PHYSICAL REVIEW D, VOLUME 70, 087302

Nonminimal coupling, exponential potentials and the w <�1 regime of dark energy

Fabio C. Carvalho*Instituto de Fısica, Universidade de Sao Paulo, CP 66318, 05315-970 Sao Paulo, SP, Brazil

Alberto Saa†

IMECC, Universidade Estadual de Campinas, CP 6065, 13083-859 Campinas, SP, Brazil.(Received 31 July 2004; published 29 October 2004)

*Electronic†Electronic

1550-7998=20

Recent observations and theoretical considerations have motivated the study of models for darkenergy with equation of state characterized by a parameter w � p=� < �1. Such models, however, areusually believed to be inviable due to their instabilities against classical perturbations or potentiallycatastrophic vacuum decays. In this Brief Report, we show that a simple quintessential model withpotential V��� � Ae�� and a gravitational coupling of the form �1� j�j�2�R can exhibit, for largesets of initial conditions, asymptotic de Sitter behavior with w < �1 regimes. Nevertheless, the modelis indeed stable at classical and quantum level.

DOI: 10.1103/PhysRevD.70.087302 PACS numbers: 98.80.Cq, 98.80.Bp, 98.80.Jk

I. INTRODUCTION

The nature of the dark energy component responsiblefor the accelerated expansion of the Universe [1,2] is oneof the most profound problems of physics (for a recentreview, see [3]). The simplest way to describe dark matteris by means of a cosmological constant �, which acts onthe Einstein equations as an isotropic and homogeneoussource with equation of state �p� � �� � �.Straightforward questions about possible fluctuations ofdark energy leads naturally to the introduction of a field��x� (the quintessence [4]) instead of the cosmologicalconstant �. Besides the issue of fluctuations, the fielddescription is preferable since for some models (theones, for instance, provided with tracker solutions [5])the appearance of an accelerated expansion (de Sitter)phase is a generic dynamical behavior, avoiding, conse-quently, problems with fine-tuning of initial conditions inthe early Universe. Quintessential equations of state are,in general, of the type p� � w��, where the parameter wcan vary with time. For minimally coupled models withusual kinetic terms, one has

w �p�

���

_�2 � 2V���_�2 � 2V���

� �1: (1)

According to the Einstein equations, cosmological mod-els with accelerated expansion phases require sourceterms for which w < �1=3.

Recent observational questions [6] and theoreticalspeculations [7] have motivated the analysis of the pos-sibility of having realistic models for which, at leasttemporally, w < �1. Indeed, recent data from HubbleSpace Telescope of Type Ia supernova at high redshifts(z > 1) [8] favor scenarios with slowly evolving w andrestrict its values to the range w � �1:02��0:13

�0:19�. They are

address: fabiocc@fma.if.usp.braddress: asaa@ime.unicamp.br

04=70(8)=087302(4)$22.50 70 0873

compatible with a simple cosmological constant (w ��1), but if, however, one really has w < �1, both cos-mological constant and minimally coupled scalar quin-tessence descriptions for dark energy are ruled out.Realistic models accommodating also the regime w <�1 are, of course, welcome.

In [9], Carroll, Hoffman and Trodden consider theviability of constructing realistic dark energy modelswith w < �1. The so-called phantom fields, i.e., mini-mally coupled scalar fields with ‘‘negative’’ kinetic en-ergy, are usually invoked to construct such models.Despite that the dominant energy condition (in fact, allusual energy conditions) is violated in models with w <�1, the authors are able to introduce a classically stablemodel involving a phantom field. However, due to thepeculiar kinetic energy of the phantom, the model isunstable against any quantum process involving it. Inparticular, as the phantom potential is unbounded frombelow, there are catastrophic vacuum decays [9].Furthermore, some general results [10] suggest that anyminimally coupled theory with w < �1 has spatial gra-dient instabilities that would be ruled out by cosmicmicrowave background observations. These results putsevere doubts on the viability of dark energy modelsbased on phantom fields.

In this Brief Report, we notice that certain quintessen-tial models can exhibit a generic asymptotic de Sitterphase with many solutions for which w is slowly evolvingand lesser than �1, without the introduction of any(classical or quantum) instability. The quintessential fieldis assumed to be nonminimally coupled to gravity

S �Z

d4x��������g

pfF���R � @a�@a� � 2V���g; (2)

where F��� � 1� ��2, � < 0, and to have an exponen-tial self-interaction potential

V��� � Ae��: (3)

02-1 2004 The American Physical Society

BRIEF REPORTS PHYSICAL REVIEW D 70 087302

Exponential potentials have been used recently in cos-mology, mainly in connection with tracker fields [11].They appear naturally in higher dimensional theoriesand string inspired models [12]. Nonminimally coupledmodels are also commonly adopted. In [13], for instance,a quintessential model with � � 1=6 (conformal cou-pling) and V��� � m

2 �2 � 4 �4 was introduced. With

such potential, however, conformally coupled modelsnever exhibit asymptotic de Sitter phases. Moreover,models with coupling F���R are generically singular onthe hypersurfaces F��� � 0 [14], precluding the viabilityof quintessential models with conformal coupling. Someproposals to circumvent this singularity with the inclu-sion of higher order gravitational terms in the action hasbeen suggested [15], but the viability of the resultingmodels is still unclear. In this work, we avoid this singu-larity by choosing hereafter � < 0. Nonminimal cou-plings of the type F��� � 1� j�j��2 � �2

0� has beenconsidered in [16] to study tracking behavior for poten-tials V��� � M4��=��, � > 0. The case of F��� �j�j�2 was considered in [17]. As we will see, despitethat the model given by (2) is known to violate weakenergy condition [18], the phase space for the comovingframe presents large regions of stability.

II. THE MODEL

The Einstein equations obtained from the action (2) are

FGab � ra�rb� �gab

2�rc�rc� � 2V � 2�F�

�rarbF; (4)

while the Klein-Gordon equation is

�� � V 0 �1

2F0R � 0: (5)

A relevant issue here is how to define an equation of statefor the field � from the Eqs. (4) and (5). The right-handside (RHS) of (4) does not correspond to a covariantlyconserved energy momentum tensor due to the presenceof F��� in its left-hand side (LHS). In order to define thepressure and energy for the field � in a consistent waywith the continuity equation, one needs a covariantlyconserved energy momentum tensor, and from (4) wecan get Gab � Tab with [19]

Tab � ra�rb� �gab

2�rc�rc� � 2V � 2�F�

�rarbF � �1� F�Gab: (6)

Assuming an isotropic and spatially flat universe,

ds2 � �dt2 � a2�t��dx2 � dy2 � dz2�; (7)

we get, from the temporal component of the EinsteinEq. (4), the energy constraint

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3H�FH � F0 _�� �_�2

2� V���; (8)

where H � _a=a. From the spatial components, one getsthe modified Friedmann equation

�2F1_H � 3 F � 2�F0�2�H2 �

1� 2F00

2_�2 � V

��H _� � V 0�F0; (9)

where F1��� � F��� � 32 F

0����2. For the metric (7), theKlein-Gordon equation reads

�� �G��; _�; H�

F1

_� � V 0eff��� � 0; (10)

where

G��; _�; H� � 3F1H �1

2�1� 3F00�F0 _� (11)

and

V 0eff��� �

1

F1�FV0 � 2F0V�: (12)

The fixed points of the Eqs. (8)–(10) are the constantsolutions ��t� � �� and H�t� � �H, corresponding to deSitter solutions a�t� / e �Ht for which w � �1. Despitethat the potential (3) has no equilibrium points, thanksto the nonminimal coupling, the model has indeed thefixed points � ���;� �H�� and � ���;� �H��, where

��� � �2

�1�

�����������������������1� 2=4�

q� (13)

and

�H 2� �

V� ����

3F� ����: (14)

The fixed points exist, of course, only for � � �2=4.The next section is devoted to the study of the phase spaceof this model. As we will see, the fixed point � ���; �H�� isan attractor; large sets of solutions tend spontaneously tothis de Sitter phase, irrespective of their initial condi-tions.When approaching the de Sitter point, solutions canhave w < �1, without any induced instability.

We finish this section with the definition of the pressurep� and energy ��. These quantities are defined from theenergy momentum tensor as Tab � �� � p�uaub � pgab,where ua is a globally timelike vector.With the hypothesisof isotropy and homogeneity, we get, in the comovingframe, �� � 3H2 and p� � ��2 _H � 3H2�, leading to

w � �

�1�

2

3

_H

H2

�: (15)

As one can see, the ratio w defined in (15) is not subject tothe restriction w � �1. Note that the continuity equation_�� � 3H��� � p�� � 0, which is a direct consequence

of the Bianchi identities for the Einstein equations, holdshere.

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FIG. 2. Aspect of the potential (solid line) Veff��� for �� � � 1. The contours correspond to the lines of constantL��; _�� (17). Since they are closed around the fixed point���, the condition _L < 0 along solutions implies that they

tend asymptotically to ���.

BRIEF REPORTS PHYSICAL REVIEW D 70 087302

III. THE PHASE SPACE

We study the phase space of the model by using thesame semianalytical approach used in [13]. The phasespace is three dimensional ��; _�; H�, but due to theenergy constraint (8) the dynamics are restricted to atwo dimensional submanifold. No restrictions are im-posed on the ��; _�� phase portrait, but for the ��; H�one, only the region H2 � V=3F1 is dynamically allowed(see Fig. 1). Since H � 0 is not allowed by the dynamics,the trajectories are confined to semispaces H > 0 andH < 0. We are concerned here only with H > 0. Weremind that real trajectories move on the two dimensionalmanifold defined by the energy constraint (8). Therefore,in the projection on the plane ��; H�, each point on theallowed region corresponds, in fact, to two possible val-ues for _� (two ‘‘sheets’’), with the exception of the linesH2 � V=3F1, where only one value for _� is allowed.Solutions always ‘‘cross’’ from one sheet to another tan-gentially to the lines H2 � V=3F1.

The aspect of Veff��� obtained from (12) is crucial tothe identification of the attractor points in the phase space.From (12), one has

V 0eff��� � A�e�� �� � ������ � ����

1� ��1� 6���2 ; (16)

which can be integrated by parts in terms of theExponential Integral function Ei�x� [20]. From Fig. 2, itis clear that the fixed point corresponding to ��� isunstable, while ��� should correspond to a stable one.However, despite the clear fact that ��� is a minimum ofVeff , one cannot conclude safely that it indeed corre-sponds to an attractor due to the function G��; _�; H�

given by (11). If G��; _�; H� � 0, the solutions of theKlein-Gordon equation around the point ��� will besimple dumped oscillations. This can be checked by in-troducing the function

L��; _�� �_�2

2� Veff���; (17)

and noticing that _L � �G _�2 along the solutions of theKlein-Gordon Eq. (10). Provided that G��; _�; H� � 0,

FIG. 1 (color online). The fixed points for the Eqs. (8)–(10)on the plane ��; H�. The shaded region (H2 < V=3F1) isdynamically inaccessible. For this graphic, �� � � 1.

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the function L��; _�� is a Lyapunov function [13] forthe fixed point ���, assuring its stability. From theEq. (9), we see that when ��t� approaches ���, _H ! 0and H�t� approaches �H�, establishing the attractor char-acter of the fixed point � ���; �H��.

From (11), we have

G��; _�; H� � 3�F���H �

1� 6�6

F0��� _��

�9

2 F0����2H; (18)

and from the energy constraint (8), one has that F���H �

F0��� _� > 0 on the semispace H > 0, implying the pos-itivity of G and, hence, establishing the attractive char-acter of the fixed point � ���; �H��, for � > �5=6.However, this is a very conservative lower bound for �.Our exhaustive numerical simulations suggest that it canbe considerably smaller. We could verify the attractivecharacter of the fixed point � ���; �H�� even for � < �100,suggesting that eventual amplifications forthcoming fromthe G��; _�; H� < 0 regions are not enough to win thepotential Veff around ���. A typical phase portrait isdis-

FIG. 3 (color online). Typical phase portrait ��; H�, corre-sponding to the case �� � A � � 1. Because of the sym-metries of Eq. (8)–(10), the H < 0 trajectories are obtainedfrom the H > 0 ones by a reflection on H � 0 and reversion ofthe arrows (time reversal operation).

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FIG. 4. The parameter w along some solutions presented inthe phase portrait of Fig. 3. According to (15), solutions havew < �1 when _H > 0, and w > �1 when _H < 0.

BRIEF REPORTS PHYSICAL REVIEW D 70 087302

played in Fig. 3, corresponding to the case �� � A � � 1. The attraction basin is considerably larger thanthe conservative estimative base on the closed lines ofconstant L around the fixed point. Note that all solutionsstarting with � > ��� are runaway solutions. Figure 4shows the curves w�t� for some solutions presented inFig. 3. According to (15), solutions approaching the fixedpoint from below have w < �1, while the ones doingfrom above have w > �1.

IV. CONCLUSION

We show here that nonminimally coupled quintessen-tial models with exponential potentials can exhibit

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asymptotic de Sitter phases for large sets of initial con-ditions. Some of these phases are characterized by aslowly evolving parameter w � p�=�� < �1, compati-ble, in principle, with the recent observational data. Ouranalysis is based on the existence of a Lyapunov functionfor the Klein-Gordon equation that can be used to esti-mate the attraction basin of the relevant fixed points. As itwas already mentioned, real attraction basins are typi-cally much larger than these estimations. Our results arein agreement with the linearized analysis recently pro-posed in [21].

We stress that the model presented here is free fromthe instabilities that are usually associated to phantommodels. Classically, since F��� � 0 , the model is notplagued with the anisotropic singularities described in[14]. Besides, since F��� is always positive and V��� isbounded from below, the model is also free from thequantum instabilities described in [9] for phantom fields.

We conclude, therefore, that it is possible, in principle,to construct realistic models for dark energy with w <�1. Relevant issues now are the introduction of matterfields and the study of the role played by the strong non-minimal coupling regime [17,22] in the model. Thesepoints are under investigation.

ACKNOWLEDGMENTS

This work was supported by FAPESP and CAPES.

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