Baryogenesis from the inflaton field

Baryogenesis from the inflaton field

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Citation

Hertzberg, Mark P., and Johanna Karouby. “Baryogenesis from the

Inflaton Field.” Physics Letters B 737 (October 2014): 34–38.

As Published

http://dx.doi.org/10.1016/j.physletb.2014.08.021

Publisher

Elsevier

Version

Final published version

Citable link

http://hdl.handle.net/1721.1/91217

Terms of Use

Creative Commons Attribution

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Baryogenesis

from

the

inflaton

field

Mark

P. Hertzberg

∗

,

Johanna Karouby

CenterforTheoreticalPhysicsandDept. ofPhysics,MassachusettsInstituteofTechnology,Cambridge,MA02139,USA

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory: Received27May2014

Receivedinrevisedform29July2014 Accepted11August2014

Availableonline14August2014 Editor:S.Dodelson

Inthisletterweshowthattheinflatoncangeneratethecosmologicalbaryonasymmetry.Wetakethe inflatontobeacomplexscalarfieldwithaweaklybrokenglobalsymmetryanddevelopanewvarianton theAffleck–Dinemechanism.Theinflationaryphaseisdrivenbyaquadraticpotentialwhoseamplitude ofB-modesisinagreementwithBICEP2data.Weshowthataconservedparticlenumberisproduced inthelatterstageofinflation,whichcanlaterdecaytobaryons.Wepresent promisingembeddingsin particle physics,includingthe useofhigh dimension operators fordecayorusing acoloredinflaton. Wealsopointoutobservationalconsequences,includingapredictionofisocurvaturefluctuations,whose amplitudeisjustbelowcurrentlimits,andapossiblelargescaledipole.

©2014TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/3.0/).FundedbySCOAP3_.

1. Introduction

One of the outstandingchallenges of modern particle physics andcosmology isto explain the asymmetry betweenmatter and anti-matterthroughouttheuniverse.Thisasymmetryisquantified bythebaryon-to-photonratio

η

,whichshowsanover-abundance of matter at the level of

η

obs

≈

6

×

10−10, as measured in [1].

Onemight tryto dismiss thisproblemby assuming theuniverse simplybegan withtheasymmetry. However,such a proposal ap-pearsbothunsatisfyingandunlikelyduetocosmologicalinflation; aphaseofexponential expansionintheearlyuniversethat helps to explain the large scale homogeneity, isotropy, and flatness, as wellasthedensityfluctuations[2].Suchaphasewouldwipeout anyinitialbaryon number.Itisusually thoughtthatthisrequires newfieldstoenterafterinflationintheradiation(ormatter)eras to generate the asymmetry (for reviews see [3]), such asat the electroweak phase transition(e.g.,see [4]). Since we have yetto see new physics beyondthe Standard Model at the electroweak scale[5],itisentirelypossiblethatbaryogenesisisassociatedwith much higher energies, and inflation is a probe into these high scales.

In this letter, and accompanying paper [6], we show that al-thoughinflationwipesoutanyinitialmatter/anti-matter asymme-try,theasymmetrycanstillbegeneratedbytheinflatonitself.The key reasonthisis possibleis thatthe inflatonacquires atype of vevduring inflationandthisinformationisnot wipedout bythe

*

Correspondingauthor.

E-mailaddresses:mphertz@mit.edu(M.P. Hertzberg),karoubyj@mit.edu (J. Karouby).

inflationaryphase.Inordertoconnectthistobaryogenesis,wewill putforwardanewvariationontheclassic Affleck–Dine[7] mech-anismforbaryogenesis,whichusesscalarfielddynamicstoobtain a net baryon number.Inthe original proposal, Affleck–Dine used a complexscalarfield, usually thoughttobe unrelatedto the in-flaton but possibly a spectator field during inflation, to generate baryonsintheradiationormattereras.Variousversions,often in-cludingconnectionstosupersymmetry,havebeenfoundforthese Affleck–Dinemodels,e.g.,see[8].

In this letter we propose a new model where the aforemen-tionedcomplex scalarfield is theinflaton itself.In the accompa-nying paper[6], we develop andprovidedetails ofthisproposal, includingbothparticlephysics andcosmologicalaspects,and dis-cusscurrentobservationalconstraints.Ourkeyideas andfindings are summarized as follows: We propose that the inflaton is a complex scalar field with a weakly broken global U

(

1

)

symme-try. For simplicity, we consider inflation driven by a symmetric quadraticpotential,plusasub-dominantsymmetrybreakingterm. The quadratic potential establishes tensor modes in agreement withrecentBICEP2results[9].Giventheserecentcosmological ob-servations, it is very important to establish a concise, predictive model as we do here. We show that a non-zero particle num-berisgeneratedinthelatterstageofinflation. Afterinflationthis candecayintobaryonsandeventuallyproduceathermaluniverse. We propose two promising particle physics models forboth the symmetry breakingandthe decayintobaryons:(i) Utilizinghigh dimension operators fordecay,which ispreferable iftheinflaton isagaugesinglet.(ii) Utilizinglowdimensionoperatorsfordecay, which isnatural iftheinflaton carriescolor. We findthat model (i) predicts the observed baryon asymmetry if the decay occurs http://dx.doi.org/10.1016/j.physletb.2014.08.021

0370-2693/©2014TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/3.0/).Fundedby SCOAP3_.

throughoperators controlledby

∼

GUT scaleandthisisprecisely theregimewheretheEFTapplies,whilemodel(ii)requiressmall couplings to obtain the observed baryon asymmetry. We find a predictionofbaryon isocurvaturefluctuationata levelconsistent withthelatestCMBbounds,whichispotentiallydetectable.

Insummary,ournewresultsbeyondtheexistingliterature in-clude: (a) thedirect comparison to the latestdata; this includes the latest bounds on tensor modes, scalar modes, and baryon asymmetry, (b) the development of a broad framework to iden-tify inflation with the origin of baryon asymmetry, without the detailedrestrictionsofsupersymmetry,(c) specificmodelbuilding examples including the cases of a singlet inflaton anda colored inflaton, (d) predictions for isocurvature modes and compatibil-itywithexisting bounds, whilestandard Affleck–Dinemodels are ruledoutifhigh-scaleinflationoccurred,(e) predictionsofalarge scaledipole.

2. Complexscalarmodel

Consideracomplexscalarfield

φ

,withacanonicalkinetic en-ergy

|∂φ|

2,minimallycoupledtogravity,withdynamicsgoverned by the standard two-derivative Einstein–Hilbert action. Our free-dom comes from the choice of potential function V

(φ,

φ

∗

)

. It is useful to decompose the potential into a “symmetric” piece Vs

and a “breaking” piece Vb piece, with respect to a global U

(

1

)

symmetry

φ

→

e−iα

φ

,i.e., V

(φ,

φ

∗

)

=

Vs

(

|φ|) +

Vb

(φ,

φ

∗

)

. In

or-derto describeinflation we assume that thesymmetricpiece Vs

dominates, even at rather large field values where inflation oc-curs.Forsimplicity,we take thesymmetric piecetobe quadratic Vs

(

|φ|)

=

m2

|φ|

2.Itiswellknownthatapurelyquadraticpotential

willestablishlargefield,or“chaotic”inflation [10].Thisisa sim-plemodelofinflationthatwillprovideausefulpedagogicaltoolto describeourmechanismforbaryogenesis.Suchamodelisingood agreement withthe spectrum of densityfluctuationsin the uni-verse[1],itisinagreementwiththemeasuredtensormodesfrom BICEP2data[9],andismotivatedbysimplesymmetryarguments

[11].Generalizingtoothersymmetricpotentialsisalsopossible. The global symmetry is associated with a conserved particle number.Sotogenerateanon-zeroparticlenumber(thatwill de-cay into baryons) we add a higher dimension operator that ex-plicitlybreakstheglobalU

(

1

)

symmetryVb

(φ,

φ

∗

)

= λ(φ

n

+ φ

∗n

)

,

withn

≥

3.Weassumethatthebreakingparameter

λ

isverysmall sothattheglobalsymmetryisonlyweakly broken.This assump-tionofvery small

λ

ismotivatedby two reasons:Firstly, since

λ

isresponsibleforthebreakingofasymmetry,itistechnically nat-uralforittobesmallaccordingtotheprinciplesofeffectivefield theory.Secondly,thesmallnessof

λ

isanessentialrequirementon anyinflationarymodelsothatsuchhigherordercorrectionsdonot spoiltheflatnessofthepotential Vs.Wealsonotethatourmodel

carriesadiscrete

Z

n symmetrythatmakesitradiativelystable.

3. Particle/anti-particleasymmetry

We assume the field begins at large field values (

|φ| 

MPl)

and drives inflation. The field exhibits usual slow-roll and then redshifts to small values at late times,where it exhibits elliptic motion. This evolution is seen in Fig. 1 for two different initial conditions.Sincen

≥

3,thenatlatetimestheinflaton

φ

becomes small, the

φ

→

e−iα

φ

symmetry violating term becomes negligi-ble,andthesymmetry becomes respected. ByNoether’stheorem this is associated with a conserved particle number. In an FRW universewithscalefactora

(

t

)

andcomovingvolumeVcom,thisis



Nφ

=

Nφ

−

N¯φ

=

i Vcoma3



φ

∗

˙φ − ˙φ

∗

φ



.

(1)

Fig. 1. Fieldevolutioninthecomplexφ-planeforn=3 andλMPl/m2=0.006,with

initialconditionρi=2 √

60 MPl.Leftiszoomedoutandshowsearlytimebehavior

duringslow-rollinflation.Rightiszoomedintoφ=0 andshowslatetimeelliptic motion.Blue(upper)curveisforinitialangleθi=π/2 andred(lower)curveisfor

initialangleθi= −5π/12.

Tobeself-consistent weignorespatialgradients,andtheequation ofmotionfor

φ

is:

¨φ +

3H

˙φ +

m2

φ

+ λ

n

φ

∗n−1

=

0,whereH

= ˙

a

/

a istheHubbleparameter.

Forsmall

λ

we canreducethecomplexity oftheproblem sig-nificantly. By using the equation of motion, we can obtain an integral expression for



Nφ which is proportional to

λ

. This al-lowsustocomputetheevolutionofthefieldtozerothorderin

λ

, whichimpliesradialmotioninthecomplexplane.Werewritethe zeroth order motionofthe field inpolar co-ordinatesas

φ

0

(

t

)

=

eiθi

_ρ

₍

_t

_)/

√

_2,where

_θ

iistheinitialangleofthefieldatthe

begin-ningofinflation.Theproblemthenreducestosolvingonlyasingle ordinarydifferentialequation.Atfirstorderin

λ

,



Nφ issimply



Nφ

(

tf

)

= −λ

Vcomn 2n2−1 sin

(

n

θ

i

)

tf



ti dt a

(

t

)

3

ρ

0

(

t

)

n

.

(2)

Here

ρ

0 is a real-valued function satisfyingthe quadratic

poten-tial version of the equation of motion

ρ

¨

0

+

3H0

ρ

˙

0

+

m2

ρ

0

=

0,

with corresponding Friedmann equation (we assume flat FRW)

H₀2

=

ε

0

/

3M2_Pl and energy density

ε

0

= ˙

ρ

02

/

2

+

m2

ρ

20

/

2, where MPl

≡

1

/

√

8

π

G is the reduced Planck mass. So by solving for a single degree of freedom in a quadratic potential, we have an expression for the particle number in the small

λ

regime. We note that for particular values of the initial angle

θ

i, such that

θ

i

=

pnπ

|

p

∈ Z

,noasymmetry isgeneratedduetothe

∼

sin

(

n

θ

i

)

factor.Since we are interested inbaryogenesis, we consider

θ

i to

beatypicalgenericvalueratherthanthesespecialones.

TheintegrandinEq.(2)isplottedinFig. 2usingdimensionless variables

τ

≡

mt and

ρ

¯

≡

ρ

0

/

MPl.Inthelimitinwhichwetake

τ

i

veryearlyduringslow-rollinflationandwetake

τ

f verylateafter

inflation,thentheintegralinEq.(2)becomesindependentofboth

τ

i and

τ

f.The dominantcontributionto theintegral,andinturn

thedominantproductionof

φ

particles(oranti-particles)occursin thelatterstageofinflation.ThisisnicelyseeninFig. 2.Itcanbe shownthatfortheparametersofthefigure,theendofinflationis

Fig. 2. TheintegrandgivingNφ(Eq.(2)),withrespecttodimensionlessvariables τ=mt,ρ¯=ρ0/MPl.Inthisplotwehavetakenn=3 andinitialconditionsρ¯i=

2√60,ai=1.Thelargepeakisinthelatterphaseofinflation; sothisiswhere

mostoftheφ(oranti-φ)particlesareproduced.

τ

≈

18,whichispreciselyattheendofthesharpriseandfall of theintegrand.Thisisshiftedtoslightlyearliertimesforhigher n. 4. Dimensionlessasymmetry

Although



Nφ is dimensionless, it is extrinsic, depending on the size of the universe. It is useful to define a related intrin-sic quantity, which provides a measure of the asymmetry A

≡



Nφ

/(

Nφ

+

N¯φ

)

. The denominator of A can be related to the energydensitystoredinthefield,becauseafterinflation

φ

is effec-tively agasofnon-relativistic

φ

andanti-

φ

particles withenergy density

ε

0

=

m

(

nφ

+

n¯φ

)

. Wefindthat thisasymmetryparameter takesonthesimpleform

A

= −

cn

λ

Mn_Pl−2

m2 sin

(

n

θ

i

).

(3)

Numerically solvingthe dimensionlessordinary differential equa-tionfor

ρ

0 andthenintegrating,leadstothefollowingresultsfor

thecoefficientcn forthefirstfewn

c3

≈

7

.

0

,

c4

≈

11

.

5

,

c5

≈

14

.

4

,

c6

≈

21

.

8

,

c7

≈

34

.

8

,

c8

≈

59

.

3

,

c9

≈

107

,

c10

≈

201

.

(4)

Inour companionpaper[6]we provethat forhighn,the coeffi-cientsaregivenby

cn

≈ ˜

c 2n/23−n/2n

Γ

1

2

(

n

/

2

),

(5)

where

˜

c isacoefficientgivenby

˜

c

≈

6

.

64 and

Γ

aistheincomplete

gamma function. We find this result to be surprisingly accurate evenforsmalln.

5. Baryonasymmetry

Recall that the baryon asymmetry is defined as the ratio of baryon difference to photon number at late times

η

≡ (

Nb

−

N_b¯

)

f

/(

Nγ

)

f,where f indicatesthelate time,or“final”value,

af-terdecayandthermalization.Weassociatewitheach

φ

particlea baryonnumberbφ;forinstancebφ

=

1 orbφ

=

1

/

3 insimple mod-els.Weassumethatthedecayof

φ

andallsubsequentinteractions is baryon number conserving, so we can relate the final num-berto theinitialnumberasfollows:

(

Nb

−

Nb¯

)

f

=

bφ

(

Nφ

−

N¯φ

)

i,

where i indicatesthe early time, or “initial” value, before decay andthermalization(but well afterthe baryon violating processes havestopped).

At earlytimeswe can relatethe numberof

φ

particlesto the Hubble parameter as

(

Nφ

+

N¯φ

)

i

=

3M_Pl2Vcom

(

a3H2

)

i

/

m. At late

timeswecanrelatethenumberofphotonstothetemperatureas

(

Nγ

)

f

=

2

ζ (

3

)

Vcom

(

a3T3

)

f

/

π

2. Byassumingthat the

thermaliza-tionisrapid,wecansimplyevaluateboththe“initial” and“final” quantitiesaroundthetimeofdecay.Denotingthedecayrateof

φ

as

Γ

φ,then thermalizationoccursaround H

≈ Γ

φ [12],which al-lowsustosolveforthereheattemperatureintermsofthenumber ofrelativisticdegreesoffreedom g_∗.Puttingallthistogethergives thefollowingexpressionforthebaryon-to-photonratio

η

≈

β

π

7/2g 3/4 ∗ bφ 27/4₃1/2₅3/4 A

Γ

_φ1/2M1_Pl/2 m

,

(6)

where

β

isan

O(

1

)

fudgefactorfromthedetailsofthetransition fromthe

φ

eratothethermalera.

6. Constraintsfrominflation

An important constraint is that the symmetry breaking term in the potential

λ(φ

n

+ φ

∗n

)

be subdominant during inflation. Since this contribution to the potential goes negative at large field values, we obviously need it to be small during inflation. For quadratic inflation, it can be shown that the field value is

ρ

i

≈

2

√

NeMPl duringinflation;thisleadstotheconstraint

λ

 λ

0

≡

m2

2n/2_Nn/2−1 e MnPl−2

.

(7)

Wenowusethethresholdvalue

λ

0 andtheabovesetofequations

toimposeaconditiononthedecayrateinordertoobtain

η

obs

Γ

φ,req

≈

10−7eV

×

2n+1Nne−2c−n2



λ

0

λ



×



m 1013_GeV



2



g3_∗/4 30

β

bφ



sin

(

n

θ

i

)





−2

.

(8)

Toprovideconcretequantitativeresultsfortherequireddecayrate, weassumethatthecoupling

λ

isafactorof10smallerthanits in-flationary upperbound

λ

0,

β

bφ

|

sin

(

n

θ

i

)

| ≈

1,m

≈

1

.

5

×

1013GeV

(requiredforthecorrectamplitude offluctuationsinquadratic in-flation), Ne

≈

55,g∗

≈

102,andwe insertthecn fromEq.(4).We

findthat

Γ

φ,reqincreaseswithn;afewexamplesare

n

=

3

⇒ Γ

φ,req

≈

4

×

10−5eV

,

n

=

4

⇒ Γ

φ,req

≈

2

×

10−3eV

,

n

=

8

⇒ Γ

φ,req

≈

9

×

103eV

,

n

=

10

⇒ Γ

φ,req

≈

107eV

.

(9)

In all thesecases,thecorresponding reheat temperatureis much biggerthan

∼

MeV,thecharacteristictemperatureofbigbang nu-cleosynthesis(BBN).Wenowexaminetowhatextentthesedecay ratescanberealizedintwoparticlephysicsmodels.

7. Highdimensionoperators

In the simplest case one can take

φ

to be a gauge singlet. Withoutfurtherrefinement,thiswouldallow

φ

todecayinto non-baryonicmatter,suchasHiggsparticles,throughoperatorssuchas

∼ φ

H†H .A natural wayaround this problemis to suppose that theglobalU

(

1

)

symmetryisalmostan exactsymmetryofnature

(orat leastinthe

φ

sector).Ofcourse globalsymmetries cannot beexact. Ifnothingelse, they must bebrokenby quantum grav-ityeffects.Usually thisimpliesthe breaking ofthe symmetry by some high dimension operator. For high n, the breaking param-eterwill need tosatisfy

λ

 (

few

/

√

G

)

4−n to be consistent with inflation. This is compatible with quantum gravity expectations. Anotherwaytoarguethisissimplytoimposeadiscrete

Z

n

sym-metry.

SowecaneitherimaginethattheU

(

1

)

symmetrybreaking oc-cursat dimensionn

≥

8 operators oronly operators that respect the

Z

nsymmetry.Thenalllowdimensionoperatorsthatbreakthe

symmetry,such as

∼ φ

H†H , wouldbeforbidden. Since

φ

carries baryon number,then upto dimension 7it could onlydecayinto quarksthroughoperatorsoftheform



L

∼

c

Λ

3

φ

∗q q q l

+

h.c.

,

(10)

wherewearesuppressingindicesforbrevity.Herewehave intro-ducedanenergy scale

Λ

thatsets thescale ofnewphysics (and thecutoff onthe field theory)andc issome dimensionless cou-pling.

Atlargeamplitude,itispossibleforparametricresonanceto oc-cur[12].However,onecanfindasensibleparameterregimewhere standardperturbativedecayratesapply.Weshallassumethishere, andleave theotherregime forfuturework[13].Theperturbative decayrateassociatedwiththisoperatorisroughly

Γ

φ

(φ

→

q

+

q

+

q

+

l

)

∼

c2

8

π

m7

Λ

6

.

(11)

We now compare this to the required decay rates from Eq. (9). Form

≈

1

.

5

×

1013_{GeV andc}

₌

_O(

₁

₎

_{,wefindthatthemodelhas}

therequireddecayratefor

Λ

intherange1015_–1016_GeV,forn

₌

8

,

10

,

12, whichis intriguingly around the GUT scale. Also, since thisscalesatisfies:Hi

 Λ



MPl,thenthisispreciselywithinthe

regimeofvalidityoftheEFT.

Onthe other hand, lower values ofn do havetheir own ad-vantage:Theytendtoleadlowervaluesofthereheattemperature, whichmayberelevanttoavoidpotentialproblemswithsphaleron washout.

8. Coloredinflaton

Anotherpossibility is to allow the inflaton to carry color. So lets give

φ

a color index, i

=

r

,

w

,

b, and allow for “up”

φ

u and

“down”

φ

d versions and different generations labelled by g. We

can construct U

(

1

)

violating terms in the potential that respect the SU

(

3

)

c symmetry. For instance, at dimension n

=

3, we can

introducethebreakingterm Vb



φ, φ

∗



= λ

gg g

ε

ii i

φ

ugi

φ

i dg

φ

i dg

+

h.c.

,

(12)

where

ε

i jk is the totally anti-symmetric tensor, and we have

summed overcolor indices anddifferent generations. Thisis the leading U

(

1

)

violating operator, but this can be generalized to higheroperators. Wenote that we are not especiallysensitive to correctionsfromgluonsduetoasymptoticfreedom[14].

Since

φ

carries color,we can readilybuild operators that me-diate

φ

decayintoquarks,while respectingthe globalsymmetry, suchasthefollowingdimension 4operator

L

∼

y

φ

i∗qi

¯

_f

₊

_h.c.,

where f issome colorneutralfermionand y isatypeofYukawa coupling.Thisdecayrateisroughly

Γ

φ

(φ

→

q

+ ¯

f

)

∼

y2m

/

8

π

.For highscaleinflation, suchasquadratic inflationthat we discussed earlier,theinflatonmassislargem

∼

1013GeV,soone would re-quireanextremelysmallvalueofy toobtaindecayrates compara-bletotherequiredvalueswecomputedearlierinEq.(9).Incertain

settings,suchassupersymmetry(whichwouldprovideextra moti-vationfortheexistenceofsuchcoloredscalars,or“squarks”),one could examineifsomenon-renormalizationtheoremmayhelpto stabilize y at such smallvalues. Anotherpossibility wouldbe to take

λ

muchsmallerthan

λ

0,whichwouldallowforhighervalues

of y.

Inthecaseofacoloredinflaton,onewouldlikeanexplanation astowhytheinflationarypotentialissufficientlyflatforinflation to occur. Acharged inflatonwilltend to leadto loop corrections thatsteepenthepotential.Thoughthisispotentiallyavoidable. 9. Isocurvaturefluctuations

Quantumfluctuations frominflation providean excellent can-didate for the origin of density fluctuations in the universe. In simple single field models,only acurvature (“adiabatic”) fluctua-tionisgenerated,duetofluctuationsintheinflaton.Formulti-field inflationarymodels,anisocurvature(“entropic”)fluctuationisalso generated [15]. This is due to quantum fluctuations in the field orthogonal to the classicalfield trajectory, whichleaves the total densityunchanged.Here thecomplex(twofield)modelwill gen-erate(baryonic)isocurvaturefluctuationsin

φ

.

In our companion paper [6] we derive the following ratio of isocurvaturefluctuationsto total(primarilyadiabatic)fluctuations intheCMB

α

II

≈

32

γ

2 5

Ω

_b2

Ω

m2 n2M2_Pl



sr

ρ

_i2 cot 2

₍

_n

_θ

i

),

(13)

where

γ

=

O(

1

)

fromde Sitterrandom walk and



sr is the first

slow-rollparameter.Planckdatarevealsthatthebaryon-to-matter ratio is

Ω

b

/Ω

m

≈

0

.

16. Let’s take

γ

∼

2, cot

(

n

θ

i

)

∼

1, and

spe-cialize to the case ofquadratic inflation with



sr

≈

1

/(

2Ne

)

and

ρ

i

≈

2

√

NeMPl, in agreement with BICEP2data [9]. Then setting

Ne

≈

55,wehaveourpredictionfortheisocurvaturefraction

α

II

∼

3

×

10−5n2

.

(14)

RecentPlanckresultshaveprovidedanupperboundoncolddark matterisocurvaturefluctuationsof[1]

α

II

<

3

.

9

×

10−2at95%

con-fidence,andweshallusethisasaroughboundonbaryon isocur-vature fluctuations. For the lowest value ofn, namely n

=

3, we predict

α

II

∼

3

×

10−4,i.e.,twoordersofmagnitudebelowthe

cur-rentbound.Formoderatelyhighvalue ofn,such asn

=

8

,

10

,

12 (as motivatedearlier), then our prediction is

α

II

∼

3

×

10−3,i.e.,

onlyoneorderofmagnitudebelowthecurrentbound.Thisisquite exciting as it is potentially detectable in the next generation of data.

10. Discussion

1. In this letter we have proposed a way to directly unify earlyuniverseinflationandbaryogenesis,withmotivationfromthe Affleck–Dine mechanism. These models intertwine parameters of high energy particle physics and inflation in an interesting way (forrelateddiscussions, see[16]).Forinstance,forthecolored in-flatonmodel,higher valuesofm are preferredinorderto obtain

η

obs,sothisfavorshigh Hi andappreciabletensormodesasseen

byBICEP2[9].Whileforthegaugesingletmodel,wefounda con-nectiontothe

∼

GUT scale.

2.Since

η

∝ −

sin

(

n

θ

i

)

,thenforinhomogenous

θ

i,thisleadsto

a large scalebaryon dipole in theuniverse,which could be rele-vanttoCMBanomalies[17].Also,itcould leadtoamultiverseof different baryon number. Furthermore,the quantum isocurvature fluctuations are potentially detectable, but consistent with con-straints. In more standard Affleck–Dine models, where

φ

is not theinflaton,theisocurvaturefluctuationcanbe solargethatmany

modelsarealreadyruledout.ThiscanbeseenfromEq.(13)dueto thefactorof1

/

ρ

2

i.Forfieldsotherthantheinflaton,theirvev’sare

typicallysub-Planckian,thusleadingtoahugeisocurvature fluctu-ation.Incontrast,forourinflatonmodels,asmall,butpotentially detectable,isocurvaturefluctuationisnatural.

3. Further work includes extension to other inflation models, further detailed embedding in particle physics, and to examine parametricresonanceafterinflation[13].

Acknowledgements

WewouldliketothankMustafaAmin,MichaelDine,AlanGuth, Jesse Thaler, Tanmay Vachaspati, and Frank Wilczek for discus-sion, and we would like to acknowledge support by the Center forTheoreticalPhysics atMIT.Thiswork issupportedby theU.S. DepartmentofEnergyundercooperativeresearch agreement Con-tractNo. DE-FG02-05ER41360.J.K. is supportedby an NSERCPDF fellowship.

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