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P

HYSICAL

J

OURNAL

C

Regular Article – Theoretical Physics

Anisotropic perturbation of de Sitter space

W.F. Kaoa

Institute of Physics, Chiao Tung University, Hsinchu, Taiwan Received: 19 July 2007 /

Published online: 16 October 2007−© Springer-Verlag / Societ`a Italiana di Fisica 2007

Abstract. A model-independent expression for the Friedmann equation in Bianchi type spaces is derived. In addition, a model-independent stability analysis of the higher curvature de Sitter solution is discussed. Stability conditions of the de Sitter solution are derived explicitly for a cubic model with interesting effects. It is known that quadratic terms do not contribute to this de Sitter background solution. Higher curvature terms are all critical to the stability of the de Sitter space.

PACS. 98.80.Cq; 04.20.-q; 04.20.Cv

1 Introduction

Our universe is known to be homogeneous and isotropic [1, 2]. Such an universe is described by the well-known Friedmann–Robertson–Walker (FRW) metric [3–6]. There have been, however, some cosmological problems associ-ated with the standard big bang model responsible for the evolution of our present universe. Inflationary models pro-vide resolutions to these problems [7–10]. It is therefore important to find out whether the de Sitter background is a stable final state for any candidate model.

Higher curvature terms should be relevant to the sta-bility of the inflationary physics at the high energy re-gion [11, 12]. Higher curvature terms are effective theories as quantum corrections of matter fields [13–16]. Therefore, the higher curvature effect on the stability of inflation de-serves more attention [17–19]. In order to survey possible constraints on the existing models, a model-independent method has been very helpful for the stability analysis of pure gravity theories [14–16, 20, 21].

The stability problem has been discussed for general relativity with a scalar field [22]. Pure gravity models with quadratic curvature terms have also been discussed previ-ously [23–27]. Solutions that do not approach a de Sitter space were found in [23–25]. Instead, we will focus on the stability of de Sitter space against anisotropic perturb-ations. For simplicity, we will also focus on the stability problem of higher curvature theory with a scalar field. A cubic curvature model will be presented as a simple demonstration. Quadratic terms are known to be irrele-vant to the de Sitter solution expansion scale (H = H0) in

de Sitter space for pure gravity theories. These quadratic terms are, however, important to the stability of the de Sit-ter space.

a e-mail: gore@mail.nctu.edu.tw

Note that anisotropic perturbation equations of FRW space are identical to the perturbation equation of aniso-tropic Bianchi spaces. In addition, relative equations are similar for all Bianchi spaces [14–16, 20, 21]. The latest ob-servation also indicates that the physical universe is a flat space. Therefore, we will focus on the perturbation equa-tion of Bianchi type I (BI) space in this paper.

Field equations will be derived in Sect. 2. The perturba-tion equaperturba-tion and model-independent stability condiperturba-tions will be shown in Sect. 3. In Sect. 4, we will focus on the ef-fect of a model with both cubic and quadratic curvature terms. Finally, we will draw some conclusions.

2 Field equations in BI space

The latest observation indicates that our universe is close to the flat FRW space. Therefore, we will focus on the sta-bility analysis of flat FRW space. A canonical derivation of the Einstein equations for a quadratic model is shown in the appendix. In fact, there is an alternative approach to derive the complete set of field equations by treating the system as a constrained system. Indeed, we are interesting in the field equation in the presence of BI space with the following metric:

ds2=−dt2+ a21(t) dx2+ a22(t) dy2+ a23(t) dz2. (1) The isotropic limit of this space with a1= a2= a3 is the

flat FRW space. Note that this is a constrained system with gab= gab(ai), as shown above. Therefore, the field

equations can be derived from the variational δai

equa-tion as a constrained system via δgab= (δgab/δai)δai. The

only nontrivial thing is that the δg00 equation, known as

(2)

approach if the lapse function b2(in dt2= b2(t) dt2) is re-stored explicitly in the above metric. The lapse function is a cyclic variable with hidden information. In fact, it is known that the Friedmann equation has a smaller differen-tiation order than the other δaiequation. This is why the

Friedmann equation is known as a constraint and nonre-dundant equation.

In order to study the anisotropic perturbations, the Friedmann equation in BI space will be adopted. General-izations to different anisotropic spaces are straightforward. In fact, it turns out that perturbation equations are identi-cal for all Bianchi spaces when we take the de Sitter space as the background space. All nonvanishing components of the curvature tensor can be shown to be [28]

Rtiti= ˙Hi+ Hi2, (2)

Rijij= HiHj, (3)

for all i, j in cyclic order and its proper permutations. Here Hi≡ ˙ai/ai.

The Friedmann equation and the δai equations of the

pure gravity model L can be shown to be [28] DL≡ L + Hi  d dt+ 3H  Li− HiLi− ˙HiLi= 0 , (4) DiL≡ L +  d dt+ 3H 2 Li−  d dt+ 3H  Li= 0 . (5)

Here we have defined the reduced Lagrangian L =√gL = L(ai(t)) of a pure gravity model in BI spaces by

L = VL  Rtjti, R ij kl  = VL  Hi, ˙Hi  , (6)

where V ≡ a1a2a3is the volume measure of the BI space.

In addition, Li≡ δL/δHi, Li≡ δL/δ ˙Hi, and 3H≡iHi.

The Bianchi identity shows that the perturbation equation associated with the aiequation becomes redundant in the

de Sitter background. Also, for convenience,L will be writ-ten as L from now on.

The Friedmann equation shown above is in fact a uni-versal formula, which holds for all Bianchi type spaces. In-deed, the lapse function b2in the metric ds2=−b2(t) dt2+

gijdxidxj can be chosen as b = 1 by a redefinition of t for

convenience. b is known, however, to be a cyclic variable that hides the nonredundant Friedmann equation Gtt= Ttt as a nontrivial constraint of the system. The Fried-mann equation is known to be nonredundant following the Bianchi identity. Therefore, a compact formula for the Friedmann equation may serve as a better tool for a model-independent analysis of any gravitational system.

Fortunately, we can always derive the hidden Fried-mann equation by the variational principle with respect to δb, or equivalently δB, once the cyclic variable b2is

re-stored in the effective Lagrangian L. In order to derive a model-independent formula for the Friedmann equation in terms of the variables Hiand ˙Hi, we need to replace the

effect of δL/δB and δL/δ ˙B as an equivalent formula de-pending only on δL/δHi and δL/δ ˙Hi. The proof follows

from the observation that ˙B always shows up as a combi-nation of ˙BHi+ 2B( ˙Hi+ Hi2) or BHiHjin the Lagrangian

of all Bianchi type spaces when the lapse function b2(t)≡ 1/B2(t) is restored. Explicitly, δL/δ ˙B = HiδL/[2δ ˙Hi].

Here we have set B = 1 whenever it will not affect the final result. Moreover, the summation over repeated indices is not written explicitly. In addition δL/δB = HiδL/[2δHi] +

˙

HiδL/δ ˙Hiif L = L(B(aiH˙i+ aijHiHj)) for arbitrary

“con-stant” coefficients aiand aij. In fact, B ˙Hiwill always show

up together with B1HiHj, as can be seen from a

dimen-sional analysis. Therefore the Friedmann equation derived above is a universal formula for all Bianchi spaces.

3 Higher derivative gravity model

with a scalar field

With a scalar field (with Lagrangian Lφ) coupled to the

pure gravity Lagrangian Lg, we have

L = Lg+ Lφ≡ Lg

 Hi, ˙Hi



−12∂µφ∂µφ− V (φ) , (7)

with V (φ) the scalar potential of the scalar field. The Fried-mann equation can be written as

DLg=

1 2 ˙

φ2+ V (φ). (8)

In addition, the scalar field equation can be shown to be ¨

φ + 3H ˙φ + V= 0. (9)

We will focus on the stability of an inflationary de Sitter background solution characterized by a constant Hubble parameter Hi= H0 in addition to a slow roll-over scalar

field φ. Equivalently, we will write Hi= H0+ δHi and

φ = φ0+ δφ as the anisotropic perturbation against the de

Sitter background space. Consequently, we have a set of zeroth order equations:

DLg(Hi= H0) = V (φ0) , (10)

V(φ0) = 0. (11)

The constraint V(φ0) = 0 can be realized at two

differ-ent stages: (i) in the inflationary phase where φ = 0 as a local maximum of some SSB potential V , (ii) in the fi-nal state where φ = φmapproaches the local minimum of

V . A model with a specific V will be shown shortly. This fi-nal state is expected to be a stable vacuum. As a result, the following stability equation can be derived from perturbing DLgdefined in (4): δ(DLg) =  HiLijδ ¨Hj  + 3HHiLijδ ˙Hj  + 3H HiLji+ L jδH j +HiLi δ(3H)− HiLijδHj . (12)

Here H≡iHi/3 = ˙V /(3V ). In addition, the notation

AiBi≡ AiBi ≡3i=1AiBi is for the summation over

i = 1 to 3 for repeated dummy indices. In addition, Lji≡ δ2L

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indexiand lower indexjdenoting variation with respect to ˙ Hiand Hj, respectively. Defining DgδH≡ H0 L02δ ¨H + 3H0L02δ ˙H + (6L01+ 3H0L11− L20)δH , (13) the stability equation of (8), with Hi= H0+ δHi and φ =

φ0+ δφ, can be shown to be

DgδH = V(φ0)δφ = 0 . (14)

Here

Lab≡ δa+bL/δHiiδHi2· · · Hiaδ ˙Hj1δ ˙Hj2· · · δ ˙Hjb|Hi→H0. Note that, as claimed earlier, this equation is exactly the same as the isotropic perturbation equation of the flat FRW solution [14–16, 20, 21]. Therefore, (14) becomes

δ ¨H + 3H0δ ˙H + KH02δH = 0 , (15)

with

K≡6L01+ 3H0L11− L20 L02H02

,

in BI space. An explicit expression for K can be derived for any given model. As a result, the values of K are criti-cal to the stability of the corresponding de Sitter universe. General selection rules can hence be obtained in a straight-forward way. We will focus on the cubic models in the following section for a simple demonstration. We will dis-cuss the model-independent stability conditions for the de Sitter background in this section.

Similarly, the perturbation of the scalar field equation (9) can be shown to be

δ ¨φ + 3H0δ ˙φ + V0δφ = 0 . (16)

In fact, the scalar field equation can be solved in the de Sit-ter background Hi= H0and V0≡ V(φ0) = 0. Indeed, the

solution to the equation ¨φ + 3H0φ˙∼ 0 is

φ∼ φ0+

˙ φ0

3H0

[1− exp(−3H0t)] . (17)

This result indicates that the scalar field does change very slowly. Such a behavior is exactly identical to the behavior of a slow roll-over scalar field.

Assuming that δH = exp[hH0t]δH0 and δφ =

exp[pH0t]δφ0for some constants h and p, one can write the

above equations as (h2+ 3h + K)δH = 0, (18)  p2+ 3p +V  0 H2 0  δφ = 0. (19)

Note that the δH equation is the same as the pure grav-ity model, independent of the scalar field. The effect of the

scalar field is minor both in the inflationary phase and the final stage. Therefore, the de Sitter space can hopefully be a stable background in both stages with V(φ)∼ 0. This is a positive sign: a stable de Sitter space as a final state is what we need. The difference between these two stages is that φ cannot stay constant forever when the initial φ is close to the local maximum of V . φ will slide off the local maximum according to the slow roll-over equation (17). Therefore, the inflationary phase is not a stable state for φ. On the other hand, φ will oscillate with a damping term around the local minimum of V and eventually settle down to the local minimum. Therefore, the final state is a stable final state for φ.

As a result, we can have a stable mode for δH in both states of φ (at V= 0) with a similar structure given by the stability condition (18). The only difference is that H = H0in the inflationary phase and H = Hmin the final

state. Here H0 and Hmare both constants characterized

the Hubble expansion scale of these different states. Hence inflation will be ended once the scalar field rolls off the initial phase V0= 0. When it rolls down to the local min-imum, V(φm) = 0, the evolution of the de Sitter solution

will be similar to the inflationary phase solution discussed here.

Indeed, (18) and (19) indicate that there are two decay-ing modes for δH and δφ with

2h =−3 ±√9− 4K, (20)

2p =−3 ± 

9− 4V(0)/H2

0. (21)

We need at least a stable δH solution with negative h, so that inflation is possible along this stable direction. It will be even better if both h solutions are negative. In addition, we need at least one unstable solution requiring either p or h to be positive. This unstable mode will end the infla-tionary phase automatically.

Explicitly, K > 0 will make both h solutions negative. K > 9/4 will make √9− 4K imaginary and hence turns exp[hH0t] = exp[−3H0t] cos

√

4K− 9H0t + θ1



into an os-cillatory solution with a constant phase θ1. Note that this

will also make δH a stable mode. On the other hand, we will have a stable mode and an unstable mode if K < 0. The case K = 0 gives us one negative and one zero h solution. Both (K = 0) modes are stable again. In summary, the con-dition K≥ 0 implies two stable modes of δH. One stable mode of δH is enough for inducing inflation. But the two stable modes of δH will further ensure that the anisotropy will not grow out of control in this model.

An appropriate effective spontaneously symmetry breaking potential V of the following form:

V (φ) =λ 4(φ

2

− v2)2+ V

m, (22)

with arbitrary coupling constant λ, can be shown to be a good candidate of such models. Here Vmis a small

cosmo-logical constant dressing the SSB potential. When the scalar field eventually rolls down to the minimum of V at φ = v, the system will oscillate around this local minimum with a friction term related to the effective Hubble con-stant Hmat this stage. A reheating process is expected to

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take away the kinetic energy of the scalar field. The scalar field will eventually become a constant background field and lose all its kinetic energy.

H0can be chosen to induce enough inflation for a brief

moment as long as the slow roll-over scalar field remains close to the initial state φ = φ0. This de Sitter phase will

hence remain stable and drive the inflationary process for a brief moment. Explicit models with a cubic coupling term will be discussed as an example in the next section.

Explicitly, the solutions for p are p = p±=−3/2 ±  9− 4V0/H2 0/2. Hence p±=−3/2 ±  9 + 4λv2/H2 0/2

for the SSB φ4 potential model. Therefore, it is easy to find that the solutions p+> 0 (unstable mode) and

p< 0 (stable mode) exist for this model. In addition, with properly chosen parameters, the unstable mode p+=



9 + 4λv2/H2

0/2− 3/2 can be made small enough to

in-duce 60 e-folds of inflation during the inflationary phase. When φ→ v, the scalar field will remain a stable mode perturbatively. The final de Sitter space will remain stable against anisotropic perturbations as long as K(Hm)≥ 0

accommodates two stable modes.

4 Cubic model

In this section, we will study the higher derivative gravity model with a coupled scalar field:

L = −R − αR2 − βRµ νR ν µ+ γR µν βγR βγ σρR σρ µν −1 2∂µφ∂ µφ− V (φ) ≡ Lg+ Lφ. (23) Here Lg=−R − αR2− βRµνR ν µ+ γR µν βγR βγ σρR σρ µν and Lφ= −1 2∂µφ∂

µφ−V (φ) denote the pure gravity Lagrangian and

scalar field Lagrangian, respectively. We will also write L1=−R, L2=−αR2−βRµνRνµand L3= γR

µν βγR

βγ σρRµνσρfor

convenience. The cubic term is shown to be the two-loop ef-fect of super gravity [14–16]. Note also that this is the most general covariant quadratic gravity model. The quadratic term Rab

cdR cd

ab is related to the α and β terms by the Euler

invariant.

In a moment, we will show that quadratic terms (1) do not contribute to the expanding parameter H0, and (2) will

affect the stability of the de Sitter phase [28]. We can write the Lagrangian (23) explicitly as

L = 6H + 2H˙ 2− 36α H + 2H˙ 2

2

− 12β H˙2+ 3 ˙HH2+ 3H4

+ 24γ ( ˙H + H2)3+ H6 , (24) when we set Hi→ H. The leading order equation of

Fried-mann equation reads

DLg(Hi= H0) = V (φ0)≡ V0, (25)

in the de Sitter background with Hi= H0and φ = φ0.

Ex-plicitly, we have

V0= 6



1− 4γH04H02. (26) Note that quadratic terms do not contribute to H0 as

promised earlier. This result is a general property associ-ated with the conformal structure of de Sitter space. In fact, this result also follows from the fact that a3( ˙H +

H2)H2= d[a3H3]/[3 dt] is a total derivative. Therefore,

( ˙H + H2)H2 will not affect the field equation. Hence the only effects of the quadratic terms come from the remain-ing Lagrangian L2=−12(3α + β) ˙H2. This term will

con-tribute to (25) in the de Sitter background. As a result, quadratic terms will not affect the expansion scale H0.

Quadratic terms will, however, affect the linear order per-turbation equation and consequently the stability of de Sitter solution. Note that when γ = 0, (26) implies that V0= 6H02. None of the quadratic terms is affecting the

ex-pansion rate H0. This indicates that the γ term does affect

the expansion rate in a very complicated way. Fortunately, (26) can be solved and served as an useful tool in the forth-coming analysis.

In addition, the coefficient K for the δH perturbation equation can be shown to be

K0= 1− 12γH4 0 2H2 0[6γH02− 3α − β] (27) for this model, which has two different decaying modes h = [−3 ±√9− 4K]/2.

Writing x = H2

0, the polynomial equation (26) can be

solved to give x = x1and x = x±with

x1=−  1 3γ 1/2 cosθ0 3, (28) x±=  1 3γ 1/2 cosθ0∓ π 3 . (29)

Here cos θ0≡√3γV0/2≤ 1. The notation x± is defined

such that x≤ x+. In addition, these two solutions become

degenerate when 3γV2 0 = 4.

Similarly, when the system settles close to the final de Sitter phase at φ→ v, similar solutions hold for this state. Explicitly, we have Km= 1− 12γH4 m 2H2 m[6γHm2 − 3α − β] , (30)

for this model, which has two different decaying modes hm=  −3 ±√9− 4Km  /2. Writing y = H2

m, the polynomial equation (26) can be

solved to give y = y1and y = y±with

y1=−  1 3γ 1/2 cosθm 3 , (31) y±=  1 3γ 1/2 cosθm∓ π 3 . (32)

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Here cos θm≡√3γVm/2≤ 1. The notation y± is defined

such that y≤ y+. In addition, these two solutions become

degenerate when 3γV2 m= 4.

Different regions of physical solutions are shown in Fig. 1. Indeed, a physical solution exists for−π/2 ≤ θ0≤

π/2. As a result, we have π/6≤ (θ0+ π)/3≤ π/2 when

x = x+. There is another set of solutions for x = x− with

−π/2 ≤ (θ0− π)/3 ≤ −π/6. There are also two similar

sets of solutions for y. Note that Vm< V0 implies that

|θm| > |θ0|. In addition, H0 Hm is a physical

assump-tion of the inflaassump-tionary model. Therefore, the only way to get a physical solution for H2

0 Hm2 is to take x = x+

and y = y→ 0. This can be shown by a simple observa-tion; see (29) and (32). The only way to make Hm small

as compared to H0and to make θmobey|θm| > |θ0| is to

choose (θm− π)/3 → 0. This can be done by writing (θm−

π)/3 = m−π/2 for some small constant m. Consequently,

we have θm=−π/2 + 3mand Hm2 =  1 3γ 1/2 cosθm− π 3 =  1 3γ 1/2 cos−π 2+ m  ∼  1 3γ 1/2 m. (33)

This result shows that 12γH4

m∼ 2m. In addition, we can

either write (θ0+ π)/3 = π/6 + 0 or (θ0− π)/3 = −π/6

− 0 for some small constant 0. This is equivalent to

chosing θ0=−π/2 + 30or θ0= π/2− 30. This will make

|θm| > |θ0| and will make H0as big as possible. As a result,

we have H02=  1 3γ 1/2 cosθ0± π 3 =  1 3γ 1/2 cos−π 6± 0  ∼  1 3γ 1/2 1 2∓ √ 3 2 0  . (34)

The result shows that 12γH04∼ 1 ∓ 2

30. In summary, we

have

12γH04∼ 1 ∓ 2√30, (35)

12γHm4 ∼ 2m, (36)

Fig. 1. cos θ is plotted with two ranges θ > π/6 and θ <−π/6 specified. (θm− π)/3 is expected to be close to −π/2

when the coupling constants are properly chosen. Here we have assumed that 0 1 and m 1 in order to observe

the physical pattern of these solutions.

As shown earlier, writing δH = exp[hH0t]δH0and δφ =

exp[pH0t]δφ0for some constants h and p, the perturbation

equations (18)–(19) become h2+ 3h + 1− 12γH 4 0 2H2 0[6γH02− 3α − β] = 0 , (37)  p2+ 3p +V  0 H2 0  = 0 . (38)

As a result, the solution to the equation for h (37) is h = h±=−3(1 ± δ2)/2 with δ22= 1 + 2 12γH4 0− 1 {9H2 0[6γH02− 3α − β]} .

In addition, the solution to the equation for p (38) is p = p±=−3/2 ±9− 4V0/H2

0/2 =−3/2 ±



9 + 4λv2/H2 0/2

for the SSB φ4potential model. Here p

+> 0 and p< 0

in-dicate an unstable mode and a stable mode for this model. Properly chosen coupling constants α, β and γ allow the unstable p-mode to have a long enough ∆t in the infla-tionary phase. As a result, inflation of 60 e-folds can be induced. Indeed, this is the amount to require that p+≤

1/60, or equivalently, λv2∼ 0.052¯7H2 0.

We can also write the perturbative solution for Hiand φ

as

Hi= H0+ Ai+exp[h+H0t] + Aiexp[hH0t] , (39)

φ = φ0+ Bi+exp[p+H0t] + Biexp[pH0t] , (40)

with Ai± and Bi± some constant coefficients determined

by the initial perturbations. These linear solutions become oscillatory solutions if the discriminant δ2 becomes pure

imaginary. In such case, these equations take the following form:

Hi= H0+ Aiexp[−3H0t/2] cos[3|δ2|H0t/2 + θAi] , (41) with Aiand θAisome constant coefficients.

We have shown that the h perturbation have two stable modes only when K≥ 0. Explicitly, these inequalities will hold when either

(1) 2(3α + β)H02< 12γH04< 1 , (42)

(2) 1 < 12γH04< 2(3α + β)H02, (43) holds. Consequently, the de Sitter background can remain stable with properly chosen coupling constants α, β and γ. Note that the above inequalities imply that the L1, L2and

L3 Lagrangians are competing for physical solutions. For

example, condition (1) in the above equation states that “L1> L3> L2”. Therefore, coupling constants have to be

chosen carefully to accommodate a physical solution. Note that similar solutions for p and h also exist when the scalar field rolls down the local minimum of the SSB potential:

(1) 2(3α + β)Hm2 < 12γHm4 < 1 , (44) (2) 1 < 12γHm4 < 2(3α + β)Hm2 . (45)

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The second set of solutions of Hm is clearly

inconsis-tent with (36). Therefore the only consisinconsis-tent Hmsolution

is (44). In summary, a consistent solution exists only when 2(3α + β)H02< 12γH04< 1 , (46) 2(3α + β)Hm2 < 12γHm4 < 1 , (47)

or

2(3α + β)Hm2 < 12γHm4 < 1 < 12γH04

< 2(3α + β)H02, (48)

hold separately. Equation (26) implies that 1 > 4γH4 0

4γH4

m. Therefore the set of solutions (46)–(47) imply that

2(3α + β)Hm2 < 12γHm4< 2(3α + β)H02

< 12γHm4< 12γH04< 1. (49) Here the term 12γHm4

can be in either position for consis-tency. Consequently, physical solutions that approach the de Sitter space as a final stable state can be found in this model.

Consider the special case where γ = 0; then the condi-tion for a stable de Sitter final state is

2(3α + β)Hm2 < 2(3α + β)H02< 1, or (50) 2(3α + β)Hm2 < 1 < 2(3α + β)H02. (51)

Similarly, for the case α = β = 0, the condition for a stable de Sitter final state is

12γHm4 < 12γH04< 1, (52) or 12γHm4 < 1 < 12γH04. (53)

In addition, there is the special case when L02= 2H02(6γH02

− 3α − β) or L02= 2Hm2(6γHm2 − 3α − β). In such cases,

the perturbative equation for h becomes δH = 0. This means that the corresponding de Sitter solution is abso-lutely stable against any anisotropic perturbation.

In summary, the presence of a scalar field makes the system far more complicated than the system without scalar field, especially in higher curvature gravity models. A scalar field can take care of the ending of the inflationary phase in a natural way. It also introduces a strong con-straint on the system. Fortunately, a consistent and phys-ical solution can always be found to support the de Sitter space as a stable final state.

5 Conclusion

The existence of a stable de Sitter background is closely related to the choices of the coupling constants. We have shown that, for gravity models with an additional scalar field, the flat FRW de Sitter background space can be a background if the coupling constants are chosen prop-erly. The ending of the inflationary process is due to the unstable mode of a slow roll-over scalar field with a SSB potential.

An explicit model with a spontaneously symmetry breaking φ4 potential is presented as a simple demon-stration. It is also shown explicitly that quadratic terms will not affect the de Sitter solution characterized by the Hubble parameters H0 and Hm. In particular, the

sim-ple observation that the effective quadratic Lagrangian L2=−12(3α + β) ˙H2has been shown explicitly. Quadratic

terms play, however, a critical role in the stability of the de Sitter background. Indeed, with properly chosen coupling constants, the anisotropy can only grow mildly. Implica-tions of these stability condiImplica-tions deserve more attention in the search for physical models.

Appendix: Field equations

The field equation of the Lagrangian L =−R − αR2

β (Ra b)

2

− ∂aφ∂aφ/2− V can be derived by the variation of

gab. The result is 1 2Rgab− Rab+ 1 2gab  αR2+ β(Rcd)2− Lφ  = 2 (αR Rab+ βRacRcb)− 2α gabD2− DaDb R −β2gabD2R− βD2Rab+ 2βDaDcRbc+ 1 2∂aφ∂bφ . (A.1) In addition, the scalar equation can be shown to be

D2φ = V. (A.2)

In order to derive the field equation in a covariant way, we may write the variation of the Riemann curvature tensor as δRd

cba=−DaδΓbcd + DbδΓacd as if δΓbca is a type

T (1, 2) tensor. The derivation has nothing to do with whether δΓa

bcis a tensor or not. Rather, by imagining δΓbca

is a tensor and using all related properties of a tensor, it helps in reducing the effort in deriving these equations, especially when integration-by-parts is required. In add-ition, we have also used the Bianchi identity DcDDRacdb=

D2Rab− D

cDaRbcin converting the differentiation of the

Riemann tensor into a differentiation of the Ricci ten-sor. The field equation of the cubic term can be derived similarly.

In summary, the reduced formulae shown in this pa-per can be helpful in extracting some useful information without going into the details of the field equations. For example, the existence of the inflationary solution H = H0

has to do with the leading order equations. It can be done by ignoring any term like f (H) ˙H, with f (H) an arbitrary function of H. On the other hand, the stability of the inflationary solution has to do with those leading order terms linear in the time differentiation of δH. We can freely ignore terms like ˙H2. In particular, ( d/ dt)(f (H)δH) =

f (H)δ ˙H can be used to skip unrelated terms, with f (H) an arbitrary function of H, with the closed formula shown in this paper.

Acknowledgements. This work is supported in part by the Na-tional Science Council of Taiwan.

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數據

Fig. 1. cos θ is plotted with two ranges θ &gt; π/6 and θ &lt; −π/6 specified. (θ m − π)/3 is expected to be close to −π/2

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