String inspired explanation for the super-acceleration of our universe
String inspired explanation for the super-acceleration of our universe
Ahmad Sheykhi 1,2,3∗ and Bin Wang 1†
Department of Physics, Fudan University, Shanghai 200433, China
Department of Physics, Shahid Bahonar University, Kerman, Iran
Nematollah Riazi ‡
Physics Department and Biruni Observatory, Shiraz University, Shiraz 71454, Iran
We investigate the effect of the bulk content in the general Gauss-Bonnet braneworld on the
evolution of the universe. We find that the Gauss-Bonnet term and the combination of the dark
radiation and the matter content of the bulk play a crucial role in the universe evolution. We show
that our model can describe the super-acceleration of our universe with the equation of state of the
effective dark energy in agreement with observations.
PACS numbers: 98.80.Cq; 98.80.-k
I. INTRODUCTION
One of the most dramatic discoveries of the modern cosmology in the past decade is that our universe is currently
accelerating [1]. A component that causes the accelerated expansion of the universe is referred to as “dark energy”.
Although the nature of such dark energy is still speculative, an overwhelming flood of papers has appeared which
attempt to describe it by devising a great variety of models. Among them are cosmological constant, exotic fields such
as phantom or quintessence, modified gravity, etc, see [2][3] for a recent review. Available models of dark energy differ
in the value and variation of the equation of state parameter w during the evolution of the universe. The cosmological
constant with w = −1, is located at a central position among dark energy models both in theoretical investigation
and in data analysis [4]. In quintessence [5], Chaplygin gas [6] and holographic dark energy models [7], w always
stays bigger than −1. The phantom models of dark energy have w < −1 [8]. However, following the more accurate
data analysis, a more dramatic result appears showing that the time varying dark energy gives a better fit than a
cosmological constant and in particular, w can cross −1 around z = 0.2 from above to below [9]. Although the galaxy
cluster gas mass fraction data do not support the time-varying w [10], theoretical attempts towards the understanding
of the w crossing −1 phenomenon have been started. Some dark energy models, such as the one containing a negative
kinetic scalar field and a normal scalar field [11], or a single scalar field model [12] and interacting holographic dark
energy models [13] have been constructed to gain insight into the occurrence of the transition of the dark energy
equation of state and the mechanism behind this transition. Other studies on the w = −1 crossing have been carried
out in [14].
Independent of the challenge we deal with the dark energy puzzle, in recent years, theories of large extra dimensions,
in which the observed universe is realized as a brane embedded in a higher dimensional spacetime, have received a lot
of interest. According to the braneworld scenario the standard model of particle fields are confined on the brane while,
in contrast, the gravity is free to propagate in the whole spacetime. In these theories the cosmological evolution on the
brane is described by an effective Friedmann equation that incorporates non-trivially with the effects of the bulk onto
the brane. An interesting consequence of the braneworld scenario is that it allows the presence of five-dimensional
matter which can propagate in the bulk space and may interact with the matter content in the braneworld. It has
been shown that such an interaction can alter the profile of the cosmic expansion and lead to a behavior that would
resemble the dark energy. The cosmic evolution of the braneworld with energy exchange between brane and bulk has
been studied in different setups [15][16][17][18]. In these models, due to the energy exchange between the bulk and
the brane, the usual energy conservation law on the brane is broken and consequently it was found that the equation
of state of the effective dark energy may experience the transition behavior (see e.g [17, 18]).
On the other hand, in string theory, in addition to the Einstein action, some higher derivative curvature terms have
been included to derive the gravity. In order to obtain a ghost-free theory, the combination of quadratic terms called
Gauss-Bonnet term is usually employed as curvature corrections to the Einstein-Hilbert action [19]. From a geometric
point of view, the combination of the Einstein-Hilbert and Gauss-Bonnet term constitutes, for 5D spacetimes, the
∗ asheykhi@mail.uk.ac.ir
† wangb@fudan.edu.cn
‡ riazi@physics.susc.ac.ir
http://arxiv.org/abs/0704.0666v4
most general Lagrangian to produce second-order field equations [20]. The Gauss-Bonnet correction significantly
changes the bulk field equations and leads to modifications in the braneworld Friedmann equations. Therefore, the
study of the effects of the Gauss-Bonnet correction term on the evolution of the universe in the braneworld scenario
is well motivated. Influences of the Gauss-Bonnet correction on the DGP braneworld have been studied in [21, 22].
The purpose of the present work is to investigate the effects of the bulk content in the general Gauss-Bonnet
braneworld on the evolution of the universe. Although the effects of the Gauss-Bonnet correction term on the late
time universe is small, we will see that it still plays an important role in the cosmic evolution. Besides we will
show that the combination of the dark radiation term and the matter content of the bulk plays the role of the dark
energy on the brane and influences the evolution of the universe. In our model, in contrast to the previous models
([15, 16, 17, 18]), we do not need to break down the standard energy momentum conservation law on the brane,
although our model can allow such assumption if one is interested. We will show that by suitably choosing model
parameters, our model can exhibit accelerated expansion of the universe. In addition, we will present a profile of the
w crossing −1 phenomenon which is in good agreement with observations.
The paper is organized as follows. In Section II, we present a braneworld model to describe the accelerated expansion
and the effective equation of state of dark energy in the presence of the Gauss-Bonnet correction term in the bulk. In
Section III, we study the cosmological consequences of the model and in particular, its effect on the evolution of the
universe. The last section is devoted to conclusions and discussions.
II. THE MODEL
The theory we are considering is five-dimensional and has an action of the form
−g (R− 2Λ + αLGB) +
−gLmbulk +
−g̃(Lmbrane − σ), (1)
where Λ < 0 is the bulk cosmological constant and LGB is the Gauss-Bonnet correction term
LGB = R2 − 4RABRAB +RABCDRABCD . (2)
Here g and g̃ are the bulk and brane metrics, respectively. R, RAB, and RABCD are the scalar curvature and Ricci
and Riemann tensors, respectively. Throughout this paper we choose the unit so that κ2 = 1 as the gravitational
constant in five dimension. We have also included arbitrary matter content both in the bulk and on the brane through
Lmbulk and Lmbrane respectively, and σ is the positive brane tension. The field equations can be obtained by varying the
action (1) with respect to the bulk metric gAB. The result is
GAB + ΛgAB + 2αHAB = TAB, (3)
where HAB is the second-order Lovelock tensor
HAB = RRAB − 2RACRBC − 2RCDRACBD
CDERBCDE − 14gABLGB .
For convenience and without loss of generality, we can choose the extra-dimensional coordinate y such that the brane
is located at y = 0 and bulk has Z2 symmetry. We are interested in the cosmological solution with a metric
ds2 = −n2(t, y)dt2 + a2(t, y)γijdxidxj + b2(t, y)dy2, (4)
where γij is a maximally symmetric 3-dimensional metric for the surface (t=const., y=const.), whose spatial curvature
is parameterized by k = -1, 0, 1. The metric coefficients n and b are chosen so that, n(t, 0) = 1 and b(t, 0) = 1, where t
is cosmic time on the brane. The total energy-momentum tensor has bulk and brane components and can be written
TAB = TAB |brane +TAB |σ +TAB |bulk . (5)
The first and the second terms are the contribution from the energy-momentum tensor of the matter field confined
to the brane and the brane tension
TAB |brane = diag(−ρ, p, p, p, 0)
, (6)
TAB |σ = diag(−σ,−σ,−σ,−σ, 0)
, (7)
where ρ, and p, being the energy density and pressure on the brane, respectively. In addition we assume an energy-
momentum tensor for the bulk content of the form
TAB |bulk =
T 00 0 T
0 T ijδ
T 05 0 T
 . (8)
The quantities which are of interest here are T 55 and T
5, as these two enter the cosmological equations of motion. In
fact, T 05 is the term responsible for energy exchange between the brane and the bulk. Integrating the (00) component
of the field equations (3) across the brane and imposing Z2 symmetry, we have the jump across the brane [23]
1 + 4α
a′ 2+
(ρ+ σ), (9)
where 2a′+ = −2a′− is the discontinuity of the first derivative. H = ȧ0/a0 is the Hubble parameter on the brane. Eq.
(9) is a cubic equation for the discontinuity a′+/a0, which has only one real solution, the other two being complex.
Therefore, if we require our cosmological equations to have the right α → 0 limit we are left with only one solution.
However, this real root is too lengthy and complicated to present here. Since we are interested to study the effect of
the Gauss-Bonnet correction term on the evolution of the universe in the late time so it is reasonable to choose the
Gauss-Bonnet coupling constant α to be small, namely 0 < α < 1. Using this fact we can expand the real solution
for a′+/a0 versus α powers. The result for k = 0 up to order α is
(ρ+ σ) +
(ρ+ σ)
108H2 − (ρ+ σ)2
+O(α2). (10)
In a similar way, integrating the (ij) component of the field equations (3) across the brane and imposing Z2 symmetry,
we can obtain the discontinuity in the metric function n′+/n0, which for k = 0 can be written up to O(α) in the following
(2ρ+ 3p− σ) + α
−2H2 (2 ρ+ 3 p− σ)
(ρ+ σ)
(8 ρ+ 9 p− σ) + 4 Ḣ (ρ+ σ)
+O(α2), (11)
where dots denote time derivatives and primes denote derivatives with respect to y. At this point we find it convenient
to absorb the brane tension σ in ρ and p with the replacement ρ + σ → ρ and p − σ → p. Therefore the junction
conditions (10) and (11) can be simplified
108H2 − ρ2
, (12)
(2ρ+ 3p) +
−2H2 (2 ρ+ 3 p) + ρ
(8 ρ+ 9 p) + 4 Ḣρ
. (13)
Substituting the junction conditions (12) and (13) into the (55) and (05) components of the field equations (3), we
obtain the modified Friedmann equation and the semi-conservation law on the brane (up to order α)
ρ (2ρ+ 3p)
Ḣ +H2
1 + 4α
(ρ+ 3p) +
ρ3 (2ρ+ 3p) =
Λ− T 55
, (14)
ρ̇+ 3H(ρ+ p) = −T, T ≡ 2T 05
1− 4α
H2 − ρ
. (15)
We shall assume an equation of state p = wρ to hold between the energy density and pressure of matter on the brane.
Therefore we have
ρ2 (2 + 3ω)
Ḣ +H2
1 + 4α
(1 + 3ω) +
ρ4 (2 + 3ω) =
Λ− T 55
, (16)
ρ̇+ 3Hρ(1 + ω) = −T. (17)
One can easily check that in the limit α → 0, Eqs. (10)-(17) reduce to the corresponding equations of the braneworld
model without Gauss-Bonnet correction term [15].
Remarkably, we can show that the Friedmann equation (16) is equivalent to the following equations
2αH4 +
, (18)
with χ satisfying
χ̇+ 4H
1− 4α
H2 − ρ
Ṫ 55
. (19)
Using the definition for T in Eq. (15), the latter equation up to order α can be written as
χ̇+ 4H
T 05 ρ
1− 8α
H2 − ρ
Ṫ 55
. (20)
Eq. (18) is the modified Friedmann equation describing cosmological evolution on the brane. The auxiliary field
χ incorporates non-trivial contributions of dark energy which differ from the standard matter fields confined to the
brane. The bulk matter contributes to the energy content of the brane through the bulk pressure terms T 55 that
appear in the right hand side of the Friedmann equation. In addition, the bulk matter contributes to the energy
conservation equation (15) through T 05 which is responsible for the energy exchange between the brane and bulk.
The functions T 55 and T
5 are functions of time corresponding to their values on the brane. The energy-momentum
conservation ∇ATAB = 0 cannot fully determine T 55 and T 05 and a particular model of the bulk matter is required
[18]. In the limit α → 0, Eqs. (18) and (20) reduce to ( after replacement ρ → ρ+ σ)
(ρ+ σ)2
, (21)
χ̇+ 4H
(ρ+ σ) +
Ṫ 55
. (22)
If we invoke the usual definitation β ≡ 1/36, λ ≡ (Λ + σ2/6)/6 and γ ≡ σβ, we get (κ2 = 1)
H2 = βρ2 + 2γρ+ λ+ χ−
χ̇+ 4H
= 4T 05 (βρ+ γ) +
Ṫ 55
, (23)
which is noting, but the general set of the equations in RS II braneworld model with bulk matter content plus
brane-bulk energy exchange (see for example [18]).
Returning to the general Friedman equation (18) with Gauss-Bonnet correction term, we can show that this equation
has the solution for H of the form
H2 = − 1
729 + 12αρ2(27 + αρ2) + 972α(6χ+ Λ− 2T 55)
. (24)
The upper solution (+) has correct α → 0 limit. Indeed, if we expand this solution verses α we get (up to O(α))
+ 2χ+
2T 55
6χ+ Λ− 2T 55
, (25)
and Eqs. (15) and (20) become
ρ̇+ 3Hρ(1 + ω) = −2T 05
1− 4α
, (26)
χ̇+ 4H
T 05 ρ
1− 8α
Ṫ 55
. (27)
Therefore, until now we have obtained the set of equations describing the dynamics of our universe (Eqs. (25)-(27))
in the general Gauss-Bonnet braneworld with both bulk matter content and bulk-brane energy exchange provided
that the Gauss-Bonnet coupling constant α is chosen sufficiently small. It is worth noting that although α is small,
it has a dramatic effect on the dynamic behavior of the cosmic evolution. Besides the appearance of the ρ4 term on
the right hand side of Eq. (25) shows that in high energy scale the Gauss-Bonnet correction term plays an important
role.
0.2 0.4 0.6 0.8 1z
Figure 1: Evolution of weff(z) (bold line) and q(z) (dashed line) versus z for ν = 0.34 and α = 0.
0.2 0.4 0.6 0.8 1z
Figure 2: Evolution of weff(z) versus z for 0 < ν ≤ 0.17. α = 0.01 (bold line), α = 0.1 (continuous line), and α = 0.9 (dashed
line).
0.2 0.4 0.6 0.8 1z
Figure 3: Evolution of q(z) versus z for 0 < ν ≤ 0.17. α = 0.01 (bold line), α = 0.1 (continuous line), and α = 0.9 (dashed
line).
0 0.2 0.4 0.6 0.8 1z
Figure 4: Evolution of weff(z) versus z for 0.18 ≤ ν ≤ 0.34. α = 0.01 (bold line), α = 0.1 (continuous line), and α = 0.9
(dashed lined).
0.2 0.4 0.6 0.8 1z
Figure 5: Evolution of q(z) versus z for 0.18 ≤ ν ≤ 0.34. α = 0.01 (bold line), α = 0.1 (continuous line), and α = 0.9 (dashed
line).
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
Figure 6: The parameter space of the function α(ν,A) for 0 < ν ≤ 0.17. A = 18 (bold line), A = 30 (continuous line), and
A = 50 (dashed line).
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
Figure 7: The parameter space of the function C(ν,A) for 0 < ν ≤ 0.17. A = 18 (bold line), A = 30 (continuous line), and
A = 50 (dashed line).
0.180.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34
Figure 8: The parameter space of the function α(ν,A) for 0.18 ≤ ν ≤ 0.34. A = −6 (bold line), A = −10 (continuous line),
and A = −20 (dashed line).
0.180.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34
Figure 9: The parameter space of the function C(ν,A) for 0.18 ≤ ν ≤ 0.34. A = −6 (bold line), A = −10 (continuous line),
and A = −20 (dashed line).
III. COSMOLOGICAL CONSEQUENCES
In this section we are going to explore some cosmological consequences of our model. To do this, first we separate
back the matter energy density and the brane tension as usual form with the replacement ρ → ρ+ σ. Therefore Eqs.
(25) and (27) become
+ 2χ+
6χ+ Λ− 2T 55
, (28)
χ̇+ 4H
1− 8α
Ṫ 55
. (29)
We are interested in the scenarios where the energy density of the brane is much lower than the brane tension, namely
ρ ≪ σ. Assuming the Randall-Sundrum fine-tuning Λ + σ2/6 = 0 holds on the brane and defining the parameter
γ ≡ σ/36, Eqs. (28) and (29) can be simplified in the following form
H2 = 2γρ+ χ−
σ2 + 18
, (30)
χ̇+ 4H
= 4γ T 05
1− 8α
Ṫ 55
. (31)
Now, one may adopt several strategies to find solutions of Eqs. (26), (30) and (31). For example, one may take
a suitable ansatz for the time dependent functions T 05 and T
5 and using Eq. (31) to find the function χ. Then
substitute χ, T 05 and T
5 into Eq. (26) one can try to obtain ρ, and finally one may find Hubble parameter H through
Eq. (30). In the following we are interested in the case in which the energy momentum conservation law on the
brane holds, which is usually assumed in the braneworld scenarios. Indeed, we want to consider the effect of the
bulk content on the evolution of the universe without brane-bulk energy exchange, therefore we set T 05 = 0. The
case with brane-bulk energy exchange in the general Gauss-Bonnet braneworld will be addressed elsewhere. It was
argued that the energy exchange between the bulk and brane T 05 will lead to the effective dark energy equation of
state crossing −1 [16,17]. Here we will show that without the energy exchange, the effect of T 55 and the combined T 55
and the Gauss-Bonnet correction have the same role.
Inserting the condition T 05 = 0 in Eq. (26), it reduces to ρ̇+3Hρ(1+ω) = 0. This equation has well known solution
ρ = ρ0a
−3(1+w), where ρ0 is the present matter density of the universe and we have omitted the “o” subscript from
the scale factor on the brane for simplicity. Then, consider a general ansatz T 55 = Da
ν for the bulk pressure [18],
where D and ν are two arbitrary constants, one can easily check that Eq. (31) has a solution of the form
χ = Ca−4 +Baν , (32)
where C is a constant usually referred to as dark radiation term and B ≡ D(ν + 2)/(3ν + 12). Finally, inserting ρ
and χ into Eq. (30), we can rewrite it in the standard form
(ρ+ ρeff) , (33)
where GN = 3γ/4π is the 4-dimensional Newtonian constant and ρeff represents the effective dark energy density on
the brane
ρeff =
Ca−4 +Aaν
Ca−4 + Aaν
σ2 + 18
Ca−4 +Aaν
, (34)
where A ≡ −2D/(3ν+ 12). The equation of state parameter of the effective dark energy on the brane can be defined
by [25]
weff = −1−
d ln δH2
d ln a
, (35)
where δH2 = (H2/H20 )−Ωma−3 accounts for terms in the Friedmann equation except the brane matter with equation
of state wm = 0. Now, if we use the redshift parameter 1 + z = a
−1 as our variable, we can easily show that
ωeff(z) = −1 +
4C (1 + z)
4 −Aν (1 + z)−ν
36A (1 + z)
+ 36C (1 + z)
C (1 + z)
+A (1 + z)
−ν − α
A (1 + z)
+ C (1 + z)
σ2 + 18A (1 + z)
+ 18C (1 + z)
. (36)
The corresponding late time deceleration parameter can be written
q(z) ≡ − 1
[Ωm + (1 − Ωm) (1 + 3ωeff(z))] , (37)
where Ωm = Ωm0 (1 + z)
is all part of the matter on the braneworld and we take its present value as Ωm0 = 0.28±0.02.
In the rest of the paper, we will obtain constraints on the parameters such as C, A, ν, α and σ in our model. Indeed,
we want to show that under what parameter space constraints our model can describe the accelerated expansion of
the universe with the equation of state of the effective dark energy ωeff crossing −1, as suggested by observations.
A. Special case with α = 0
Let us begin with the special case, in which the Gauss-Bonnet coupling constant α is equal to zero. In this case we
have the usual Randall-Sundrum II braneworld model and Eq. (36) reduces to
weff(z) = −1 +
4C (1 + z)
ν+4 −Aν
C (1 + z)
. (38)
Therefore, we are left with three parameters C, A, ν, and two of them are independent. Requiring that at the present
moment weff(z = 0) = −1.06 and w crossed −1 around z = 0.2 as indicated by extensive analysis of observational
data [9], we can obtain
C = 0.039A, ν = 0.34, A = A. (39)
For these value of parameters and Ωm0 = 0.28, from Eq. (37) we have q(z = 0) = −0.64 and in addition q(z) crosses
0 around z = 0.33 which is in good agreement with recent observational data [1][24]. In figure 1 we plot weff(z) and
q(z) for the above value of the parameters versus redshift parameter z.
B. General case with α 6= 0
Next, we consider the general Gauss-Bonnet braneworld with bulk matter content. In this case we have five
parameters only four of which are independent. Considering that the value of σ does not affect the general profile
of our model and further according to the Randall-Sundrum fine-tuning relation it should be small, we first fix
σ = 10−3. Thus, we have now four parameters and among them three are independent. Numerical calculations
show that the functions weff and q are well behaved for z ≥ 0, provided that 0 < ν ≤ 0.34. Employing the present
value of the equation of state parameter of dark energy ωeff(z = 0) = −1.06 and the moment it crossed −1, namely
ωeff(z = 0.2) = −1, we get
α = α(A, ν), C = 0.12Aν (1.2)− ν , A = A. (40)
If we impose the condition 0 < α < 1 which was used in deriving our equations, we can get constraint on the
free parameter A. In numerical calculations we find that for 0 < ν ≤ 0.17 we should have A > 17.97, while for
0.18 ≤ ν ≤ 0.34 we should have A < −5.23 to satisfy the condition on α. In figures 2 and 3 we plot weff(z) and q(z)
for 0 < ν ≤ 0.17 versus redshift parameter z for different value of the Gauss-Bonnet coupling constant α. From these
figures we observe that at large z, the weff(z) increases with the increase of α, while q(z) decreases with the increase
of α. This qualitative behavior is quite opposite when 0.18 ≤ ν ≤ 0.34 as one can see from figures 4 and 5. Finally
we plot in figures 6-9 the parameter space for the functions α = α(A, ν) and C = C(A, ν). We find that in the case
0 < ν ≤ 0.17, α and C increase with the increase of ν while, in contrast, for 0.18 ≤ ν ≤ 0.34, α and C decrease with
the increase of ν.
IV. CONCLUSIONS AND DISCUSSIONS
In this work we have generalized the Randall-Sundrum II braneworld with both bulk matter content and bulk-brane
energy exchange by adding the Gauss-Bonnet curvature correction term in the bulk action. We have investigated the
effects of the bulk content in the general Gauss-Bonnet braneworld on the evolution of the universe and found that
although the effect of the Gauss-Bonnet correction term in the late time universe is small, it still plays an important
role in the universe evolution.
In contrast to the previous models ([15, 16, 17, 18]), in our study we kept the energy momentum conservation
law on the brane as usual and found that the combination of the dark radiation term and the matter content of
the bulk can play the role of the dark energy on the brane and influence the evolution of the universe. By suitably
choosing parameter space in our model, we can describe the super-acceleration of our universe with the behavior of
the effective dark energy equation of state in agreement with observations. In [22], it was argued in a Gauss-Bonnet
brane world with induced gravity that the Gauss-Bonnet term and the mass parameter in the bulk play a crucial
role in the evolution of the universe. Here in our general model, we confirmed their argument. It is easy to see
from Eqs.(25)-(27) that the Gauss-Bonnet correction influences the dynamics of our universe, especially in the early
universe with high energy scale. Phenomenon on the Gauss-Bonnet role has been disclosed in Figs.2-5. We observed
that although the Gauss-Bonnet effect is not clear at the present moment, it influences the universe evolution in the
past and was more important in the earlier period.
In this work we just restricted our numerical fitting to limited observational data. Giving the wide range of
cosmological data available, in the future we expect to further constrain our model parameter space and test the
viability of our model.
Acknowledgments
This work was partially supported by NNSF of China, Ministry of Education of China and Shanghai Educational
Commission and also by Shiraz University.
[1] A.G. Riess, et al., Astron. J. 116 (1998) 1009; S. Perlmutter, et al., Astrophys. J. 517 (1999) 565; S. Perlmutter, et al.,
Astrophys. J. 598 (2003) 102; P. de Bernardis, et al., Nature 404 (2000) 955.
[2] T. Padmanabhan, Phys. Rep. 380, (2003) 235; P. J. E.Peebles, B. Ratra, Rev. Mod. Phys. 75, (2003) 559; V. Sahni,
astro-ph/0403324 and references therein.
[3] Edmund J. Copeland, M. Sami and Shinji Tsujikawa, hep-th/060305.
[4] S. Weinberg, Rev. Mod. Phys. 61, (1989) 1; N. Straumann, astro-ph/0203330; T. Padmanabhan, hep-th/0406060.
[5] R. R. Caldwell, R. Dave and P. J. Steinhardt, Phys. Rev. Lett. 80, 1582 (1998); P. J. E. Peebles and A. Vilenkin, Phys.
Rev. D 59, 063505 (1999); P. J. Steinhardt, L. M. Wang and I. Zlatev, Phys. Rev. D 59, 123504 (1999); M. Doran and J.
Jaeckel, Phys. Rev. D 66, 043519 (2002); A. R. Liddle, P. Parson and J. D. Barrow, Phys. Rev. D 50, 7222 (1994).
[6] A. Y. Kamenshchik, U. Moschella and V. Pasquier, Phys. Lett. B 511, (2001) 265, gr-qc/0103004.
[7] M. Li, Phys. Lett. B 603, (2004) 1; Q.G. Huang and M. Li, JCAP 0408, (2004) 013.
[8] R. R. Calwell, Phys. Lett. B 545, 23 (2002); R. R. Caldwell, M. Kamionkowski and N. N. Weinberg, Phys. Rev. Lett. 91,
071301 (2003); J. M. Cline, S. Y. Jeon and G. D. Moore, Phys. Rev. D 70, 043543 (2004).
[9] U. Alam, V. Sahni and A. Starobinsky, JCAP 0406 (2004) 008 [astro-ph/0403687]; Y.G. Gong, Class. Quant. Grav. 22
(2005) 2121 [astro-ph/0405446]; Y. Wang and M. Tegmark, Phys. Rev. D71 (2005) 103513 [astro-ph/0501351]; Y. Wang
and P. Mukherjee, Astrophys. J. 606 (2004) 654 [astro-ph/0312192]; R. Daly and S. Djorgovski, Astrophys. J. 612 (2004)
652 [astro-ph/0403664]; U. Alam, V. Sahni, T. Saini and A. Starobinsky, Mon. Not. Roy. Astron. Soc. 354, (2004) 275
[astro-ph/0311364]; T. Choudhury and T. Padmanabhan, Astron. Astrophys. 429, (2005) 807 [astro-ph/0311622].
[10] G. Chen and B. Ratra, Astrophys. J. 612, (2004) L1.
[11] B. Feng, X. L.Wang and X. M. Zhang, Phys. Lett. B 607 (2005) 35; W. Hu, Phys. Rev. D 71, (2005) 047301; Z.K. Guo,
Y.S. Piao, X.M. Zhang and Y.Z. Zhang, Phys. Lett. B 608, (2005) 177; X.F. Zhang, H.Li, Y.S. Piao and X. Zhang,
astro-ph/0501652.
[12] M.Z. Li, B. Feng and X.M. Zhang, hep-ph/0503268.
[13] B. Wang, Y. Gong, E. Abdalla, Phys. Lett. B 624, (2006) 141; B. Wang, C. Y. Lin, E. Abdalla, Phys. Lett. B 637, (2006)
357; B. Wang, J.D. Zang, C.Y. Lin, E. Abdalla and S. Micheletti, astro-ph/0607126.
[14] S. Nojiri and S.D. Odintsov, Phys. Lett. B 562, (2003) 147; S. Nojiri and S. D. Odintsov, Phys. Rev. D 70 (2004) 103522;
A. Vikman, Phys. Rev. D 71, (2005) 023515; A. Anisimov, E. Babichev and A. Vikman, JCAP 0506, (2005) 006; S. Nojiri,
S.D. Odintsov and S. Tsujikawa, Phys. Rev. D 71 (2005) 063004; P. Singh, gr-qc/0502086; R.G. Cai, H.S. Zhang and A.Z.
Wang, hep-th/0505186; H. Stefancic, astro-ph/0504518; E. O. Kahya and V. K. Onemli, gr-qc/0612026; Tomi Koivisto,
David F. Mota, Phys. Lett. B 644, 104 (2007), astro-ph/0606078; Tomi Koivisto, David F. Mota, Phys. Rev. D 75 023518
(2007), hep-th/0609155.
[15] E. Kiritsis, G. Kofinas, N. Tetradis, T. N. Tomaras and V. Zarikas, JHEP 0302 (2003) 035; E. Kiritsis, N. Tetradis and
T. N. Tomaras, JHEP 0203 (2002) 019; P. S. Apostolopoulos and N. Tetradis, Phys. Rev. D 71 043506 (2005); P. S.
Apostolopoulos and N. Tetradis, Phys. Lett. B 633 409 (2006); E. Kiritsis, JCAP 0510 014 (2005); K. I. Umezu, K. Ichiki,
T. Kajino, G. J. Mathews, R. Nakamura and M. Yahiro, Phys. Rev. D 73 063527 (2006).
[16] G. Kofinas, G. Panotopoulos and T.N. Tomaras, hep-th/0510207.
[17] R.G. Cai, Y. Gong and B. Wang, JCAP 0603, (2006) 006 ; P. S. Apostolopoulos and N. Tetradis, hep-th/0604014.
[18] C. Bogdanos and K. Tamvakis, hep-th/0609100; C. Bogdanos, A. Dimitridis, K. Tamvakis, hep-th/0611094.
[19] B. Zwiebach, Phys. Lett. B 156, 315 (1985); D. G. Boulware and S. Deser, Phys. Rev. Lett. 55, 2656 (1985).
[20] D. Lovelock, J. Math. Phys. 12, 498 (1971). x
[21] R. Brown, R. Maartens, E. Papantonopoulos, V. Zamarias, gr-qc/0508116.
[22] R.G. Cai, H.S. Zhang and A. Wang, hep-th/0505186.
[23] G. Kofinas, R. Maartens, E. Papantonopoulos, JHEP 0310, 066 (2003) hep-th/0307138.
[24] V.F. Cardone, A. Troisi, S. Capozziello, Phys. Rev. D, 69, 083517, 2004;
[25] E. V. Linder and A. Jenkins, Mon. Not. Roy. Astron. Soc. 346, 573 (2003) astro-ph/0305286.
http://arxiv.org/abs/astro-ph/0403324
http://arxiv.org/abs/astro-ph/0203330
http://arxiv.org/abs/hep-th/0406060
http://arxiv.org/abs/gr-qc/0103004
http://arxiv.org/abs/astro-ph/0403687
http://arxiv.org/abs/astro-ph/0405446
http://arxiv.org/abs/astro-ph/0501351
http://arxiv.org/abs/astro-ph/0312192
http://arxiv.org/abs/astro-ph/0403664
http://arxiv.org/abs/astro-ph/0311364
http://arxiv.org/abs/astro-ph/0311622
http://arxiv.org/abs/astro-ph/0501652
http://arxiv.org/abs/hep-ph/0503268
http://arxiv.org/abs/astro-ph/0607126
http://arxiv.org/abs/gr-qc/0502086
http://arxiv.org/abs/hep-th/0505186
http://arxiv.org/abs/astro-ph/0504518
http://arxiv.org/abs/gr-qc/0612026
http://arxiv.org/abs/astro-ph/0606078
http://arxiv.org/abs/hep-th/0609155
http://arxiv.org/abs/hep-th/0510207
http://arxiv.org/abs/hep-th/0604014
http://arxiv.org/abs/hep-th/0609100
http://arxiv.org/abs/hep-th/0611094
http://arxiv.org/abs/gr-qc/0508116
http://arxiv.org/abs/hep-th/0505186
http://arxiv.org/abs/hep-th/0307138
http://arxiv.org/abs/astro-ph/0305286
	Introduction
	The model
	COSMOLOGICAL CONSEQUENCES
	Special case with =0 
	General case with =0 
	Conclusions and Discussions
	Acknowledgments
	References
