Saddle point inflation from f(R) theory

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Physics Letters B

www.elsevier.com/locate/physletb

Saddle point inflation from f ( R ) theory

Michał Artymowski

^b,∗

, Zygmunt Lalak

^a

, Marek Lewicki

^a,c

aInstituteofTheoreticalPhysics,FacultyofPhysics,UniversityofWarsaw,ul.Pasteura5,02-093Warsaw,Poland bInstituteofPhysics,JagiellonianUniversity,Łojasiewicza11,30-348Kraków,Poland

cMichiganCenterforTheoreticalPhysics,UniversityofMichigan,AnnArborMI48109,USA

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

Articlehistory:

Received2September2015

Receivedinrevisedform28September 2015

Accepted30September2015 Availableonline3October2015 Editor:M.Cvetiˇc

Weanalyseseveralsaddlepointinflationaryscenariosbasedonpower-law f(R)models.Weinvestigate inflationresultingfrom f(R)=^R+αnM²⁽¹⁻ⁿ⁾Rⁿ+αn+¹M⁻²ⁿRⁿ⁺¹and f(R)=l

nαnM²⁽¹⁻ⁿ⁾Rⁿaswell asl→ ∞^{limitofthelatter.Inallcaseswehavefoundrelationbetween}αncoefficients andchecked consistencywith the PLANCKdata as wellas constraints comingfrom the stabilityof themodels in question.EachofthemodelsprovidessolutionswhicharebothstableandconsistentwithPLANCKdata, howeveronlyinpartsoftheparameterspacewhereinflationstartsontheplateauofthepotential,some distancefromthesaddle.AndthusallthecorrectsolutionsbearsomeresemblancetotheStarobinsky model.

©^{2015TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense} (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP³.

1. Introduction

Cosmicinflation [1–3] isa theoryof theearly universewhich predicts cosmicacceleration andgeneration ofseeds ofthe large scalestructureofthepresentuniverse.Itsolvesproblemsofclassi- calcosmologyanditisconsistentwithcurrentexperimentaldata [4]. Thefirst theory ofinflation was the Starobinsky model[5,6], whichisan f(R)theory[7]withR+R²/6M²Lagrangiandensity.

In such a model the acceleration of space–time is generated by thegravitationalinteractionitself,withoutaneedtointroduceany newparticles orfields.The embeddingofStarobinsky inflationin no-scaleSUGRAhasbeendiscussedinRef.[8].Recentlythewhole classofgeneralisationsoftheStarobinskyinflation havebeendis- cussedin the literature [9–15], also inthe context of thehigher ordertermsinStarobinskyJordanframepotential[16–18].

ThetypicalscaleofinflationissetaroundtheGUTscale,which isoftheorderof(10¹⁶GeV)⁴.Suchahighscaleofinflationseems tobe a disadvantage ofinflationary models.First of all inflation- aryphysicsisveryfarawayfromscaleswhichcanbemeasuredin accelerators and other high-energy experiments. The other issue is, that high scale of inflation enables the production of super- heavyparticlesduring thereheating[19].Thoseparticlescouldbe inprincipleheavierthantheinflatonitself, soparticleslikemag- netic monopoles, which abundant existence is inconsistent with

*

Correspondingauthor.

E-mailaddress:michal.artymowski@uj.edu.pl(M. Artymowski).

observations,couldbeproducedafterinflation.Anotherargument, which supportslow-scale inflation is the Lythbound [20], which istherelationbetweenvariationoftheinflatonduringinflationin Planckunits(denotedasφ)andtensor-to-scalarratior,namely

φ 



N

0



r

8dN

,

(1)

which fornearly scale-invariant power spectrum gives φ <Mp forr<0.002.Smallφ seemstobe preferablefromthepoint of view ofthenaturalnessprinciple, since M_p isthecut-off scaleof thetheory.The value ofr determinesthescale ofinflation,since V/r (where V isthepotentialoftheinflaton)atthescaleofinfla- tionissetby thenormalisationofCMBanisotropies.Thereforein ordertoobtainsmallr oneneedsalow-scaleinflation,whichmay beprovidedbyapotentialwithasaddlepoint.

Aseparateissuerelatedwith f(R)inflationisrelatedwithloop correctionstothe f(R)function.InordertoobtainquasideSitter evolution ofspace–timeone needs arange ofenergies forwhich theR²M⁻² termdominatestheLagrangiandensity.Thiswouldre- quireallhigherordercorrections(suchasR³,R⁴,etc.)[21]tobe suppressed by a mass scale much bigger than M. One naturally expects all higher order correction to GR to appear at the same energyscaleifonewantstoavoidthefine-tuningofcoefficientsof allhigherorderterms.Fromthisperspectiveitwouldbebetterto generateinflation in f(R)theory withouttheStarobinskyplateau, whichinprinciplecouldbeobtainedinthesaddlepointinflation.

http://dx.doi.org/10.1016/j.physletb.2015.09.076

0370-2693/©^{2015TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense}(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP³.

inhomogeneities

The f(R) theory is one of the simplest generalisations of general relativity (GR). It is based on Lagrangian density S=

1 2

d⁴√

−^{g f}(R) andit canbe expressedusingthe so-calledaux- iliaryfield

ϕ

defined by

ϕ

=^F(R):=^df_dR^{. In such a casethe Jor-} danframe(JF)actionisequalto S= √−^g(

ϕ

R/2−^U(

ϕ

),where U = (^{R F}− ^f)/2 is the JF potential. For F=^{1 one recovers GR,} sothe GRvacuumofthe JFpotential ispositioned at

ϕ

=^1.The samemodelcanbeexpressedintheEinsteinframe(EF),withthe metrictensordefinedbygμν˜ =

ϕ

_{gμν .}Thisispurelyclassicaltrans- formation of coordinates and results obtained in one frame are perfectly consistent with the ones from another frame.¹ The EF actionis equalto S= 

−˜^g( ˜R/2+ (∂μφ)²/2−^V(φ)),where R,˜ φ:=√

3/2 log F and V:= (^{R F}−^f)/(2F²)are theEFRicciscalar, field and potential respectively. The EF potential should have a minimumattheGRvacuum,whichispositionedatφ=^0.

IntheEFthegravityobtainsitscanonicalformandthisiswhy theEFisusually usedfortheanalysisofinflationandgeneration ofprimordialinhomogeneities.Thecosmicinflationproceedswhen both slow-roll parameters

and

η

are much smallerthan unity.

Theseparametersare,asusualgivenby

=

¹ 2



V_φ V



2

, η =

^Vφφ

V

,

(2)

whereV_φandV_φφ arethefirstandthesecondderivativeoftheEF potentialwithrespecttoφ.Duringinflation

and

η

canbeinter- pretedasdeviationfromthede Sittersolution forFRWuniverse.

Duringeach Hubble time theEF scalarfield produces inhomoge- neous modeswith an amplitude of the orderof the Hubble pa- rameter.Fromthemandfromthescalarmetricperturbationsone constructs gaugeinvariant curvatureperturbations, which are di- rectlyrelatedtocosmicmicrowavebackgroundanisotropies.Their power spectrum P_R^{, their spectral indexn}s and their tensor to scalarratioareasfollows

PR



^V

24

π

²

^,

^ns

^

¹

⁻

⁶

+

²

η ,

r



¹⁶

.

(3) In the low scale inflation one obtains

 |

η

|^{, which for}

η

<0 gives1−ⁿs²|

η

_|.

3. Saddlepointinflationinpower-law f(R)theory 3.1. Saddlepointwithvanishingtwoderivatives

As mentioned inthe introduction,the loop corrections tothe Starobinsky model are of the form_∞

n=²

α

nRⁿM²⁽¹⁻ⁿ⁾, where M

1 DifferencesbetweenEinsteinandJordanframeinloopquantumcosmologyare describedinRef.[23].

whereF=^{f andprimedenotesthederivativewithrespecttothe} Ricciscalar.Letusassumethefollowingformof f(R)

f

(

R

) =

^R

+ α

2

R² M²

+ α

n

Rⁿ

M²⁽ⁿ⁻¹⁾

+ α

n+1

Rⁿ⁺¹

M²ⁿ

,

(5)

where n>2 is a given number. Insuch a casethe saddle point appearsfor

R

=

^Rs

=

^M2

 (

n

−

²

) α

n

n



⁻¹

n−1

,

α

n+1

= −

 (

n

−

²

) α

n

n



ⁿ

n−1

.

(6)

Equations above are

α

2 independent, becauseany R² cansatisfy Eq. (4). In order to keep Rs and

α

n+¹ real we need to assume that

α

n>0 and

α

n+¹<0. Then for sufficiently big R one finds F<0 andthegravitybecomesrepulsive.Thisinstabilitybecomes anissueforR^M2_n^α₊ⁿⁿ₁(

α

n(n−²)/n)ⁿ⁻¹⁻ⁿ,whichistypicallyofthe same order ofmagnitude as Rs.By redefining M we canalways set one of

α

n to be anygiven constant.For negative

α

n one can satisfy Eq. (4)forn<2.Nevertheless the saddlepoint wouldlie in therepulsivegravity regime, where F<0.Thus inthefollow- ing analysisn<2 isexcluded.Notethatfornon-zerovalue of

α

2 the valueof M grows with

α

2.Thiscomes fromthefact thatfor

α

2

α

n one obtains inflationary plateaufollowed by the saddle point,duetogrowing valueof Rs withrespectto

α

2.The

α

2 de- pendenceofM isshowninFig. 1.Big

α

2termmeansthatthelast 60 e-foldsofinflation happenon theStarobinsky plateau,so one doesnotobtainsignificantdeviationsfromtheR² model.

The modeldescried in Eq.(5)can be generalisedinto f(R)= R+

α

2R²M⁻²+

α

nRⁿM²⁽¹⁻ⁿ⁾+

α

mR^mM^2(1−m). Then, for

α

n=¹ onefinds

Rs

=

^M2

 (

n

−

²

)(

m

−

ⁿ

)

m

−

¹



_−n−¹₁

,

α

m

= − (

n

−

¹

)(

n

−

²

) (

m

−

¹

)(

m

−

²

)

 (

n

−

²

)(

m

−

ⁿ

)

m

−

¹



ⁿ_n⁻₋^m₁

(7)

The EF potential around the saddlepoint (up to the maximal allowedvalueofφ)for f(R)=^R+

α

2R²/M²+^R3/M⁴+

α

4R⁴/M⁶ has beenshowninFig. 1.We haverescaled M to obtain

α

3=^1.

The R² term in not necessary to obtain a saddle point, butwe includeittocombinetheinflationontheStarobinskyplateauwith thesaddlepointinflation. FromEq.(6)onefindsthevalue of

α

4, normalisationofinhomogeneitiesgivesM asafunctionof

α

2.

2 SomehigherordercorrectionstoStarobinskyinflationdiscussedherehaveal- readybeenanalysedinRefs.[24–26],howevernotinthecontextofthesaddle-point inflation.Inaddition,inouranalysiswetakeintoaccountatleast2higherorder terms,whichisnotthecaseinpaperscitedabove.

Fig. 1. Leftpanel:theEinsteinframepotentialforthe(5)modelaroundthesaddlepointforn=^3,α3=^{1 anddifferentvaluesof}α2.Theα4coefficientissetfromEq.(6)for n=^{3.Themaximalallowedvalueof}φisveryclosetothesaddlepoint.Rightpanel:thescaleofnewphysicsM asafunctionofα2.DottedgreenlinerepresentStarobinsky limit.(Forinterpretationofthereferencestocolourinthisfigurelegend,thereaderisreferredtothewebversionofthisarticle.)

Fig. 2. Tensortoscalarratior andspectralindexnsasafunctionofα2forthe(5)modelwithn=3.OnecanfitthePLANCKdataforα2100,whichmeansthatthesaddle pointisprecededbytheinflationaryplateau.

Fig. 3. Bothpanelspresentns(n)forthemodelfromEq.(5)forN=^{50 and}α2=^{0.Ifn isanaturalnumberonecannotfitthePlanckdataduetotoosmalln}s.Inallof thosecasesasignificantcontributionoftheR²termisneededinordertoobtainns⁰.958.Ontheotherhandforn−²^{10−2oneobtainn}s⁰.96.Thecaseofn² seemstobeespeciallyinterestingsinceitallowstoreconstructStarobinskyresultsitthepresenceofhigherorderterms.

3.2.Saddlepointwithvanishingk derivatives

Ingeneralone candefine thesaddlepointwithfirstk deriva- tives vanishing, which was analysed in Ref. [27]. In that case 1−ⁿs _N(^2k_k−1) ^{when freeze-out of primordial} inhomogeneities happensclosetothesaddlepoint.Thus,forsufficientlybigk one canfitthePlanckdata.Inourcaseall ^d_d^k_φ^V_k =^{0 atthesaddlepoint} areequivalent to ^d_dR^kf_k =^{0 fork}>2.The f(R) modelfromEq.(5) cannotsatisfytheseequations,soinordertoobtainasaddlepoint with vanishing higher order derivatives one needs to introduce

moretermsto f(R)function(seealsoFigs. 2–5).Thusletusnow consider

f

(

R

) =

^R

+ α

₂^R

2

M²

+

l n=3

α

_n ^R

n

M²⁽ⁿ⁻¹⁾

,

(8)

wherel>4 isan evennaturalnumber.Again,withoutanylossof generality one can choose

α

3 to be anypositive constant, so for simplicityweset

α

3=^{1.ThenonecansatisfyEq.(4)and f(n)}=⁰ (forn= {³,4,. . . ,l−²}^{andanyvalueof}

α

2)andthesaddlepoint appearsat

Fig. 4. Leftpanel:Numericalresultsforthemodel(11)forN=50 andN=60 (redandbluedotsrespectively).Rightpanel:EFpotentialforthemodel(11)forl=6,l=8, l=10,l=12 andl=14 (orange,green,red,brownandbluelinesrespectively).Thesaddlepointliesclosetotherightedgeofthepotential,beyondwhichoneobtainsa secondbranchofV ,whichleadstorepulsivegravity.(Forinterpretationofthereferencestocolourinthisfigurelegend,thereaderisreferredtothewebversionofthis article.)

Fig. 5. Numericalresultsforthemodel(11)forN=50 andN=60 (redandbluedotsrespectively).Allvaluesofr obtainedinthisanalysisareconsistentwithPLANCK, butnsfitsthePLANCKdataonlyforN60.(Forinterpretationofthereferencestocolourinthisfigurelegend,thereaderisreferredtothewebversionofthisarticle.)

Fig. 6. Numericalresultsforthemodel(12)forN=^{60 Then}sfitsthePLANCKdatafor0<α<1.4 andforα2^34,whentheα2termdominatestheinflationaryevolution.

R

=

^Rs

= √

p M²

,

where p

=

(

l

−

¹

)



l 2

−

¹



.

(9)

The

α

n coefficientssatisfy

α

n

= (−

1

)

^{n−1 2}

(

l

−

³

) ! (

l

−

n

)!(

n

−

1

)!

^p

3−n

2 for n

= {

3

, . . . ,

l

} .

(10)

Note that Eqs. (9) and (10) are completely independent of

α

2. Since

α

l<0 one obtains F<0 for sufficiently big R. Alike the modelfromEq.(5)thebiggest allowedvalue ofR is slightlybig- gerthan Rs.UsingEq.(8)and(10)oneobtains

f

(

R

) =

R

+ α

2

M²R²

+

^R



lM²

√

p R

+

^2M4p2



1

−

_M2^R√_p



l

−

¹



− (

^l

−

¹

)

R²



M⁴p

−

^M2

√

p R

.

(11) 3.3. Thel→ ∞^limit

Numerical analysis showsthat in orderto obtain correct nor- malisation of primordial inhomogeneities one needs M=^M(l). Nevertheless forl→ ∞ ^{oneobtains M}→^Mo (where Mo∼¹⁰⁻⁵ for

α

2=^{0), which implies R}s→ ∞ ^{for l}→ ∞^{.Hence forl1} one cannot obtain inflation closetosaddlepoint. Forl→ ∞ ^one obtains

Fig. 7. LeftPanel:TheEinsteinframepotentialasafunctionoftheRicciscalar.TheGRminimumatR=^{0 appearstobe}meta-stable,withapossibilityoftunnellingtoanti deSittervacuum.RightPanel:TheEinsteinframepotentialV asafunctionoftheEinsteinframefieldφforthemodel(12).Twobranchesofpotentialcorrespondtotwo solutionsofϕ=F(R).InordertoavoidovershootingtheminimumatR=0 onerequiresα20.7.

Fig. 8. Theminimalvalueofα2,whichallowstoavoidovershootingthemeta-stable GRminimum.

f

(

R

) =

^R

e⁻

√2R M2o

+

√

2

+ α

2

M²_o R



.

(12)

The

α

2 maybe again used tostabilise the GR vacuumat φ=^0.

Thenumericalresultsfor N=^{60 areplottedin}Figs. 6 and7.As expected,for

α

^{1 valuesofM}/√

α

,r andnsobtainthelimitof theStarobinsky theory.AsshowninFig. 7thepotentialshavetwo branches, whichsplit atsome φ= φm, where φm isthe minimal valueofφ.The

α

2 terminnecessaryinorderto stabilisetheGR vacuum.For

α

2=^{0 oneobtains twobranchesofpotential which} growfromφ=^{0.Both ofthemexistonlyfor}φ >0 withnomin- imum.Whileincreasing the valueof

α

2 thesplittingof branches moves towards φ <0 and the inflationary branch obtains mini- mum at φ=^{0. We} investigated the stability of minimum from the perspective of classical evolution of the Einstein frame field.

Namely, we considered the slow-roll initial conditions at φ= φ fordifferent valuesof

α

2 andchecked whetherthe minimum is deepenoughtostopthefieldbeforeitwouldreachφm.Wepost- ponetheissueofquantumtunnellingtotheanti-de Sittervacuum forfuturework.

4. Conclusions

In this paper we considered several f(R) theories with sad- dlepointintheEinsteinframepotential.AllmodelsconsistofGR term R,Starobinskyterm

α

2R² andhigherordertermswhichare thesource ofthe saddlepoint. InSubsection 3.1we investigated

two additionaltermsproportionalto Rⁿ andRⁿ⁺¹.We foundan- alytical relation between their coefficients and Rs, which is the valueoftheRicciscalaratthesaddlepoint.Thepotentialbecomes unstablefor R slightlybiggerthan Rs –thesecond branchofthe auxiliaryfieldequation

ϕ

=^F(R)becomesphysical,whichleadsto thesecond branchofpotentialandasaconsequencetorepulsive gravity.Significantcontributionofthe R² termextendtheplateau beforethe saddlepointandpushesaway theinstability fromthe inflationaryregion.Forn≥^{3 itisimpossibletoobtaincorrectn}s, howeverforn slightlybiggerthan2 onecanfitthePLANCKdata.

InSubsection3.2we investigatedTheEinstein framepotential withzero value of the firstl−2 derivativesatthe saddle point, where l≥^{6 is an even natural number. Toobtain such a saddle} point we considered f(R)=^R+

α

2R²+l

n=³

α

nM²⁽¹⁻ⁿ⁾Rⁿ. We foundanalyticalformulaeforRsandforall

α

n coefficients,aswell asthe explicit value of f(R) aftersummation. Unfortunately the result is slightly disappointing, because the saddle point moves away from the scale offreeze-out of primordialinhomogeneities withgrowingl. ThusbringingusclosertotheStarobinsky caseas l getsbigger.Wealsoobtainednumericalresultsforn_s,r andfor thesuppressionscaleM asafunctionofl.Thefinalresultstrongly dependson N,andtherefore onthe thermalhistoryofthe uni- verse. Onecan fit thePLANCK dataforl^{20 and N}^{60 even} for

α

2=^{0.Again,forR slightlybigger than R}soneobtains anin- stability ofpotential,whichforbigl isorders ofmagnitudeaway fromthefreeze-outscale.

In Subsection 3.3 we considered the limit l→ ∞^{, which re-} sultedin f(R)=^R(e^−√2R/M2o+(√

2+

α

2)R/M²_o),whichisbasically Starobinsky model plus an exponentially suppressed correction.

In such a casethe saddlepoint (and thereforethe instability for R>Rs)movestoinfinityandinflationhappensfarawayfromthe saddlepoint. The

α

2 termis necessary tocreate themeta-stable minimumofthe Einsteinframe potential(Fig. 8). Onecanfit the PLANCKdatafor0.7

α

2¹.4 and

α

2^34.

Acknowledgements

This work was partially supported by the Foundation for Pol- ish Science InternationalPhDProjectsProgramme co-financedby the EU European Regional Development Fund and by National Science Centre under research grants DEC-2012/04/A/ST2/00099 and DEC-2014/13/N/ST2/02712. ML was supported by the Pol- ish National Science Centre under doctoral scholarship number 2015/16/T/ST2/00527.MAwassupportedbyNationalScienceCen- tregrantFUGAUMO-2014/12/S/ST2/00243.

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