Slow-roll inflation in the presence of a dark energy coupling

Philippe Brax,1 Carsten van de Bruck,2 Lisa M.H. Hall,2 and Joel M. Weller2

1Institut de Physique Theorique, CEA, IPhT, CNRS, URA 2306, F-91191 Gif/Yvette Cedex, France2Department of Applied Mathematics, University of Sheffield, Hounsfield Road, Sheffield S3 7RH, United Kingdom

(Received 15 January 2009; published 8 May 2009)

In models of coupled dark energy, in which a dark energy scalar field couples to other matter

components, it is natural to expect a coupling to the inflaton as well. We explore the consequences of

such a coupling in the context of single-field slow-roll inflation. Assuming an exponential potential for the

quintessence field we show that the coupling to the inflaton causes the quintessence field to be attracted

toward the minimum of the effective potential. If the coupling is large enough, the field is heavy and is

located at the minimum. We show how this affects the expansion rate and the slow-roll of the inflaton field,

and therefore the primordial perturbations generated during inflation. We further show that the coupling

has an important impact on the processes of reheating and preheating.

DOI: 10.1103/PhysRevD.79.103508 PACS numbers: 98.80.Cq

I. INTRODUCTION

The nature and origins of dark energy, the energy com-ponent which is responsible for the observed acceleratedexpansion of the Universe, remain a mystery. Although theobservations can be accounted for by the cosmologicalconstant, scalar fields [1,2] and modified gravity theories[3] have also been suggested (for reviews and referencessee e.g. [4–6]). One of the major aims of modern cosmol-ogy is to determine the properties of dark energy, many ofthe parameters of which (equation of state, matter cou-plings, etc.) can be constrained by considering data fromsupernovae at high redshifts, observations of anisotropiesin the cosmic microwave background (CMB) radiation,and the large scale structures (LSS) in the Universe. Ateven higher redshifts and in the early Universe, constraintsfrom varying constants and big bang nucleosynthesis(BBN) give further constraints on the dark energy evolu-tion [7,8]. Some models of dark energy can even be testedin the laboratory [9–13]. In this paper, we study the impactof couplings of a quintessencelike scalar field to the in-flaton field, the scalar field responsible for an acceleratedexpansion in the very early Universe. In this importantepoch, the seeds for the structures we observe in theUniverse were created. It is usually assumed that darkenergy is not important during inflation. In the case inwhich dark energy and the inflaton field are not coupled,the vacuum expectation value (VEV) of the dark energyfield is driven by quantum fluctuations to large field values,but otherwise there are no consequences for the inflation-ary dynamics ([14]; see also [15]). We show that this is notnecessarily the case if the inflaton field couples to the darkenergy field. In models such as coupled quintessence([16,17]; see [5] for a recent review) or quintessencemodels with a growing matter component [18], dark energycouples to at least one species, which is thought to be thedecay product of the inflaton field. Therefore, it is naturalin these types of models to consider a coupling between

dark energy and the inflaton field as well. We will show inthis paper that for large enough couplings (to be specifiedbelow) one can expect modifications to the predictions ofthe spectral index, its running, and the tensor-to-scalarratio. We find also that the details of the physics of reheat-ing and preheating are affected by the presence of a cou-pling between dark energy and the inflaton. The paper isorganized as follows: in Sec. II we describe our model andstudy the inflationary epoch and discuss the effect of thequintessence field on the primordial perturbations. InSec. III we discuss the consequences for reheating andpreheating. We conclude in Sec. IV.

II. SLOW-ROLL INFLATION IN THE PRESENCEOF COUPLED DARK ENERGY

The theory we consider in this paper is specified by theaction

S ¼Z

d4xffiffiffiffiffiffiffi�g

p �M2

Pl

2RþLQ þL�

�

with

LQ ¼ � 1

2g��ð@�QÞð@�QÞ � VðQÞ

L� ¼ �A2ðQÞ2

g��ð@��Þð@��Þ � AðQÞ4Uð�Þ:

Here, R is the Ricci scalar, � is the inflaton field, Q isanother scalar field, possibly playing the role of darkenergy, and g is the determinant of the metric tensor. Theappearance of the coupling function AðQÞ follows directlyfrom the fact that we focus on a scalar-tensor theory wherematter, i.e. the inflaton field here, couples to the rescaledmetric ~g�� ¼ A2ðQÞg��. While these equations are valid

for any potential Uð�Þ, we will use as an example thestandard chaotic inflation potential

PHYSICAL REVIEW D 79, 103508 (2009)

1550-7998=2009=79(10)=103508(10) 103508-1 � 2009 The American Physical Society

Uð�Þ ¼ 1

2m2�2: (1)

For the coupling function we choose AðQÞ ¼expð�Q=MPlÞ, (as in, for example, [16,17,19]). After afield redefinition, the (effective) mass of the inflaton fieldis nothing but AðQÞm, which grows as Q rolls down alongthe potential toward large values. Additionally, we choosethe potential for the quintessence field to be an exponentialpotential, i.e.

VðQÞ ¼ M4Pl expð��Q=MPlÞ; (2)

with � positive.

A. Slow-roll inflation

Let us now consider the inflationary period. Since Qcouples to�, the inflationary dynamics will be modified bythe coupling. The coupling can potentially ruin inflationand therefore we first consider the conditions on the pa-rameters in order to obtain a period of slow-roll inflation.

The equations of motion for the fields � and Q in ahomogeneous and isotropic universe are

€Qþ 3H _Qþ @V

@Q¼ A2�ð _�2 � 4A2UÞ; (3)

€�þ 3H _�þ A2 @U

@�¼ �2� _Q _� : (4)

The Friedmann equation reads

H2 ¼ 1

3M2Pl

�A2

2_�2 þ 1

2_Q2 þ V þ A4U

�: (5)

The Q-field moves in an effective potential, which is givenby the bare potential and a part coming from the couplingto �. Taking � positive, the sign of � determines whetherthe effective potential has a minimum or not. For positive�, a local minimum exists, whereas there is not one in thecase of a negative �. If the minimum does not exist, theeffective potential is of a runaway form and the discussionof the field behavior will be similar to that in [14]. In thispaper, we will assume the existence of a minimum, that is,we consider the case of a positive �. Assuming then slow-roll of the scalar field �, this determines the value of Q atthe minimum of the effective potential to be

Qmin ¼ 1

4�þ �ln

��M4

Pl

4�Uð�Þ�; (6)

together with the condition for the minimum

�VðQminÞ ¼ 4�A4Uð�Þ: (7)

For later use, we also state AðQminÞ during slow-roll in-flation:

AðQminÞ ¼�

�M4Pl

4�Uð�Þ��=ð4�þ�Þ

: (8)

During inflation, in which we assume that both of thefields are rolling slowly, using Eq. (7) the Friedmannequation takes the form

3H2 � 1

M2Pl

ðA4Uþ VÞ � A4U

M2Pl

�1þ 4�

�

�: (9)

We will justify below the assumption that both fields rollslowly. Note that the Q-field contributes to the expansionrate with an amount depending on� and �. The mass of theQ-field can be found to be

m2eff ¼

@2V

@2Q� 2A2�2 _�2 þ 16A4�2U: (10)

In the case of a slowly rolling �-field, this gives

m2eff �

@2V

@2Qþ 16A4�2U � @2V

@2Qþ 48�2H2

1þ 4��

; (11)

where in the last line we have used the Friedmann equa-tion. For large enough �, the field is rather heavy (m2

eff >H2) and will therefore settle into the minimum of theeffective potential. To get a rough idea on how small �can be, we can solve the equation m2

eff=H2 ¼ 1 and obtain

�crit ¼ 112� (we remind the reader that � has to be positive

for a minimum to exist, since we are assuming � positive).This equation implies that, even when� is rather small, theQ-field sits at the minimum of the effective potential. Wefind numerically that Q settles into the minimum even for� as small as 0.05 (see Sec. II C).It is possible to obtain a degree of analytical insight into

the behavior of the system during the inflationary period bystudying the slow-roll regime and deriving the slow-rollparameters. Firstly, we show that the extra friction term inEq. (4) is negligible during inflation. Throughout thisanalysis, we will assume that the value of � is such thatQ settles into the minimum of the effective potential. Thusone can write _Q ¼ _Qmin. Using Eq. (6) we find (

0 ¼ d=d�)

_Q min ¼ � 1

4�þ �

U0

U_� ¼ � 2

�þ 4�

� _�

�

�; (12)

where Eq. (1) was used in the last step. Therefore we find

j2� _Qj ¼ 4�

�þ 4�

� _�

�

�� _� � H: (13)

In general, we see that the extra damping term is propor-

tional to _�2 and we can therefore neglect it:

€�þ 3H _�þ @U

@�A2 � 0: (14)

This means that the condition for slow-roll differs from thestandard case by a factor of A2:

BRAX, VAN DE BRUCK, HALL, AND WELLER PHYSICAL REVIEW D 79, 103508 (2009)

103508-2

_� � �U0A2

3H: (15)

The first slow-roll parameter is defined by � � � _HH2 and

in terms of the fields can be found to be

� � M2Pl

2A2

�U0

U

�2�1þ 4�

�

��2; (16)

where Eqs. (15) and (9) were used. We can find the �parameter in a similar way. Differentiating Eq. (15) yields

€� � � 1

3�U0A2 � U00

3H_�A2 þ 2

3

U02A2

HU

�

4�þ �_�: (17)

The requirement that j €�j � jH _�j gives���������� �þ 4�

��

��������� 1: (18)

where

� � U00

U

M2Pl

A2

�1þ 4�

�

��1: (19)

B. Cosmological perturbations

We now derive the relevant equations for the cosmologi-cal perturbations. We show first that the perturbations inthe Q-field are much smaller than the perturbations in the�-field. This is because the Q-field is a heavy field (i.e.meff >H) and it is well known that for these fields pertur-bations are suppressed. The fluctuations of a massive scalarfield during inflation satisfy an equation composed ofoscillatory and nonoscillatory parts [20]. The expansionof the Universe causes the wavelength of the perturbationsto be stretched, so that the former can be neglected as theaverage contribution of oscillations averages to zero. Theamplitude of the resulting power spectrum is suppressed by(H=meff) and decreases rapidly for large wavelengths. Forcompleteness, however, we will study numerically theperturbations for both fields. The inflaton field � is light(m<H) and its perturbations are not suppressed.

In the longitudinal gauge and in the absence of aniso-tropic stress, the scalar perturbations of the Friedmann-Robertson-Walker (FRW) metric can be expressed as (seee.g. [21])

ds2 ¼ �ð1þ 2�Þdt2 þ a2ð1� 2�Þ�ijdxidxj: (20)

The perturbed Einstein equations give

3Hð _�þH�Þ þ k2

a2� ¼ � 1

2M2Pl

�; (21)

_�þH� ¼ � 1

2M2Pl

�q; (22)

€�þ 4H _�þ ð3H2 þ 2 _HÞ� ¼ 1

2M2Pl

�p; (23)

where

� ¼ ð _Qð�QÞ: �� _Q2Þ þ �A2 _�2�Q

þ A2ð _�ð��Þ: �� _�2Þþ

�@V

@Q�Qþ 4�A4U�Qþ A4 @U

@���

�; (24)

�q ¼ �ð _Q�Qþ A2 _���Þ; (25)

�p ¼ ð _Qð�QÞ: �� _Q2Þ þ �A2 _�2�Q

þ A2ð _�ð��Þ: �� _�2Þ�

�@V

@Q�Qþ 4�A4U�Qþ A4 @U

@���

�: (26)

The perturbed field equations are

ð��Þ:: þ ð3H þ 2� _QÞð��Þ: þ�k2

a2þ A2 @

2U

@�2

���

¼ �2�

�A2 @U

@��Qþ _�ð�QÞ:

�þ 4 _� _��2�A2 @U

@�;

(27)

ð�QÞ:: þ 3Hð�QÞ: þ�k2

a2þ @2V

@Q2� 2�2A2 _�2

þ 16A4�2U

��Q

¼ �2�

�@V

@Qþ 4A4U�

�þ 4 _� _Qþ2A2� _�ð��Þ:

� 4A4�@U

@���: (28)

Integrating these equations numerically shows the per-turbations of the Q-field are indeed suppressed. A typicalplot of the evolution of �� and �Q is shown in Fig. 1. Asalready said above, the difference in the behavior of thetwo quantities is due to the large effective mass of theQ-field. Thus, the inflaton perturbations dominate and wecan ignore the perturbations of the quintessence field. Wehave checked numerically that, in the regime we are inter-ested in, the perturbations in Q are always suppressedrelative to the perturbations in �. There is an intermediateregime, in which the quintessence mass is smaller but oforderH and contributes to the cosmological expansion by areasonable amount. In this case, the quintessence field willnot sit in the minimum of the effective potential, but isattracted to it and its perturbations cannot be ignored.However, in this paper we do not deal with this case.With this in mind, the calculations proceed in the stan-

dard way, taking into account the modifications of thebackground evolution, as discussed above. The power

SLOW-ROLL INFLATION IN THE PRESENCE OF A DARK . . . PHYSICAL REVIEW D 79, 103508 (2009)

103508-3

spectrum of curvature perturbations is

P RðkÞ ¼�H_�

�2�H

2

�2 � 1

242M4Pl

UA6

�

�1þ 4�

�

�:

(29)

The spectral index is found by calculating the derivativeof this quantity with respect to lnðkÞ, where k is the wavenumber. Using the slow-roll condition one finds

d

d lnk¼ �M2

Pl

U0

UA2

�1þ 4�

�

��1 d

d�: (30)

The derivatives of the slow-roll parameters are:

d�

d lnk¼ �2��þ 4�2

�1þ 3�

�

�; (31)

d�

d lnk¼ ��2 þ 2��

�1þ 2�

�

�; (32)

where

�2 � M4Pl

A4

U000U0

U2

�1þ 4�

�

��2: (33)

So we find the spectral index to be

ns � 1 ¼ 2�� 6�� 8�

��: (34)

The running of the spectral index is found to be

dnsd lnk

¼ �2�2 þ 16��� 24�2

þ�24�

���� 104�

��2 � 96�2

�2�2�: (35)

For the tensor perturbations, we have

P grav ¼ 2

M2Pl

�H

2

�2: (36)

Writing P grav / kng , we can show that

ng ¼ �2�; (37)

whereas the tensor-to-scalar ratio is, using the slow-rollcondition (15),

r � P grav

PR¼ 4�

A2¼ � 2ng

A2¼ �2ng

�4�U

�M4Pl

�2�=ð4�þ�Þ

:

(38)

Note that while the prediction for the scalar-tensor ratio r ismodified from the standard case, the expression for thetensor spectral index ng remains the same.

C. Consequences

Having derived the relevant equations describing slow-roll inflation with a coupled dark energy scalar field, wewill now discuss the consequences and predictions. Thefirst difference from the standard case is that the expres-sions for the slow-roll parameters have changed. The originof the modifications are two-fold: firstly, the slow-rollcondition for the inflaton field is modified and, secondly,the expansion rate is enhanced by a factor A4ð1þ 4�=�Þ.As a result, both slow-roll parameters contain an additionalfactor A�2 when compared to the standard expression; seeEqs. (16) and (19). Additionally, they are modified by afactor ð1þ 4�=�Þ�2 (for �) and ð1þ 4�=�Þ�1 (in the caseof �). Slow-roll inflation ends when maxf�; �g ¼ 1. Thepresence of the factor of A�2 means that the end of inflationis delayed in this model, in the sense that smaller values of� are reached in the slow-roll phase. An immediate con-sequence of this is that the oscillations of the inflatonaround its minimum—responsible for the production ofparticles in the reheating phase—will have a smaller am-plitude than in the standard case.Another important difference is in the expressions for

the cosmological perturbations. Apart from the spectralindex of the gravitational wave power spectrum, all ex-pressions for the perturbations have changed: the ampli-tude of scalar perturbations is different and the expressionsfor the spectral index ns and its running include additionalfactors which depend on the ratio �=�. However, becausethe actual values of � and � during the last 60 e-foldschange from their values in the standard chaotic inflation-ary scenario, it is not obvious whether ns will be bigger orsmaller than in the standard case.

FIG. 1. The evolution of the field perturbations as a function ofN ¼ lna. The upper panel shows the perturbations of the inflatonand the lower panel shows the perturbations of the Q-field. Here� ¼ 0:5, � ¼ 10, m ¼ 7� 10�7, and k ¼ 0:001. As one cansee, the perturbations in Q drop to a value several orders ofmagnitude below that of the perturbations in �, due to theheaviness of the Q-field relative to �.

BRAX, VAN DE BRUCK, HALL, AND WELLER PHYSICAL REVIEW D 79, 103508 (2009)

103508-4

To illustrate the effect of the dark energy field, let usconsider as an example the case of a massive inflaton fieldwith potential energy given by Eq. (1). The theory has threefree parameters, namelym, �, and �. We can therefore useEq. (29) and the Cosmic Background Explorer (COBE)normalization to fixm as a function of � and �. The resultsare shown in Fig. 2(a). At the same time, we have to makesure that � is big enough to ensure that the Q-field sits inthe minimum of the effective potential. We have checkednumerically that for � � 0:05, the field sits indeed in the

minimum of the effective potential (see Fig. 3). As one cansee, the mass of the inflaton field is smaller than in thestandard case (m � 10�5MPl). This is because of the factorA6 in the expression for the power spectrum and, since A �1, a smaller mass m is sufficient to obtain the correctamplitude for the power spectrum. The predictions forthe spectral index ns are shown in Fig. 2(b). As one cansee, the spectral index ns is larger than in the standard casewithout couplings. The current constraint from theWilkinson Microwave Anisotropy Probe (WMAP) aloneon the spectral index is ns ¼ 0:963þ0:014

�0:015, assuming a

�CDM cosmology [22]. Note that in order to comparethe theory to data, one would have to study the evolution ofcosmological perturbations in the presence of dark energycouplings in the subsequent epoch and make assumptionsabout the couplings to different matter species. The cou-pling � we have discussed so far is the coupling of darkenergy to the inflaton field and is a priori not the same asthe coupling to dark matter or neutrinos. If dark energy issubsequently only coupled to dark matter, then currentconstraints give � � 0:95 and j�j � 0:055 at 95% confi-dence level [19]. Assuming that the dark energy couplingto the inflaton field is the same as to dark matter, thisimplies that the predictions of the dark energy couplingto the primordial perturbations are only slightly modifiedfrom the standard chaotic inflation predictions. On theother hand, in the case of theories with a mass-growingcomponent [18], which requires larger values for � and �,modifications to the primordial power spectrum are pre-dicted. In any case, when comparing the theory to data,assumptions about the subsequent composition of theUniverse and new interactions between dark energy andother matter forms have to be made. The results presented

1

2

3

4

5

10

15

20

λ

-12-10-8-6

1

2

3β λ β

4

5

10

15

20

0.965

0.97

1

2

3

FIG. 2 (color online). Predictions for a chaotic inflationary potential:U ¼ m2�2=2; the left plot shows logðm=MPlÞ as a function of �and �, and the right plot shows ns as a function of � and �. The values of the inflaton mass (as obtained by normalizing the powerspectrum to COBE) and spectral index vary with the parameters � and �. In these plots, only the data for � > 4� (i.e. the region inwhich the inflaton is the dominant component during inflation) are plotted.

FIG. 3. These plots show the Q-field as a function N ¼ lna(dotted lines). Even if the initial value of the field changes, thefield is drawn toward its minimum (solid line).

SLOW-ROLL INFLATION IN THE PRESENCE OF A DARK . . . PHYSICAL REVIEW D 79, 103508 (2009)

103508-5

in this section will be useful when constructing a theory ofinflation, dark matter, and dark energy based on particlephysics.

III. REHEATING AND PREHEATING IN THEPRESENCE OF COUPLED QUINTESSENCE

In our scenario, when the slow-roll conditions are vio-lated, the value of� is smaller than in the standard case andthus the oscillations of the field around its minimum are ofsmaller amplitude. This can affect the processes of reheat-ing and preheating. In the following we will discuss indetail the effect that the presence of coupled quintessencehas on these important epochs.

A. Reheating

Any successful model of reheating relies heavily on theoscillatory behavior of the inflaton. The oscillations aredamped due to the expansion of the Universe by a frictionterm proportional to H and a term describing the decay ofthe inflaton field into radiation. Following the standardtreatment of reheating (cf. [23]), the equation of motionfor the inflaton field reads

€�þ ð3H þ �rÞ _�þ A2m2� ¼ �2� _Q _�; (39)

where �r is the decay rate of the process. In the standardsingle-field case the value of the Hubble parameter isproportional to the square root of the total energy density,which consists of inflaton energy density and radiationenergy density. In the present case, however, there is alsothe energy density of the Q-field, which contributes to theHubble friction. Just at the end of inflation, for example,the ratio of the energy densities of quintessence and theinflaton field is roughly 4�=�, from Eq. (7). This ratio canbe potentially quite big (up to 10�1, or so), so the Q-fieldcontributes significantly to the Hubble damping. Thiscould cause heavy damping of the inflaton oscillationsthat would reduce the efficiency of the reheating process.

In the model of elementary reheating, the equation ofmotion for the energy density of radiation r is

_ r þ 4Hr ¼ �rð� þ p�Þ: (40)

In the standard scenario without quintessence, the energydensity of the radiation produced from the decay of theinflaton quickly increases to a maximum value, then de-creases as the decay products are diluted by the expansionof the Universe. This continues until H � �r, when theinflaton decays away rapidly and the radiation-dominatedera begins with temperature TRH. In our model, we want toavoid the dominance of the Q-field, and it is thereforenecessary either to produce enough radiation so that½rad�MAX is greater than Q or to dissipate the energy of

the Q-field quickly, so that it becomes smaller than r.

½r�MAX depends on the decay rate �r and therefore on thedetails of the decay processes. For example, if the fielddecays to two light scalar particles with coupling gA, thehighest decay rate is �r mA [21]. This means that theamount of radiation produced is limited by the (effective)inflaton mass, the value of which, as was seen in Sec. II,must be chosen to be consistent with the COBE normal-ization for particular values of � and�. Figure 4 shows thatif one uses constraints on the quintessence potential from[19], it is not possible to get successful reheating in thismodel as r is always less than Q.

We are thus forced to reduce the energy density of theQ-field. One method would be to letQ decay into radiationin a similar way to �, with a decay rate proportional to theeffective mass of the field. This would introduce a termrQ, representing the energy density of the radiation pro-

duced in this manner. In this case the equations for Q andrQ are

€Qþ ð3Hþ �QÞ _Qþ @V

@Q¼ A2�ð _�2 � 4A2Uð�ÞÞ; (41)

_ rQ þ 4HrQ ¼ �QðQ þ pQÞ: (42)

However, numerical calculations (see Fig. 5) show that theradiation produced is still not sufficient. This is partly dueto the small magnitude of the term driving the productionof radiation: �QðQ þ pQÞ ¼ �Q

_Q2. ( _Q is small as the

effective potential is very flat at this stage.)

FIG. 4 (color online). The log of the ratio of the radiationenergy density to the total energy density for different values of� and �. It is not possible to get radiation dominance usingconstraints on the parameters from [19]. We have restrictedourselves to the case where the coupling � is large enough tomake Q stay in the minimum of its effective potential.

BRAX, VAN DE BRUCK, HALL, AND WELLER PHYSICAL REVIEW D 79, 103508 (2009)

103508-6

A favorable alternative is to vary the parameter � thatcontrols the steepness of the potential of the Q-field. Alarger � means that the field rolls more quickly to largervalues. This has a two-fold effect: ½rad�MAX is increased(as the damping is reduced, so the source term in Eq. (40)

does not decrease as quickly as in the previous case) andQ is decreased, so ½rad�MAX does not have to be very

large for radiation to dominate the Universe. It can be seenin Fig. 6 that using a large value of � does indeed allowradiation to dominate the Universe. It is interesting to notethat this is true even if �r is much less than the maximumpossible decay rate. In this case, as can be seen in the lowerpanel of Fig. 6, the total amount of radiation produced isdramatically reduced, leading to a smaller reheatingtemperature than in the case of large �r. The same result

is true in the standard single-field case, where TRH �0:55g�1=4

ð ffiffiffiffiffiffiffi8

pMPl��Þ1=2 [23]. The difference is that in

our model if the decay rate is too small, one cannot satisfythe condition r > Q.

It is clear that if our model is to be consistent, theparameters of the quintessence field must be constrainedby this theoretical consideration and �must be large. It hasbeen shown [18] that it is possible to achieve a realisticcosmology with large � in the case with a growing mattercomponent. As Fig. 6 shows, it is possible for radiation todominate the Universe after the decay of the inflaton usingvalues that satisfy this lower bound.

B. Preheating

We have seen that reheating is considerably affected bythe presence of a coupled quintessence field. In particular,the process is less efficient than in the standard case,because the Hubble damping is much larger. In the follow-ing, we discuss the process of preheating, in which theinflaton energy is converted into other particles in a veryefficient way. We consider the possibility that the energydensity of these decay products could counterbalance theeffect of theQ-dominance and lead to successful reheating.Following the standard description of preheating, as

discussed in [24–26], we assume that the inflaton fielddecays to a light scalar field � of mass m�. To describe

the interaction between� and �, we will add the followingterm in the Lagrangian:

�Lint ¼ �gA��2: (43)

The conformal factor A arises because our analysis isperformed in the Einstein frame. This term describes athree-legged interaction involving one inflaton particleand two � particles. Another common choice of interactionis of the form � 1

2g2�2�2. As similar behavior arises in

both cases and we are interested in the more generalconsequences of the presence of the Q-field on the mecha-nism of reheating, we will concentrate our attention on theinteraction in Eq. (43). We assume that the effective cou-pling constant gA is less than the frequency of the inflatonoscillations mA, so that the interaction is not modified toomuch by quantum corrections. Thus, we have the limit g <m. The vacuum expectation value of the �-field is zero, so

FIG. 6 (color online). These plots show the evolution of �,Q, and r (on a log scale) for large �. Here, � ¼ 10, � ¼ 0:5,

and m ¼ 7� 10�7. In the upper panel, the decay rate is large(�r ¼ 0:9 mA) so radiation domination occurs very quickly. Inthe lower panel, although the decay rate is much smaller (�r ¼0:001 mA), the radiation can still dominate over the Q-field butthis process is of longer duration. The reheating temperaturewould be smaller in this latter case.

FIG. 5 (color online). In this simulation, the model was modi-fied to allow the Q-field to decay to radiation in a manneranalogous to � by adding a friction term �Q meff to Eq. (3).

The radiation produced in this manner contributes a fraction ofthe total energy density. These plots shows the log of the fractionof the energy density accounted for by the total amount ofradiation (i.e. including also that from the decay of the inflaton).

SLOW-ROLL INFLATION IN THE PRESENCE OF A DARK . . . PHYSICAL REVIEW D 79, 103508 (2009)

103508-7

the Friedmann equation and the classical equation of mo-tion for the inflaton will be unaffected. Expanding �around zero, we find the following equation for the pertur-bations of �, which are interpreted as particles after quan-tization:

€� k þ 3H _�k þ�k2

a2þm2

� þ 2gA�

��k ¼ 0: (44)

To study the resonance, we neglect the expansion of spaceand introduce a sinusoidal ansatz with which to describethe oscillations of the inflaton, � ¼ �sinðmAtÞ. In thiscase, Eq. (44) can be written in the form of a Mathieuequation, i.e.

�00k þ ½Ak � 2q cosð2zÞ��k ¼ 0; (45)

with

Ak �4ðk2 þm2

�Þm2A2

; (46)

q � 4g�

m2A; (47)

z � mAt=2; (48)

where prime indicates a derivative with respect to z. Theseare the standard equations with the addition of the factors Ain the expressions for Ak, q, and z (cf. [21] or [24]). As wehave also seen in Sec. II, the inflaton mass is smaller than inthe standard theory once the theory is normalized to COBEand the amplitude� of the inflaton field is smaller. We notethat for a given value of g, the parameter q, which deter-mines the behavior of the solutions to the equation above,can be considerably bigger than in the standard case.

A similar equation can be derived for the Q-particlesproduced, since the Q-field couples to the inflaton field aswell. We obtain (Qk is the Fourier component of theperturbation of the Q-field around its VEV)

Q00k þ ½AðQÞ

k � 2qðQÞ cosð2zQÞ�Qk ¼ 0; (49)

with

AðQÞk � ðk2 þm2

QÞm2A2

þ 6

5qðQÞ; (50)

qðQÞ � 5

2�2�2A2; (51)

zQ ¼ mAt: (52)

Since mQ is much bigger than mA, we see that AðQÞk is

dominated by the first term and therefore AðQÞk � qðQÞ. This

means that the periodic term in the equation for Qk doesnot play an important role and there is no resonanceproduction of Q particles. Therefore we concentrate onthe production of the light � particles in the following.Let us consider first the case of narrow resonance, in

which q � 1. Solutions to Eq. (45) falling within particu-lar resonance bands are exponentially unstable and take theform �k expð�kzÞ. The most important band is the firstone, for which the resonance reaches its maximum at�k �q=2. The number density of particles with momentum k is

nk ¼ !k

2

�j _�j2!2

k

þ j�j2�� 1

2; (53)

where !2k ¼ Ak � 2q cosð2zÞ. From this, one can see that,

while narrow resonance continues, the number of particleswith k � mA=2 increases exponentially, with nk expðqzÞ (as � is, by design, much lighter than the inflatonand the resonance occurs at Ak 1).Narrow resonance will only be an important decay

mechanism as long as it is more efficient than the pertur-bative decay, which corresponds to the elementary theoryof reheating discussed in the last section. In narrow reso-nance, the number of particles in the resonance band kincreases exponentially as nk expðqzÞ expðqmAt=2Þso the decay rate can be approximated by qmA. In theregime where perturbative decay itself is efficient (i.e.�� > H), we find that the condition for narrow resonance

to be the leading effect is

qmA> ��; (54)

where �� ¼ g2A=ð8mÞ is the decay rate as obtained by

standard quantum field theory methods. The second phe-nomenon that affects the time scale of narrow resonance isthe redshift of momenta out of the resonance bands. Theexponential nature of the resonance means that the rate ofproduction of � particles depends on the number of parti-cles present already. If the modes do not spend enough timewithin the resonance band, nk will be small and the processwill be inefficient.The width of the resonance band at k ¼ mA=2a is kq. If

we assume that the resonance is most efficient in themiddle of this band we have �k ¼ mAq=2. We wish tofind the time �t that the mode spends in the region �k.Using �k ¼ j dkdt j�t ¼ kH�t, this is �t ¼ qH�1. During

this interval, the number of particles increases asexpðqmA�t=2Þ ¼ expðq2mA=2HÞ. So the second condi-tion required for decay by narrow resonance to be efficientis

q2mA>H: (55)

We can write the inequalities in terms of the inflatonamplitude �:

BRAX, VAN DE BRUCK, HALL, AND WELLER PHYSICAL REVIEW D 79, 103508 (2009)

103508-8

�>gA

32; (56)

�>m

ffiffiffiffiffiffiffiffiffiffiffiffimAH

p4g

: (57)

A small value ofmmeans that parametric resonance cancontinue unhindered for long enough to produce largeamounts of � particles, which subsequently decay toradiation.

If, instead, the parameter q is greater than one, we are inthe regime of broad resonance. Here, the mass term inEq. (45) is dominated by the sinusoidal behavior of theinflaton. When this term is zero, the number of particlesgiven by Eq. (53) is not conserved and one observesresonance in a broad range of modes. This can be seen inFig. 7. The small value of m required to match the COBEnormalization means that the parameter q is very large inthis case, leading to a large increase in the number of �particles, despite the small amplitude of the oscillations.

The theory governing the energy density of the radiationproduced by this method and the backreaction of theparticles on the evolution of the inflaton is dependent onthe model with which one is working and the free parame-ters therein. We can assume, at the very least, that theenergy density of the products of parametric resonancewill not exceed that of the inflaton prior to decay. As, at

the start of the reheating era, Q � ð4�� Þ�, Hubble damp-

ing of the inflaton can reduce the energy density of the

inflaton field to a value less than that of the Q-field beforethe field can complete many oscillations about zero. Thisrapid damping is exhibited in Fig. 7. Because any model ofparametric resonance assumes the inflaton is oscillating, bythe time a significant quantity of �-particles are produced,the inflaton is incapable of producing enough radiation forit to dominate over the Q-field. Therefore, even if weemploy parametric resonance, we still have to use a largevalue of � to reheat the Universe in our model of coupledquintessence.

IV. CONCLUSIONS

We have described a model of coupled quintessence inwhich the scalar field responsible for dark energy iscoupled to the inflaton field. To be specific, we have usedthe exponential potential in Eq. (2) and assumed that thecoupling between the fields is of the form AðQÞ ¼expð�Q=MPlÞ with � positive, so that the effective poten-tial has a minimum. The large effective mass of theQ-fielddrives it into this minimum for a large range of � and �values, and in Sec. II A we used this to evaluate the slow-roll parameters. We found that the expressions for � and �are modified by factors of A�2ð1þ 4�=�Þ�1 and A�2ð1þ4�=�Þ�2 respectively, meaning that the end of inflation isdelayed in this model, so that the inflaton field has asmaller field amplitude during its oscillations.We further discussed the modification to the cosmologi-

cal perturbations. The equations for the spectral index andits running (in terms of the new slow-roll parameters) varywith the ratio �=�. Normalizing the power spectrum toCOBE, we found that the mass of the inflaton field isrequired to be smaller than in the standard case, whilethe effect of the presence of a dark energy coupling is toincrease the value of ns, for a given �. The parameters �and � can be constrained by observations if one assumesthat the coupling between the inflaton and dark energy hasthe same equal magnitude as the coupling between darkmatter (or neutrinos) and dark energy.We found that the presence of the Q-field during reheat-

ing leads to an unnaturally large amount of Hubble damp-ing in the equation for the inflaton oscillations duringreheating. We showed that if reheating is to be successful,� must be large. The dark energy field rolls faster down itspotential, which in turn reduces the Hubble damping andallows the radiation produced during reheating to becomethe dominant component of the Universe. We also consid-ered how the mechanism of parametric resonance duringpreheating might work in this model. We found that para-metric resonance could still produce large amounts ofradiation as the smaller value of m required to match theCOBE normalization decreases the lower bound on theamplitude of the inflaton oscillations required for reso-nance. However, if � is small, the energy density of theinflaton quickly becomes less than that of the Q-field so alarge value of � is still required to reheat the Universe in

FIG. 7 (color online). The oscillations of the �-field are stillcapable of driving parametric resonance, despite their smallamplitude. In this plot, � ¼ 0:1 and � ¼ 1. g is set to 0:9m,where m is the inflaton mass, which is chosen to be 4� 10�8 tomatch the COBE normalization. In the upper plot we show thebehavior of �, whereas the lower plot shows the number densitynk of the � particles. The dotted lines on the upper plot show thebounds given in Eqs. (56) and (57), below which narrow reso-nance is not possible.

SLOW-ROLL INFLATION IN THE PRESENCE OF A DARK . . . PHYSICAL REVIEW D 79, 103508 (2009)

103508-9

our model. The fact that a large � is required for successfulreheating has interesting consequences. For example, it isnot compatible with the condition � < 0:95 stated in [19].Such models would be only consistent if the quintessencefield does not couple to the inflaton field. Alternatively, thedark energy field could couple to a subdominant species,such as neutrinos, as e.g. in [18].

ACKNOWLEDGMENTS

J.W. is supported by EPSRC. C. v. d. B. and L.H.M.H.are supported by STFC. C. v. d. B. thanks the Institut dePhysique, Saclay, and Christof Wetterich and the Institutfur Theoretische Physik of the University of Heidelberg forhospitality while parts of this work have been completed.

[1] C. Wetterich, Nucl. Phys. B302, 668 (1988).[2] Bharat Ratra and P. J. E. Peebles, Phys. Rev. D 37, 3406

(1988).[3] Sean M. Carroll et al., Phys. Rev. D 71, 063513 (2005).[4] Edmund J. Copeland, M. Sami, and Shinji Tsujikawa. Int.

J. Mod. Phys. D 15, 1753 (2006).[5] Jerome Martin, Mod. Phys. Lett. A 23, 1252 (2008).[6] Shin’ichi Nojiri and Sergei D. Odintsov, in Proceedings of

the ECONF, C0602061, 2006 (2006).[7] James P. Kneller and Gary Steigman, Phys. Rev. D 67,

063501 (2003).[8] Thomas Dent, S. Stern, and C. Wetterich, J. Cosmol.

Astropart. Phys. 01 (2009) 0389.[9] Philippe Brax, Carsten van de Bruck, and Anne-Christine

Davis, Phys. Rev. Lett. 99, 121103 (2007).[10] Philippe Brax, Carsten van de Bruck, Anne-Christine

Davis, David Fonseca Mota, and Douglas J. Shaw, Phys.Rev. D 76, 124034 (2007).

[11] Philippe Brax, Carsten van de Bruck, Anne-ChristineDavis, David F. Mota, and Douglas J. Shaw, Phys. Rev.D 76, 085010 (2007).

[12] Dejan Stojkovic, Glenn D. Starkman, and Reijiro Matsuo,Phys. Rev. D 77, 063006 (2008).

[13] Eric Greenwood, Evan Halstead, Robert Poltis, and DejanStojkovic, arXiv:hep-ph/08105343.

[14] Michael Malquarti and Andrew R Liddle, Phys. Rev. D 66,023524 (2002).

[15] Jerome Martin and Marcello A. Musso, Phys. Rev. D 71,063514 (2005).

[16] Christof Wetterich, Astron. Astrophys. 301, 321(1995).

[17] Luca Amendola, Phys. Rev. D 62, 043511 (2000).[18] Luca Amendola, Marco Baldi, and Christof Wetterich,

Phys. Rev. D 78, 023015 (2008).[19] Rachel Bean, Eanna E. Flanagan, Istvan Laszlo, and Mark

Trodden, arXiv:0808.110.[20] Antonio Riotto, arXiv:hep-ph/021016.[21] V. Mukhanov, Physical Foundations of Cosmology

(Cambridge University Press, Cambridge, England, 2005).[22] J. Dunkley et al., arXiv:0803:0586.[23] E.W. Kolb and M. S. Turner, The Early Universe,

Frontiers in Physics (Addison-Wesley, Reading, MA,1990).

[24] Lev Kofman, Andrei D. Linde, and Alexei A. Starobinsky,Phys. Rev. D 56, 3258 (1997).

[25] Lev Kofman, Andrei D. Linde, and Alexei A. Starobinsky,Phys. Rev. Lett. 73, 3195 (1994).

[26] Jean Lachapelle and Robert H. Brandenberger,arXiv:0808.0936.

BRAX, VAN DE BRUCK, HALL, AND WELLER PHYSICAL REVIEW D 79, 103508 (2009)

103508-10