Exit from inflation with a first-order phase transition and a gravitational wave blast

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Exit

from

inflation

with

a

first-order

phase

transition

and

a gravitational

wave

blast

Amjad Ashoorioon

Institutionenförfysikochastronomi,UppsalaUniversitet,Box803,SE-75108Uppsala,Sweden

a

r

t

i

c

l

e

i

n

f

o

a

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a

c

t

Articlehistory:

Received27April2015

Receivedinrevisedform11June2015 Accepted11June2015

Availableonline15June2015 Editor:M.Trodden

Indouble-fieldinflation,whichexploitstwoscalarfields,oneofthefieldsrollsslowlyduringinflation whereastheotherfieldistrappedinameta-stablevacuum.Thenucleationratefromthefalsevacuum tothetrueonebecomessubstantialenoughthattriggersafirstorderphasetransitionandendsinflation. Werevisitthequestionoffirstorderphasetransitioninan“extended”modelofhybridinflation,realizing thedouble-fieldinflationaryscenario,andcorrectlyidentifytheparameterspacethatleadstoafirstorder phasetransitionattheendofinflation.We computethegravitationalwaveprofilewhichisgenerated duringthisfirstorderphasetransition.Assuminginstantreheating,thepeakfrequencyfallsinthe1GHz to10GHzfrequencybandandtheamplitudevariesintherange10−11_{ }

GWh210−8,dependingon

thevalueofthecosmologicalconstantinthefalsevacuum.Foranarrowbandofvacuumenergies,the firstorderphasetransition canhappenafterthe endofinflationviatheviolationofslow-roll,witha peakfrequency thatvaries from1 THzto 100THz. Forsmallervalues ofcosmologicalconstant, even though inflation can endvia slow-roll violation, the universe gets trapped ina falsevacuum whose energydrivesasecondphaseofeternalinflation. Thisrangeofvacuumenergiesdonotleadtoviable inflationarymodels,unlessthevalueofthecosmologicalconstantiscompatiblewiththeobservedvalue,

M_∼10−3_eV.

©2015TheAuthor.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3_.

1. Introduction

The mechanism that ends inflation is still an open question incosmology. In factit was the Achilles heelofthe old inflation model[1],astheuniversewouldneverrecoverfrominflation. Ac-celerated expansion of the inflationary false vacuumbackground pushesthewallsoftruevacuumbubblesexponentiallyapartsuch that theynever manageto coalesceandendinflation. New infla-tion [2], with the evolution of the inflaton, allowed for another mechanism of termination of inflation. Fast-roll evolution of the scalar field at the endof inflation can violate the slow-roll con-ditionandendinflation. Howeverthere still remainedtoaddress howtheuniversehasbeenreheatedfromthesupercooledphase. Forthat onehastwoassume thattheinflatoniscoupledtoother degrees of freedom and decay of the inflaton to these degrees offreedomhappenseitherperturbativelyornonperturbatively[3]. Forthistohappenoneoftenhastofine-tunethecouplingsorthe bare-massesofthesenewdegreesoffreedom.

E-mailaddress:amjad.ashoorioon@physics.uu.se.

End ofinflation through bubblenucleation hasthis advantage thatreheatingtheuniversehappensfromthecollisionoftrue vac-uum bubble wallsand their natural conversion to radiation. The ideasofnewandoldinflationwerecombinedin“double-field” in-flation[4,5]inwhichoneofthefieldsrollsduringinflation,asin slow-roll inflation,andthe secondfield isinitially trappedin the meta-stable vacuum(ascanbe seeninourexample,thisvacuum could betheonlyexisting minimuminthebeginningofinflation. Thetruevacuumcoulddevelop asinflationproceeds).Asthefirst field rolls, the nucleation rate from the meta-stable vacuum to the trueone becomes large enough that bubblesof truevacuum canindeedpercolateandendinflation. Thisisvery similartothe false-vacuumdominatedHybridinflation[6,7]wherethetachyonic instabilityofthewaterfallfieldisreplacedwithafirstorderphase transition.Aprototypeofsuchapotentialtakestheform

V

(φ, ψ )

=

V0

+

V1

(φ)

+

V2

(φ, ψ ),

(1.1)

where V0 is the vacuumenergy which is constant and

φ

is the

rollingfieldwhosepotentialV1

(φ)

alongwiththevacuumenergy

drivesinflation.

ψ

isthefieldwhichfacilitatesthefirstorderphase transitiontothetruevacuum.

http://dx.doi.org/10.1016/j.physletb.2015.06.022

0370-2693/©2015TheAuthor.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3_.

firstorderphasetransition,thespectrumhasaninvertedV-shape whichisdeterminedby thepeakfrequencyandtheamplitude at thepeak frequency. The peak frequency alsodepends onthe re-heatingtemperatureafterthefirstorderphasetransition,butifall theenergyinthefalsevacuumphaseisconvertedtoradiation,the peakfrequencyfallssomewhereintheGHzrange,whichisinthe frequencyrange probed by Chongqing University [13] butbelow itscurrentsensitivitylimitby10−2_–10−5_{.Nonetheless,itishoped}

thatfutureimprovementsofthe detectorsimprovetheir sensitiv-ityandclosethegapbetweenthepredictedsignalandthecurrent sensitivitylimit.Theshapeoftheproducedgravitationalspectrum canbeusedtodistinguishthemechanismofterminationof infla-tionfromtheparametricresonance[3]whichleadstogravitational wavespectrumwithdifferentprofile[14,15].

We also investigate the gravitational profile from inflationary modelsthatexitfrominflationaryphasethroughviolationof slow-rollbutaftertheterminationofinflationthenucleationratefrom

meta-stable vacuum to the true vacuum becomes large enough

thatfirstorderphase transitionoccurs.Ifthereheatingis instan-taneousandefficient,thepeak frequencyfromsuchphase transi-tionsliesintheteratopeta-Hertz bandanditsamplitudeismuch smallerthanthemodelsthatexitfrominflationthroughfirstorder phase transition. At the moment there is no planned probe that aimstothedetectionofsuchhighfrequencygravitationalwaves.

Inthiswork,wechosetherollingpotentialtobethequadratic potential for simplicity. Also, the large-scale predictions of the modelwerevery closetom2

_φ

2 _{predictionin then}

S

−

r planein thelimit ofvanishingvacuumenergy.Thispartofthepredictions ofthemodelwasstill withinthe 2

σ

limitofPlanck2013results

[16].AftertherevelationofPlanck2015results[17],theprediction ofthemodelnowfallsoutsidethe2

σ

region inthens

−

r plane excluding the running ofscalar spectral indexfrom the parame-ters.Nonetheless,theregionclosetothepredictionpointofm2

φ

2 is still within the 3

σ

region. Allowing for the running of scalar spectralindex, thisregioncomes back tothe2

σ

C.L. region.Itis expectedthattakingtherollingpotentialtobealowenergyscale model,likethehilltop model[18] ortheStarobinsky model[19], thepredictedvaluesforr forthe modelsexiting inflation witha firstorderphasetransitionhavea largeroverlapwiththe2

σ

re-gionofPlanck2015data.

Theoutlineofthepaperisasfollows.Firstwewillexplainthe setupthat canrealizetheidea ofdouble-fieldinflation.We iden-tifytheparameterspaceforwhichinflationcanendthroughafirst orderphase transitionandthen calculatethe power spectrumof gravitationalwavesproducedduring such phase transitions.Then weidentifytheregionofparameterspacewherephasetransition happensafter the termination ofinflation through slow-roll vio-lation.We alsocomputethegravitationalsignatureofsuch phase transitions.Attheend,weshowthattheuniversegetstrappedin themeta-stablevacuumifthevacuumenergyislessthana

thresh-inflation. For large values of

φ

, the potential has one minimum in both

φ

and

ψ

direction. As inflation proceedsand

φ

rolls to-warditsvacuum,secondminimumalongthe

ψ

directiondevelops if

γ

2

_>

₄

_αγ

_at

φ

_inflection2

=

M2

γ

2

₋

₄

_α

_λ

4

λ



λ

(2.2)

Thetwo minimaareseparatedby abarrierthat allowsforafirst order phase transition from the meta-stable vacuumto the true one ifthe nucleation rateis substantial andwhen it is energeti-callyfavorable.Mappingthepotentialto(1.1),V1

(φ)

isthevanilla

chaotic model m2

φ

2 and theexpression in the second lineplays the roleof V2

(φ,

ψ)

,whichcouples theinflatonto thefield that

creates the false vacuum in the

ψ

direction and facilitates the phasetransition.Inprinciple,V1

(φ)

determinesthepredictionsof

themodelatlargescales inthelimitingcasethatthe cosmologi-calconstantgoestozeroandcanbechosensuchthatthemodelis compatiblewiththeCMBobservablesatcosmologicalscales[16].

The probability of phase transition is givenby the nucleation ratedividedbythe4-dimensionalHubblevolume

p

=



H4 (2.3)

where



isthenucleationrate[20]



=

A

exp

(

−

SE

)

(2.4)

whereSE istheEuclideanfour-dimensionalactionforthesolution thatinterpolatesbetweenthetwominima.Forafirstorderphase transition witha quartic polynomial potential, the numerical re-sultswerefitby[21]tohavetheform

SE

=

4

π

2

3

λ

(

2

− δ)

−3

₍

_α

1

δ

+

α

2

δ

2

+

α

3

δ

3

) ,

(2.5)

where

α

1

=

13

.

832,

α

2

= −

10

.

819,

α

3

=

2

.

0765,and

δ

isa

func-tionof

φ

2,

δ

=

9

λ

α

γ

2

+

9

λλ



φ

2

γ

2_M2

.

(2.6)

The allowed range has 0

< δ <

2. Prefactor

A

has dimension mass4.In[8],thisprefactoristakentobeequaltothe cosmologi-calconstantinthefalsevacuum, λM₄4.Howeveraspointedoutby

1 _{ForthestatusofhybridinflationwithquadraticrollingpotentialafterPlanck} 2013datarelease,pleasesee[9].Inparticularthepaperdiscussesaninteresting scenarioinwhichthefirstpartoftherequirednumberofe-folds,neededtosolve theproblemsofBigBang,isprovidedfromthechaoticphaseandthesecondpart isresultedfromthevacuumdominatedphaseofhybridinflation.Themodelcan renderlowerscalarspectralindicesbutlargeramountoftensor-to-scalarratioin comparisonwiththepurelyquadraticpotential.

Fig. 1. The behavior of m (left) and nS(right) as a function of M in the region of parameter space that inflation ends with a first order phase transition.

Lindein[10],ifM2

ψ

>

2H2,thepre-exponentialfactor

A

iscloser to M4_ψ.As wewill see,uponthischange,theparameter spacein whichexitfrominflation can happenthrough afirst orderphase transitionexpandssubstantially.2

Forillustrationwefocusonthefollowingsetofparameters

α

=

0

.

01

,

λ

=

1

,

&

λ



=

1

.

(2.7)

Allmassiveparameters are inunitof Planckmass,mP

=

G−1/2

₌

1

.

2209

×

1019 GeV.Forthepotential (2.1)tohaveazerovacuum constantafterthephasetransition,thefollowingrelationbetween theparametersshouldhold

γ

2

=

−

α

3

₊

₉

_α

_λ

2

_{+ (}

_α

2

₊

₃

_λ

2

₎

3/2

3 (2.8)

Asin[8],wetaketherequirednumberofe-foldstosolvethe stan-dardBigBang cosmologyproblemstobe55.Inflationcanalsoend throughendofslow-rollwhichisparameterizedby



≡

m 2 Pl 16

π



_V 1 V1



2

=

m4

φ

2m2P

π

(λ

M4

₊

_2m2

_φ

2

₎

2

=

1 (2.9)

whichhasthebiggestsolution

φ

2

=

m2m2_Pl

+

mmP



m2_m2 Pl

−

8

π

λ

M4

−

4

π

λ

M4 8

π

m2

,

(2.10)

asthephysicallyviablesolution.

In order for inflation to end through phase transition rather thanslow-rollviolation,

φ

pt

> φ

 .Thenumberofe-foldingscould

beobtainedasafunctionoftheinflatonfield.Asstatedabove,we taketherequirednumberofe-foldsequalto55:

N

(φ

∗

, φ

pt

)

= −

8

π

m2_P φpt



φ∗ V1 V₁ d

φ

=

2

π

λ

M 4 m2_m2 P ln

φ

∗

φ

pt

+

2

π

m2_P

(φ

2 ∗

− φ

2pt

)

(2.11)

From theabove one canobtain

φ

_∗.The scalarspectral indexand thetensoroverscalarratiothencouldbeobtainedat

φ

∗ using

nS

−

1

= −

6



+

2

η

;

(2.12)

r

=

16



,

(2.13)

2 _IfM2

ψ<2H

2_{,theprefactorisoforderH}4_[11].

where



isgivenaboveand

η

≡

m2P 8

π

V V

=

m2m2_P 2

π

(λ

M4

₊

_2m2

_φ

2

₎

.

(2.14)

3. Endofinflationwithafirstorderphasetransition

Inthissectionwefocusontheregionofparameterspacewhere beforetheviolationoftheslow-rollparameter,thenucleationrate becomes substantialenoughthat leadstopercolationoftrue vac-uumbubblesintheseaofinflatingfalsevacuum.Weassumethat the pre-exponential factor,

A

is of order M4

ψ, as in [10], which makesourcalculationsdistinctfromtheanalysisof[8].Aswewill see,thisassumptionwillexpandtheregionofparameterspacein whichendofinflationhappensthroughafirstorderphase transi-tion. Thispart ofparameter spaceispart ofthe regionin which thevacuumenergycontributiontothepotentialiscomparableto the energy densityof the inflaton field

φ

. Detailed computation showsthatonlyintheregionwhere

4

.

97

×

10−4



M



2

.

66

×

10−3

,

(3.1)

first order phase transition precedes the slow-roll violation. For

M



2

.

66

×

10−6, there is no solution to the density perturba-tionamplitudenormalizationandhencenoviablemodel.Onecan matchtheamplitudeofdensityperturbations withtheCOBE nor-malizationwhich fixes themassparameter ofthe scalarfield,m.

IntheleftplotofFig. 1,wehavegraphedhowthemassparameter variesasafunctionofM.Theobtainedrangeofm is

1

.

24

×

10−6



m



6

.

33

×

10−6

.

(3.2)

Withintherange(3.1),nS andr varyinthefollowingranges

0

.

91



nS



0

.

97

,

(3.3)

0

.

15



r



0

.

55

,

(3.4)

asintherightgraphinFig. 1andleftplotofFig. 2.Wehavealso plotted the predictions of this region of parameter space in the

nS

−

r plane.As can be seen fromthe plots, withincreasing M, initially nS decreases andr increases.However there are turning points: around M



2

.

2

×

10−3_,n

S starts to increase andaround

M



2

.

4

×

10−3,r startstodecrease.Thelociofpredictionsofthe modelare designatedinthenS

−

r planeintherightplotofFig. 2. ForsmallvaluesofM,thepredictionsofthemodelsforlargescale fluctuationsare veryclosetothechaoticquadraticpotential.

Withthechoiceofparametersasin(2.7),thetrueminimumin the

ψ

directionappearsverylate,i.e. towardstheendofinflation.

Fig. 2. The behavior of r vs. M (left) and vs. nS(right) in the region of parameter space that inflation ends with a first order phase transition.

Thereis nominimumfor

ψ

that abubble oftruevacuumforms andleads toopeninflationaryscenario,asin[12].

If the energy stored in the inflaton potential is completely transformed to the radiation,i.e. reheating is instantaneous, one cancalculatethereheatingtemperatureattheendofinflation

T_∗

=



90H2_f 8

π

3_g ∗



(3.5)

where g_∗ is the total number of relativistic degreesof freedom attemperatureT_∗ which we take tobe g_∗



106. Wehave plot-tedT_∗ vs. M intheleftplotofFig. 3.Ifreheatingisnotefficient,

T_∗ is smaller than what is given in (3.5). This temperature de-termines the peak frequencyof the gravitationalwave spectrum generatedthroughthefirstorderphasetransition. Ifoneassumes instantreheating,thereheatingtemperaturevariesalmostlinearly intherangeof 1015GeV and1016GeV vs.M intherange(3.1).

3.1.Gravitationalwavespectrumfrommodelswithfirstorderphase transitionattheendofinflation

Sinceinflationsupercoolstheuniverse,onecanexploitthe for-malism of gravitational wave generation from first order phase transition at zero temperature. The numerical computations for bubblecollisionsfroma firstorderphasetransitionwere initially done by [24] for two bubbles andgeneralized for more bubbles in[25].The spectrumhasthe shapeofasymmetric ofinvertedV aroundapeakfrequency, fm,whichdecayslike f2.8 and f−1

re-spectivelyatsmallerandlargerfrequencies.Thepeakfrequency fm today,afterthepost-inflationaryredshiftingistakenintoaccount, isgivenby[24] fm

=

3

×

10−10



_g∗ 100

1/6



_T∗ 1 GeV

 

β

Hf



,

(3.6)

andtheamplitudeatthispeakfrequencyisgivenby



GWh2

(

fm

)

=

10−6



_g∗ 100

1/3



_Hf

β



2

,

(3.7) where

β

is

β

=

dSE dt

=

dSE d

φ

d

φ

dt

.

(3.8)

β

−1 isameasureofhowfastthefirstorderphasetransitiontakes tocomplete. Forour computations tobe reliable,we expect this

time tobemuchsmallerthantheexpansion rateoftheuniverse,

i.e.

β/

Hf



1,where Hf is theHubbleparameter atthetime of phase transitionwhich coincides withthe endof inflation. dφ_dt is thevelocityofthescalarfield,

φ

,whichcanbefoundduring infla-tionthroughthefollowingrelation[26]

˙φ

2

₌

2



(φ)

V1

(φ)

3

−



(φ)

(3.9)

where



isthefirstslow-rollparameter.Wehaveplottedlog



β Hf

vs. M intherange(3.1)ofM thatleadstoexitfrominflationwith a firstorder phasetransition. As can beseen fromthe rightplot ofFig. 3,forsmallervaluesofM intherange,phasetransitionis quite fastincomparisonwiththeexpansion rateofthe universe,

β/

Hf

∼

few

×

100.As M increases, phasetransitiontakeslonger to complete in comparisonwith the expansion time of the uni-verse.ForthemaximumvalueofM intherange,M

=

2

.

66

×

10−3,

β/

Hf

=

6

.

11, which is nonetheless fast enough to validate our computations.Theintensityofthegravitationalwavesatthepeak frequencyis onlydependent on thisparameter,

β/

Hf.The faster the phase transition, the smaller the amplitude of the produced gravitationalwaves.Wehaveplottedlog

(

GWh2

)

vs. M which in-creasesintherange(3.1)from1

.

07

×

10−11_to2

_.

₆₃

_×

₁₀−8_,seethe

rightplotinFig. 4.Howeverthepeakfrequency,besides

β/

Hf de-pendsonthereheatingtemperaturetoo.Assumingthatthe reheat-ingisinstantaneousandthetotalenergydensityofpotentialatthe endofinflation transformstoradiation,wehaveplottedthepeak frequency asa function of M inthe range (3.1),see left plot of

Fig. 4.ForthesmallestvalueofM intherange,M



5

×

10−4,peak frequency, fm is around 1

.

63

×

1010Hz whereas for the largest valueof M, fm



1

.

75

×

109 Hz.Such agravitationalwave profile liesin thefrequencyband ofChongqing High Frequency Gravita-tional (HFGW) probe but belowits currentsensitivity limit by a factorof10−2–10−5 [13].Itis expectedthatthe improvementof thedetectorinfutureclosethegapbetweentheexpectedsignals andthecurrentsensitivitylimit.

If the reheating coming from bubble wall collision is not in-stantaneous anda phase ofnon-radiation domination intervenes the end of first order phase transition and the radiation domi-nation, T_∗ willbe lowered.Ifthisefficiencyfactor isassumedto be

χ

, where

χ



1, the amplitude at the peak frequency will be lowered by a factorof

χ

4 _{[30]. For exampleif}

_χ

_

₀

_.

_01–0

_.

_1,

thefrequencyrangewillbeshifted withinthesensitivityband of Birmingham HFGW probe [27]. If the reheatingtemperature isa factorof

χ



10−7_–10−10_{smaller,thepeakfrequencywillshiftto}

Fig. 3. The behavior of T∗and_Hβ

f

vs. M in the region of parameter space that inflation end with a first order phase transition.

Fig. 4. The peak frequency of the gravitational wave spectrum and its amplitude as a function of M for models that exit inflation with a first order phase transition.

thesensitivitybandofDECIGO[28] andBBO[29].Howeverinall thesecases,theamplitude ofthesignalwillbeloweredsuchthat thesignalcouldnotbeobservedbyanyoftheseprobes.

With the choice of V1

(φ)

as the quadratic potential, most of the parameter space of inflationary models that exit from infla-tion with a first order phase transition is ruled out. There is a smallregionofparameterspacewhichhaspredictionsvery close to m2

φ

2. This region inparticular is ofinterest forthe CMB po-larization probes likethe BICEP2[22] orthefuture oneslikethe CMBPol[23].Sincethesemodelsexitinflation throughfirst order phase transition which is accompanied by bubble collision, they canleave anextra signatureofgravity wavesathigherfrequency scales. We expect that the predictions of the model for r could

beloweredifonewouldreplace V1 withanothermodel,likethe

hilltopmodel,[18],ormodelswithsmallerenergyscalesthat nat-urallypredictalowervalueforr.

4. Firstorderphasetransitionaftertheendofslow-rollinflation Itispossiblethateventhough inflationendsthroughviolation oftheslow-rollinflation, thefirstorderphase transitionhappens after the end of inflation, when

φ

becomes closer to the meta-stablevacuum,

φ

=

0.Infactif

3

×

10−6



M



4

.

88

×

10−4 (4.1)

phase transitionhappensafterthe endofslow-roll inflation. The lowerboundoftheaboveintervalisintriguinglyveryclosetothe

scalar fieldmassin therollingdirection. Tobeable todetermine thisrangeofM,wesolvedtheequationsofmotionandcomputed theevolutionofthescalarfieldafterinflationnumerically.3_As

be-fore,wehavetopickuponlythe

φ

solutiontothenucleationrate equation that 0

< δ <

2.We haveplottedthevariation ofm asa function of M in the range (4.2), see the left plot in Fig. 5. As-suming that thereheatingis instantaneous, wehave alsoplotted the reheating temperaturein thisrange of M, see the rightplot inFig. 5.Inbothcases,parametersm and T_∗ arealmost constant for smaller values of M inthe range,but they gradually rise as

M increases. As expected, contrary to the behavior of r, nS de-creaseswiththeriseofM, pleaseseeFigs. 5 and 6.Wehavealso plotted the behavior of nS vs. r as M increases, see Fig. 6. The behavior of log



_Hβ

f

vs. M, where Hf is the Hubble parameter when phasetransitioncompletescan beseen intheright plotin

Fig. 7.Ascanbeseenfromtheplot,withtheincreaseofM,phase transition becomes slower.Nonetheless, in general, phase transi-tions happen much faster after inflation incomparison with the inflationary models that endviafirst orderphase transition. This

3 _Intherange

4.88_×10−4_M4.97×10−4_{, (4.2)}

theconstraintsonthenumberofe-folds,Ne=55,withendofinflationgivenbythe

slow-rollviolationdidnotyieldarealsolutionform.Onecansatisfytheconstraint equationswithlessnumberofe-foldsthough.

Fig. 5. m (left) and nS(right) vs. M for the models for which first order phase transition happens after the end of inflation.

Fig. 6. r vs. M (left) and nS(right) for the models for which first order phase transition happens after the end of inflation.

Fig. 7. T∗(left) and logβHf 

(right) vs. M for the models for which first order phase transition happens after the end of inflation.

ispartlyduetothefactthat theHubbleparameterafterinflation ends,issmaller than itscorresponding value at theendof infla-tion.As M enhances inthe interval, (4.2),

β/

Hf decreases from 4

.

21

×

106 to 90321

.

6.We havecheckedthat inthisrangeof M,

theeffectivemassof

ψ

field,M2

ψ

>

2H2andthusphasetransition happensviaColeman-deLucciainstantontransitions[20].

Using Eqs. (3.6) and (3.7), we have calculated the peak fre-quency of the gravitational wave spectrum and its intensity,



GWh2,as a function of M, please see the plotsin Fig. 8.As M

increases, fm variesfrom1

.

8

×

1014 to 4

.

95

×

1012Hz,which is well outside thefrequency band ofany currentlyplanned probe. Of course this frequency range is obtained assuming that the reheating is instantaneous and efficient. If the reheating tem-perature is smaller than its instantaneous value by a factor of

χ



10−3–10−4,thepeak frequencyrangeismovedto the sensi-tivitybandofChongqinghighfrequencygravitationalwave probe.

Fig. 8. Thepeakfrequency(left)andtheamplitudeatthepeakfrequency(right)vs.M forthemodelsofinflationinwhichfirstorderphasetransitionhappensafterinflation.

However,theintensitywillthengetssuppressedfurtherbyafactor of10−12–10−16whichmakesthesignaltoosmalltobedetected. 5. Trappinginthemeta-stablevacuumandeternalinflation

Forsmallervaluesof M, M

<

3

×

10−6_{,the nucleationrateas}

givenby Coleman–DeLuccia transitionrate,is toosmalltoallow forthefirstorderphasetransitiontocompleteasthescalarfield,

φ

,passesthroughtheminimumofthepotential, V1.Graduallyas

theeffective massofthe

ψ

field decreases whilethe

φ

field ap-proachesthemetastablevacuum,thereisachancethatM2

ψ

≤

2H2. In particular thiscan happen for smaller values of

α

and M. In thissituation,Hawking–Mossphasetransition[31] appearswhere aninhomogeneousbubbles whoseradiusisgreaterthande-Sitter spaceradius, H−1,willformintheflatspace-time.Thetunneling probabilityperunit fourvolumeisoforder[31]



=

H4exp

(

−

B

),

(5.1) where B

=

1 8



1 V

(

0

,

0

)

−

1 V

(

0

, ψ

max

)

(5.2)

is the difference between the combined gravitational and scalar fieldactionofthe

ψ

= ψmax

,whichisanotherhomogeneous solu-tionapartfrom

ψ

=

0. V

(

0

,



max

)

isthelocalmaximumof V on

afour-sphere ofradius, H−_max1 ,where H_max2

=

8

π

V

(

0

,

ψ

max

)/

3.As

ψ

= ψmax

isunstable,thescalarfieldrunsdownhillfrom

ψ

max,to

theglobalminimum

ψ

min,afterwards. Forthepotential (2.1),one

cancalculatetheexponentB andshowthat

B



α

3

3

λ

2_M4 (5.3)

for

α

1.Thereforefor

M



α

3/4

31/4

_λ

1/2 (5.4)

theHawking–Mossphasetransitionratefromthemeta-stable vac-uumto thetruevacuumisverysmall.Inourexamplewherewe took,

λ

=

1 and

α

=

0

.

01,for M



6

.

08

×

10−5 such phase tran-sition takes a lot of time to complete and basically leads to a self-reproductionregimelikeoldinflation[1].AftertheHawking– Moss transition completes, the

φ

field will be homogeneous on scalaroforder H−₀1,with H0

=



8πV(0,0)

3m2 P

.Nonetheless,itwillbe stochasticallydistributedwiththedispersion[11]

φ

_rms2

≡ φ

2

 =

3H 4 0

8

π

2_m4

,

(5.5)

whichinourexampleisabout



10−5_{andthustoosmalltodrive}

anotherphaseofinflation.

In [8], the authors claim that they have been able to find a branch that corresponds tothe vacuum-dominatedregime of hy-bridInflation[7]intheExtendedHybridinflation, potential(2.1). Howeverinhybridinflation,thisregimeisobtainedassumingthat thetachyonicinstabilityinthewaterfallfieldendsinflation.Inthis case, howeverthewaterfield masssquared,

α

M2

_{+ λψ}

2 _isalways

positive andnever becomes tachyonic. For such energy scales in the vacuum dominated regime, the slow-roll can never get vio-lated as the first slow-roll parameter decreases asinflation pro-gresses.We alsoshowedthat Hawking–Moss phasetransitionfor such smallvacuum energiesis not substantialenough to end in-flation. Therefore, itis notpossible to realizethevacuumenergy dominated regime of hybrid inflation in such extended models. The scalar field gets trapped in the metastable vacuum with no gracefulexitandthereforethisregionofparameterspacedoesnot yieldaviableinflationarymodel.Ontheotherhand,fromthe phe-nomenological perspective,ifM isoforder



10−30,thisvacuum energycanbe responsibleforthecurrentaccelerationofthe uni-verse.

6. Conclusion

Terminatinginflationwithafirstorderphasetransitionhasthe benefit ofreheatingthe universe fromthe supercoolingphase of inflation through the collisions of bubbles of true vacuum with-out invoking and fine-tuning of the couplings of the inflaton to the otherfields.Onecan achieve thisscenario, modifyingtheold inflation scenario, with time-dependent nucleation rate which is small in the beginning and becomes substantial at the end of inflation. One specific realization ofthis scenariois extended in-flation[32],inwhichthegravity sectorofthe theoryismodified to Jordan–Brans–Dicketheory.Anotherwayofachievingthis sce-nario, is having two scalar fields, where one of the fields rolls and the other one is trapped in a meta-stable vacuum [4,5]. As the rolling field evolves, the nucleation rateat the false vacuum becomes large enough that the condition for percolationof true vacuumbubblesholdsandinflationends.Were-examinedamodel ofextendedhybridinflationwhichprovidesuswithsuch asetup. We noticedthat thepre-exponential factorinthe nucleationrate plays a crucial role incorrectlydetermining the parameterspace that allows for a first order phase transition in the model. For models that exit inflation with a first orderphase transition, we computedthepeak frequencyandtheamplitudeatthepeak fre-quency, which are respectively in the ranges fm



109–1010Hz and



GWh2



10−11–10−8.Thesignatureisinthefrequencyrange

thesignal ofsuch a phase transitionisweak andout ofthe fre-quencybandoffutureprobes.

Thelarge-scalepredictionsofthemodelareverydependenton thepotentialoftherollingfield.Herewiththechoiceofquadratic potential for the rolling field, the predictions of the model ap-proach thepredictionsofm2

_φ

2_inthen

S

−

r plane,whichwasstill withinthe2

σ

limitofPlanck2013results[16].Asthismodel pro-ducesgravitationalwavesat theCMBscales, the analyzedmodel producesthe observablegravitationalwavesatbothsmallandhigh frequencyrangeofthespectrum:thesmallfrequencygravitational waveshave quantum originbut the high frequency gravitational waveshaveclassicalorigin. Aftertheexposure ofPlanck2015 re-sults[17],major prediction of themodel is now outsidethe 2

σ

regioninthens

−

r planeexcludingtherunningofscalarspectral indexfromtheparameters.Theregionclosetothepredictionpoint ofm2

φ

2isstillwithinthe3

σ

regionthough.Ifoneallowsforthe runningofscalarspectralindex,thisregioncomesbacktothe2

σ

confidenceregion.Oneshouldalsonotethatthepredictionsofthe modelatlargescalesare verysensitivetotheinitialcondition for fluctuations.Asitwas shownin[35],choosingexcitedinitial con-ditionfor cosmologicalperturbations generally tends to suppress thetensor-to-scalarratio.

Itisexpectedthatifonetakestherollingfieldpotentialtobea lowerenergyscalemodel,likethe hilltop[18] ortheStarobinsky

[19]model,thepredictedvaluesforr forthemodelsexiting infla-tionwithafirstorderphasetransitionhavea largeroverlapwith the2

σ

confidenceregionofPlanck2015results.Thisissomething thatIwillpostponetoafuturepublication.

Asthepeakfrequencyofthegravitationalspectrum,produced fromthe first orderphase transition, depends verymuch on the Hubble parameter at the time of phase transition, it is also in-terestingtoinvestigateinflationarymodelsthatexitinflation with smaller Hubble parameter and, hence, peak frequency that falls within the sensitivity bands of BBO, DECIGO or even Advanced LIGO.

Acknowledgements

I am indebted to Andrei Linde for helpful discussions. I also thank A. Abolhasani, R. Allahverdi, M. Cortes, K. Freese and A. Lopezforcommentsanddiscussions.

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