Initial Conditions for Critical Higgs Inflation

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Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Initial

conditions

for

critical

Higgs

inﬂation

✩

Alberto Salvio

CERN,TheoreticalPhysicsDepartment,Geneva,Switzerland

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received12December2017 Receivedinrevisedform2March2018 Accepted2March2018

Availableonline6March2018 Editor:M.Trodden

Keywords:

Inﬂation Higgsboson StandardModel

It hasbeen pointed outthatalarge non-minimal couplingξ betweenthe Higgsand theRicci scalar cansource higherderivativeoperators,whichmaychangethepredictionsofHiggsinflation.A variant, calledcriticalHiggsinflation,employsthenear-criticalityofthetopmasstointroduceaninflectionpoint in the potential and lower drastically the value of ξ. We here studywhether criticalHiggs inflation canoccur evenifthepre-inflationaryinitialconditionsdonotsatisfythe slow-rollbehavior (retaining translationandrotationsymmetries).A positiveanswerisfound:inflationturnsouttobeanattractor and therefore no fine-tuning of the initial conditions is necessary. A very large initial Higgs time-derivative(ascomparedtothepotentialenergydensity)iscompensatedbyamoderateincreaseinthe initialfieldvalue.TheseconclusionsarereachedbysolvingtheexactHiggsequationwithoutusingthe slow-rollapproximation.Thisalsoallowsustoconsistentlytreattheinflectionpoint,wherethestandard slow-rollapproximationbreaksdown.HerewemakeuseofanapproachthatisindependentoftheUV completionofgravity,bytakinginitialconditionsthatalwaysinvolvesub-planckianenergies.

1. Introduction

SofarnoclearevidenceforphysicsbeyondtheStandardModel atthescalesexploredbytheLHChasbeenfound.Inthissituation, itis usefultolook forcomplementarytests. The extrapolationof theSM atenergies muchabovethosereachableatcollidersoffers a newway to look forfurther evidence of newphysics (besides thealreadyestablishedones,suchasneutrinooscillationsanddark matter).

Inﬂationisanaturalarenatoperformthesetests.Itwasfound thattheHiggs oftheSM mightplaythe roleofthe inﬂaton pro-videdthat asizablenon-minimalcoupling

ξ

withtheRicciscalar

R is introduced. Ref. [1] considered originally the caseof a very large

ξ,

which corresponds to the SM living well inside the so calledstability region.1 In thissetup two different scales appear, thereducedPlanckmass M

¯

Pl

2.435

×

1018 GeVandM

¯

Pl

/ξ

and

a violationof perturbative unitarity at M

¯

Pl

/ξ

has been found by

consideringscatterings ofparticles viewedasﬂuctuationsaround theEWvacuum [2,3].Thisleadstothenecessityofnewphysicsor strongcouplingmethods toanalyze thatphysicalsituation.While thisdoesnot undoubtedlyexcludeHiggsinﬂation (HI)asthe

rel-✩ _Report_number:_{CERN-TH-2017-261.}

E-mailaddress:alberto.salvio@cern.ch.

1 _This_is_the_region_of_parameter_space_where_the_electroweak_(EW)_vacuum_is

theglobalminimumoftheSMeffectivepotential.

evantexpansion inthat caseisaround alargeHiggsﬁeld [4],the issuemaybeavoidedbylivingveryclosetotheboundarybetween stabilityandmetastability,2 thesocalledcriticality.Indeed,a dras-ticdecreaseof

ξ

occursatcriticality [5–7],leadingtoasinglenew physicsscale, M

¯

Pl,wherequantumgravityeffectsare expectedto

emerge.

Moreover, in [8] another issue of the large-ξ HIwas pointed out.Atthequantumlevel,itisnecessarytotunethehighenergy values of some parameters in order to preserve the inﬂationary predictions:ifthisisnotdonelargehigherderivativetermsinthe effectiveaction,suchas R2_,_are_generated,_changing_the_output_of

the model (see also [9]). However, Ref. [8] did not consider the criticalHIcase,which,asstatedabove,doesnotrequirealarge

ξ.

TheaimofthisarticleistoinvestigatewhethercriticalHI suf-fersfromanytuninginthechoiceofthehighenergyparameters. This will includein particular the analysisof the dependence of criticalHI onthe initial (pre-inﬂationary) conditions.Indeed,any

slow-roll model of inflation, such as HI, should provide a mech-anism that drives generic initial conditions to slow-rolling con-figurations, i.e.an inflationary attractor. Ifsuch an attractordoes not existafine-tuning oftheinitialconditionsis required,which makes the whole idea of inflation less attractive, given that its

2 _The_{metastability}_region_is_the_region_of_parameter_space_where_the_lifetime_of

theEWvacuumexceedstheageoftheuniverse. https://doi.org/10.1016/j.physletb.2018.03.009

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main purpose is to solve ﬁne-tuning problems (the horizon and ﬂatnessproblems).

The paper is organized as follows. In Sec. 2 details ofHI are given,includingtheclassicalanalysisofinflationandadescription ofthequantumcorrections;there,wewilladdressthequestionof whetherafine-tuningofthehighenergyconditionsoftherunning parametersisrequiredincriticalHI.InSec.3wewillconsider ini-tialconditionsviolatingtheslow-rollbehavior inordertoestablish theexistenceofaninflationaryattractorinHI;bothanalyticaland numericalargumentswillbeused.Finally,Sec.4providesthe con-clusions.

2. Themodel

LetusdeﬁnetheHiggsinﬂationmodel[1].Theactionis

S

=

d4x

√

−g

L

SM

−

¯

M2_Pl 2

+ ξ|H|

2

R

,

(1)

whereH istheHiggsdoublet,

ξ

isarealparameterand

√

−

g

L

SM

istheSMLagrangianminimallycoupledtogravity.Thepartofthe action that depends on the metric and the Higgs ﬁeld only (the

scalar-tensorpart)is Sst

=

d4x

√

−

g

|∂

H

|

2

−

V

−

¯

M2_Pl 2

+ ξ|

H

|

2

R

,

(2)

where V

= λ(|

H

|

2

₋

_v2

_/2)

2 _is_the_classical _Higgs_potential,_and_v

isthe EW Higgsvacuumexpectationvalue. We assume asizable non-minimalcoupling,

ξ >

1,becausethisisrequiredbyinﬂation aswewillsee.

2.1. Classicalanalysis

The

ξ

|

H

|

2R termcanbeeliminated throughaconformal

trans-formation(a.k.a.Weyltransformation): gμν

→

−2gμν

,

2

=

1

+

2

ξ

|

H

|

2

¯

M_Pl2

.

(3)

Theoriginalframe,wheretheLagrangianhastheforminEq. (1),is calledtheJordanframe,whiletheonewheregravityiscanonically normalized(obtainedwiththetransformationabove)iscalledthe Einsteinframe.Intheunitarygauge,wheretheonlyscalarﬁeldis theradialmode

φ

≡

2

|

H

|

2_,_we_have_(after_having_performed_the

conformaltransformation) Sst

=

d4x

√

−

g

K

(∂φ)

2 2

−

V

4

−

¯

M2_Pl 2 R

,

(4) and K

=

−4

2

+

3 2

d

2 d

φ

2

.

(5)

The non-canonical Higgs kinetic term can be made canonical throughtheﬁeldredeﬁnition

φ

= φ(

χ

)

deﬁnedby

dχ d

φ

=

−2

2

₊

3 2

d

2 d

φ

2

,

(6)

withtheconventional condition

φ (

χ

=

0)

=

0.Notethat

φ (

χ

)

is invertiblebecauseEq. (6) tellsusd

χ

/

d

φ >

0.Thus,onecanextract thefunction

φ (

χ

)

byinvertingthefunction

χ

(φ)

deﬁnedabove.

Notethat

χ

feelsapotential U

≡

V

4

=

λ(φ (

χ

)

2

−

v2

)

2

4

(

1

+ ξφ(χ

)

2

/ ¯

M2_Pl

)

2

.

(7)

Letusnowrecall howslow-roll inﬂationemerges inthis con-text. From (6) and (7) it follows [1] that U is exponentially ﬂat when

χ

¯

MPl,whichisakeypropertytohaveinﬂation.Indeed,

forsuchhighﬁeldvaluesthequantities

U

≡

¯

M2 Pl 2

1 U dU dχ

2

,

η

U

≡

¯

M2 Pl U d2_U dχ2 (8)

areguaranteedtobesmall.Therefore,theregioninﬁeld conﬁgura-tionswhere

χ

¯

MPl(orequivalently[1]

φ

¯

MPl

/

√

ξ)

corresponds

toinflation.InSec.3we willinvestigatewhethersuccessful slow-roll inflation emergesalso forlarge initialfield kinetic energy. In this subsection we simply assume that the time derivatives are small.Inthiscase,duringthewholeinflationtheslow-roll param-eters

U and

η

U aresmallandtheslow-rollapproximationcanbe used.

Alltheparameters ofthemodelcan bedeterminedwithgood accuracythroughexperimentsandobservations,including

ξ

[1,10].

ξ

can be ﬁxed by requiring that the measured curvature power spectrum(athorizonexit3 q

=

aH )[11],4

PR

(

q

)

(

2

.

14

±

0

.

06

)

×

10−9

,

(9)

isreproducedforaﬁeldvalue

φ

= φ

b corresponding toan

appro-priatenumberofe-folds[10]:

N

=

φb

φe U

¯

M2_Pl

dU d

φ

₋1

dχ d

φ

2 d

φ

59

,

(10)

where

φe

is the ﬁeld value atthe endof inﬂation, computedby requiring

(φ

e

)

1

.

(11)

Intheslow-rollapproximation(usedinthissubsection)such con-straintcanbeimposedbyusingthestandardformula

PR

(

k

)

=

U

/

U 24

π

2_M

¯

4

Pl

.

(12)

ForN

=

59,thisprocedureleadsto

ξ

(

5

.

02

∓

0

.

06

)

×

104

√

λ,

(

N

=

59

)

(13) where the uncertainty corresponds to the experimental uncer-taintyquotedinEq. (9).Notethat

ξ

dependsonN:

ξ

(

4

.

61

∓

0

.

06

)

×

104

√

λ

(

N

=

54

),

(14)

ξ

(

5

.

43

∓

0

.

06

)

×

104

√

λ

(

N

=

64

).

(15) Giventhat

λ

∼

0.1,

ξ

hastobemuchlargerthanoneatthe classi-callevel.Theneedofaverylarge

ξ

canbeavoidedwhenquantum correctionsareincluded[5–7], aswe willseeinthenext subsec-tion.

3 _We_use_a_standard_notation:_{a is}_the_cosmological_scale_factor,_H_{≡ ˙}_a_/_{a and}_a

dotdenotesthederivativewithrespectto(cosmic)time,t.

4 _See_for_instance_Table₃_of_the_second_paper_in_{Ref. [}₁₁_{] ( P}

R isdenotedwith Asinthattable).Thevaluequotedherecorrespondstotheonewiththesmallest uncertaintyinthattable.

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Wecanalsoextractthescalarspectralindexnsandthe tensor-to-scalarratio r: in the slow-roll approximation we are usingin thissubsectiontheformulæarer

=

16

U andns

=

1

−

6

U

+

2

η

U. Theseparametersareofinterestastheyareconstrainedby obser-vations.

2.2.Quantumcorrections

Here we take into account quantum corrections to the Higgs potential. We would like to include both the large-ξ inﬂation-ary scenario of [1] and the critical Higgs inﬂation proposed in [5–7], whichemploys a value ofthe top massclose to the fron-tier between stability and metastability of the electroweak vac-uum.Thelattercaseallows fora drasticdecreaseoftherequired valueof

ξ

withrespecttotheresultoftheclassical analysis(see Eqs. (13)–(15).). Thisindicatesthat wecannot relyonlarge-ξ ap-proximations to analyze this case. We thereforedo not use such approximationshere.However,wedoassumeinthefollowingthat

ξ >

1 asthisconditionispresentbothintheoriginalformulation ofHIandincriticalHI.

Notethat Eqs. (3)–(6) holdalsoif

ξ

isfield-dependent,as dic-tated by quantum corrections [12]. A second step we should do nowisthecomputation ofthe effectivepotential.Indefining the quantumtheorytherearewell-knownambiguities[13,14,6,15,16]. WefollowhereRef. [6] andchoose todeterminetheloop correc-tionstotheeffectivepotential–a.k.a.Coleman–Weinbergpotential – in the Einstein frame (after having performed the conformal transformation (3)); the effective potential is also RG-improved byusing theRGEs5 of theSM properlymodified to take into ac-count

ξ.

Theway

ξ

affectstherunningisthroughtheappearance ofafactors thatsuppressesthecontributionofthephysicalHiggs ﬁeldtotheRGEs[17].Ref. [17] found

s

=

1

+ ξφ

2

_{/ ¯}

_M2 Pl 1

+ (

1

+

6

ξ )ξ φ

2

_{/ ¯}

_M2 Pl

.

(16)

Forsmallenough

φ

onehass

≈

1,while inthelarge-φ limit s

≈

1/(1

+

6ξ ).Thisresultdoesnotreallydependonthesizeof

ξ,

but, ofcourse,thelarger

ξ

isthemoreeffectivethesuppressionis.

Suchproceduretocomputetheeffectivepotentialisknownas PrescriptionIanditleads tothefollowingvalue ofthe renormal-izationgroupscale

¯

μ

(φ)

=

φ/

κ

1

+ ξφ

2

_{/ ¯}

_M2 Pl

,

(17)

where

κ

isanorderonefactor.Thisformulafollowsfromthefact thattheloopcorrectionstotheeffectivepotentialaredetermined intheEinsteinframe.6 Thefunction

μ

¯

(φ)

can alsobeinvertedto obtain

φ (

μ

¯

)

andusedin(16) toexpresss asafunctionof

μ

¯

only, asappropriatefortheRGEs.

TheSMRGEsmodiﬁed bythes-factorcanbefoundinthe ap-pendixofRef. [19], wherethe RGEof

ξ

isalsoprovided.Wewill employtheseformulæinthenumericalcalculationofSec.3.2.

Furthermore,wewillusetheRG-improvedpotentialneglecting theloopcorrections:thismeansthatwewilltakeaseffective po-tentialtheonein(7) withtheconstants

λ

and

ξ

substitutedwith thecorrespondingrunningparameters. Thereare goodreasonsto

5 _We_use_dimensional_{regularization}_(DR)_to_regularize_the_loop_integrals_and_the

modiﬁedminimalsubtraction(MS)schemetorenormalizeawaythedivergences. This,asusual,leadstoarenormalizationscale,μ¯.

6 _The _explicit _detailed _expression _of _the _1-loop _correction _can _be _found _in

Ref. [18].

use thisapproximation. First, our main objective is to see if the initialconditionswithlargetime-derivativesoftheHiggsﬁeldare attractedtowardsa slow-rollregime. Inorderto knowifthere is thisqualitative behavior wedonotneedaveryprecise determina-tion ofthe effective potential (which is beyondthe scope of the presentwork). Note, moreover, that takinginto account theloop correctionstothepotentialwouldonlybemorepreciseif supple-mentedbytheloopcorrectionstothekinetictermoftheinﬂaton; suchcorrectionshavenotbeenincludedintheanalysisofHIand are expectedtobecomparableto theloopcorrectionsto the po-tentialformoderatevaluesof

ξ,

unlike whathappensforlarge

ξ

[14]:thelargevalueof

ξ

allowed [14] toshowthatthecorrections to the kinetic termin theeffective action are negligible,butthe smallervalue of

ξ

ofcriticalHIdoesnot permitto trustthis ap-proximationanymore.AnotherreasontoemploytheRG-improved potential is its gauge independence, which is not shared by the loopcorrectionstotheeffectivepotential.Therefore,theuseofthe RG-improved potential allowsfora moretransparent physical in-terpretation.

AsboundaryconditionstosolvetheRGEsoftheSMcouplings we usethe currentlymostprecise determinationsof their values atthetoppolemass Mt,whichwerecomputedinRef. [20].These values are functions of some observables: Mt, the Higgsand W polemassesMh and MW,respectively,and

α

3

(

MZ

).

ForMW and

α

3

(

MZ

)

we take the same values quoted in Ref. [20], while for

MhwetakethemoreprecisedeterminationpresentedinRef. [21], that is Mh

=

125.09

±

0.21

±

0.11 GeV. The boundary condition for

ξ

isinsteadﬁxedto reproducetheexperimentalvaluesofthe inﬂationary observables. The top pole mass is a variable in this work.

Now, Eqs. (8), (10) and (12) of Sec. 2.1 are still validas long asoneisintheslow-rollregime,butoneshouldnowinterpretU

astheeffectivepotential,not justastheclassical potential. How-ever, aswe will see inSec. 3, incritical Higgs inflation, because ofthepresenceofan inflectionpointinthepotential(see Fig.2), the standard slow-roll condition may not be always satisfied; in particular it can break down around the inflection point, where the inertial term in the inflaton equation may not be negligible withrespecttothefrictionterm[24,12,22,23].Wewilldiscuss fur-therthispointinSec.3.Nevertheless,alreadyatthislevelwecan observethat theslow-rollcondition isviolated alsoatthe begin-ningof the inflatonpath if we start frominitial conditions with large timederivative ofthe Higgsfield. Therefore,theframework to analyze the pre-inflationary dynamics withsuch initial condi-tion (whichwill be discussedin Sec.3) willbe applicable to the periodwhentheHiggscrossedtheinflectionpointtoo.

Havingdeterminedtheeffectivepotentialwecannowestimate the relevant inﬂationary scales. In HI the energy density during inﬂation UI is roughly given by7 UI

∼ λ ¯

M4_Pl

/ξ

2, as clear from Eq. (7) andthe discussion below that formula. This is relatedto the inﬂationary Hubblescale HI through the Einstein equations,

HI

∼

√

UI/ ¯MPl.Thescale HI ismuchlowerthanthenewphysics scaleM

¯

Pl

/ξ

incriticalHIthankstothesmallnessof

λ

atthe

inﬂa-tionaryscale.Furthermore,thesmallnessof

λ

alsoleadstoasmall

UI inPlanckunits,whichallows ustotreat gravityclassically, as we will discussin Sec. 3. For example,in Fig. 2, UI

∼

10−9M

¯

4_Pl, whichgives HI

∼

10−5M

¯

Pl,aHubblescalemuchsmallerthanthe

newphysics scale M

¯

Pl

/ξ

forthecorresponding value ofthe

non-minimalcoupling: M

¯

Pl

/ξ

∼

10−1M

¯

Pl.Therefore,althoughnew

de-7 _One_thing_we_learn_then_is_that,_once_the_observed_P

Risreproduced,arelation

betweenλandξemerges;itisthisrelationthatreducesthevalueofξincritical Higgsinﬂation:whatmattersinreproducingPRisonlytheoverallconstant,λ/ξ2,

(4)

Fig. 1. Running of the SM couplings and ξ for Mt≈171.04 GeV with the s-insertions.(Forinterpretationofthecolorsintheﬁgures,thereaderisreferred tothewebversionofthisarticle.)

grees offreedom orstrong coupling should eventually appear at

¯

MPl

/ξ

,therelevantinﬂationaryscaleHI islowerthan M

¯

Pl

/ξ.

Wenow recallan importantresultofRef. [8]:alarge

ξ

natu-rally induces large higherdimensionaloperators that can inturn changethephysical predictions.Thecoeﬃcient

α

ofaradiatively induced

√

−

g R2 termintheLagrangianwas showninRef. [8] to besubjecttothefollowingstrongnaturalness8 _bound

|α|

ξ

2

8

π

2

.

(18)

Alarge valueof

ξ

is necessaryattheclassical level(see Eq. (13) andthe corresponding discussion there). However,incriticalHiggs inﬂation

ξ

doesnothavetobeverylargeandavalueof

ξ

oforder10 ispossible [5–7]. Suchvaluewouldleadtothetrivial9_bound

_α

_1. Therefore,thecriticalHiggsinﬂationdoesnotsufferfromaﬁne-tuning ofthehighenergyconditionfortherunningcouplings.

3. Pre-inﬂationarydynamics

Letusnow analyzethedynamicsofthissysteminthe homo-geneouscasewithoutmakinganyassumption ontheinitial value ofthetime derivative

χ

˙

.Thisanalysishasbeenperformedin[8] forHiggsinﬂationattheclassicallevel.Herewetakeintoaccount quantumcorrectionstothepotential.Wewillfocusonthecritical Higgsinﬂationforthereasonswediscussedabove.

IntheEinsteinframe Sst is

Sst

=

d4x

√

−

g

(∂

χ

)

2 2

−

U

−

¯

M2 Pl 2 R

(19) (U istheRG-improvedEinsteinframeeffectivepotential).

Let usassume a universe with 3-dimensional translation and rotation symmetries, that is a homogeneous and isotropic FRW

8 _Naturalness_implies_that_condition_because_if_one_starts_from_an_|_α_|_much

be-lowthatthresholdtherenormalizationgroupﬂowgeneratesan|α|comparableto it,unlessveryﬁne-tunedinitialconditionsarechosen.

9 _This_bound_is_trivial_in_view_of_the_fact_that_an_α_of_order₁_leads_to_negligible

correctionstoEinsteingravityforenergiesmuchbelowthePlanckscale(instead, forenergiesapproachingthisvaluewecannothaveamodelindependentargument becauseEinsteingravitybreaksdown).

geometry.Wedonotregardthisasafine-tuningoftheinitial con-ditionasit isimplied bytherequirementofhaving anenhanced symmetry (translation and rotation symmetries in this case). In otherwords,itisnaturaltoassumethatinitiallytherewasneither any special point in space nor any preferred direction. Further-more,we willneglect thespatial curvatureintheFRWmetricas the energy density is expected to be dominated by the inflaton duringinflation.

Then the equations for the spatially homogeneous ﬁeld

χ

(

t

)

anda

(

t

)

are

¨

χ

+

3

χ

˙

2

₊

_6U

√

2M

¯

Pl

˙

χ

+

U

=

0 (20) and

˙

a2 a2

=

˙

χ

2

₊

_2U 6M

¯

2_Pl

.

(21)

From(20) and(21) onecanalsoderivetheuseful

˙

H

= −

χ

˙

2

2M

¯

2_Pl

.

(22)

Notethat,oncewehaveasolutiontoEq. (20) wecanimmediately determinea

(

t

)

throughEq. (21).Therefore,ourjobnowistosolve Eq. (20) withappropriateinitialconditions

(¯

t

)

=

χ

(¯

t

)

=

χ

,

(23) where

¯

t issomeinitialtimebeforeinﬂationand

χ

and

arethe initialconditionsforthedynamicalvariables10_at_t

_{= ¯}

_t.

Sincewe donotwanttocommitourselvestoanyUV comple-tion of Einstein gravity, we conﬁne our attention to the regime wherequantumgravitycorrectionsareexpectedtobesmall,

U

¯

M4_Pl

,

χ

˙

2

¯

M4_Pl

,

(24)

such that we canignore the details ofthe UV completion. How-ever, wedonot requireheretobe initially inaslow-roll regime. Theconditionsin(24) comefromtherequirementthatthe energy-momentumtensorbesmall(comparedtothePlanckscale)sothat a large curvature is not generated. The first condition in (24) is automaticallyfulfilledbytheHiggsinflationpotential:thequartic coupling

λ

issmall[25,20,26,27];notethat

λ

isparticularlysmall inthecriticalHI[5–7],whichisourmaininteresthere.Thesecond condition in (24) is implied by the requirementof starting from an(approximately)deSitterspace,whichismaximallysymmetric; thereforewedonotconsiderthatasﬁne-tuningoftheinitial con-ditions.Indeed,indeSitterwe havetoset H

˙

=

0,whichleads to

˙

χ

=

0 givenEq. (22).

However, note that we cannot start from an exact de Sitter space given Eq. (20): the potential U is indeed not exactly flat. Given that the extra symmetries of de Sitter space (besides ro-tation and translation symmetries) are anyhow broken, there is no remainingsymmetry that forcesthefield kineticenergyto be smallcomparedtothe potentialenergyorthatforcestheinertial termintheinflatonequation tobe negligiblewithrespect tothe friction term. This motivates our studyof initial conditions with generickineticenergy.

10 _The_initial_condition_for_{a is}_not_needed_as_the _{normalization}_of_{a does}_not

haveaphysicalmeaningforvanishingspatialcurvature:indeed,givenasolutiona

(5)

3.1.Analyticdiscussion

Sincewedonotknowifandwhenexactlyaninflationary slow-rollphaseoccursitisessentialforourpurposestohavea descrip-tion of the inflationary path that does not rely on the slow-roll approximation.Inflationingeneraltakesplacewhen

≡ −

H

˙

H2

<

1

,

(25)

whichgeneralizes the usual deﬁnitionof theslow-roll parameter

U (given in Eq. (8)) to situations where the slow-roll does not occur.Inadditionto

,onecanintroduceanotherparameter

δ

≡ −

χ

¨

H

χ

˙

.

(26)

If

δ

1 one can neglect the inertial term in the inflaton equa-tion(20) andreducetheproblemtoasinglefirstorderdifferential equation. The necessary and sufficient condition for inflation is only

<

1. The condition

δ

1 is by no means necessary, al-though it leads to great simpliﬁcations. As we mentioned before andwewilldiscussinmoredetailinSec.3.2,inthecriticalHiggs inﬂationthecondition

δ

1 isnotalwayssatisfiedbecauseofthe presenceofan inflection point inthe potential.Both because we willconsiderinitialconditionswithlargekineticenergiesand be-causeoftheinflectionpoint,itisthereforeimportanttosolvethe exactequationin(20) withoutusingtheslow-rollapproximation. Notethat this alsomeans that we cannot usenow the slow-roll formula N

=

φb

φe U

¯

M2_Pl

dU d

φ

₋1

dχ d

φ

2 d

φ

(27)

that we used before in Eq. (10) to compute the number of e-folds N.Wewilluseinsteadtheexactformula

N

=

te

tb

dt H

(

t

),

(28)

whereteisthetimeattheendofinﬂationandtbisthetimewhen theinﬂationaryobservablesPR,nsandr aremeasured.

Oneof the main purposes ofthe presentwork is to consider largeinitialkineticenergies,

χ

˙

2

_{U ,}_and_study_whether_inﬂation

isanattractor.Thisproblemcanbetreatedanalyticallyduringthe ﬁrst phase of the inﬂaton motion when

χ

˙

2

_{U ,} _such _that _the

potentialenergycanbeneglected.Inthiscase,combiningEqs. (21) and(22) gives

˙

H

+

3H2

=

0

,

(

χ

˙

2

U

),

(29) whichleadsto H

(

t

)

=

H

¯

1

+

3H

¯

(

t

− ¯

t

)

,

(

χ

˙

2

_U

_),

₍₃₀₎

whereH

¯

≡

H

(¯

t

)

and

¯

t isagainsomeinitialtime.Byinsertingthis resultintoEq. (22) weﬁnd

˙

χ

2

=

6M

¯

2 PlH

¯

2

1

+

3H

¯

(

t

− ¯

t

)

2

,

(

χ

˙

2

U

).

(31) Thismeans that the kinetic energy densityscales as1/t2 _if _one

takesintoaccountthetimedependenceofH .Thisresult[28] tells usthat an initial condition withlarge kinetic energy is attracted

Fig. 2. SMeffectivepotential(asdeﬁnedinthetext)withtheξ-couplingchosenas inFig.1.

towards one with smaller kinetic energy, but it also shows that neglecting the potential energy cannot be a good approximation forverylarge times.Moreover, noticethatEqs. (30) and(31) im-ply

χ

¨

= −

3H

χ

˙

, so the dynamics is not approaching the usually assumedslow-rollcondition

δ

1 thatallowstodroptheinertial terminthe inﬂatonequation ofmotion.Therefore,the argument aboveisnotconclusive andoneneedstosolvetheequations tak-ingintoaccountU inordertoseeifinﬂationisreallyanattractor. Wewilldosonumericallyinthenextsubsection.

3.2. Numericalstudies

Inthis section wepresentthe numericalstudies. Weset here thereferencevalue

κ

≈

2 fortheparameterappearinginthe opti-malvalueof

μ

¯

fortheRG-improving,Eq. (17).Aswewillsee,this leadstoarealisticinﬂation.

InFig.1wegivetherunningofthelargestSMcouplingsand

ξ

withthes-insertions(see thediscussionaroundEq. (16)).Inthat plot we use a value of Mt close to criticality Mt

≈

171.04 GeV: thequarticcoupling

λ

nearlyvanishesathighenergies.There the minimum of

λ

occurs at around 0.15M

¯

Pl. In Fig. 2 we provide

the corresponding SM effective potential including the effect of

ξ.

We see that by varying Mt by only 10 keV around the criti-cal valuethepotential changessignificantly.Thisgivesusanidea of the level of adjustment of Mt required to have an inflection point inthepotential, whichis an importantissueofthe critical Higgs inflation. We could regard thiseither asa drawback or as an attractivefeature ofthe model,dependingon whetherwe re-gardthisadjustmentasafine-tuningorapredictionofthemodel. A possible problem here is the tension between the critical Mt and the measured value: Mt

=

172.51

±

0.50 GeV (ATLAS) and

Mt

=

172.44

±

0.49 GeV (CMS) [29], which is, separately, at the 2–3

σ

level.11 _If _future _measurements _and _calculations_will con-ﬁrmthisdifference,newphysicscouldbeinvokedtoreconcilethe twovaluesofMt,suchasthewell-motivatedscenarioofRef. [30]. We studied numerically the exact Higgs equation in (20). In Fig. 3 we provide the canonically normalized ﬁeld

χ

asa func-tion of time. We observe that even if start from a large kinetic

11 _If _we_had _quantized_the _system_in_the _Jordan_frame_before_performing_the

conformaltransformation(adeﬁnitionofthetheoryknownasprescriptionII)we wouldhavefoundanevenstrongertension [13,14].Thisis ourmainreason to chooseprescription I.TheextensionofthesecalculationstoprescriptionIIis, there-fore,beyondthescopeofthisarticle,butweexpectthatsimilarqualitative proper-tiescanbefoundwiththealternativeprescription.

(6)

Fig. 3. Canonicallynormalizedﬁeldχasafunctionoftime.Thevaluesofthe pa-rametersthatarenotquotedintheplot arechosenasinFigs.1and2andas explainedinthetext.

Fig. 4. Theparameters andδduringaperiodofinﬂation.Thevaluesofthe param-etersthatarenotquotedintheplotarechosenasinFigs.1and2andasexplained inthetext.

energy12_the_field_quickly_reaches_the_slow-roll _regime;_the ultra-slow-roll regime [31,23] quoted in that plot corresponds to the period when the Higgs passed through the inflection point (see alsoFig.2)wherethepotentialisflatter.InFig.4theparameters

and

δ

areshownduringaperiodofinﬂation.Themainpointof thatplotistoshow that,becauseoftheinﬂection point [12] and thelargeinitialkineticenergy,theparameter

δ

isnotalwaysvery small,whichindicatesthatonecouldnotalwaysneglectthe iner-tialterminthe inflatonequation duringthe wholeinflation. The inflection point is reached atthe time when

δ

=

0 asit can be checkedbylookingatFigs.2and3.

Fig.3alreadyindicatesthatinﬂation isanattractorinthe crit-ical SM. We performed a more general analysis by varying the initialmomentum

inFig.5.There,aswellasinFigs.3and4,the

12 _In_that_plot

= −10−3_M_¯2

Pl,thereforethekineticenergy 2

/2 ismuchlarger thanthepotentialenergy,asitcanbecheckedbylookingatFig.2.

Fig. 5. InitialconditionsχandforthecanonicallynormalizedHiggsfieldχand itsmomentum≡ ˙χrespectively.Thevaluesoftheparametersthatarenotquoted intheplotarechosenasinFigs.1and2andasexplainedinthetext.Thevaluesof inflationaryparametersN e-foldsbeforetheendofinflationarealsoprovided(the valuesofN insidethebracketsindicateinsteadthetotalnumberofe-foldssince theearliesttime,whentheinitialconditionsχandaregiven).

initial conditionsfor

havebeenchosen tobe negative because positive valuesfavor inﬂationevenrespectto thecasewherethe initial kinetic energy is much smaller than the potential energy: thisisbecausethepotential,Eq. (7),isanincreasingfunctionof

χ

(for

χ

v). Weobservethat averylarge initialkinetic energy13 can be compensatedby avery modestincrease (not evenofone orderofmagnitude)intheinitialfieldvalueoftheHiggs.This con-firmsthatinflationisastrongattractorinthismodel.Thesituation issimilar(andevenslightlybetter)thentheoneofclassicalHiggs inflation[8] inthisrespect.Inthesameplotwealsoshowthatthe inflationary observablesns,r and PR arewithin theobservational bounds [11].

Therefore, the critical Higgs inﬂation does not suffer from a ﬁne-tuningproblemfortheinitialconditions.

4. Conclusions

InthispaperwehavestudiedwhetherHiggsinflation (HI) suf-fers froma fine-tuningof thehighenergyvalues ofthe parame-ters. Inparticular, it hasbeen investigatedthe dependence ofHI on theinitial (pre-inflationary) conditions.Inouranalysis we as-sumed a spatially homogeneous and isotropic geometrypointing out the naturalnessofthischoice. As shownin [8],although the large-ξ HI [1] doesnot sufferfromanytuning oftheinitial con-ditionsattheclassicallevel,atthequantumlevelafine-tuningof thehighenergyvaluesofsomerunningparametershastobe per-formed, asdiscussed at the end of Sec. 2.2. For this reason the mainfocusofthisarticlehasbeencriticalHI [5–7], whichallows a drastic decrease of

ξ.

Moreover, critical HI, unlike the large-ξ original version, has a single cut-off scale, M

¯

Pl, where quantum

gravity effectsare expectedto emerge, and is free from a much lowerscale,whereperturbativeunitaritytheorybreaksdown.

We pointedout thatcriticalHIdoesnot sufferfromany ﬁne-tuning ofthehighenergyparameters, suchasthe oneoflarge-ξ HInotedin [8].ThemainresultofthispaperwasthatcriticalHIenjoys

13 _In_that_plot_we_considered_values_of_up_to₋₀_.₀₅_M¯2

Plwhichcorrespondstoan

initialkineticenergydensityoforder10−3_M¯4

Pl.Weregardthisvalueasthemaximal

kineticenergydensityallowedtohavenegligiblequantumgravitycorrections(see also(24)).

(7)

arobustinflationaryattractor:evenstartingfromalargekineticenergy densityoftheHiggsfield(ascomparedtothepotentialenergydensity), theinflatonrapidlyreachestheslow-rollbehavior.

Acknowledgements

IthankJ.Garcia-BellidoandM.Shaposhnikovforuseful discus-sions.Thisworkwassupportedbythegrant669668–NEO-NAT– ERC-AdG-2014.

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Initial Conditions for Critical Higgs Inflation

Physics

Letters

B

Initial

conditions

for

critical

Higgs

inﬂation

✩

Alberto Salvio

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

ξ

ξ,

¯

×

¯

/ξ

¯

/ξ

ξ

¯

ξ.

=

√

−g

L

−

¯

+ ξ|H|





,

ξ

√

−

L

=

√

−

|∂

|

−

−

¯

+ ξ|

|





,

= λ(|

|

−

/2)

ξ >

ξ

|

|

→ 

,



₋

_/2)

→

=

=

₊