχ

>

p

eq

χ ,then thedarkmatter momen-tumredshiftsaspχ

∝

T .Theregion(C.A.)isboundedfrombelow bytheconditionthatthedarkmatterisdecoupledwitha relativis-ticenergy.

For each regions, we can obtain the approximate formulafor thefinalyieldvaluebysolvingtheEq.(7).Sinceweareinterested inthecasewhere pχ Treh

<

m2χ ,we parameterizetheannihilation andelasticscatteringcross-sectionsinthefollowingway:

σ

annvrel



χ

=

α

2 ann E2 χ



2p2_χ E2 χ



kann

,

σ

elvrel



χ,T

=

α

2 el m2 χ



E2 χT2 m4 χ



kel

.

(9)

Fig. 2. Illustrativeplotshowingthedomainsofdifferentmechanisms todetermine therelicdensityoftheWIMPdarkmatter.Theelasticscatteringisactiveinthered coloredarea:el(Treh,Eχ)>Hreh.Thedarkmatterpairannihilationisactiveinthe greencoloredarea:ann(Treh,Eχ)>Hreh.Themagentalinesaretheboundaryof thefollowingthreeregions.Region(N.A.):noannihilationafterdirectproductionat T=Treh.Region(I.A.):instantaneousthermalizationandannihilation.Region(C.A.): theelasticscatteringbecomesinactivewhenthedarkmatterisrelativistic,butstill thepairannihilationhappensafterreheating.(Forinterpretationofthereferencesto colorinthisfigurelegend,thereaderisreferredtothewebversionofthisarticle.)

kel and kann are the integers determined by the nature of the

interactions, such as spin of the initial and final particles, and CPviolatingeffects, etc.When thedarkmatteris non-relativistic, forkann

=

0,the s-waveannihilationdominates.Forkann

=

1,the

p-waveannihilation dominates.Itiscommonthatkel

=

1 forthe

elasticscattering. Iftheelastic scatteringismediatedby a vector boson,kel

=

0 isalsopossible.Inthispaper,wefocusonthecases

withkel

=

1 andkann

=

0,1.

Beforemovingforward,letusdefine usefulquantitiesthat are independentofthedarkmattermomentum;

σ

annvrel



0

≡

α

2

ann

m2

χ

,



ann



0

≡ σ

annvrel



0nrehχ

,



σ

elvrel



0

≡

α

2 elTreh2 m4 χ

,



el



0

≡ 

σ

elvrel



0 Trehnrehγ mχ

.

(10)

Inordertoobtainthefinalyieldvalue,theEq.(7)shouldbe eval-uated.TheintegralpartofEq.(7)canbewrittenas



ann



0 Hreh 1



0 du

σ

annvrel



χ

σ

annvrel



0

.

(11)

In a naive estimation, comparing



ann



0 with Hreh is the only

importantcriterion.Fig. 3showsthetimedependenceofthe inte-grand,



σ

annvrel



χ

/



σ

annvrel



0.Forkann

=

0,theannihilation

cross-section approaches to



σ

annvrel



0 as the momentum of the dark

matter decreases. However, for kann

=

1, there is a sharp peak

around T

Treh intheregion (I.A.),whose height is1/2 andthe

widthis



u

Hreh

/



el



0



1. Therefore,its contributionto the

Eq.(11)is of

O(

ann

/ 

el

).

A simple interpretationis asfollows.

If the elastic scattering rate is large enough, the dark matter is quicklythermalizedbeforethedarkmatterstartstoannihilate,so thatthepeakcontributionissmall,andmostofannihilation hap-penswithathermalaveragedannihilationcross-sectionas

Yχ

(

t0

)

∼

Hreh

σ

annvrel



Tsreh

=

Hreh

σ

annvrel



0sreh

mχ

6Treh

.

(12)

Fig. 3. Timedependenceofσannvrelχ/σannvrel0(kann=0,1)forthemomentum evolutiondescribedinFig. 1.Foraslowvaryingg∗,uT/Treh.ForTreh=0.2GeV, thekineticdecouplingtemperatureisaboutTkdTreh/3.

Intheoppositelimit,largepairannihilationcanhappenbeforethe darkmatteriscompletelythermalized.Thecorresponding yieldis dominantlydeterminedbythepeakcontributionas

Yχ

(

t0

)

∼



el



0



ann



0

Yχ

(

treh

)

=



el



0

σ

annvrel



0sreh

.

(13)

If Hreh



el



0,the production mechanism is lying in either

thedomain(N.A.)or(C.A.)withtheyieldvalue,

Yχ

(

t0

)

min



Yχ

(

treh

),

Hreh



σ

annvrel



0sreh

c0Hreh



el



0



1/3



,

(14)

where c0 is an

O(

1) numerical constant. The enhancement

fac-tor

(

Hreh

/



el



0

)

1/3 is interpreted as Edec_χ

/

mχ ,where Edec_χ is the

decoupling energy at Treh. The reason of this factor is that the

number density is mostly determined by



σ

annvrel



0nχ

=

H

(

T∗

),

where T_∗ is the temperatureat whichthe dark matter becomes non-relativistic.

If Hreh

 

el



0, the dark matter is completely thermalized

at Treh, and the yield is specified by either (N.A.) or (I.A.). For

kann

=

0,theyieldissimply

Yχ

(

t0

)

=

min



Yχ

(

treh

),

Hreh

σ

annvrel



0sreh



.

(15)

However, forkann

=

1,the formulaisrather complicatedbecause

the annihilationrateishighly sensitive tothe momentum evolu-tionevenforthenon-relativisticdarkmatter.Theyieldvalueis

Yχ

(

t0

)

min



Yχ

(

treh

),

Hreh



σ

annvrel



0sreh



c0Hreh



el



0

+



3

−

T 2 kd T2 reh



Treh mχ



₋1

⎤

⎦ .

(16)

In the expression, the contribution of

O(

Hreh

/



el



0

)

is coming

fromthepeakaroundu

1.Thisalsocanberephrasedintermsof thekinetic decouplingtemperature.After



t

=

1/



el



0,the

elas-ticscatteringratescales as



el

∝

T6,whiletheHubbleratescales

asH

∝

T2_{.Therefore,fromH}

₍

_T kd

)

= 

el

(

Tkd

),

wefind Hreh



el



0

Tkd Treh



4

.

(17)

As thereheatingtemperature islower, the peak contribution be-comesmoreimportantbecause Tkd isnearlyindependentofTreh.

The remaining contribution of

O(

Treh

/

mχ

)

is for u

<

1

− 

u,

whichgives



σ

annvrel



Trehnχ

∼

Hrehifthepeakcontributionis ne-glected.

Theanalytic formulaeare matchedwitheach otherata naive boundarybetween(C.A.)and(I.A.),c0Hreh

= 

el



0.Thevaluec0 is

numericallydeterminedtobec0

0.4,asitisshowninFig. 5. 5. Darkmatterconstraints

5.1. Relicdensity

Nowwetrytofittheaboveresultstothepresentdarkmatter relicabundance[21],

χ

h2

=

0

.

11



mχ 100 GeV



_Yχ

₍

_t0

₎

4

×

10−12



,

(18)

for

α

=

α

ann

=

α

el,andfordifferent choicesofmχ and Treh. For

theWIMPdarkmatter,taking

α

ann

=

α

el isa reasonable

assump-tion.

α

hasanupperboundfromunitarityandperturbativity con-dition.Herewetake

α

<

1 asthecriterionforbothconditions.The initialyield Yχ

(

treh

)

alsohasan upperbound.The direct

produc-tionfromheavyparticledecaysgivesEreh_{χ n}reh_χ

=

Brχ

ρ

reh φ ,sothat Yχ

(

treh

)

=

3g_∗

(

Treh

)

4g_∗S

(

Treh

)

Brχ Brγ Treh Ereh χ

.

(19)

ForBrχ



Brγ , Yχ

(

treh

)

isboundedbyTreh

/

Erehχ .

InFigs. 4 and5,westudytheallowedparameter spaceinthe plane ofmχ

− 

σ

annvrel



0 fordifferent choicesof reheating

tem-perature. For each figures, the green dotted lines stand for the contour to satisfy the present relic density with the condition, 0.4Hreh

= 

el



0.Forkann

=

0,thepresentdarkmatterabundance

isproportional to



σ

annvrel



₀−4/3T_reh−7/3m2χ inthe (C.A.)regionthat corresponds to the diagonal line above the boundary (0.4Hreh

=



el). Below the boundary line, the production mechanism is in

the (I.A.) region, and the corresponding

χ h2 is proportional to



σ

annvrel



−01T− 1

rehmχ .Thereforetheslopeisslightlychanged.

For kann

=

1, the diagonal line on the right hand side is the

same as that of the region (C.A.) with kann

=

0. However there

is adrastic change around the boundary. The verticalline corre-spondstothe(I.A.)regionwherethecontributionisdominatedby thetermc0Hrel

/



el



0 inEq.(16).Consequently,

χ h2

∝

Treh3

/

m2χ , and the relic density does not explicit depend on the cross-section. For a quite small Hrel

/



el



0,

χ h2 is proportional to



σ

annvrel



−01T− 2

rehm2χ ,sothattheslopeischangedagain.The numer-ical calculation smoothes the analytic lines around theboundary between(C.A.)and(I.A.).Themorecorrectboundarylineisgiven asHreh



el



0.

Fig. 4. Thecontourplotforkann=0,withlinessatisfyingχh2=0.11 for differ-entTrehs;10 MeV(blue),50 MeV(black),100 MeV(red)intheplaneofdarkmatter massandσannvrel0.σvth≡3×10−26cm3/s.Thegraycoloredregionisexcluded byconservativeperturbativeandunitarycriterion(α>1).Mostofregionsare al-readyexcluded byindirect detectionconstraintsfromtheFermi-LAT [22](green coloredregion).ThedashedlinesarefortheanalyticapproximationinEqs.(14), (16).(Forinterpretationofthereferencestocolorinthisfigurelegend,thereader isreferredtothewebversionofthisarticle.)

Fig. 5. Thecontourplotforkann=1.Theredcoloredregionisexcludedbydark matterdirectdetectionexperiment,PandaX-II[23]andLUX2016[24].(For inter-pretationofthereferencestocolorinthisfigurelegend,thereaderisreferredto thewebversionofthisarticle.)

5.2. Direct/indirectdetection

AtalowtemperaturebelowGeV,thequarksarenolongerlight degreesoffreedom,andtheinteractionsbetweenthedarkmatter and leptons are more crucial to determine the darkmatter den-sity.Ontheother hand,theinteractions betweenthedarkmatter andquarksaremoreimportantforthedirect/indirectdetection.In orderto givea ratherstrongcorrelation,herewe takeleptophilic darkmatter.

For kann

=

0, the present dark matter annihilation is

domi-nated by the s-wave contribution, which is strongly constrained by the Fermi-LAT data [22] as in Fig. 4. Therefore kann

=

1 is

moreviable becausetheannihilationrateatthepresentUniverse is quitesuppressed bythe square ofthepresentdark matter ve-locity(v2

χ0

∼

10−6)comparedto



σ

annvrel



0.

Forthedirectdetectionofthedarkmatter,the1-loopor2-loop inducedinteractionbetweenthedarkmatterandnucleuscan gen-eratethe sizablesignal. Onecanthink of various effective

opera-tors for

χ χ

ll withcomplex couplingsandvarious spin structure. As a benchmark example, we assume that the dark matter is a Majoranafermion,andthat the interactionis mediated by areal scalar. Using fourcomponent spinor notation, the relevant effec-tiveoperatorisgivenas

L

eff

=

(

χ χ

¯

)(¯

ll

)



2

.

(20)

Weconsiderleptonflavoruniversalcouplingsinordernotto gen-erateanyflavor problem.Aftermatchingtheeffectiveoperatorof Eq.(20)tothat forthescatteringcross-sectionofEq.(9),we can applytheconstraintsfromdirectdetectionexperiments[23,24].

ForEq. (20), the first non-vanishing dark matter nucleus (N) elasticscatteringcross-sectionisgeneratedatthetwo-loop level;

[25]

σ

χN

=

O

(

1

)



μ

2 NZ2

π



_e4_Z 192

π

2



2



2

μ

Nvχ0 ml



2

≡

μ

2N

μ

2 n A2

σ

χn

,

(21)

wheren isthenucleon. The

O(

1)uncertainties are coming from thetwo-loopinducednucleusformfactor,whoseevaluationis be-yond the scope of this paper. Z is the atomic number, A is the mass number of the target nucleus. Z

=

54, A

=

131 for 131Xe.

μ

N

=

mχ mN

/(

mχ

+

mN

)

isthereducedmassforthedarkmatter

andthe nucleus, and

μ

n is the reducedmass for the dark

mat-terandanucleon.

μ

Nvχ0 isthetypical recoilmomentum ofthe nucleus,andtheformulaisvalidfor

μ

Nvχ0

=

O(

MeV)



ml. Tak-ing all those uncertainties asa factor

O(1),

in Fig. 5we get the excluded region of the cross-section (red color) for a given dark mattermass.Eventhoughitisgeneratedata two-looplevel,the strongconstraintexistsfortherangethatsatisfiesthedarkmatter density.Moreaccurateconstraintsconsideringallthe

O(

1) coeffi-cientscorrectlywillbediscussedinfuturework.

6. Conclusions

Non-thermalhistoryoftheearlyUniversecanbenaturally ob-tained in new physics beyond the Standard Model, and it also provides various interesting effects which cannot be simply cap-turedbythestandardthermalhistoryoftheUniversewithahigh reheatingtemperature,includingtheworks[26–29].

In this work, we have studied the effect of the elastic scat-teringbetweenthe WIMPdarkmatterandbackgroundradiations when the Universeis reheated at a low temperature. This effect iscrucialiftheamountofnon-thermallyproduceddarkmatteris sizable,and the reheatingtemperature is well below the freeze-out temperature. We specified the three conceptual domains for the determination of the dark matter abundance, and presented theanalyticandnumericalsolutionstotheBoltzmannequation.

When the reheating temperature is low enough, the elastic scatteringrateis not effective tocompletely thermalize thedark matter. The dark matter particles decouple from thermal plasma whenthey are still relativistic, andthe annihilationcould persist untiltheybecome non-relativistic.Inthiscase,we show thatthe final abundance of the dark matter could depend on the elastic scatteringrate. Evenin the caseof instantaneous thermalization, the relative size between the elastic and annihilation rates can changethefinalabundanceforthep-waveannihilatingdark mat-ter.Ontheotherhand,thenon-thermalWIMPmechanismrequires largeannihilationcross-sectiontoexplainthepresentdarkmatter relicdensity.Westudiedtheconstraintsfromdirect/indirect detec-tionexperimentsbyconsideringtheleptophilicdarkmattermodel

asa specific example,andshowed that wide rangeofparameter spaceisseverelyconstrained.

Those strong constraints can be avoided ifthe darkmatter is “DarkWIMP”inwhichthedarkmatteristhermalizedbyand anni-hilatestodarkradiations.Themechanismsthatwehavediscussed canalsobegeneralizedtothedarkWIMPscenario.Insuchacase, therecouldbemoreinterestingconnectionbetweenthehistoryof theearlyUniverseandthesignaturesimprintedonthecosmic mi-crowavebackgroundandlargescalestructure.

Acknowledgements

WewouldliketothankJinn-OukGong,KyuJungBae,Jong-Chul Park, and Seodong Shin for useful discussions. HK is supported by IBS under the project code, IBS-R018-D1. JH is supported by World Premier International Research Center Initiative (WPI Ini-tiative), MEXT, Japan. CSS is supported in part by the Ministry ofScience,ICT& FuturePlanning andby theMax PlanckSociety, Gyeongsangbuk-DoandPohangCity.

Appendix A. TheBoltzmannequationforthemomentum expectationvalueofdarkmatter

Inthisappendix,wederivethefullBoltzmannequationforthe momentumexpectationvalueofdarkmatterincludingthe annihi-lationeffect.Thiseffectcouldbe importantbecausethe annihila-tioncross-sectiondependsonthemomentum,sothedarkmatter withdifferentmomentumwillannihilatewithadifferentrate.

TheBoltzmannequationforthedistributionis

dfχ

(



p

)

dt

=

Cel

[

fχ

] +

Cann

[

fχ

].

(A.1)

Thecollisiontermsaregivenas

Cel

[

fχ

] =

1 Ep



d

qd

pd

q

(

2

π

)

4

δ

(4)

(

pμ

+

qμ

−

pμ

−

qμ

)

|

M

el

|

2

[

fχ

(



p

)

fγ

(

q





)

−

fχ

(

p



)

fγ

(

q



)

],

Cann

[

fχ

] =

1 Ep



d

pd

qd

q

(

2

π

)

4

δ

(4)

(

pμ

+

pμ

−

qμ

−

qμ

)

|

M

ann

|

2

[

fγ

(



q

)

fγ

(



q

)

−

fχ

(



p

)

fχ

(



p

)

],

(A.2) where d

pi

= (

2

π

)

− 3

_(2E

pi

)

− 1_d3

_

_p

i. IntegratingEq.(A.1) over the

phasespaceof

χ

,wecouldobtainequationsfordarkmatter num-berdensityandmomentumexpectationvalue.Assumingnχ

neqχ , theequationfornumberdensityis

˙

nχ

+

3Hnχ

= −

σ

annvrel



χn2χ

≡ −

annnχ

,

(A.3) andtheequationformomentumexpectationvalueis

˙



p



_χ

+

H



p



_χ

= −σ

elvrel



p



χTnγ

+ σ

annvrel



χ



p



χnχ

− σ

annvrelp



χnχ

≡ −(

el

− 

annS

)



p



χ

.

(A.4) Here p

= |

p

|

,and



p isthechangeofdarkmattermomentumfor singleelasticscattering.S isdefinedas

S

=

1

−

σ

annvrelp



χ



p



χ

σ

annvrel



χ

.

(A.5)

Itisobviousthattheannihilationaffectsthemomentumevolution, anditrelies on



annS.

Itisdifficulttoobtainanexactanalyticformof S forarbitrary distributionfunctions.Butitispossibletoapproximate S as

S

σ

2 sd,χ



p



2 χ

,

(A.6)

whenthedistributionfunctionhas smallvariancecomparedtoits meanvalue;

σ

2

sd,χ

= 

p2



χ

− 

p



2χ

 

p



2χ .

At the end of reheating, dark matter particles are produced from the decay of long-lived heavy particle,

φ.

If they are pro-ducedbytwo-bodydecays,thedarkmattermomentumwouldbe centeredat



p



χ

mφ

/2,

andthevariance (orwidth)of distribu-tion would be given as

σ

2

sd,χ



2

φ. Thus, we see that S

(

treh

)

(

φ

/

mφ

)

2



1,andthattheannihilationdoesnotchangethe mo-mentumevolutionaslongasS

< 

el

/ 

ann.

Thisargumentisonlyvalidattheendofreheatingbecausethe elasticscatteringspreadsthemomentumdistributionofdark mat-ter.Tomakesurethatwecansafelyignoretheannihilationofdark matterforits momentum evolution,itis necessaryto investigate thetimeevolutionof S.

Letus consider a casewhere H

< 

el

< 

ann att

=

treh. This

isthecasewheretheannihilationmayplaya roleindetermining momentum evolution. In other cases, the effect of



ann on

mo-mentum evolution is safely ignored. Just after dark matters are produced,thewidthwillevolve dominantlyby elasticscatterings. When

σ

χ isstillsmallerthan



p



χ , itisstraightforwardtoderive theBoltzmannequationfor S.Itis

dS

dt

=

2



elS

+ 

el



p



_χ



p



χ

+

O

(

S2

).

(A.7)

ForasmallinitialvalueofS

(

treh

)

= (φ

/

mφ

)

2,thesecondtermof RHSbecomesthesourceofS.Thensolutionbecomes

S



t



el



p



χ



p



χ

,

(A.8)

where



t

≡

t

−

treh

<

1/ el.Ontheother hand, for1/ ann

|

Treh

<



t

<

H−_reh1,thesolutiontoEq.(A.3)isnχ

(

σ

annvrel



χ χ



t

)

−1 so that



ann

1/t.InsertingthesesolutionstoEq.(A.4)gives

˙



p



_χ

+

H



p



_χ

−



el

− 

el



p



χ



p



_χ





p



_χ

.

(A.9)

When the dark matter is non-relativistic at the center of mass frame for

χ

–

γ

collision system,



p

/

p



1. Thismeans that the momentum evolvesdominantlyby elasticscatteringuntil the re-laxationtime,



t

∼

1/ el.

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