Quantum phase of inflation

Arjun Berera*

School of Physics and Astronomy, University of Edinburgh, Edinburgh EH9 3JZ, United Kingdom

Raghavan Rangarajan†

Theoretical Physics Division, Physical Research Laboratory, Navrangpura, Ahmedabad 380009, India(Received 11 March 2010; revised manuscript received 16 November 2012; published 5 February 2013; corrected 13 February 2013)

Inflation models can have an early phase of inflation where the evolution of the inflaton is driven by

quantum fluctuations before entering the phase driven by the slope of the scalar field potential. For a

Coleman-Weinberg potential this quantum phase lasts 107–8 e-foldings. A long period of fluctuation

driven growth of the inflation field can possibly take the inflaton past ��, the value of the field where our

current horizon scale crosses the horizon; alternatively, even if the field does not cross ��, the inflaton

could have high kinetic energy at the end of this phase. Therefore, we study these issues in the context of

different models of inflation. In scenarios where cosmological relevant scales leave during the quantum

phase, we obtain large curvature perturbations of Oð10Þ. We also apply our results to quadratic curvaton

models and to quintessence models. In curvaton models we find that inflation must last longer than

required to solve the horizon problem, that the curvaton models are incompatible with small field inflation

models, and that there may be too large non-Gaussianity. A new phase of thermal fluctuation driven

inflation is proposed, in which during inflation the inflaton evolution is governed by fluctuations from a

sustained thermal radiation bath rather than by a scalar field potential.

DOI: 10.1103/PhysRevD.87.043514 PACS numbers: 98.80.Cq

I. INTRODUCTION

In the early stages of inflation the evolution of theinflaton field is dominated by quantum fluctuations ratherthan by the slope of the potential, V0ð�Þ. In the initial

stages of inflation, V 0 is small, and _� is not given by

�V 0=ð3HÞ but by the change inffiffiffiffiffiffiffiffiffiffih�2ip

due to quantumfluctuations leaving the horizon, freezing out, and becom-ing part of the large wavelength background condensate[1,2]. For a Coleman-Weinberg new inflation potential thisinitial phase can last for as long as 107–8 e-foldings, afterwhich V 0ð�Þ dominates over the fluctuations and one getsstandard evolution.

In this article we discuss the quantum evolution ofthe inflaton field in the context of new inflation, hilltopinflation, inflection point inflation, chaotic inflation, andnatural inflation, and then warm (new) inflation in the weakand strong dissipative regimes. A long quantum fluctuationdriven phase for the inflaton can drive it to the bottom of itspotential or past��, the value of the inflaton field when ourpresent horizon scale crosses the horizon. This will neces-sarily change our understanding of inflation. Alternatively,even if the inflaton does not cross ��, the inflaton kineticenergy at the end of the fluctuation driven phase can belarge, thereby precluding standard inflation thereafter.While the quantum phase of inflation has been known fora long time, these possibilities have not been discussedin the literature so far. The thermal fluctuation driven phase

is new and, for the first time, is being proposed in thispaper.We first consider new inflation with a quartic potential,

V ¼ V0 � 14��

4, which mimics the inflationary part of a

Coleman-Weinberg potential when the field is far from thepotential minimum [1,2]. We initially presume that ourcurrent horizon scale leaves the horizon during the stan-dard classical rolling phase (which fixes the coupling � tobe�10�14) and then check for the viability of this scenarioafter a long period of quantum evolution. We find that thefield is still high up on its potential at the end of thequantum phase, and its kinetic energy is not dominant.Therefore, the standard new inflation scenario can followan initial quantum fluctuation driven phase.We then carry out the same analysis of quadratic new

inflation, inflection point inflation, chaotic inflation, andnatural inflation. In chaotic, inflection point, and naturalinflation, we find that, unlike in new inflation, quantumfluctuations do not play a significant role in the evolution ofthe inflaton, i.e., classical evolution effectively dominatesfrom the beginning of inflation.We next consider scenarios where our current horizon

scale crosses the horizon during the quantum phase ofinflation. These are new scenarios that, to our knowledge,have not been considered earlier. We consider twoscenarios—�� >�Q >�e and �Q >�e, where �Q is

the value of the inflaton at the end of the quantum phaseand �e is the value of the inflaton field at the end ofinflation. We do this for new inflation and hilltop inflationmodels. The expressions for �� must be rederived forthese scenarios in which the current horizon scale leaves

*ab@ph.ed.ac.uk†raghavan@prl.res.in

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during the quantum phase. Since _� in is not given by

_� ¼ �V 0=ð3HÞ but byffiffiffiffiffiffiffiffiffiffih _�2i

q, the expression for the

comoving curvature perturbation, Rk, must also berederived.

For quartic new inflation models and �� >�Q >�e,

we find that consistency of the scenario requires the cou-pling � to be greater than�10�2. But this then implies thatRk � 13 for modes leaving during the quantum phase.Thus, this scenario is ruled out. This scenario for quadraticnew inflation and hilltop inflation is also ruled out due toconflict with the observed density perturbations.

For the scenario where the quantum phase lasts theentire inflationary epoch, i.e., �Q >�e, we find that it

requires a very large value of H �MPl in new inflationmodels. This scenario also generates too large curvatureperturbations.

We then apply our results for modes leaving during thequantum phase to quadratic curvaton models [3,4], quin-tessence models, and to a new model of dark energy [5]where the dark energy is a condensate of quantum fluctua-tions generated during inflation of a very light field. Ifquantum fluctuations of the curvaton determine the valueof the curvaton field at the end of inflation, then we findthat the duration of inflation must be orders of magnitudelarger than the usual number of e-foldings required to solvethe horizon problem. We also find that the slow rollparameter �H is determined in these models and has avalue larger than those compatible with small field inflationmodels such as new inflation, small field natural inflation,and some hybrid inflation models with a concave down-ward potential. In the context of certain alternate expres-sions in the literature for quantum fluctuations duringquadratic chaotic inflation, the non-Gaussianity in thecurvaton models increases to levels in conflict with obser-vations. In the dark energy model we investigate whetherthe large curvature perturbations associated with the darkenergy condensate during inflation are compatible with thecosmic microwave background (CMB) constraints on thetotal curvature perturbation at decoupling.

We then consider the above issues in the warm infla-tionary scenario in which the continuous decay of theinflaton during inflation creates a thermal bath whichsurvives throughout the inflationary phase [6]. In warminflationary dynamics, h�2i can grow initially due to fluc-tuations of the inflaton field in this thermal background.Westudy the quartic new inflation potential in the context ofboth weak dissipation and strong dissipation of the inflatonfield. We investigate whether the inflaton reaches the mini-mum of its potential during the fluctuation driven phaseitself. We find that in the weak dissipative regime thereis no thermal fluctuation dominated phase. Therefore, thepotential driven phase is preceded by a quantum fluctuationdriven phase, as in cold inflation. In the strong dissipativeregime there is a thermal fluctuation driven phase whichlasts for 108 e-foldings. This is a new phase of warm

inflation not considered earlier. The further requirementthat the thermal fluctuations do not take the field to thebottom of the potential requires that the scale of inflation isless than 1014 GeV. This is two orders of magnitude lessthan earlier constraints [7]. For both dissipative regimes,after the fluctuation driven phase is over, standard warminflation can follow, indicating the consistency of the warminflation scenario.Below we summarize the paper. In Sec. II we investigate

the consistency of cold inflation models preceded by aquantum phase. In Secs. III and IV we consider scenariosin which the current horizon scale leaves the horizonduring the quantum phase of inflation. In Sec. V we studythe quantum phases of curvaton and quintessence fields. InSec. VI we apply our ideas to warm inflation and inves-tigate a thermal fluctuation driven phase. Section VII con-tains our conclusions.

II. COLD INFLATION

We first present the relevant equations for studying thequantum evolution phase in cold inflation models. Thereare three scenarios that we consider: (i) the quantum phaseends before our current horizon scale crosses the horizon attime t�, (ii) the quantum phase ends after t� but before theend of inflation at te, and (iii) the quantum phase lasts forthe entire duration of inflation. In the first scenario ourfocus is on determining whether the inflaton ends thequantum phase with conditions suitable for a subsequentperiod of classical inflation (thereby confirming the valid-ity of existing models of inflation which allow a quantumphase). For the other two cases, we investigate (in Secs. IIIand IV) the feasibility of models where the current horizonscale leaves during the quantum phase

�Q <��:

During the quantum evolution phase in cold inflation,field fluctuations about an initial value �0 grow as

h��2i¼ 1

ð2�Þ3Z aH

Hd3kj�kj2¼

�H

2�

�2�NðtÞ; (1)

where k is the comoving momentum, NðtÞ ¼ Hðt� t0Þ isthe number of e-foldings since the beginning of inflation att0, and we have only integrated over modes outside thehorizon which can act as part of the homogeneous back-ground field. For small field inflation models we ignore anyinitial �0.

1 (We have also assumed an effectively masslessfield, meff � H.)

1For standard classical inflation one argues that, due to quan-tum fluctuations, the initial value of the inflaton should not beless than H=ð2�Þ. This argument is not relevant when one isstudying quantum fluctuation driven evolution. Nevertheless, forsmall field models, presuming �0 >H=ð2�Þ, we impose �� >H=ð2�Þ. �0 actually depends on the inflaton dynamics as theUniverse approaches the inflationary epoch.

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Therefore,

� ¼ �0 � ðH=2�ÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiHðt� t0Þ

q(2)

(we take �0 > 0), and _� during the quantum phase is thengiven by Eq. (8.3.12) of Ref. [1] as

_�q ¼ H2

4�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiHðt� t0Þ

p : (3)

The period of quantum evolution lasts while

j _�qj � j _�Vj ¼��������� V 0

3H

��������; (4)

that is, for values of � satisfying

3H4

8�2� j���0ÞjjV0ð�Þj: (5)

As we shall see, this condition is satisfied only for smallfield inflation models such as new inflation.

To confirm that the Universe is potential energy domi-nated during and at the end of the fluctuation driven epoch,

we compare h _�2i with Vð�Þ. Now

h _�2i ¼ 1

ð2�Þ3Z aH

Hd3kj _�kj2: (6)

Ignoring the gradient (k2=a2) term in the equation ofmotion for �k,

€�k þ 3H _�k þ V00ð�Þ�k ¼ 0 (7)

and

_�k � �V 00ð�Þ3H

�k: (8)

Then

h _�2i ¼�V 00ð�Þ3H

�2 1

ð2�Þ3Z aH

Hd3kj�kj2 (9)

¼�V 00ð�Þ3H

�2h��2i: (10)

We now apply the above results to specific models ofinflation.

A. Quartic new inflation

For a Coleman-Weinberg potential, modelled as V ¼V0 � �

4�4 for small �, the quantum phase occurs for

�<�Q, where, using Eq. (4) and Eqs. (2) and (3),2

�Q ¼ H

2�

�60

�

�14

(11)

for �0 � 0. Using Eq. (2), the number of e-foldings until

tQ is NQ HðtQ � t0Þ ¼ 8=ffiffiffiffi�

p. For the scenario with

�Q <��, �� 10�14 for grand unified theory (GUT)

scale inflation, and one can see that NQ � 108, i.e., 108

e-foldings occur before one gets to classical evolution inthis new inflation model, �Q � 103H. For �Q <�� the

number of e-foldings of inflation after the inflaton fieldcrosses �� is given by Eq. (8.62) of Ref. [8] as

N ð�� ! �eÞ ¼ 8�

M2Pl

Z �e

��

V

�V0 d� (12)

¼ 3H2

2�

�1

�2�� 1

�2e

�; (13)

and taking �e � ðV0=�Þ1=4 we get �� � 106H, which islarger than �Q as presumed. (Typically, one uses the slow

roll condition jV00j � 9H2 to obtain �e ¼ ð3=�Þ1=2H [8].

But then Vð�eÞ< 0. We instead take �e � ðV0=�Þ1=4.)Vð�QÞ ¼ V0 �H4=�4, and with V0 � 0:1H2M2

Pl,

Vð�QÞ � V0. Using Eq. (10), h _�2i ¼ �2�6=H2, and for

� �Q, h _�2i � V0, i.e., conditions for inflation remain

valid during the quantum phase and at the end of thequantum phase.

The quantum evolution is statistical, and thus � ¼ðH=2�Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Hðt� t0Þp

reflects the magnitude of the displace-ment due to quantum fluctuations averaged over many

Hubble volumes. Even for the case where j _�qj � j _�V jon average, there will be some regions where the classicaldynamics dominates the quantum. However, the number ofsuch regions will be relatively small.

B. Quadratic new inflation

We now consider a quadratic model of new inflation. Forquadratic new inflation with V ¼ V0 �m2�2=2, the slow

roll conditions jV 00j � 9H2 and jV 0MPl=Vj � ð48�Þ1=2[8] give m2 � 9H2 and � � ðH2=m2ÞMPl. We take

�e ¼ V12

0=m, as in Ref. [9]. From Eqs. (2)–(4), �Q ¼0:2ðH=mÞH. Then from Eq. (2), NQ ¼ 1:6H2=m2.

For quadratic new inflation the WMAP bounds on1� ns imply GUT scale inflation [9]. Then for H ¼10�6MPl, N ¼ 60, and using Eq. (12)

�� ¼ �ee�20m2=H2

: (14)

The condition that �Q <�� then gives m=H < 0:8.

Setting Rk¼H2=ð2� _�Þ� ¼5�10�5 and using _� ¼�V 0=3H, one gets m=H¼0:03 or 0.35. But 1�ns¼2m2=ð3H2Þ [9] and WMAP observations imply that

2Our value of �Q differs slightly from that obtained inRef. [1], possibly because of some factors in Eq. (8.3.12) ofRef. [1].

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0:025< 1� ns < 0:049 (at 68% C.L.) [10], thereby allow-ing only m=H ¼ 0:03. Then �Q ¼ 7H and NQ ¼ 2000.

Using Eq. (10) we can also note that Vð�QÞ � V0 � h _�2iduring the quantum phase, and so the Universe is poten-tial energy dominated during and at the end of the fluc-tuation driven phase. Thus, the scenario with �Q <�� isconsistent.

C. Other models of inflation

We have also investigated the quantum phase of inflationfor inflection point inflation, chaotic inflation, and naturalinflation. A quantum phase during inflection point inflationabout the saddle point ��0 is mentioned in Refs. [11,12](though the criterion for quantum evolution is a bit differ-ent than ours), and it is assumed that the field is out of therange for quantum evolution for the cosmologically rele-vant phase of inflation. We did not consider this scenariofurther. The existence of the saddle point requires a certainfine-tuned relation between parameters in the Lagrangian.References [13,14] consider deviations from this relation.In Ref. [14] the deviation is parametrized by a variable�. For weak scale supersymmetry withm�100–1000GeV,one has ��0 � 1014–15 GeV and V0 � 1032–34 GeV4, andCMB observations require ��Oð10�10Þ [14]. (Also seeRefs. [12,13].) For such values the quantum phasecondition in Eq. (5) with �0 replaced by ��0 is valid onlyfor j�� ��0j & 10�6 GeV, whereas H � 10�2;�3 GeV.Therefore, one can ignore the quantum phase for thesemodels, as the quantum phase range is much smaller thanH. Reference [14] also considers scenarios without fine-tuning of �, but again j�� ��0j � H.

For quadratic and quartic chaotic inflation there is noquantum phase of inflation for �<�QG where �QG is the

value of the � for which quantum gravity effects becomeimportant (Vð�QGÞ ¼ M4

Pl). Natural inflation with a poten-

tial of the form V ¼ �4½1þ cosð�=fÞ� [15,16] where �lies between 0 and 2�f, and we take�<�f, also does nothave a quantum phase of inflation (unless the field value isextremely small, �=f < 10�6).

D. The quantum condition

Another approach to identifying the quantum phase is tocompare the evolution of the inflaton due to quantumfluctuations and classical slow roll at an instant in time,over a time interval �t ¼ H�1 [1,17], i.e., to check if

��q ¼ H

2�> ��V ¼ _�VH

�1 (15)

¼ V 0ð�Þ3H2

: (16)

This approach checks if instantaneously the quantumjump overrides the classical evolution, while the condition

in Eq. (2) is a measure of whether quantum evolutiondominates over classical evolution averaged overlonger time durations. One notices that the condition in

Eq. (15) is similar to that for eternal inflation K ��q=½0:61 _�VH

�1� * 1 [18].

Comparing the above approach and that in Eqs. (2) and(3), we see that the quantum evolution of the field in

Eq. (2) goes as �qðtÞ �ffiffiffiffiffiffiffiffiffiffiffiffiN ðtÞp ¼ ffiffiffiffiffiffi

Htp

, which means_�q � 1=

ffiffiffiffiffiffiffiN

p � 1=ffiffit

p. In other words, _�q decreases over

time. Thus, if one is comparing this motion with theclassical motion, and if the latter has approximately con-stant velocity as expected in a slow roll regime, then itimplies that at late time eventually the classical motionalways dominates. Since the quantum kicks on the scalarfield are a random process, it might seem that there shouldbe no dependence on the past history of the quantumevolution that should enter in comparing whether quantumor classical motion dominates at a given time. In fact at anygiven instant the root-mean-square (RMS) quantum kick inEq. (15) is of order H, and so the RMS velocity at anyinstant from quantum kicks is of order H2. So from thispoint of view, one might think the relevant quantities tocompare is whether this RMS velocity from quantumevolution, which is independent of past history, is thecorrect quantity to compare against the classical velocity,as in Eq. (15). However, the history of the evolution shouldenter the comparison. Even though at any given instant, thevelocity from quantum kicks is some approximately con-stant value, the direction is random. As such, with increas-ing time steps, there is an ever increasing number ofpossible paths that the � field could have followed. Ifone asks what is the net motion from the quantum kicksafter some interval of time, on average it increases only asffiffit

p. So a time derivative over an increasing time interval

actually is decreasing as 1=ffiffit

p. On the other hand, the

classical evolution of the field, even if slow, is steadilyalways moving in the same direction. As such, the longeronewaits, the greater the chance that the� field has arrivedat increasing field values due to classical rather than quan-tum evolution.

III. �e > �Q > ��We now consider the scenario where the quantum fluc-

tuations take the inflaton beyond ��, and so our currenthorizon scale leaves the horizon when inflaton evolution isdominated by quantum fluctuations. In this section weconsider the scenario in which the fluctuations do nottake the inflaton past �e. Since in the previous sectionwe found that the quantum phase is important only forsmall field inflation models, here we investigate only newinflation and hilltop inflation models.To obtain �� we break the evolution from �� to �Q and

from �Q to �e. Then

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N ð�� ! �eÞ ¼Z te

t�Hdt ¼

Z �e

��Hd�_�

(17)

¼Z �Q

��Hd�_�þZ �e

�Q

Hd�_�

(18)

¼ 8�2

H2

Z �Q

���d�þ 8�

M2Pl

Z �e

�Q

Vð�Þ�V0ð�Þ d� (19)

¼ 4�2

H2ð�2

Q ��2�Þ þ 8�

M2Pl

Z �e

�Q

Vð�Þ�V 0ð�Þd�; (20)

where we have used Eqs. (2) and (3) for _� for �<�Q.

The comoving curvature perturbation is given by

R ¼ c com ¼ c þH��; (21)

where c is the curvature potential, �� ¼ ��= _� is the timedisplacement to go from a generic slicing with generic ��to the comoving slicing with ��com ¼ 0 [19]. Evaluatingthe rhs in the flat gauge in which c ¼ 0 gives

Rk ¼ H��k

_�: (22)

We first evaluate the rhs at horizon crossing. In the flat

gauge ��k is given byH=ð2�Þ. _� represents the motion ofthe background condensate of long wavelength modes.

Therefore, _� is given by Eq. (10). Then

Rk ¼ 3H3

2�jV 00ð��Þj��; (23)

where �� is given by the value at horizon exit obtainedfrom Eq. (20). (The sign of Rk is suppressed, as therelevant quantity is the 2-point function �jRkj2.)

While considering times after horizon exit the back-ground should only include those superhorizon modesthat are of longer wavelength than the mode being inves-

tigated. Therefore, after horizon exit both ��k and _� inEq. (22) evolve as in the standard classical roll picture, andone gets constancy of Rk outside the horizon as usual.(Quantum evolution of � will continue as more and moremodes cross the horizon and add to the condensate offluctuations. But the condensate value of �� relevant inEq. (23) will not include modes that leave the horizon afterthe mode being investigated.)

We point out an interesting fact that both ��k and _� inRk are quantum variables in the quantum phase of infla-tion. Therefore, in the quantum phase we expect that therewill be an increase in the cosmic variance because of thestochastic nature of the evolution of �.

A. Quartic new inflation

For quartic new inflation Eq. (20) implies

�2� ¼ �2Q �

�NH2

4�2� 3H2

2ð60�Þ12 þ3

8�2�12

H4

V12

0

�: (24)

The third term in the square brackets is larger thanthe second only for H > 2MPl. Note that limits such as

V1=40 < 1016 GeV and H < 1013 GeV are derived for

classical inflation and not valid in the quantum phase.Nevertheless,H > 3MPl implies Vð�Þ>M4

Pl which would

put us in the realm of quantum gravity. We take H <MPl

and ignore the third term in the square brackets above.To check for consistency of this scenario, the quantity

in square brackets should be positive. We also impose�� >H=ð2�Þ. These conditions give a lower bound andan upper bound on � respectively, namely,

60

N 2< �<

240

ðN þ 1Þ2 : (25)

(Imposing �Q <�e only gives a weak bound H < 2MPl.)

For GUT scale inflation with N � 60 this implies

0:02< �< 0:06: (26)

The cosmologically relevant scales leave inflation overeight e-foldings. For all these scales to leave during thequantum phase for GUT scale inflation, the condition is0:02< �. For inflation models with scales varying fromthe electroweak scale to the Planck scale, N ranges from30–65, and we get a lower bound of ð1� 7Þ � 10�2 for �for all these models. Clearly these values of � are interest-ing, but one has to also check for constraints from the thepower spectrum.From Eq. (23)

Rk ¼ 1

2�

H3

��3�: (27)

Using the expression for �� in Eq. (24) in Eq. (27) we get

Rk ¼ 4�2

�½2ð60� Þ12 �N �32 : (28)

For GUT scale inflation with N ¼ 60 we take� ¼ 4� 10�2 consistent with Eq. (26) and get

Rk � 13: (29)

Such a large value of the curvature perturbation is ruled outby observations. Therefore, this scenario, despite its mildconstraint on the inflaton coupling, namely, �� 10�2, isnot feasible.

It may be surprising that though in the quantum phase _�is larger than the classical value, one gets such a large value

of Rk. However note that while _�q � _�V ,ffiffiffiffiffiffiffiffiffiffih _�2i

q, which

enters the expression for Rk, is 3 _�V . Now the expression

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for Rk for classical evolution, using Eq. (22) and_�V ¼ �V 0=ð3HÞ, is

Rk ¼ 1

2�

3H3

��3�: (30)

Comparing this with Eq. (27) one sees that the expressionsare similar. Yet, though � in Eq. (27) is large compared to �in Eq. (30), Rk;q � Rk;cl because it happens that �� is

much smaller than in the standard classical evolution sce-nario. As shown earlier, the value of �� for the classicalcase for � ¼ 10�14 and N ¼ 60 is ��;cl ¼ 106H.

However, for the value of � ¼ 4� 10�2 and N ¼ 60one gets, from Eq. (24), ��;q ¼ 0:7H.

To understand why �� is much smaller for the quantum

evolution scenario, note that (i) �e � 1=ffiffiffiffi�

pis much

smaller in the quantum scenario than for the classical

scenario as � is large, and (ii) a larger _�q in Eq. (18)

requires a larger displacement in � to obtain the requirednumber of e-foldings. Both these effects push the inflatonto a much smaller value of� at the time of horizon exit fora given value of N .

B. Quadratic new inflation

For quadratic new inflation Eq. (20) implies

�2� ¼ �2Q �

�NH2

4�2� 3H4

4�2m2ln

��e

�Q

��: (31)

Using expressions for �e and �Q from Sec. II B we get

�2� ¼ �2Q �

�NH2

4�2� 3H4

4�2m2ln

�1:7MPl

H

��: (32)

The constraint that �Q >�� implies

m2

H2>

3

Nln

�1:7MPl

H

�; (33)

while the requirement that �� >H=ð2�Þ implies

1:6

N þ 1þ 3

N þ 1ln

�1:7MPl

H

�>

m2

H2: (34)

Consistency of the above constraints implies

H > 1:7MPle�Nþ1

2 : (35)

For GUT scale inflation with N ¼ 60 this requiresH > 106 GeV.

To obtain the curvature perturbation we use Eq. (23)with �� given by Eq. (31).

Rk ¼ 3

2�

H3

m2��: (36)

Now for �Q >�� the curvature perturbation satisfies

Rk >3

2�

H3

m2�Q

¼ 2H

m: (37)

For this to be consistent with observations would requirem=H > 105. But such a large value of m=H is in conflictwith the upper bound in Eq. (34). Therefore, this scenariois not feasible. (For electroweak scale inflation the lowerbound on H is too large [3� 1012 GeV], and so we do notconsider it.)

C. Hilltop inflation

For the potential

V ¼ V0 � ��p

pMp�4p > 2; (38)

N is obtained from Eq. (20) as

N ¼ 4�2

H2ð�2

Q ��2�Þ þ 8�

M2Pl

V0Mp�4

�ðp� 2Þ�

1

�Qp�2

� 1

�p�2e

�;

(39)

and so

�2� ¼�2Q�

�NH2

4�2� H2

4�2

8�

M2Pl

V0Mp�4

�ðp�2Þ�

1

�p�2Q

� 1

�p�2e

��:

(40)

Using H ¼ ½ð8�=3ÞV0=M2Pl�1=2

1

�p�2Q

� 1

�p�2e

¼ 1hH2

M2Pl

V0Mp�4

�

ip�2p

� 1hV0M

p�4

�

ip�2p

: (41)

Considering H2 � M2Pl we ignore 1=�

p�2e . For p > 4 and

H � M [the potential in Eq. (38) is an effective potentialfor �<M, and we only consider �>H=ð2�Þ], one getsfrom Eq. (23) (using Mathematica)

Rk ¼ 0:51

�1p

�M

H

�p�4p; (42)

which is greater than 1.

IV. �Q > �e

We now consider the scenario where evolution in theinflationary phase is entirely dominated by quantum fluc-tuations, i.e., �Q >�e. Then

N ð�� ! �eÞ ¼ 4�2

H2ð�2

e ��2�Þ; (43)

and so

�� ¼��2

e � N4�2

H2

�12: (44)

A. Quartic new inflation

For quartic new inflation, as discussed earlier, �e �ðV0=�Þ1=4. The condition �Q >�e implies that

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H > 1:8MPl: (45)

(This is close to the limit H < 3MPl for classical gravity tobe valid.) From Eq. (44) �2

e >NH2=ð4�2Þ. [Imposing�� >H=ð2�Þ gives a similar bound with N ! N þ 1.]For Planck scale inflation we take N ¼ 65. This thenimplies

H <0:02�2ffiffiffiffi

�p MPl: (46)

Combining the above two bounds one finds that for infla-tion to be in the quantum phase during its entire durationrequires a large Hubble parameter during inflation and onlya reasonable upper bound on �, i.e.,

H > 1:8MPl and � < 1� 10�2: (47)

The comoving curvature perturbation is given by Eq. (27)with �� as in Eq. (44). Once again we obtain a large valueof Rkð>17Þ which rules out this scenario.

B. Quadratic new inflation

The condition that �Q >�e implies

H > 1:7MPl (48)

and the condition that �2e >NH2=ð4�Þ in Eq. (44)

implies that

3

2NM2

Pl >m2: (49)

For Planck scale inflationN ¼ 65, and 0:15MPl >m. Thecurvature perturbation in Eq. (36) satisfies

Rk >3

2�

H3

m2�e

: (50)

Using Eqs. (48) and (49) we get Rk > 26 which is inconflict with observations.

V. OTHER SCALAR FIELDS

A. Curvaton

We now consider scenarios where a field other than theinflaton evolves due to its quantum fluctuations duringinflation. In the curvaton scenario quantum fluctuationsof a field � with a flat potential are responsible for thedensity perturbations in the Universe [3,4]. (Also seeRefs. [20–23].) The curvature perturbation in these scenar-ios is given by

¼ 4rr þ 3��4r þ 3�

(51)

� r�; (52)

where the subscript r refers to radiation, and the variable ris the ratio of the curvaton energy density � to the total

energy density just before the curvaton decays. Onepresumes that r is negligible, and it is ignored.The curvature spectrum due to the curvaton field is

obtained from the expression for � at the time when thecurvaton starts to oscillate after inflation when H falls tom� at tosc.

�;k ¼ 2

3

��k

�

��������osc: (53)

Since ��k is constant outside the horizon, we take��kðtoscÞ to be H=ð2�Þ. We take �ðtoscÞ to be the value�e at the end of inflation at te. �e ¼ �clðteÞ � ��qðteÞ. If�e is determined by quantum fluctuations then

� ¼ 2

3

��k

��ðteÞ : (54)

For a curvaton mass m�, ��ðteÞ is either ½H=ð2�Þ�N12e or

3H4=ð8�2m2�Þ, depending on whether the condensate of

curvaton fluctuations does not or does attain the asymptoticvalue by te. (Ne is the total number of e-foldings ofinflation.) If it does not, then

� ¼ 2

3

1

N12e

: (55)

Now � ¼ =r where ¼ 5� 10�5, and 10�2 < r < 1.(The lower limit on r comes from the relation between rand the non-Gaussianity parameter fNL, namely, fNL ¼5=ð4rÞ [4], and the upper limit on fNL ofOð100Þ [24].) Thisthen implies

2� 104 <Ne < 2� 108: (56)

Therefore, curvaton models where quantum fluctuationsdetermine �e and where �� has not reached the asymp-totic value by te require more e-foldings of inflation thanstandard models of inflation. For quartic new inflationmodels the upper limit on Ne is of the same order as theduration of the quantum phase.The condition that the curvaton has not reached its

asymptotic limit (for a quadratic curvaton potential) is [25]

2m2�

3H2Ne � 1: (57)

Combining this with the lower bound on Ne above gives

m2�

H2� 8� 10�5: (58)

Now the scalar power spectrum spectral index in the cur-vaton scenario is [4]

ns ¼ 1� 2�H þ 2���; (59)

where �H ¼ � _H=H2 and ��� ¼ M2Pl=ð8�Þ @

2V=@�2

V . For a

quadratic curvaton potential, ��� ¼ m2�=ð3H2Þ. Then for

ns ¼ 0:963 [24]

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m2�

3H2� �H � 0:02: (60)

Therefore, the above upper bound on m2�=H

2 furtherimplies that in quadratic curvaton models where quantumfluctuations determine �e and �� has not reached theasymptotic value by te, �H ¼ 0:02.3 Small field inflationmodels such as new inflation, small field natural inflation,and some hybrid inflation models with a concave down-ward potential would not satisfy this criterion [26,27].

If Ne � ð3H2Þ=ð2m2�Þ one uses the asymptotic value for

the curvaton fluctuations in Eq. (54). Then using theobserved value of and the bounds on r, one finds

9� 10�5 <m�

H< 9� 10�3: (61)

This is in conflict with the bound m�=H * 10�4 obtainedin Sec. IIIB of Ref. [28] from CMB constraints. (Webelieve there is an error in the derivation of the bound inRef. [28].) Once again, Eq. (60) implies that �H ¼ 0:02,which is incompatible with small field inflation models.The lower bound on H=m� from Eq. (61) and the lowerbound on Ne above also implies that inflation must lastmuch longer than 104 e-foldings.

If �e is determined by �clðteÞ rather than by quantumfluctuations, then ��qðteÞ � �e and

� � 2

3

��k

��ðteÞ ; (62)

and one can only conclude that Ne < 2� 108 andm�=H > 9� 10�5, respectively, in the scenarios wherethe curvaton fluctuations do not and do attain their asymp-totic value.

As an aside, we comment on certain existing results inthe literature which may be relevant for the curvatonscenario. The asymptotic value of the quantum fluctuationsfor the curvaton

h�2i ¼ 3H4

8�2m2�

(63)

may be used in Eq. (53) as in Refs. [28]. However, it maybe shown (via Eq. (4.11) of Ref. [29], Eq. (13) of Ref. [30],and Eq. (14) of Ref. [31]) that on including the timevariation of H during quadratic chaotic inflation, theasymptotic value of the fluctuations during inflation ofany light scalar field goes as

h�2i ¼ 3H4

16�2m2; (64)

where m is the inflaton mass. If one uses Eq. (64) thenone gets

� ¼�2

3

�32 m

H� 1:6� 10�2; (65)

which would need r � 3� 10�3 to agree with theobserved density perturbations. Such a small value of rwill imply fNL � 400 which is in conflict with boundson fNL [24]. In models with nonquadratic terms in thecurvaton potential, fNL can be small even when r is small[32–35] because the coefficient of the term proportional to1=r in fNL in the presence of nonquadratic terms can bevery small. These models are then not ruled out (thoughRef. [35] indicates that very small values of r are notfavored).

B. Quintessence

In quintessence models the dark energy is associatedwith a slowly rolling background field Q. Since the poten-tial must be flat so that potential energy dominates thekinetic energy (and the pressure is negative), we verifywhether quantum evolution can dominate over the slowclassical evolution. Consider the potential

V ¼ V0

�MPl

Q

�p: (66)

The condition in Eq. (4) implies that quantum evolutiondominates for

H2

4�ffiffiffiffiffiffiffiN

p >p

8�

�MPl

Q

�HMPl: (67)

(We have presumed that the de Sitter result for the fluctua-tions is valid in our current epoch of acceleration.) ForQ�MPl [36] and H � MPl the above inequality is notsatisfied. Note that any perturbations generated in thecurrent accelerating phase leave the horizon and cannotbe detected.In the model of quintessential inflation [37] the quintes-

sence field plays the role of the inflaton at early times.The potential is as in Eq. (66) with p ¼ 4. The value of Qtoday is approximately its value at the beginning of theradiation dominated era and is about 8MPl. Once again, forH � MPl quantum evolution does not dominate during thecurrent accelerated phase.In Ref. [5] the dark energy today is associated with a

frozen condensate of fluctuations of a field ’ generatedduring inflation. The field, which has a quadratic potential,is almost massless during inflation and evolves due toquantum fluctuations, similar to the evolution of the infla-ton in the quantum phase that has been studied above.As the Universe evolves after inflation the superhorizon

fluctuations remain frozen as a condensate at a value ’c ¼½3H4

I =ð8�2m2Þ�1=2 and dominate the energy density of theUniverse today (HI is the Hubble parameter during infla-tion). The condensate behaves like a slowly rolling quin-tessence field today with the equation-of-state of darkenergy. Inflation in this model occurs at a low scale of3This was also pointed out to us by K. Dimopoulos.

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5 TeV. The curvature perturbation generated during infla-tion due to quintessence field fluctuations is given byEq. (23) which implies

Rk ¼ 3H2

m2

�’

’(68)

¼ ffiffiffi6

pHI=m � 1: (69)

CMB fluctuations are influenced by the curvatureperturbation at decoupling. The contribution of the darkenergy to the curvature perturbation at decoupling will beRkðtdecÞ � f where

f ¼ ðþ pÞ’ðþ pÞ

��������dec(70)

¼ _’2

dm

��������dec(71)

¼ m2H4I

9�H4decM

2Pl

; (72)

and we have used _’ ¼ �m2’=ð3HÞ and approximated ’by the value ’c above (as also in Ref. [5]). To obtain thevalue of the curvature perturbation at decoupling, oneassumes that �’=’ remains constant during slow rollbecause both �’ and ’ have the same equation of motion(as in curvaton models). Once the field starts oscillatingwhen H �m, then Rk � �=j’ � �’=’� constant.

Therefore, RkðtdecÞ ¼ffiffiffi6

p ðHI=mÞðm=HIÞ2. Then

fRkðtdecÞ ¼ffiffiffi6

pm3H3

I

9�H4decM

2Pl

(73)

� 5� 10�5; (74)

where m & H0, the Hubble parameter today, as indicatedin this model. Thus, the curvature perturbation at decou-pling due to the quintessence condensate will not give riseto a large net curvature perturbation. Therefore, the curva-ture perturbation due to the quintessence field condensatein this model is not in conflict with CMB observations.

VI. WARM INFLATION

In warm inflation [6] dissipative effects are importantduring inflation so that radiation production occurs con-currently with inflationary expansion. The basic equationfor describing the evolution of an inflaton field that dis-sipates energy is of a Langevin form [38,39]

€�þ ½3H þ�� _�� 1

a2ðtÞ r2�� @V

@�¼ : (75)

In this equation � _� is a dissipative term, and is afluctuating force. Both are effective terms, arising due tothe interaction of the inflaton with other fields. In general,

these two terms are related through a fluctuation-dissipationtheorem, which would depend on the statistical state of thesystem and the microscopic dynamics. Although the statis-tical state can be quite general, all studies so far havefocused on the thermal state, and we will restrict our con-sideration here to that also. Thus, the evolution of theinflaton field has to be calculated in a thermal background.The above dynamics need not be restricted just to the

period when there is a potential driven inflation period. Theabove dynamics could also occur previous to such a period.This point leads to a new type of inflation phase in whichwhile inflation occurs, the inflaton, rather than being gov-erned by the potential V, is instead governed by the thermalbackground which produces large fluctuations in the infla-ton field. This is similar to the quantum fluctuation driveninflation discussed in the previous sections, except now thefluctuations are thermal rather than quantum.To examine the initial period during this thermal fluc-

tuation dominated inflation phase, we first need to evaluate�T which is the value of � when evolution due to thefluctuations is no longer dominant. Following the sameprocedure as done for the cold inflation case, we need toevaluate the equivalent of Eq. (8.3.12) of Ref. [1] and

Eq. (3.11) of Ref. [2] to obtain _� due to fluctuations anddue to the potential and ascertain until when the formerdominates. Fluctuations in warm inflation are obtainedfrom a Langevin equation derived using a real time formal-ism of thermal field theory [39,40].In treating warm inflation one caveat is important. In

general, the inflaton dynamics is a nonequilibrium prob-lem. Whether the case of cold or warm inflation certainassumptions are already being made about this dynamicswhen one writes down the evolution equation. In coldinflation, the basic assumption is the inflaton is evolvingat effectively zero temperature, and interactions with otherfields are negligible. In the case of warm inflation, one isassuming there is a thermal state, and the inflaton inter-action with other fields is significant. In this case there willbe a point in time, td, when these conditions are realized.Previous to that time, in principle, one would need tocalculate the full quantum field dynamics and determinethe statistical state and its evolution. This is beyond thescope of this paper. Here what we will assume is eitherdissipation effects are important and evolve as Eq. (75), orelse they are not important, and to a good approximationthe evolution is the same as the cold inflation case. If td issufficiently early, then the entire fluctuation dominatedera can be calculated based on the evolution Eq. (75).However, if td occurs fairly close to the onset of the slowroll period, then this can allow for possibilities whichcombine both quantum and thermal fluctuation regimesbefore potential driven inflation.There are two dissipative regimes that must be consid-

ered depending on whether in Eq. (75) � 3H, which iscalled the weak dissipative regime, or �> 3H, which is

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called the strong dissipative regime. For all these cases theLangevin equation for the modes of the inflaton field can besolved, as was done in Ref. [41]. Our interest here is inthe long wavelength modes, for which Ref. [41] finds thesolution coincides with the homogeneous solution, as ifthe effect of the noise force had no effect. Thus, calculationof h��2i in both these cases is found to have the samegeneral form as for cold inflation and, followingEqs. (7.3.10–7.3.12) of Ref. [1],

h��2i¼ 1

ð2�Þ3Z aH

Hd3kj�kj2¼

Z aH

H

dk

kP�ðkÞ; (76)

where only the inflaton power spectrum P�ðkÞ k3j�kj2=ð2�2Þ is different for the various cases. The limitsof the integral admit only those modes that have left thehorizon during inflation. Below we consider a quartic newinflation potential.

A. Weak dissipation

In the scenario in which �< 3H the slow roll of theinflaton is because of the Hubble damping term in theequation of motion, so, in particular, the inflaton massm< 3H must hold in order that a slow-roll regime iseventually achieved. The analysis for this case is verysimilar to the cold inflation case, except the expression forthe inflaton fluctuation is different. In this regime the infla-

ton fluctuation at freeze-out gives P�ðkÞ ¼ ð3�=4Þ1=2HT

[38,39,41] and

h��2i ¼�3�

4

�12HT � NðtÞ: (77)

[This may be compared with Eq. (1), h��2i ¼ ðH=2�Þ2 �NðtÞ.] Then, for �0 � 0,

� ¼�3�

4

�14ðHTÞ12

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiHðt� t0Þ

q; (78)

and the evolution rate of the inflaton in the thermal fluctua-tion driven phase is

_�th ¼ 1

2

�3�

4

�14H

32T

12

1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiHðt� t0Þ

p : (79)

Thermal fluctuations will dominate the evolution of � aslong as

1

2

�3�

4

�14H

32T

12

1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiHðt� t0Þ

p � � V 0

3H¼ ��3

3H: (80)

Defining�T as the largest value for which thermal fluctua-

tions dominate, �T � H34T

14=�

14.

The temperature during the thermal phase must bedetermined. Any residual radiation from initial conditionswill rapidly redshift away during any inflation epoch, andso in order for a thermal fluctuation dominated phase ofinflation to exist, there must be a source of radiationproduction. Although, in general, this is a problem of

nonequilibrium quantum field theory, following our state-ments at the start of this section, we will assume theradiation is produced by the background component ofthe scalar field, �, which is controlled by Eq. (75) withoutthe noise force, for which the radiation produced is

r ¼ � _�2

4H: (81)

In the weak dissipative regime, _� ¼ �V0=ð3HÞ, and forthe quartic new inflation potential of Sec. II this gives

r ¼ �2��6

36H3: (82)

Since the radiation energy density increases with �, anestimate of the largest radiation energy density producedwill be for ���T . Equating the expression Eq. (82) withr � g�T4 the temperature can be expressed in terms of the

other quantities, and we find T � 0:2�1=5�2=5H3=5=g2=5� �H. The latter inequality follows since � � 1, and in theweak dissipative regime�<H. Thus, we conclude that inthe weak dissipative case for the new inflation quarticpotential there is never a thermal fluctuation driven regimeduring inflation. The dynamics before potential driveninflation will be the same as for the quantum fluctuationdominated phase in cold inflation.

B. Strong dissipation

In the strong dissipation case P�ðkÞ ¼ffiffiffiffiffiffiffiffiffi�=4

p ð�HÞ12T[39,41] (see Eq. (34) of Ref. [41]). Results will now becalculated during the fluctuation era for the quartic newinflation model. Using the above expression for P�ðkÞ inEq. (76), h��2i in the fluctuation dominated regime can beobtained as

h��2i ¼��

4

�12ð�HÞ1=2T � NðtÞ: (83)

Then, for � � 0,

� ¼��

4

�14ð�HÞ14T1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiHðt� t0Þ

q; (84)

and so

_�th � ð�HÞ14T12H

1

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiHðt� t0Þ

p : (85)

The thermal fluctuations will dominate the � evolutionfrom the potential as long as

ð�HÞ14T12H

1

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiHðt� t0Þ

p � ��3

�: (86)

If fluctuations dominate evolution until �T then Eq. (86)implies that

�T ¼ ð�HÞ3=8T1=4=�1=4: (87)

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The temperature during the thermal fluctuation domi-nated phase must now be determined using Eq. (81), where

in the strong dissipative regime _� ¼ �V 0=�. To estimatethe maximum the radiation energy density will be duringthe thermal fluctuation inflation regime, Eq. (87) is used,from which we find

T � �1=5

g2=5�ð�HÞ1=2: (88)

In order for this regime to be thermal dominated requiresT >H, which implies the condition

� *g4=5��2=5

H: (89)

In this regime, from Eqs. (84) and (87)

NT ¼ ð�HÞ14ð�TÞ12 : (90)

The curvature power spectrum for strong dissipation isgiven by [39,41]

P12

R ¼ H�

jV0jP12

� (91)

¼ 2�14ð�TÞ12ðH�Þ14 N

32

k; (92)

with Nk � H�=ð2��2kÞ. (The expression for Nk is that for

the weak dissipative regime with 3H replaced by �, as can

be surmised from Eq. (17).) Setting Nk ¼ 60 and P12

R ¼10�5 gives NT ¼ ðH�Þ14=ðT�Þ12 ¼ 108. One can verify that

Vð�TÞ � V0 � ð�=HÞ32TH3=4� V0 ��1=5ð�HÞ2=ð4g2=5� Þ.In order that Vð�TÞ � V0 and so the fluctuations do nottake the inflaton to the bottom of the potential, it requires

the condition 10V0=ð�7=20MPlÞ4 � 1, which holds for

V1=40 & 1014 GeV. This is two orders of magnitude less

than earlier constraints based on CMB density fluctuationmeasurements [7].

In this regime, we also must confirm that the kineticenergy of the field fluctuations does not dominate thepotential during the fluctuation driven epoch. The kineticenergy of the field fluctuations is obtained from

h _�2i ¼Z akF

kF

dk

kj _��ðkÞj2; (93)

where ��ðkÞ ¼ ðk3=2�2Þ12�k and kF ¼ ð�HÞ1=2 is thefreeze-out scale in the strong dissipative regime [39].

Using Eq. (38) of Ref. [41] for _��ðkÞ,_��ðkÞ ¼ �

��

4

�14 k2

�a2ð�HÞ14T1

2; (94)

we get from Eq. (93),

h _�2i � �1=2H5=2T; (95)

which is much smaller than V0 ¼ 0:1H2M2Pl since T and �

are much smaller than MPl. Also from Eq. (88) it followsthat r is much less than V0.This result has demonstrated a new thermal fluctuation

inflation phase. There can be other variations to the abovescenario. In the strong dissipative regime slow roll motiononly requires the condition on the inflaton mass m<�;thus, in general, the inflaton mass can be much bigger thanthe Hubble parameter. This feature, combined with thepresence of a radiation component and dissipative dynam-ics can lead to other possible dynamics previous to thepotential driven inflation phase. One example is if thedissipation dynamics is not initially active, previous totime td, the inflaton field now is massive and evolves like

the cold inflation case, in that only a 3H _� term damps itsevolution. In such a case for a massive inflaton field theestimate for h��2i differs from that in Eq. (1), and rather itis Eq. (7.3.13) of Ref. [1].

VII. CONCLUSION

In this article we have investigated the evolution of theinflaton due to quantum fluctuations and studied its pos-sible consequences. If the field travels far on its potentialdue to quantum fluctuations or acquires a large kineticenergy, then standard inflation where the field rolls dueto the slope of its potential will not subsequently occur.This will dramatically alter the inflationary scenario andsignificantly affect the density spectrum. For a givenpotential a priori one cannot predict the impact of quantumfluctuations on the evolution of the inflaton. For example,for GUT scale inflation the quantum phase lasts for108 e-foldings for a Coleman-Weinberg potential (approxi-mated by a quartic potential). In contrast, for quadratic newinflation the quantum phase lasts for 2000 e-foldings. Forchaotic inflation, inflection point inflation, and for naturalinflation, the quantum phase is negligible, and classicalrolling is important from the beginning of inflation. Onemight have expected quantum fluctuations to be relevantfor all small field inflation models, but it is not so for smallfield natural inflation.For new inflation models, if one assumes that our current

horizon scale left the horizon during classical slow roll,then the earlier quantum phase ends with the inflaton farfrom the minimum of its potential and with subdominantkinetic energy. This allows for the standard classical roll-ing inflationary phase to follow. If cosmologically relevantscales leave the horizon during the quantum phase, whichis subsequently followed by a classical phase, then forquartic new inflation this requires that the coupling � isgreater than 10�2. We have derived the expression for thecurvature perturbation which is valid for the quantumphase. We get a large value for the curvature perturbationfor modes that leave during the quantum phase. This thenrules out this scenario. We also consider a scenario where

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the quantum phase lasts for the entire inflationary epochwhich is also ruled out because of the large curvatureperturbations. Our conclusions are similar for quadraticnew inflation.

We have also studied curvaton and quintessence modelswhere quantum evolution can be relevant. For curvatonmodels, if the curvaton evolves during inflation due toquantum fluctuations which determine the curvaton fieldvalue at the end of inflation, we find that the number ofe-foldings of inflation must be orders of magnitude morethan the usual minimum value required to solve the horizonproblem. We also find that the slow roll parameter �H isdetermined and has a value that is incompatible with smallfield inflation models. We further point out that newerresults for asymptotic values of scalar fields during infla-tion can lead to large non-Gaussianity in conflict withobservations. Quantum fluctations in quintessence modelsdo not lead to any inconsistencies with observations,though in the condensate dark energy model the curvatureperturbation associated with the condensate field is largeduring inflation.

We have studied quartic new inflation in the contextof warm inflation (weak dissipation and strong dissipa-tion regimes). We find that as in cold inflation about108 e-foldings of inflation occur before inflaton evolutionis driven by the slope of the potential. However, in theweak dissipative regime the fluctuation driven phase is

due to quantum fluctuations, while in the strong dissipa-tive regime it is due to thermal fluctuations. Quantumfluctuations of the inflaton in a thermal background arelarger than in vacuum, and the condition that the inflaton isnot driven to the minimum of its potential by fluctuations inthe strong dissipative regime requires that the scale ofinflation must be less than 1014 GeV. In both dissipativeregimes the kinetic energy of the inflaton at the end of thefluctuation driven phase is much less than the potentialenergy, thereby allowing for the standard warm inflationscenario with a slowly rolling inflaton field driven by thepotential to commence after the fluctuation driven phase isover. This confirms the robustness of the warm inflationscenario.

ACKNOWLEDGMENTS

A.B. would like to thank the Physical ResearchLaboratory, Ahmedabad, India, for support and hospitalitywhile researching this article. We would like to thank theorganizers of the Xth Workshop on High Energy PhysicsPhenomenology (WHEPP-X) at the Institute ofMathematical Sciences, Chennai, India, during which dis-cussions on the above work were initiated. We would alsolike to thank Mar Bastero-Gil, Karim Malik, and KostasDimopoulos for very useful discussions. We also acknowl-edge very useful comments from an anonymous referee.

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