Dynamics of quintom and hessence energies in loop quantum cosmology

Dynamics of quintom and hessence energies in loop quantum cosmology

Hao Wei*Department of Physics and Tsinghua Center for Astrophysics, Tsinghua University, Beijing 100084, China

Shuang Nan ZhangDepartment of Physics and Tsinghua Center for Astrophysics, Tsinghua University, Beijing 100084, China

Key Laboratory of Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, ChinaPhysics Department, University of Alabama in Huntsville, Huntsville, Alabama 35899, USA

(Received 30 May 2007; published 12 September 2007)

In the present work, we investigate the universe dominated by quintom or hessence energies in loopquantum cosmology (LQC). Interestingly enough, we find that there are some stable attractors in thesetwo cases. In the case of quintom, all stable attractors have the feature of decelerated expansion. In thecase of hessence, most of stable attractors have the feature of decelerated expansion while one stableattractor can have decelerated or accelerated expansion depend on the model parameter. In all cases, theequation-of-state parameter (EoS) of all stable attractors are larger than �1 and there is no singularity inthe finite future. These results are different from the dynamics of phantom in LQC, or the ones ofphantom, quintom, and hessence in classical Einstein gravity.

DOI: 10.1103/PhysRevD.76.063005 PACS numbers: 95.36.+x, 04.60.Pp, 98.80.�k

I. INTRODUCTION

Dark energy [1] has been one of the most active fields inmodern cosmology since the discovery of accelerated ex-pansion of our universe [2–8]. In the observational cos-mology of dark energy, equation-of-state parameter (EoS)wde � pde=�de plays an important role, where pde and �deare the pressure and energy density of dark energy, respec-tively. Recently, evidence for wde�z�<�1 at redshift z <0:2� 0:3 has been found by fitting observational data (see[9–16] for examples). In addition, many best-fits of thepresent value of wde are less than �1 in various datafittings with different parameterizations (see [17] for arecent review). The present data seem to slightly favor anevolving dark energy with wde being below �1 aroundpresent epoch from wde >�1 in the near past [10].Obviously, the EoS cannot cross the so-called phantomdivide wde � �1 for quintessence or phantom alone.Some efforts have been made to build dark energy modelwhose EoS can cross the phantom divide (see for examples[10,18–35] and references therein).

In Ref. [10], Feng, Wang and Zhang proposed a so-called quintom model which is a hybrid of quintessenceand phantom (thus the name quintom). It is one of thesimplest modeles whose EoS can cross the phantom divide.The cosmological evolution of the quintom dark energywas studied in Refs. [23,24]. Perturbations of the quintomdark energy were investigated in Refs. [36,37]; and it isfound that the quintom model is stable when EoS crosses�1, in contrast to many dark energy models whose EoScan cross the phantom divide [28]. Other works concerningquintom also include [31] for examples. In [18], by a newview of quintom dark energy, one of us (H.W.) and his

collaborators proposed a novel noncanonical complex sca-lar field, which was named ‘‘hessence,’’ to play the role ofquintom. In the hessence model, the phantomlike role isplayed by the so-called internal motion _�, where � is theinternal degree of freedom of hessence. The transition fromw>�1 to w<�1 or vice versa is also possible in thehessence model [18]. The cosmological evolution of thehessence dark energy was studied in [19] and then wasextended to the more general cases in Ref. [20]. The w-w0

analysis of hessence dark energy was performed inRef. [21]. In Ref. [30], the method to reconstruct hessencedark energy was proposed. We will briefly review the mainpoints of quintom and hessence energies in Sec. II.

In fact, many works by now are considered in the frame-work of classical Einstein gravity. However, it is com-monly believed that gravity should also be quantized,like other fundamental forces. As well-known, for manyyears, the string theory is the only promising candidate forquantum gravity. In the recent decade, however, the loopquantum gravity (LQG) (see e.g. Refs. [38–41] for re-views) has became a competitive alternative to the stringtheory. LQG is a leading background independent, non-perturbative approach to quantum gravity. At the quantumlevel, the classical spacetime continuum is replaced by adiscrete quantum geometry and the operators correspond-ing to geometrical quantities have discrete eigenvalues.

Loop quantum cosmology (LQC) (see e.g. Refs. [42–44] for reviews) restricts the analysis of LQG to the homo-geneous and isotropic spacetimes. Recent investigationshave shown that the discrete quantum dynamics can bevery well approximated by an effective modifiedFriedmann dynamics [45–47]. There are two types ofmodification to the Friedmann equation due to loop quan-tum effects [44,48]. The first one is based on the modifi-cation to the behavior of inverse scale factor operator*[email protected]

PHYSICAL REVIEW D 76, 063005 (2007)

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below a critical scale factor a�. So far, most of the LQCliterature has used this one. Many interesting results havebeen found, for instance, the replacement of the classicalbig bang by a quantum bounce with desirable features[43,45,49], avoidance of many singularities [50], easierinflation [51,52], correspondence between LQC and brane-world cosmology [53], and so on. However, as shown in,e.g. Ref. [54], the first type of modification to Friedmannequation suffers from gauge dependence which can not becured and thus lead to unphysical effects. In a recent paper[55], Magueijo and Singh provided very sharp results toshow that it is only for the closed model that such mod-ifications can be sensible and for flat models they make nosense.

The second type of modification to Friedmann equationwas discovered very recently [44,48,54], which essentiallyencodes the discrete quantum geometric nature of space-time. The corresponding effective modified Friedmannequation in a flat universe is given by [44,48,54,56–59]

H2 ��2�

3

�1�

��c

�; (1)

where H � _a=a is the Hubble parameter; a dot denotes thederivative with respect to cosmic time t, and a is the scalefactor; � is the total energy density; �2 � 8�G; and thecritical density reads

�c �

��3p

16�2�3G2@; (2)

where � is the dimensionless Barbero-Immirzi parameter(it is suggested that � ’ 0:2375 by the black hole thermo-dynamics in LQG [60]). Differentiating Eq. (1) and usingthe energy conservation equation

_�� 3H�� p� � 0; (3)

one obtain the effective modified Raychaudhuri equation[48,56–59]

_H � ��2

2�� p�

�1� 2

��c

�; (4)

where p is the total pressure. Actually, as shown inRef. [55], the effective modified Raychaudhuri equationcan be derived directly by using the Hamilton’s equationsin LQC, without assuming the energy conservation equa-tion. It is easy to check that only two of Eqs. (1), (3), and(4) are independent of each other, and the third one can bederived from these two. By using the second type ofmodification to Friedmann equation, the physically appeal-ing features of the first type are retained [44], for instance,resolution of big bang singularity [44,54], avoidance of bigrip and other singularities [56–58], inflation in LQC [59],correspondence between LQC and braneworld cosmology[48], and so on.

For the universe with a large scale factor, the first type ofmodification to the effective Friedmann equation can be

neglected and only the second type of modification isimportant [48], while the matter Hamiltonian HM andthe corresponding expressions for energy density and pres-sure retain the same classical forms [44,48,54,56–58,61].This is the particular case which we will consider here.

In the present work, we will investigate the universedominated by quintom or hessence in LQC. Following[48,56–58], we use the method of dynamical system[62]. After a brief review of quintom and hessence energiesin Sec. II, we consider the dynamics of quintom andhessence in Sec. III and IV respectively. Interestinglyenough, we find that there are some stable attractors inthese two cases. In the case of quintom, all stable attractorshave the feature of decelerated expansion. In the case ofhessence, most of stable attractors have the feature ofdecelerated expansion while one stable attractor can havedecelerated or accelerated expansion depend on the modelparameter. In all cases, the EoS of all stable attractors arelarger than �1 and there is no singularity in the finitefuture. These results are different from the dynamics ofphantom in LQC [57,58], or the ones of phantom, quintomand hessence in classical Einstein gravity [19,20,23,24,63].

II. QUINTOM AND HESSENCE ENERGIES

A. Quintom energy

Phenomenologically, one may consider the Lagrangiandensity for quintom [10,23,24]

L q �12�@��1�

2 � 12�@��2�

2 � V��1; �2�; (5)

where �1 and �2 are two real scalar fields and play theroles of quintessence and phantom, respectively.Considering a spatially flat Friedmann-Robertson-Walker(FRW) universe and assuming the scalar fields �1 and �2

are homogeneous, one obtains the effective pressure andenergy density for the quintom, i.e.

p � 12

_�21 �

12

_�22 � V��1; �2�;

� � 12

_�21 �

12

_�22 � V��1; �2�;

(6)

respectively. The corresponding effective EoS is given by

w �p��

_�21 �

_�22 � 2V��1; �2�

_�21 �

_�22 � 2V��1; �2�

: (7)

It is easy to see that w �1 when _�21

_�22 while w<

�1 when _�21 < _�2

2. We consider the simplest quintomwhose V��1; �2� � V1��1� � V2��2� in the presentwork. In this case, the equations of motion for �1 and�2 are given by

�� 1 � 3H _�1 �dV1

d�1� 0; (8)

�� 2 � 3H _�2 �dV2

d�2� 0: (9)

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B. Hessence energy

Following Refs. [18,19,30], we consider a noncanonicalcomplex scalar field, namely, the hessence,

� � �1 � i�2; (10)

with a Lagrangian density

L h �14�@��2 � �@��2� �U��2 ��2�

� 12�@��

2 ��2�@��2� � V��; (11)

where we have introduced two new variables ��; �� todescribe the hessence, i.e.

�1 � � cosh�; �2 � � sinh�; (12)

which are defined by

�2 � �21 ��

22; coth� �

�1

�2: (13)

In fact, it is easy to see that in terms of �1 and �2, thehessence can be regarded as a special case of quintom withgeneral V��1; �2�. Considering a spatially flat FRW uni-verse with scale factor a�t� and assuming � and � arehomogeneous, from Eq. (11) we obtain the equations ofmotion for � and �,

�� 3H _�� _�2 �dVd�� 0; (14)

�2 �� 2� _�� 3H�2� _� � 0: (15)

The pressure and energy density of the hessence are

p � 12�

_�2 ��2 _�2� � V��;

� � 12�

_�2 ��2 _�2� � V��;(16)

respectively. Equation (15) implies

Q � a3�2 _� � const: (17)

which is associated with the total conserved charge withinthe physical volume due to the internal symmetry [18,19].It turns out that

_� �Q

a3�2 : (18)

Substituting into Eqs. (14) and (16), they can be rewrittenas

�� 3H _��Q2

a6�3�dVd�� 0; (19)

p �1

2_�2 �

Q2

2a6�2� V��;

� �1

2_�2 �

Q2

2a6�2� V��:

(20)

Noting that the EoS w � p=�, it is easy to see that w

�1 when _�2 Q2=�a6�2�, while w<�1 when _�2 <Q2=�a6�2�. We refer to the original papers [18,19,30] formore details.

III. DYNAMICS OF QUINTOM ENERGY IN LQC

In this section, we consider the universe dominated byquintom energy in LQC. Following Refs. [23,24,57,58,64],we introduce these five dimensionless variables

x1 �� _�1��

6pH; x2 �

� _�2��6pH; y1 �

��V1

p��3pH;

y2 ��

��V2

p��3pH; z �

��c:

(21)

We introduce z just for convenience. It is expected that z isnot independent, because of Eq. (6). In fact, in the case ofthe universe dominated by quintom energy in LQC, theeffective modified Friedmann equation, namely, Eq. (1),can be rewritten as

�x21 � x

22 � y

21 � y

22��1� z� � 1: (22)

Thus, one can explicitly express z in terms of x1, x2, y1 andy2. Notice that 0 � z � 1 is required by the positiveness of� andH2 in Eq. (1). In addition, by using Eq. (6), we recastEq. (4) as

�_H

H2 � 3�x21 � x

22��1� 2z�; (23)

which will be used extensively. Also, Eq. (7) becomes

w �x2

1 � x22 � y

21 � y

22

x21 � x

22 � y

21 � y

22

: (24)

In this work, we consider the case of quintom withexponential potentials

V1��1� � V�1e��1��1 ; V2��2� � V�2

e��2��2 ; (25)

where �1 and �2 are dimensionless constants. Without lossof generality, we choose �1 and �2 to be positive, since wecan make them positive through field redefinition �1 !��1, �2 ! ��2 if �1 and �2 are negative. By the help ofEqs. (3), (6), and (22)–(25), the equations of motion for�1

and �2, namely, Eqs. (8) and (9), can be rewritten as anautonomous system

x01 � 3x1�A� 1� �

��3

2

s�1y2

1; (26)

x02 � 3x2�A� 1� �

��3

2

s�2y

22; (27)

y01 � 3y1

�A�

�1��6p x1

�; (28)

DYNAMICS OF QUINTOM AND HESSENCE ENERGIES IN . . . PHYSICAL REVIEW D 76, 063005 (2007)

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y02 � 3y2

�A�

�2��6p x2

�; (29)

z0 � �6z�x21 � x

22��1� z�; (30)

where a prime denotes the derivative with respect to the so-called e-folding time N � lna, and

A � �x21 � x

22��1� 2z�

� �x21 � x

22�2�x

21 � x

22 � y

21 � y

22��1 � 1�; (31)

in which we have used Eq. (22). We can obtain the criticalpoints � �x1; �x2; �y1; �y2; �z� of the autonomous systemEqs. (26)–(30) by imposing the conditions �x01 � �x02 ��y01 � �y02 � �z0 � 0. Of course, they are subject to theFriedmann constraint Eq. (22), namely � �x2

1 � �x22 � �y2

1 ��y2

2��1� �z� � 1. We present the critical points and theirexistence conditions in Table I. In fact, there are otherfour critical points with

�x 1 ��1�

22��

6p��2

2 � �21�; �x2 �

�21�2��

6p��2

2 � �21�;

�y1 � �2��rqp

; �y2 � �1��rqp

; �z � 0;

where

rq �6�2

2 � �21�6� �

22�

6��22 � �

21�

2 :

However, the existence of �y1 and �y2 requires rq � 0. In thiscase, these four critical points reduce to points (Q2p) or(Q4p).

To study the stability of the critical points for the au-tonomous system Eqs. (26)–(30), we substitute linear per-turbations x1 ! �x1 � x1, x2 ! �x2 � x2,y1 ! �y1 � y1, y2 ! �y2 � y2, and z! �z� z aboutthe critical point � �x1; �x2; �y1; �y2; �z� into the autonomoussystem Eqs. (26)–(30) and linearize them. Because of theFriedmann constraint (22), there are only four independent

evolution equations, i.e.

x01 � 3�x1A� 3� �A� 1�x1 ��6p�1 �y1y1; (32)

x02 � 3�x2A� 3� �A� 1�x2 ��6p�2 �y2y2; (33)

y1 � 3�y1

�A�

�1��6p x1

�� 3

��A�

�1��6p �x1

�y1; (34)

y2 � 3�y2

�A�

�2��6p x2

�� 3

��A�

�2��6p �x2

�y2; (35)

where

�A � � �x21 � �x2

2�2� �x21 � �x2

2 � �y21 � �y2

2��1 � 1�; (36)

A��4� �x21� �x2

2�� x21� �x2

2� �y21� �y2

2��2� �x1x1� �x2x2

� �y1y1� �y2y2� � 22� �x21� �x2

2� �y21� �y2

2��1� 1�

� � �x1x1� �x2x2�: (37)

The four eigenvalues of the coefficient matrix of Eqs. (32)–(35) determine the stability of the critical point. We presentthe corresponding eigenvalues for the critical points inTable II. It is easy to see that points (Q1m), (Q2m), (Q3),(Q4m), and (Q5) are unstable. Point (Q1p) exists and isstable under conditions �x1 > 1, �1 �x1

��6p

and

�2

��x2

1 � 1q

��6p

. Point (Q2p) exists and is stable under

conditions �1 ��6p

and �2

��9��2

1 � 3=2q

3. Point (Q4p)

exists and is stable under condition �1

��9��2

2 � 3=2q

3.The three stable attractors (Q1p), (Q2p) and (Q4p) have

the common features �y1 � �y2 � �z � 0 and �x21 � �x2

2 � 1.From Eq. (24), we get the EoS w � 1, which implies thatthe quintom behave as a stiff fluid. From Eq. (23),� _H=H2 � 3. Then, we find that H � t�1=3 (the integralconstant can be set to zero by redefining the time). Thus,we obtain a / t1=3. From Eq. (3) and w � 1, we find that

TABLE I. Critical points for the autonomous systemEqs. (26)–(30) and their existence conditions.

Label Critical point � �x1; �x2; �y1; �y2; �z� Existence

Q1p �x21 1,

��x2

1 � 1q

, 0, 0, 0 �x21 1

Q1m �x21 1, �

��x2

1 � 1q

, 0, 0, 0 �x21 1

Q2p��6p

�1,

��6�2

1� 1

q, 0, 0, 0 �1 �

��6p

Q2m��6p

�1, �

��6�2

1� 1

q, 0, 0, 0 �1 �

��6p

Q3 �1��6p , 0,

��1�

�21

6

q, 0, 0 �1 �

��6p

Q4p��1� 6

�22

q,��6p

�2, 0, 0, 0 Always

Q4m ��1� 6

�22

q,��6p

�2, 0, 0, 0 Always

Q5 0, � �2��6p , 0,

��1�

�22

6

q, 0 Always

TABLE II. The corresponding eigenvalues for the criticalpoints of the autonomous system Eqs. (26)–(30).

Point Eigenvalues

Q1p �6, 0, 3��32

q�1 �x1, 3� �2

��32 � �x

21 � 1�

qQ1m �6, 0, 3�

��32

q�1 �x1, 3� �2

��32 � �x

21 � 1�

qQ2p �6, 0, 0, 3� �2

��9��2

1 �32

qQ2m �6, 0, 0, 3� �2

��9��2

1 �32

qQ3 ��2

1, �21

2 , 12 ��

21 � 6�, 1

2 ��21 � 6�

Q4p �6, 0, 0, 3� �1

��9��2

2 �32

qQ4m �6, 0, 0, 3� �1

��9��2

2 �32

qQ5 �

�22

2 , �22, � 1

2 ��22 � 6�, � 1

2 ��22 � 6�


063005-4

� / a�6 / t�2. The universe undergoes decelerated ex-pansion and there is no singularity in the finite future.

IV. DYNAMICS OF HESSENCE ENERGY IN LQC

In this section, we consider the universe dominated byhessence energy in LQC. Similar to the case of quintom,following Refs. [19,20,57,58,64], we introduce these fivedimensionless variables

x �� _��6pH; y �

��Vp

��3pH; z �

��c;

u �

��6p

��; v �

��6pH

Q

a3�:

(38)

Again, we introduce z just for convenience, since it isexpected that z is not independent due to Eq. (20). Infact, in the case of the universe dominated by hessenceenergy in LQC, the effective modified Friedmann equation,namely, Eq. (1), can be rewritten as

�x2 � y2 � v2��1� z� � 1; (39)

which can be used to explicitly express z in terms of x, yand v. Notice that 0 � z � 1 is required by the positive-ness of � andH2 in Eq. (1). By using Eq. (20), the effectivemodified Raychaudhuri equation, namely, Eq. (4), can berewritten as

�_H

H2 � 3�x2 � v2��1� 2z�: (40)

From Eq. (20), the EoS of hessence is given by

w �p��x2 � v2 � y2

x2 � v2 � y2 : (41)

In this work, we consider the case of hessence withexponential potential

V�� V�e��; (42)

where � is a dimensionless constant. Without loss of gen-erality, we choose � to be positive, since we can make itpositive through field redefinition �! �� if � is nega-tive. By the help of Eqs. (3), (20), and (39)–(42), theequation of motion for �, namely, Eq. (19), can be rewrit-ten as an autonomous system

x0 � 3x�B� 1� � uv2 �

��3

2

s�y2; (43)

y0 � 3y�B�

��6p x

�; (44)

z0 � �6z�x2 � v2��1� z�; (45)

u0 � �xu2; (46)

v0 � 3v�B� 1�

1

3xu�; (47)

where

B � �x2 � v2��1� 2z�

� �x2 � v2�2�x2 � y2 � v2��1 � 1�; (48)

in which we have used Eq. (39). We can obtain the criticalpoints � �x; �y; �z; �u; �v� of the autonomous system Eqs. (43)–(47) by imposing the conditions �x0 � �y0 � �z0 � �u0 � �v0 �0. Of course, they are subject to the Friedmann constraintEq. (39), namely � �x2 � �y2 � �v2��1� �z� � 1. We presentthe critical points and their existence conditions inTable III.

To study the stability of the critical points for the au-tonomous system Eqs. (43)–(47), we substitute linear per-turbations x! �x� x, y! �y� y, z! �z� z,u! �u� u, and v! �v� v about the critical point� �x; �y; �z; �u; �v� into the autonomous system Eqs. (43)–(47)and linearize them. Because of the Friedmann constraint(39), there are only four independent evolution equations,i.e.

x0 � 3�xB� 3� �B� 1�x� 2 �u �vv� �v2u

��6p� �yy; (49)

y0 � 3�y�B�

��6p x

�� 3

��B�

��6p �x

�y; (50)

u0 � �2�x �uu� �u2x; (51)

v0 � 3 �vB� 13� �xu� �ux�� 3� �B� 1� 1

3 �x �u�v;

(52)

where

�B � � �x2 � �v2�2� �x2 � �y2 � �v2��1 � 1�; (53)

B � �4� �x2 � �v2�� x2 � �y2 � �v2��2� �xx� �yy� �vv�

� 22� �x2 � �y2 � �v2��1 � 1�� xx� �vv�: (54)

The four eigenvalues of the coefficient matrix of Eqs. (49)–(52) determine the stability of the critical point. We presentthe corresponding eigenvalues for the critical points in

TABLE III. Critical points for the autonomous systemEqs. (43)–(47) and their existence conditions.

Label Critical point � �x; �y; �z; �u; �v� Existence

H1 �x2 1, 0, 0, 0, ��x2 � 1p

�x2 1H2p 1, 0, 0, 0, 0 AlwaysH2m �1, 0, 0, 0, 0 Always

H3��6p

� , 0, 0, 0, ��6�2 � 1

q� �

��6p

H4 ��6p ,

��1� �2

6

q, 0, 0, 0 � �

��6p


063005-5

Table IV. It is easy to see that Point (H1) exists and is stableunder conditions �x 1 and � �x

��6p

; Point (H2p) is stableunder condition �

��6p

; Point (H2m) is unstable; Points(H3) and (H4) are stable when they exist under condition� �

��6p

.The stable attractors (H1), (H2p) and (H3) have the

common features �y � �z � �u � 0 and �x2 � �v2 � 1. FromEq. (41), it is easy to find that the EoS w � 1, whichimplies that the hessence behave as a stiff fluid. FromEq. (40), � _H=H2 � 3. Then, we find that H � t�1=3(the integral constant can be set to zero by redefining thetime). Thus, we obtain a / t1=3. From Eq. (3) and w � 1,we find that � / a�6 / t�2. The universe undergoes decel-erated expansion and there is no singularity in the finitefuture.

The stable attractor (H4) is slightly different from otherthree stable attractors. From Eq. (41), we get the EoS w ��1� �2=3 �1. Note that w �1=3 for

��2p� � ��

6p

, while w<�1=3 for � <��2p

. From Eq. (40),� _H=H2 � �2=2. Then, we find that H � 2t�1=�2 (theintegral constant can be set to zero by redefining thetime). Thus, we obtain a / t2=�

2. From Eq. (3) and w �

�1� �2=3, we find that � / a��2/ t�2. The universe

experiences decelerated expansion for��2p� � �

��6p

, oraccelerated expansion for � <

��2p

. However, the universecannot undergo superaccelerated expansion ( _H > 0) forany �. Therefore, there is no singularity in the finite futurefor any case.

V. CONCLUSION

In the framework of classical Einstein gravity, the dy-namics of phantom, quintom and hessence have been

studied in literature [19,20,23,24,63]. In the case of phan-tom, the universe will end in a big rip singularity [63]. Inthe case of quintom without direct couping between�1 and�2 [23], or with a special interaction between �1 and �2

through Vint � V1��1�V2��2��1=2 [24], the phantom-

dominated solution is the unique attractor and the big ripis inevitable. In the case of hessence [19,20], however, thebig rip can be avoided, while its attractors are slightlydifferent from the ones of the present work.

In the framework of LQC, the dynamics of phantom hasalso been studied [57,58]. It is found that there is no stableattractor in this case. Therefore, the phase trajectory is verysensitive to initial conditions. However, the big rip can beavoided and the universe finally enters oscillatory regime.This is mainly due to the quantum correction to Friedmannequation.

In the present work, we investigate the universe domi-nated by quintom or hessence energies in loop quantumcosmology (LQC). Interestingly enough, we find that thereare some stable attractors in these two cases. In the case ofquintom, all stable attractors have the feature of deceler-ated expansion. In the case of hessence, most of stableattractors have the feature of decelerated expansion whileone stable attractor can have decelerated or acceleratedexpansion depend on the model parameter. In all cases, theequation-of-state parameter (EoS) of all stable attractorsare larger than �1 and there is no singularity in the finitefuture. These results are different from the dynamics ofphantom in LQC, or the ones of phantom, quintom andhessence in classical Einstein gravity.

ACKNOWLEDGMENTS

We are grateful to Professor Rong-Gen Cai for helpfuldiscussions. We also thank Xin Zhang, Zong-Kuan Guo,Hui Li, and Sumin Tang, Shi-Chao Tang, Jian Hu, YueShen, Xin Liu, Lin Lin, Jing Jin, Wei-Ke Xiao, Feng-YunRao, Nan Liang, Rong-Jia Yang, Jian Wang, and Yuan Liufor kind help and discussions. We acknowledge partialfunding support from China Postdoctoral ScienceFoundation, and by the Ministry of Education of China,Directional Research Project of the Chinese Academy ofSciences under Project No. KJCX2-YW-T03, and by theNational Natural Science Foundation of China underProject No. 10521001.

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TABLE IV. The corresponding eigenvalues for the criticalpoints of the autonomous system Eqs. (43)–(47).

Point Eigenvalues

H1 �6, 0, 0, 3��32

q� �x

H2p �6, 0, 0, 3��32

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Dynamics of quintom and hessence energies in loop quantum cosmology