UV corrections in sgoldstino-less inflation

Contents lists available atScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

UV corrections in sgoldstino-less inflation

Emilian Dudas

^a

, Lucien Heurtier

^b

, Clemens Wieck

^c,∗

, Martin Wolfgang Winkler

^d

aCentredePhysiqueThéorique,EcolePolytechnique,CNRS,Univ.Paris-Saclay,91128PalaiseauCedex,France bServicedePhysiqueThéorique,UniversitéLibredeBruxelles,BoulevardduTriomphe,CP225,1050Brussels,Belgium

cDepartamentodeFísicaTeóricaUAMandInstitutodeFísicaTeóricaUAM/CSIC,UniversidadAutónomadeMadridCantoblanco,28049Madrid,Spain dBetheCenterforTheoreticalPhysicsandPhysikalischesInstitutderUniversitätBonn,Nussallee12,53115Bonn,Germany

a r t i c l e i n f o a b s t ra c t

Articlehistory:

Received21January2016

Receivedinrevisedform4April2016 Accepted21May2016

Availableonline25May2016 Editor:A.Ringwald

We study the embedding of inflation with nilpotent multiplets in supergravity, in particular the decoupling ofthesgoldstinoscalarfield.Insteadofbeingimposedby hand,the nilpotency constraint onthegoldstinomultipletarisesinthelowenergy-effectivetheorybyintegratingoutheavydegreesof freedom.Wepresentexplicitsupergravitymodelsinwhichalargebutfinitesgoldstinomassarisesfrom Yukawaorgaugeinteractions.Inbothcasestheinflatonpotentialreceivestwotypesofcorrections.One isfromthebackreactionofthesgoldstino,theotherfromtheheavyfieldsgeneratingitsmass.Weshow thatthesescaleoppositelywiththeVolkov–Akulovcut-offscale,whichmakesaconsistentdecoupling ofthesgoldstinonontrivial.Still,weidentifyaparameterwindowinwhichsgoldstino-lessinflationcan takeplace,uptocorrectionswhichflattentheinflatonpotential.

©^{2016TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense} (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP³.

1. Introduction

Constrained chiral multiplets or, equivalently,nilpotent super- fields and their application to cosmology have attracted a large amountofinterestinrecentyears[1–17].Onefeatureoftheories withnonlinear supersymmetry, i.e., with a constrained multiplet satisfying S²=^{0, is theabsence of a dynamical scalardegree of} freedom. The auxiliary field of S breaks supersymmetry andthe goldstinofermionistheonlypropagatingfield[18–21].Thismakes themappealingincosmologicalmodelbuildingforvariousreasons.

The connection ofsuch theories to stringtheory hasrecently been studied in [15,22–27]. Effective supergravity theories with a constrained goldstino multiplet can be shown to arise from D3-branes incertain geometries [23–27].The emergence ofnon- linear supersymmetry in string models with anti-branes was provenin[28],inthecontextofglobalstringtheoryvacua[29].In such UV embeddingsit is difficultto extract the behavior ofthe supergravityabove thecut-off scale oftheVolkov–Akulov action.

Usually there is no scale at which linear supersymmetry is re- stored and therefore the scalar component of S does not exist.

A step towardsunderstanding the connection betweenthelinear andnonlinear regimeswas recently made in [30], where it was shown explicitly that nonlinear supergravity theories are equiv-

*

Correspondingauthor.

E-mailaddress:[email protected](C. Wieck).

alent to linear supergravities with an infinitely heavy sgoldstino scalar,andthatthelimitrelatingthetwoiswell-definedviafunc- tional integration. With a few restrictions this connection was previouslyknownintherigidlimit[31].¹

Thereforeitisdesirabletostudyfieldtheoryexamplesinwhich aheavysgoldstino existssothatsupersymmetrybecomeslinearly realizedatahighscale.Insuch cases,thesgoldstinofield cannot beinfinitelyheavy.Itsmass,andhencetheVolkov–Akulovcut-off scale, must be lower than the Planck scale – and favorably be- lowtheKaluza–Kleinandstringscales.Astrongerconstraintarises fromunitaritywhichsignalsaperturbativebreakdownofthenon- lineartheory ata scale ∼ √^m3/2 in Planckunits.A UV complete theorywhichcandescribeboththelinearandnonlinearregimesis boundtoyieldcorrectionswhicharemissedbysimplyimposinga nilpotencyconstraintonthegoldstinomultipletinsupergravity.In thisletterwecomputethesecorrectionsandevaluatetheireffects in simpleinflation models previously studied in theliterature. It is our aimto prove that in a well-defined regime of the theory, correctionsareundercontrol–though inaquiteconstrainedpa- rameterspace.

Forthispurposetheclassofmodelsdevelopedin[7]isparticu- larlyinstructive.²Theyfeaturethecouplingofanilpotentstabilizer

1 Forarecentstudyregardingtheapplicabilityofnilpotencyconditionscf.[32].

2 Werecommend[33]asareviewoftheseandotherinflationmodelsinvolving nonlinearsupersymmetry.

http://dx.doi.org/10.1016/j.physletb.2016.05.072

0370-2693/©^{2016TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense}(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP³.

multiplettoaholomorphicfunctionoftheinflatonmultiplet,giv- ingrise toaplethora ofpossible potentialshapesfortheinflaton scalar.Duringinflation andinthevacuumsupersymmetryisbro- ken by the auxiliary field of the nilpotent multiplet. The setup can accommodatelow-energy supersymmetrywhich is nontrivial giventhehighscaleofinflation.Weextendthissetuptoasuper- gravitywithheavyfieldsinwhichalarge massforthesgoldstino scalarisgenerateddynamically.We expectourresultstoberele- vantinmanyothersupergravitytheorieswithnilpotentgoldstino multiplets. Thus,we hope thatthiswork isanother steptowards understandingnilpotentmultipletsandtheirroleincosmology.

2. Sgoldstinodecoupling

Thesuccessofnilpotentfieldsincosmologyhastriggeredgrow- inginterestintheirfield-theoreticalorigin.Itiswell-knownthatin spontaneously broken linear supersymmetry, the sgoldstino field acquiresalargemassthroughtheoperator

K

⊃

^c

|

^S

|

⁴



^{2 (1)}

intheKählerpotential.Inthelimitc→ ∞^{,thesgoldstinobecomes} infinitelyheavy andthe resulting theory is equivalent to nonlin- early realized supersymmetry with a nilpotent goldstino multi- plet S [30,31]. Clearly this theory is only a low-energy effective theory.Withthesgoldstinodecoupled,itviolatesperturbativeuni- tarityat theintermediate energyscale √

m3/2 [34].Requiring in- flation in the perturbative regime one obtains the generic con- straint[7]

m3/2

>

H²

,

(2)

where H denotes the Hubblescale. However, the scale ofsuper- symmetry breaking may be different during and after inflation.

Hence, nilpotent inflation models consistent withlow-energy su- persymmetrycanbeconstructed[7].

A different concern is the limit c→ ∞^{: in a} UV-complete modeltheoperator(1)arisesfromcouplingsofS toheavydegrees of freedom. As an example, we may consider the superpotential couplingW⊃ λ^{S X2 ofS totheheavyfield X withmassm}X.This couplinggeneratesaone-loopcorrection[35],

K

⊃ − λ

⁴ 16

π

²

|

^S

|

⁴

m²_X

.

(3)

The limit c→ ∞ ^{in (1)then} corresponds totaking the coupling λ to infinity or the mass mX to zero.Since both must be finite andmX mustbe largefortheeffectivefieldtheory(EFT)tomake sense,wemustconsidertheregimewherethesgoldstinohasafi- nitemass,i.e.,finite c.Intheremainderofthisletterwestriveto determine whetherinflation isstill possiblein this case. Specifi- cally,wedeterminewhethertheinflatonpotentialobtainedinthe nilpotentlimitstillholdsandcorrectionsareundercontrol.

Wewillfindthatsuchcorrectionsareoftwodifferentnatures.

Additionalheavyfieldsattheenergyscalebackreactonthein- flatonpotential,introducingcorrectionswhichvanishas→ ∞^.3 Ontheother hand,thefinitemassofthesgoldstinofield leadsto correctionswhichvanishinthelimit wherethelatterisinfinitely heavy.Thiscorrespondsto→^{0.Therefore,itisfarfromobvious} thatbothtypesofcorrectionscanbesuppressedsimultaneously.

Notethat whilewe studythisinthe classofinflation models proposed in [7], ourfindings canstraightforwardly be applied to alternativescenarioswithnilpotentmultiplets.

3 Thesewecall“UVcorrections”becausetheyarisefromembeddingthenilpotent multipletinacompletetheoryofsupergravity.

3. Sgoldstino-lessmodelsofinflation

Let usbriefly review the inflation models of[7]. Theyfeature theKählerandsuperpotential

K

=

¹

2

( + )

²

+ |

^S

|

²

,

(4a)

W

=

^f

()



1

+ √

3S



,

(4b)

wheredenotestheinflatonsuperfieldandS containsthestabi- lizer field.Thissetup isa generalizationofthemodels developed in [36,37] with built-in supersymmetry breaking by the auxil- iary field of S. The function f satisfies f(0)=^{0, f} (0)=^{0 and} f(x)=^f(−¯^x).In[7]itisassumedthat S fulfillstheboundarycon- dition S²=^{0 of a nilpotent chiralmultiplet. Thisimplies s}=⁰ foritsscalarcomponent.⁴

Thefactor√

3 ensuresthecancellationofthecosmologicalcon- stant inthe vacuum at φ=^{0. Along the} inflationary trajectory thepotentialreads

V

= 



^f



i

ϕ

√

2

 

²

,

(5)

where

ϕ

₌^√2 Imφ denotes the canonically normalized inflaton.

Twoexamplesfor f arediscussedin[7].Oneis f

() =

^f0

−

^m

2



²

,

(6)

leadingtothepotentialofchaoticinflation,V=¹₂^m2

ϕ

².Theother is

f

() =

^f0

−

ⁱ



V₀



 +

ⁱ

√

3 2 e^2i/√3

,

(7)

producingtheplateaupotentialV =^V0



1−^e−√2/3ϕ .⁵

In the following we call S the goldstino multiplet and s the sgoldstino, its scalar component. This is because s is the heavy scalarthat issupposed todecouple,anddespite thefactthat the inflaton multiplethas asub-dominant butnonvanishingauxiliary fieldduringinflation.

4. Correctionsfromthesgoldstino

Letusdiscusscorrectionsto theinflatonpotentialwhich arise ifthesgoldstinohasafinitemass.Tothisendweconsider

W

=

^f

()(

1

+ δ

^S

) ,

(8a)

K

=

¹

2

( + )

²

+ |

S

|

²

− |

^S

|

⁴



²

.

(8b)

Thedifferencecomparedtotheprevioussectionisthatwedonot imposethenilpotencyconstraintS²=^{0.Insteadweintroducethe} term|^S|⁴/² intheKählerpotentialwhichgeneratesalarge–but finite – mass forthe sgoldstino s and dynamically keeps s close to the origin.⁶ Supersymmetry breaking introduces an inflaton- dependent lineartermforthestabilizerfieldwhichslightlyshifts itawayfromtheorigin[4].Asthiseffectscalesinverselywiththe massofs,itisabsentinthenilpotentlimit.Noticethatweintro- ducedtheparameterδwhichallowsustotunethevacuumenergy

4 Weusecapitallettersforsuperfieldsandsmalllettersfortheirscalarcompo- nents.

5 AspointedoutinSection 5of[7],thefunction f canbeextendedtoinclude matterfieldslikeanMSSMsector.Tachyonicdirectionsareavoidedautomatically formatterfieldswhichappearatleastquadraticallyin f .

6 ComparedtoSection2weabsorbedtheparameterc inthedefinitionof.

tozeroattheminimumofthepotential.Duetotheshiftofs,δis closetobutnotexactly√

3.Wefind

δ = √

3

+ 

²

2

√

3

+

O

(

⁴

) .

(9)

Foracompactnotationweintroduce V0

= 



^f



i

ϕ

√

2

 

²

,

(10)

denoting the inflatonpotential in the limit where S is nilpotent andhences isinfinitelyheavy.Thegravitinomassalongtheinfla- tionarytrajectorycanbeapproximatedas

m²_3/2

=

^eK

|

^W

|

²

 



^f



i

ϕ

√

2

 

²

.

(11)

Asonlythe realpartofthestabilizerfield isdisplaced during inflation, we set ¯s=s in thefollowing. At second orderin s the scalarpotentialreads

V

=

^V0

+

^m2_3/2



²

+ √

3



2V₀

−

^4m2_3/2



s

+

^m2_s^s2

,

(12)

includingonlytermsup toO(²).⁷ Thesgoldstino massisgiven by

m²_s

=

^12m

2 3/2



²

,

(13)

which,throughm3/2,dependson

ϕ

duringinflation.Theinflaton- dependentminimumofs liesat

s

 =

^2m

2 3/2

−

^V0

m²_3/2



² 4

√

3

.

(14)

Thescalarpotentialafterintegratingouts reads V

=

^V0

1

+



1

−

^V0 4m²_3/2



²

+

O

(

⁴

) 

.

(15)

Asmentionedabove,thecorrectionsfromthesgoldstinosectorap- pear in powers of m²_3/2/m²_s and H²/m²_s, where H∼√

V0 again denotes the Hubble parameter. Corrections are under control as longasm_3/2>Hwhichisthe caseduring inflation inthe two examplesofSection3.⁸

Notethat even when the correctionsare small the sgoldstino can affect post-inflationary cosmology. If the above constraint is violated after inflation, s may no longer trace its minimum.If it gets trapped the associated potential energy can alter late-time cosmology. Thisis not necessarily problematic andmay evenin- duceinterestingsignatures.Wemerelypointoutthatdecouplings fromall dynamicsintheuniverserequirestheboundm3/2>H to be satisfied during the entire cosmologicalevolution. We will showinthe followingthatina consistentEFT cannot be arbi- trarilysmall,makingthisaverysevereconstraint.

5. CorrectionsfromUVcompletions

In the previous section we have included corrections to the inflatonpotential whicharisefromthesgoldstino sector.Thecor- rections disappear in the limit →^{0 in which the sgoldstino} becomes infinitely heavy. But there are more correctionsrelated

7 Noticethats= O(²)andm²s= O(⁻²).

8 Forthesgoldstinotobeheavier thanRe()one wouldadditionally haveto require|^f ()|,|^f ()|<|^f ()|^ontheinflationarytrajectory.Theseconditions are,however,alreadyfulfilledbyrequiringslow-rollinflation.

to the heavy fields living at the scale . Contrary to the sgold- stinocorrections,thesescalewith⁻¹ andpreventusfromtaking the limit →^{0. In the following we discuss these UV correc-} tionsintwoexamples.Inthefirstexamplethesgoldstinomassis generatedbyYukawainteractionswithheavy fields,inthesecond examplebygauge interactions.Despitetheirsimplicityweexpect thatourexamplesarerepresentativeofmoresophisticatedUVem- beddings.

5.1. Example1

ConsidertwoadditionalchiralmultipletsX,Y withavector-like massM.WeconsiderM tobelargecomparedtotheHubblescale andthe gravitino mass during inflation. We define the model as follows,

W

=

^f

()(

1

+ δ

^S

) + λ

^{S X2}

+

^{M X Y}

,

(16) K

=

¹

2

( + )

²

+ |

^S

|

²

+ |

^X

|

²

+ |

^Y

|

²

.

(17) It bearsresemblancetotheO’Raifeartaighmodel[38].The sgold- stinosuperfield S obtainsa masstermthroughits couplingto X . Theparameter δischosen suchthat thevacuumenergyvanishes attheminimumofthepotential

δ = √

3



1

+

²

π

²M²

λ

⁴



+

O

(

M⁴

) .

(18)

Thetree-levelscalarpotentialalongthedirectionx=^y=^{0 reads}

V

=

V0

+

¹²

π

²M²

λ

^{4 m}

2 3/2

+

2

√

3



V0

−

2m²_3/2



s

+ 

4V₀²

−

^2m23/2



s²

+

O

(

s³

) ,

(19)

where V0= |^f |^{2 andm2}_3/2 |^f|^{2 asbeforeand}¯^s=^{s isassumed.}

The imaginary part of s is stabilized at the origin and doesnot play a role inour discussion.The tree-level massof s isnegligi- blecomparedto theone-loop contributionduetotheinteraction with X .WeusetheColeman–Weinbergformula

V_CW

=

¹ 64

π

^2StrM

4logM²

Q²

,

(20)

whereStrM⁴=

i(−¹)^{2 Ji}(2 Ji+¹)m⁴_i isthetraceoverthefield- dependentmasseigenvaluesofstateswithspin Ji.TheColeman–

Weinbergpotentialgivesrisetoanadditionalmassterm VCW

= λ

^{4 m}

2 3/2

π

²M²s²

+ . . . .

(21)

Notethatitisthesameasthemasstermarisingfromtheequiv- alentquantumcorrection totheKählerpotential K = −|^S|⁴/² with

 =

²

√

3

π

λ

^{2 M}

.

(22)

We conclude that we obtain the model of Section 4 as a low- energy effective theory and the small shift of s does not affect inflationforsufficientlysmall.

However,wehaveyettoconsidertheeffectofinflationonthe sector of heavy fields X and Y .Inflation does not induce linear termsforthescalarcomponents x andy.However,it generatesa bilinearmasstermforx.ThemassofIm x isgivenby

m²_{Im x}



^M2

−

²

√

3

λ

m3/2

.

(23)

Table 1

Representativeexampleofthedifferentscalesappearinginsgoldstino-lessinflation.

Thevaluesduringinflationrefertothe beginningofobservable inflation,50–60 e-foldsinthepast.

m3/2 ms H 

inflation 8·10¹⁴GeV 9·10¹⁵GeV 9·10¹³GeV 7·10¹⁷GeV

vacuum 10⁵GeV 10⁶GeV ∼^{0 7}·^1017GeV

Thus, taking the limit M→^{0 to make} S nilpotent introduces a tachyonicdirectioninthefulltheory,whichmakesinflationimpos- sible.⁹ Toobtainapositivesquaredmass,weobtaintheconstraint M²

>

2

√

3

λ

m3/2

.

(24)

Takingtheexampleofchaoticinflation(6)andusingm6·10⁻⁶,

ϕ

∼^{15, this translates into M}>0.03√

λ. At the same time, to make the sgoldstino sufficiently heavy we have to require that

^{1 which is equivalent to M}⁰.09λ², cf. (22). After com- bining these two constraints there is a small window at λ¹ andM∼⁰.05,wheresgoldstino-lessinflationcanconsistentlytake place.Inthisregime theheavyfieldsremainattheir minimaand inflation does not receive corrections besidesthose of Section 4.

Notice,however,thatthesgoldstinomasscanatmostbeenhanced by an O(¹⁰) factor compared to the gravitino mass. This is il- lustrated in Table 1, where we show a possible choice of scales whichleads tosuccessfulsgoldstino-lessinflation. Inthevacuum, thegravitinomassismuchsmallerthanintheinflationaryepoch andlow-energysupersymmetrycanbeobtained.

5.2. Example2

Second,weconsideranexamplewherethesgoldstinoreceives its mass fromgauge interactions. We introduce three newchiral multiplets X , Y , which carry thechargesq(X)= −^1,q(Y)=¹ andq()=^{0 under aU(1) symmetry.10 Wefurther assume that} q(S)=^{1 anddefinethemodelby}

W

=

^f

()(

1

+ δ

^{X S}

) + λ(

^{X Y}

−

^v2

) ,

(25) K

=

¹

2

( + )

²

+ |

^S

|

²

+ |

^X

|

²

+ |

^Y

|

²

+ ||

²

,

(26) whereδ is againchosen toadjust thecosmologicalconstant. We find

δ =

√

3 v



1

−

^2v2

9

+

O

(

v⁴

)



.

(27)

Thesecond terminthesuperpotential isintroduced tobreakthe U(1)symmetry ata highscale.Forthesamereasonasbeforewe setx¯=^x, ¯y=^y, ¯ψ = ψ ^{inthefollowing. Theimaginaryparts do} notplayaroleinourdiscussion.Giventhat v ^Max(m3/2,H)the U(1)symmetryisbrokenalongthealmost D- and F -flatdirection xy

=

^v2

,

s²

−

^x2

+

^y2

=

⁰

, ψ =

⁰

.

(28) Using thesethree conditions toeliminate x, y,and ψ yields the scalarpotential

V

=

^V0

+

²

3m²_3/2v²

+ √

3



2V₀

−

^2m2_3/2



s

+

^m2_s^s2

+

O

(

s³

) ,

(29)

9 Weassumeλ>0.IntheoppositecaseRe x isthetachyon.

10 InordertoavoidanomalieswehavetointroduceanotherfieldZ withcharge q(Z)= −1.ThefieldZ canbecoupledtoanewsingletviaatermY Zinthe superpotential.WhenY breakstheU(1)symmetrythisbecomesalargevector-like masstermforZ .InthiscaseZ anddonotaffectouranalysis,andweneglect theminthefollowingdiscussion.

Fig. 1. Effectiveinflaton potential for thechaotic inflation modelwith v=⁰.03 (blue) v=⁰.1 (orange)and v=⁰.3 (green).Thebackreactionoftheheavyfields flattenstheinflatonpotential.Forv0.1 correctionsfromtheheavyfieldsareun- dercontrol,whilesgoldstinodecouplingrequiresv1.Thisleavesasmallwindow ofviableparameterspaceinwhichsgoldstino-lessinflationconsistentlyproceeds.

(Forinterpretationofthereferencestocolorinthisfigurelegend,thereaderisre- ferredtothewebversionofthisarticle.)

with m²_s

=

^9m

2 3/2

2v²

.

(30)

This resembles (12)ifwe identify =√

4/3v.¹¹ The large mass m_sdecouplesthesgoldstinoandthesmallshiftofs doesvirtually notaffectinflation.

Unfortunately, thisisnotthe endofthestory. Sofarwe have worked inthe regimem_3/2,H^{v. Weexpect additional correc-} tions ifeither m3/2 or H are close to the scale v. To findthese correctionswemusttreatx,y andψ asdynamicalfields.Weper- forma Taylorexpansionaround s=^0, ψ=^0,x=^{v, y}=^{v upto} second orderintheshiftofthefourfields.Settingthefourfields to theirnewminima,wearriveatthefollowingeffectiveinflaton potential

V

=

^V0

−

⁹ 4



1

2g²

+

¹

λ

²



_m4 3/2

v⁴

.

(31)

Notice the difference to our first example. In this case the shift of the heavy fieldsduring inflation causes a backreaction on the potential.Expression(31)onlyincludesthecorrectionsduetothe heavyfields.Inaddition,thesgoldstino-inducedcorrectionsofSec- tion4arise.

Requiring the correction to be suppressed compared to the leading-orderinflatonpotentialleadstotheconstraint

v

^m3/2

V₀^1/4

,

(32)

for λ,g∼O(¹).In the modelof chaotic inflation definedby (6), withm∼⁶·^{10−6 and}

ϕ

∼^{15,theconstrainttranslatesinto}

v

⁰

.

03

.

(33)

Even for larger v there are substantial corrections. We depict the effective inflaton potential of the example (6) in Fig. 1 for f0=^{10−14, m}=⁶·^10−6, λ=^g=^{1, and different values of v.}

Again thereisasmallwindowatv0.1 wherethebackreaction isundercontrolandsgoldstino-lessinflationcantakeplace.Asin theprevious example,choosing v toolargedecreasesthemassof thesgoldstinoscalarbeyondthepointwhereitcanbeconsistently

11 Torecovertheexactformof(12)wewouldhavetosubstitutem3/2→√ 2m3/2.

decoupled.The correctionsfromtheheavy fieldsofthe UVcom- pletioncauseaflatteningoftheinflatonpotential.

6. Discussion

Wehaveemphasized that sgoldstino decoupling incosmology isnontrivial.Workinginspontaneouslybrokenlinearsupergravity, insteadofimposing anilpotency constraintby handweassumed that the mass of the sgoldstino field is produced dynamically.

This required the inclusion of heavy degrees of freedom which coupleto the sgoldstino. We discussed two possible UV embed- dingsofnilpotentgoldstinomultiplets.Bothscenariosresultinthe sgoldstino-less inflation models of [7] as a low-energy effective theory. The sgoldstino has a large but finite mass ms∼^m3/2/

during inflation, where  is the mass scale of the heavy fields whichcoupletothesgoldstino.Asm_3/2>H thesgoldstinodecou- plesfromtheinflationarydynamics.

Thescalesetsanewcut-offatwhichthelow-energyeffec- tive theory breaks down andtheheavy fieldsbecome dynamical degrees of freedom. For inflation to take place in a controlled regime,where theheavy fieldscan beintegratedout, one hasto requirethattheHubblescaledoesnotexceed.However,aneven moresevereconstraintarisesintheclassofmodels[7]whichfea- turem_3/2 ^{H duringinflation.There inflationinduceslarge“soft} terms”whichmaydestabilize theheavy fields.Depending onthe specificUV embedding,we find that tadpole terms ∝^m23/2M²_P/

and bilinear terms ∝^m3/2MP are particularly dangerous. In all examples we findthat inflation is genericallyspoiled if ⁰.1.

Thisdoesnot leavemuch room fora completetheory belowthe Planckscalewithadecoupledsgoldstino.Still,awindowofviable parameter choicessurvives in which sgoldstino-lessinflation can successfullytakeplaceandthebackreactionontheheavyfieldsis undercontrol. Withinthiswindow we calculated thecorrections totheinflatonpotentialwhichtypicallyappearintheformofflat- teningeffects.

The constraints on  imply that the sgoldstino mass can at most be enhanced by one order of magnitude compared to the gravitinomass. Requiring thesgoldstino to decouplein thepost- inflationary cosmology as well puts strong additional constraints ontheformofthescalarpotential.

Duetothestructureofthedangeroustermsthatarise,weex- pect these results to be relevant for many other applications of constrainedmultipletsincosmology.

Acknowledgements

The authors thank G. Dall’Agata and A. Uranga for discus- sions. The work of C.W. is supported by the ERC Advanced Grant SPLE under contract ERC-2012-ADG-20120216-320421, by the grant FPA2012-32828 from the MINECO, and by the grant SEV-2012-0249 of the “Centro de Excelencia SeveroOchoa” Pro- gramme.The work ofL.H. is supported by the IISNandthe Bel- gian Federal SciencePolicy through theInteruniversity Attraction PoleP7/37 “FundamentalInteractions”. The work ofM.W.issup- portedbytheSFB-TransregioTR33“TheDarkUniverse”(Deutsche Forschungsgemeinschaft).

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