arXiv:0704.1035v1  [gr-qc]  8 Apr 2007
Anisotropic brane gravity with a confining potential
M. Heydari-Fard 1∗and H. R. Sepangi†
Department of Physics, Shahid Beheshti University, Evin, Tehran 19839, Iran
December 4, 2018
Abstract
We consider an anisotropic brane world with Bianchi type I and V geometries where the mecha-
nism of confining the matter on the brane is through the use of a confining potential. The resulting
equations on the anisotropic brane are modified by an extra term that may be interpreted as the
x-matter, providing a possible phenomenological explanation for the accelerated expansion of the
universe. We obtain the general solution of the field equations in an exact parametric form for
both Bianchi type I and V space-times. In the special case of a Bianchi type I the solutions of
the field equations are obtained in an exact analytic form. Finally, we study the behavior of the
observationally important parameters.
1 Introduction
The type Ia supernovae (SNe Ia) [1] observations provide the first evidence for the accelerating expan-
sion of the present universe. These results, when combined with the observations on the anisotropy
spectrum of cosmic microwave background (CMB) [2] and the results on the power spectrum of large
scale structure (LSS) [3], strongly suggest that the universe is spatially flat and dominated by a
component, though arguably exotic, with large negative pressure, referred to as dark energy [4]. The
nature of such dark energy constitutes an open and tantalizing question connecting cosmology and
particle physics. Different mechanisms have been suggested over the past few years to accommodate
dark energy. The simplest form of dark energy is the vacuum energy (the cosmological constant).
A tiny positive cosmological constant which can naturally explain the current acceleration would
encounter many theoretical problems such as the fine-tuning problem and the coincidence problem.
The former can be stated as the existence of an enormous gap between the vacuum expectation value,
in other words the cosmological constant, in particle physics and that observed over cosmic scales.
The absence of a fundamental mechanism which sets the cosmological constant to zero or to a very
small value is also known as the cosmological constant problem. The latter however relates to the
question of the near equality of energy densities of the dark energy and dark matter today.
Another possible form of dark energy is a dynamical, time dependent and spatially inhomogeneous
component, called the quintessence [5]. An example of quintessence is the energy associated with a
scalar field φ slowly evolving down its potential V (φ) [6, 7]. Slow evolution is needed to obtain a
negative pressure, pφ =
φ̇2+V (φ), so that the kinetic energy density is less than the potential energy
density. Yet another phenomenological explanation based on current observational data is given by
the x-matter (xCDM) model which is associated with an exotic fluid characterized by an equation
of state px = wxρx (wx < −
is the necessary condition to make a universe accelerate), where the
parameter wx can be a constant or more generally a function of time [8].
∗email: m-heydarifard@sbu.ac.ir
†email: hr-sepangi@sbu.ac.ir
http://arxiv.org/abs/0704.1035v1
Over the past few years, we have been witnessing a phenomenal interest in the possibility that
our observable four-dimensional (4D) universe may be viewed as a brane hypersurface embedded
in a higher dimensional bulk space. Physical matter fields are confined to this hypersurface, while
gravity can propagate in the higher dimensional space-time as well as on the brane. The most
popular model in the context of brane world theory is that proposed by Randall and Sundrum
(RS). In the so-called RSI model [9], the authors proposed a mechanism to solve the hierarchy
problem with two branes, while in the RSII model [10], they considered a single brane with a positive
tension, where 4D Newtonian gravity is recovered at low energies even if the extra dimension is not
compact. This mechanism provides us with an alternative to compactification of extra dimensions.
The cosmological evolution of such a brane universe has been extensively investigated and effects such
as a quadratic density term in the Friedmann equations have been found [11]-[13]. This term arises
from the imposition of the Israel junction conditions which is a relationship between the extrinsic
curvature and energy-momentum tensor of the brane and results from the singular behavior in the
energy-momentum tensor. There has been concerns expressed over applying such junction conditions
in that they may not be unique. Indeed, other forms of junction conditions exist, so that different
conditions may lead to different physical results [14]. Furthermore, these conditions cannot be used
when more than one non-compact extra dimension is involved. To avoid such concerns, an interesting
higher-dimensional model was introduced in [15] where particles are trapped on a 4-dimensional
hypersurface by the action of a confining potential V. In [16], the dynamics of test particles confined
to a brane by the action of such potential at the classical and quantum levels were studied and the
effects of small perturbations along the extra dimensions investigated. Within the classical limits,
test particles remain stable under small perturbations and the effects of the extra dimensions are not
felt by them, making them undetectable in this way. The quantum fluctuations of the brane cause
the mass of a test particle to become quantized and, interestingly, the Yang-Mills fields appear as
quantum effects. Also, in [17], a braneworld model was studied in which the matter is confined to the
brane through the action of such a potential, rendering the use of any junction condition unnecessary
and predicting a geometrical explanation for the accelerated expansion of the universe.
In brane theories the covariant Einstein equations are derived by projecting the bulk equations
onto the brane. This was first done by Shiromizu, Maeda and Sasaki (SMS) [18] where the Gauss-
Codazzi equations together with Israel junction conditions were used to obtain the Einstein field
equations on the 3-brane. In a series of recent papers a number of authors [23, 24] have presented
detailed descriptions of the dynamic of homogeneous and anisotropic brane worlds in the SMS model.
The study of anisotropic homogeneous brane world cosmological models has shown an important
difference between these models and standard 4D general relativity, namely, that brane universes are
born in an isotropic state. For the anisotropic Bianchi type I and V geometries, with a conformally
flat bulk (vanishing Weyl tensor), this type of behavior has been found by exactly solving the field
equations [25].
In this paper, we follow [17] and consider an m-dimensional bulk space without imposing the
Z2 symmetry. As mentioned above, to localize the matter on the brane, a confining potential is
used rather than a delta-function in the energy-momentum tensor. The resulting equations on the
anisotropic brane are modified by an extra term that may be interpreted as the x-matter, providing
a possible phenomenological explanation for the accelerated expansion of the universe. The behavior
of the observationally important physical quantities such as anisotropy and deceleration parameter
is studied in this scenario. We should emphasize here that there is a difference between the model
presented in this work and models introduced in [19, 20] in that in the latter no mechanism is
introduced to account for the confinement of matter to the brane.
2 The model
In this section we present a brief review of the model proposed in [16]. Consider the background
manifold V 4 isometrically embedded in a pseudo-Riemannian manifold Vm by the map Y : V 4 → Vm
such that
,ν = ḡµν , GABY
a = 0, GABN
b = gab = ±1, (1)
where GAB (ḡµν) is the metric of the bulk (brane) space Vm(V 4) in arbitrary coordinates, {Y
A} ({xµ})
is the basis of the bulk (brane) and NAa are (m − 4) normal unit vectors, orthogonal to the brane.
Perturbation of V̄4 in a sufficiently small neighborhood of the brane along an arbitrary transverse
direction ξ is given by
ZA(xµ, ξa) = YA + (LξY)
A, (2)
where L represents the Lie derivative and ξa (a = 1, 2, ...,m− 4) is a small parameter along NAa that
parameterizes the extra noncompact dimensions. By choosing ξ orthogonal to the brane, we ensure
gauge independency [16] and have perturbations of the embedding along a single orthogonal extra
direction N̄a giving local coordinates of the perturbed brane as
ZA,µ(x
ν , ξa) = YA,µ + ξ
aN̄Aa,µ(x
ν). (3)
In a similar manner, one can find that since the vectors N̄A depend only on the local coordinates xµ,
they do not propagate along the extra dimensions. The above assumptions lead to the embedding
equations of the perturbed geometry
Gµν = GABZ
,ν , Gµa = GABZ
a, GABN
b = gab. (4)
If we set NAa = δ
a , the metric of the bulk space can be written in the following matrix form
GAB =
gµν +AµcA
ν Aµa
Aνb gab
, (5)
where
gµν = ḡµν − 2ξ
aK̄µνa + ξ
aξbḡαβK̄µαaK̄νβb, (6)
is the metric of the perturbed brane, so that
K̄µνa = −GABY
a;ν , (7)
represents the extrinsic curvature of the original brane (second fundamental form). We use the
notation Aµc = ξ
dAµcd, where
Aµcd = GABN
c = Āµcd, (8)
represents the twisting vector fields (the normal fundamental form). Any fixed ξa signifies a new
perturbed geometry, enabling us to define an extrinsic curvature similar to the original one by
K̃µνa = −GABZ
a;ν = K̄µνa − ξ
K̄µγaK̄
νb +AµcaA
. (9)
Note that definitions (5) and (9) require
K̃µνa = −
. (10)
In geometric language, the presence of gauge fields Aµa tilts the embedded family of sub-manifolds
with respect to the normal vector NA. According to our construction, the original brane is orthogonal
to the normal vector NA. However, equation (4) shows that this is not true for the deformed geometry.
Let us change the embedding coordinates and set
XA,µ = Z
,µ − g
abNAa Abµ. (11)
The coordinates XA describe a new family of embedded manifolds whose members are always or-
thogonal to NA. In this coordinates the embedding equations of the perturbed brane is similar to
the original one, described by equation (1), so that YA is replaced by XA. This new embedding of
the local coordinates is suitable for obtaining induced Einstein field equations on the brane. The
extrinsic curvature of a perturbed brane then becomes
Kµνa = −GABX
a;ν = K̄µνa − ξ
bK̄µγaK̄
νb = −
, (12)
which is the generalized York’s relation and shows how the extrinsic curvature propagates as a result
of the propagation of the metric in the direction of extra dimensions. The components of the Riemann
tensor of the bulk written in the embedding vielbein {XA,α,N
a }, lead to the Gauss-Codazzi equations
Rαβγδ = 2g
abKα[γaKδ]βb +RABCDX
,δ , (13)
2Kα[γc;δ] = 2g
abA[γacKδ]αb +RABCDX
,δ , (14)
where RABCD and Rαβγδ are the Riemann tensors for the bulk and the perturbed brane respectively.
Contracting the Gauss equation (13) on α and γ, we find
Rµν = (KµαcK
ν −KcK
µν ) +RABX
,ν − g
abRABCDN
b , (15)
and the Einstein tensor of the brane is given by
Gµν = GABX
,ν +Qµν + g
abRABN
b gµν − g
abRABCDN
b , (16)
where
Qµν = −g
KγµaKγνb −KaKµνb
KαβaK
αβa −KaK
gµν . (17)
As can be seen from the definition of Qµν , it is independently a conserved quantity, that is Q
;ν = 0
[19]. Using the decomposition of the Riemann tensor into the Weyl curvature, the Ricci tensor and
the scalar curvature
RABCD = CABCD −
(m− 2)
GB[DRC]A − GA[DRC]B
(m− 1)(m − 2)
R(GA[DGC]B), (18)
we obtain the 4D Einstein equations as
Gµν = GABX
,ν +Qµν − Eµν +
(m− 2)
gabRABN
b gµν
(m− 2)
(m− 1)(m − 2)
Rgµν , (19)
where
Eµν = g
abCABCDN
,ν , (20)
is the electric part of the Weyl tensor CABCD. Now, let us write the Einstein equation in the bulk
space as
AB + Λ
(b)GAB = α
∗SAB, (21)
where
SAB = TAB +
VGAB , (22)
here α∗ = 1
(M∗ is the fundamental scale of energy in the bulk space), Λ
(b) is the cosmological
constant of the bulk and TAB is the energy-momentum tensor of the matter confined to the brane
through the action of the confining potential V. We require V to satisfy three general conditions:
firstly, it has a deep minimum on the non-perturbed brane, secondly, depends only on extra coordi-
nates and thirdly, the gauge group representing the subgroup of the isometry group of the bulk space
is preserved by it [16]. The vielbein components of the energy-momentum tensor are given by
Sµν = SABX
,ν , Sµa = SABX
a , Sab = SABN
b . (23)
Use of equation (22) then gives
RAB = −
(m− 2)
GABS +
(m− 2)
Λ(b)GAB + α
∗SAB, (24)
R = −
(α∗S −mΛ(b)). (25)
Substituting RAB and R from the above into equation (19) and using equation (23), we obtain
Gµν = Qµν − Eµν +
(m− 3)
(m− 2)
α∗gabSabgµν +
(m− 2)
Sµν −
(m− 4)(m− 3)
(m− 1)(m− 2)
α∗Sgµν
(m− 7)
(m− 1)
Λ(b)gµν . (26)
On the other hand, again from equation (21), the trace of the Codazzi equation (14) gives the “gravi-
vector equation”
Kδaγ;δ −Ka,γ −AbaγK
b +AbaδK
bδ +Baγ =
3(m− 4)
α∗Saγ , (27)
where
Baγ = g
mnCABCDN
n . (28)
Finally, the “gravi-scalar equation” is obtained from the contraction of (15), (19) and using equation
S − gmnSmn
gab =
(Q+R+W ) gab −
Λ(b)gab, (29)
where
W = gabgmnCABCDN
a . (30)
Equations (26)-(30) represent the projections of the Einstein field equations on the brane-brane,
bulk-brane, and bulk-bulk directions.
As was mentioned in the introduction, localization of matter on the brane is realized in this model
by the action of a confining potential. Since the potential V is assumed to have a minimum on the
brane, which can be taken as zero, localization of matter may simply be realized by taking equation
(22) and consider its components on the brane, in which case we may write
ατµν =
(m− 2)
Tµν , Tµa = 0, Tab = 0, (31)
where α is the scale of energy on the brane. Now, the induced Einstein field equations on the original
brane can be written as
Gµν = ατµν −
(m− 4)(m− 3)
2(m− 1)
ατgµν − Λgµν +Qµν − Eµν , (32)
where Λ = −
(m−7)
(m−1)
Λ(b) and Qµν is a conserved quantity which according to [19] may be considered
as an energy-momentum tensor of a dark energy fluid representing the x-matter, the more common
phrase being ‘x-Cold-Dark Matter’ (xCDM). This matter has the most general form of the equation of
state which is characterized by the following conditions [21]: violation of the strong energy condition
at the present epoch for ωx < −1/3 where px = ωxρx, local stability i.e. c
s = δpx/δρx ≥ 0 and
preservation of causality i.e. cs ≤ 1. Ultimately, we have three different types of ‘matter’ on the
right hand side of equation (32), namely, ordinary confined conserved matter represented by τµν , the
matter represented by Qµν which will be discussed later and finally, the Weyl matter represented by
Eµν .
3 Field equations on the anisotropic brane
In the following we will investigate the influence of the extrinsic curvature terms on the anisotropic
universe described by Bianchi type I and V geometries. We restrict our analysis to a constant
curvature bulk, so that Eµν = 0. The constant curvature bulk is characterized by the Riemann tensor
RABCD = k∗(GACGBD − GADGBC), (33)
where GAB denotes the bulk metric components in arbitrary coordinates and k∗ is either zero for
the flat bulk, or proportional to a positive or negative bulk cosmological constant respectively, corre-
sponding to two possible signatures (4, 1) for the dS5 bulk and (3, 2) for the AdS5 bulk. We take, in
the embedding equations, g55 = ε = ±1. With this assumption the Gauss-Codazzi equations reduce
Rαβγδ =
(KαγKβδ −KαδKβγ) + k∗(gαγgβδ − gαδgβγ), (34)
Kα[β;γ] = 0. (35)
The effective equations derived in the previous section with constant curvature bulk can be written
Gµν = ατµν − λgµν +Qµν . (36)
Here, λ is the effective cosmological constant in four dimensions with Qµν being a completely geo-
metrical quantity given by
Qµν =
(KKµν −KραK
ν ) +
αβ −K2
, (37)
where K = gµνKµν . To proceed further, the confined source on the brane should now be specified.
The most common matter source used in cosmology is that of a perfect fluid which, in co-moving
coordinates, is given by
τµν = (ρ+ p)uµuν + pgµν , uµ = −δ
µ, p = (γ − 1)ρ, 1 ≤ γ ≤ 2, (38)
where γ = 2 represents the stiff cosmological fluid describing the high energy density regime of the
early universe.
From a formal point of view the Bianchi type I and V geometries are described by the line element
ds2 = −dt2 + a21(t)dx
2 + a22(t)e
−2βxdy2 + a23(t)e
−2βxdz2. (39)
The metric for the Bianchi type I geometry corresponds to the case β = 0, while for the Bianchi type
V case we have β = 1. Here ai(t), i = 1, 2, 3 are the expansion factors in different spatial directions.
For later convenience, we define the following variables [25]
ai, Hi =
, i = 1, 2, 3, 3H =
Hi, ∆Hi = Hi −H, i = 1, 2, 3. (40)
In equation (40), v is the volume scale factor, Hi, i = 1, 2, 3 are the directional Hubble parameters,
and H is the mean Hubble parameter. From equation (40) we also obtain H = v̇
. The physical
quantities of observational importance in cosmology are the expansion scalar Θ, the mean anisotropy
parameter A, and the deceleration parameter q, which are defined according to
Θ = 3H, 3A =
, q =
H−1 − 1 = −H−2
Ḣ +H2
. (41)
We note that A = 0 for an isotropic expansion. Moreover, the sign of the deceleration parameter
indicates how the universe expands. A positive sign for q corresponds to the standard decelerating
models whereas a negative sign indicates an accelerating expansion in late times.
Using the York’s relation
Kµνa = −
, (42)
we realize that in a diagonal metric, Kµνa is diagonal. After separating the spatial components, the
Codazzi equations reduce to (here α, β, γ, σ = 1, 2, 3)
Kαγa,σ +K
βσ = K
σa,γ +K
βγ , (43)
Kαγa,0 +
Kαγa =
K00a, i = 1, 2, 3. (44)
The first equation gives K11a,σ = 0 for σ 6= 1, since K
1a does not depend on the spatial coordinates.
Repeating the same procedure for α, γ = i, i = 2, 3, we obtain K22a,σ = 0 for σ 6= 2 and K
3a,σ = 0
for σ 6= 3. Assuming K11a = K
2a = K
3a = ba(t), where ba(t) are arbitrary functions of t, the second
equation gives
ḃa +
K00a, i = 1, 2, 3. (45)
Summing equations (45) we find
K00a = −
3ḃav
. (46)
For µ, ν = 1, 2, 3 we obtain
Kµνa = bagµν . (47)
Assuming further that the functions ba are equal and denoting ba = b, θ =
and Θ = v̇
, we find from
equation (37) that
αβ = b2
, K = b
, (48)
Qµν = −
gµν , µ, ν = 1, 2, 3, Q00 =
. (49)
Now, using these relations and equation (36), the modified Friedmann equations become
3Ḣ +
H2i = λ−
ρ(3γ − 2) +
, (50)
(vHi) = 2β
2v−2/3 + λ−
ρ(γ − 2) +
, i = 1, 2, 3. (51)
For β = 0 we obtain the field equations for Bianchi type I geometry, while β = 1 gives the Bianchi type
V equations on the brane world. These equations are modified with respect to the standard equations
by the components of the extrinsic curvature. Such term may be used to offer an explanation for
the x-matter. In the next section we discuss the ramifications of this term on the cosmology of our
model. We also note the implicit effects of the bulk signature ε on the expansion of the universe.
For the sake of completeness, let us compare the model presented in this work to the usual brane
worlds models where the Israel junction condition is used to calculate the extrinsic curvature in terms
of the energy-momentum tensor on the brane and its trace, that is
Kµν = −
α∗2(τµν −
τgµν), (52)
where α∗ is proportional to the gravitational constant in the bulk. If we did that, we would obtain
b(t) = −1
α∗2ρ, which, upon its substitution in equation (50), gives
3Ḣ +
H2i = λ+
ρ(2− 3γ) +
ρ2(1− 3γ), (53)
which is same as equation (16) in [25]. Therefore, the emergence of a ρ2 term in the Friedmann
equations is a consequence of the discontinuity in the bulk and the brane system. The existence of
this term either does not agree with observations or requires extra parameters and fine tuning.
4 Dark energy and role of extrinsic curvature
As we noted before, Qµν is an independently conserved quantity, suggesting that an analogy with the
energy momentum of an uncoupled non-conventional energy source would be in order. To evaluate
the compatibility of such geometrical model with the present experimental data, we identify Qµν with
x-matter [21] by defining Qµν as a perfect fluid and write
Qµν =
[(ρx + px)uµuν + pxgµν ] , px = (γx − 1)ρx. (54)
Comparing Qµν , µ, ν = 1, 2, 3 and Q00 from equation (54) with the components of Qµν and Q00 given
by equation (49), we obtain
px = −
, ρx =
. (55)
Equation (54) was chosen in accordance with the weak-energy condition corresponding to the positive
energy density and negative pressure with ε = +1. Use of the above equations leads to an equation
for b(t)
. (56)
If γx is taken as a constant, the solution for b(t) is
b(t) = b0v
−γx/2. (57)
Using equation (55) and this solution, the energy density of xCDM becomes
v−γx . (58)
A brief discussion on the energy-momentum conservation on the brane would be appropriate at
this point. The contracted Bianchi identities in the bulk space, GAB;A = 0, using equation (21),
imply
TAB +
= 0. (59)
Since the potential V has a minimum on the brane, the above conservation equation reduces to
τµν;µ = 0, (60)
and gives
γρ = 0. (61)
Thus, the time evolution of the energy density of the matter is given by
ρ = ρ0v
−γ . (62)
Using the geometrical energy density for Qµν and the evolution law of the matter energy density, the
field equations (50)-(51) become
3Ḣ +
H2i = λ+
ρ0(2− 3γ)v
b20(2− 3γx)v
−γx , (63)
(vHi) = 2β
2v−2/3 + λ+
ρ0(2− γ)v
b20(2− γx)v
−γx , i = 1, 2, 3. (64)
Summing equations (64) we find
(vH) = 2β2v−2/3 + λ+
ρ0(2− γ)v
b20(2− γx)v
−γx . (65)
Now, substituting back equation (65) into equations (64) we obtain
Hi = H +
, i = 1, 2, 3, (66)
with hi, i = 1, 2, 3 being constants of integration satisfying the consistency condition
i=1 hi = 0.
The basic equation describing the dynamics of the anisotropic brane world with a constant curvature
bulk can be written as
v̈ = 6β2v1/3 + 3λv +
ρ0(2− γ)v
1−γ +
b20(2− γx)v
1−γx . (67)
Here, we note that for a stiff fluid (γ = 2), the dynamics of the matter on the brane is solely
determined by the geometrical matter (xCDM). The general solution of equation (67) becomes
t− t0 =
9β2v4/3 + 3λv2 + 3αρ0v
2−γ + 9b20v
2−γx + C
)−1/2
dv, (68)
where C is a constant of integration. The time variation of the physically important parameters
described above in the exact parametric form, with v taken as a parameter, is given by
Θ = 3H =
9β2v4/3 + 3λv2 + 3αρ0v
2−γ + 9b20v
2−γx + C
, (69)
ai = a0iv
1/3 exp
9β2v4/3 + 3λv2 + 3αρ0v
2−γ + 9b20v
2−γx +C
)−1/2
, i = 1, 2, 3, (70)
A = 3h2
9β2v4/3 + 3λv2 + 3αρ0v
2−γ + 9b20v
2−γx + C
, (71)
9β2v4/3 +
2−γ +
2−γx + 3C
9β2v4/3 + 3λv2 + 3αρ0v2−γ + 9b
2−γx + C
) − 1, (72)
where h2 =
i=1 h
i . In addition, the integration constants hi and C must satisfy the consistency
condition h2 = 2
C. For β = 0 we obtain the general solutions for Bianchi type I geometry, while
β = 1 gives the Bianchi type V solutions on the anisotropic brane world.
In a matter dominated Bianchi type I universe, γ = 1 with γx = 0, equation (68) becomes
t− t0 =
3αρ0v + 9b
2 + C
, (73)
where for later convenience, we take C =
3α2̺2
. The time dependence of the volume scale factor of
the Bianchi type I universe is given by
v(t) = e3b0(t−t0) −
, (74)
which for t = t0+
becomes zero. By reparameterizing the initial value of the cosmological
time according to e−3b0t0 = αρ0
, the evolution of the anisotropic brane universe starts at t = 0 from
a singular state v(t = 0) = 0. Therefore the expansion scalar, scale factors, mean anisotropy, and
decelerating parameter are given by
Θ(t) =
3b0(t−t0)
e3b0(t−t0) − αρ0
, (75)
ai = a0i
e3b0(t−t0) −
]1/3 [
e−3b0(t−t0)
] 2b0hi
, i = 1, 2, 3, (76)
A(t) =
3b20e
6b0(t−t0)
, (77)
q(t) =
e−3b0(t−t0) − 1. (78)
We consider λ = 0 and show that, within the context of the present model, the extrinsic curvature
can be used to account for the accelerated expansion of the universe. In figure 1 we have plotted the
behavior of the deceleration parameter for different values of γ. The behavior of this parameter shows
that when the geometrical energy density is positive and the pressure is negative the AdS5 bulk is
not compatible with the expansion of the universe. Also, this behavior is much dependent on the
range of the values that γx can take. The use of the de Sitter bulk with ρx > 0 allows us to use the
wealth of available data from the recent measurements to determine limits on the values of γx in our
geometric model. For having an accelerating universe we distinguish γx <
. As mentioned before,
q(t) > 0 corresponds to the standard decelerating models whereas q(t) < 0 indicates an accelerating
expansion at late times. Therefore, the universe undergoes an accelerated expansion at late times in
0.5 1 1.5 2
0.5 1 1.5 2
Figure 1: Left, the deceleration parameter of the Bianchi type I universe and right, the same parameter in the Bianchi
type V universe for the de Sitter bulk with γ = 1 (solid line), γ = 4/3 (dashed line), γ = 2 (dot-dashed line), γx = 0.3
and λ = 0.
the absence of a positive cosmological constant. As has been noted in [20], it should be emphasised
here too that the geometrical approach considered here is based on three basic postulates, namely,
the confinement of the standard gauge interactions, the existence of quantum gravity in the bulk
and finally, the embedding of the brane world. All other model dependent properties such as warped
metric, mirror symmetries, radion or extra scalar fields, fine tuning parameters like the tension of the
brane and the choice of a junction condition are left out as much as possible in our calculations.
To understanding the behavior of the mean anisotropy parameter in our model, let us consider it
as a function of the volume scale factor
A(v) =
9β2v4/3 + 3λv2 + 3αρ0v
2−γ + 9b20v
2−γx +C
. (79)
The behavior of the anisotropy parameter at the initial state depends on the values of γ and γx. For
an accelerating universe we obtain γx <
. From equation (79), in the limit v → 0 (singular state)
and taking γx <
, we find
A(v) =
, 1 ≤ γ ≤ 2. (80)
Therefore for a brane world scenario with a confining potential, the initial state is always anisotropic.
In our model the behavior of the anisotropy parameter coincides with the standard 4D cosmology
and is different from the brane world models where a delta-function in the energy-momentum tensor
is used [26] to confine the mater on the brane. The behavior of the mean anisotropy parameter of
the Bianchi type I and V geometries are illustrated, for γx = 0.3 and different values of γ, in figure 2.
The behavior of this parameter shows that the universe starts from a singular state with maximum
anisotropy and ends up in an isotropic de Sitter inflationary phase at late times. In figure 3 we have
plotted the deceleration parameter and the anisotropy parameter of the Bianchi type I geometry for
γ = 1 and different values of γx = 0, 0.3, 0.5.
At this point it would be appropriate to compare our model with other forms of dark energy such
as the 4D quintessence. One may consider a 4D effective Lagrangian whose variation with respect
to gµν would result in the dynamical equations (36) compatible with the embedding and with the
confined matter hypotheses [20]. In contrast to the standard model, our model corresponds to an
Einstein-Hilbert Lagrangian which is modified by extrinsic curvature terms. The resulting Einstein
equations are thus modified by the term Qµν . Since, as was mentioned before, this quantity is
independently conserved, there is no exchange of energy between this geometrical correction and the
confined matter source. Such an aspect has one important consequence however; if Qµν is to be related
to dark energy, as we did in this paper, it does not exchange energy with ordinary matter, like the
coupled quintessence models [27]. The coupled scalar field models may avoid the cosmic coincidence
problem with the available data being used to fix the corresponding dynamics and, consequently, the
scalar field potential responsible for the present accelerating phase of the universe.
0.2 0.4 0.6 0.8 1 1.2 1.4
0.2 0.4 0.6 0.8 1 1.2 1.4
Figure 2: Left, the anisotropy parameter of the Bianchi type I universe and right, the same parameter in the Bianchi
type V universe for the de Sitter bulk with γ = 1 (solid line), γ = 4/3 (dashed line), γ = 2 (dot-dashed line), γx = 0.3
and λ = 0.
0.5 1 1.5 2 2.5 3
0.5 1 1.5 2
Figure 3: Left, the deceleration parameter of the Bianchi type I geometry and right, the anisotropy parameter in the
Bianchi type I geometry for the de Sitter bulk with γx = 0 (solid line), γx = 0.3 (dashed line), γx = 0.5 (dot-dashed
line), γ = 1 and λ = 0.
5 Conclusions
In this paper, we have studied an anisotropic brane world model in which the matter is confined to
the brane through the action of a confining potential, rendering the use of any junction condition
redundant. We have shown that in an anisotropic brane world embedded in a constant curvature dS5
bulk the accelerating expansion of the universe can be a consequence of the extrinsic curvature and
thus a purely geometrical effect. The study of the behavior of the anisotropy parameter shows that
in our model the universe starts as a singular state with maximum anisotropy and reaches, for both
Bianchi type I and V space-times, an isotropic state in the late time limit. The study of this scenario
in an inhomogeneous brane will be the subject of a future investigation.
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  We consider an anisotropic brane world with Bianchi type I and V geometries
where the mechanism of confining the matter on the brane is through the use of
a confining potential. The resulting equations on the anisotropic brane are
modified by an extra term that may be interpreted as the x-matter, providing a
possible phenomenological explanation for the accelerated expansion of the
universe. We obtain the general solution of the field equations in an exact
parametric form for both Bianchi type I and V space-times. In the special case
of a Bianchi type I the solutions of the field equations are obtained in an
exact analytic form. Finally, we study the behavior of the observationally
important parameters.

Introduction
The type Ia supernovae (SNe Ia) [1] observations provide the first evidence for the accelerating expan-
sion of the present universe. These results, when combined with the observations on the anisotropy
spectrum of cosmic microwave background (CMB) [2] and the results on the power spectrum of large
scale structure (LSS) [3], strongly suggest that the universe is spatially flat and dominated by a
component, though arguably exotic, with large negative pressure, referred to as dark energy [4]. The
nature of such dark energy constitutes an open and tantalizing question connecting cosmology and
particle physics. Different mechanisms have been suggested over the past few years to accommodate
dark energy. The simplest form of dark energy is the vacuum energy (the cosmological constant).
A tiny positive cosmological constant which can naturally explain the current acceleration would
encounter many theoretical problems such as the fine-tuning problem and the coincidence problem.
The former can be stated as the existence of an enormous gap between the vacuum expectation value,
in other words the cosmological constant, in particle physics and that observed over cosmic scales.
The absence of a fundamental mechanism which sets the cosmological constant to zero or to a very
small value is also known as the cosmological constant problem. The latter however relates to the
question of the near equality of energy densities of the dark energy and dark matter today.
Another possible form of dark energy is a dynamical, time dependent and spatially inhomogeneous
component, called the quintessence [5]. An example of quintessence is the energy associated with a
scalar field φ slowly evolving down its potential V (φ) [6, 7]. Slow evolution is needed to obtain a
negative pressure, pφ =
φ̇2+V (φ), so that the kinetic energy density is less than the potential energy
density. Yet another phenomenological explanation based on current observational data is given by
the x-matter (xCDM) model which is associated with an exotic fluid characterized by an equation
of state px = wxρx (wx < −
is the necessary condition to make a universe accelerate), where the
parameter wx can be a constant or more generally a function of time [8].
∗email: m-heydarifard@sbu.ac.ir
†email: hr-sepangi@sbu.ac.ir
http://arxiv.org/abs/0704.1035v1
Over the past few years, we have been witnessing a phenomenal interest in the possibility that
our observable four-dimensional (4D) universe may be viewed as a brane hypersurface embedded
in a higher dimensional bulk space. Physical matter fields are confined to this hypersurface, while
gravity can propagate in the higher dimensional space-time as well as on the brane. The most
popular model in the context of brane world theory is that proposed by Randall and Sundrum
(RS). In the so-called RSI model [9], the authors proposed a mechanism to solve the hierarchy
problem with two branes, while in the RSII model [10], they considered a single brane with a positive
tension, where 4D Newtonian gravity is recovered at low energies even if the extra dimension is not
compact. This mechanism provides us with an alternative to compactification of extra dimensions.
The cosmological evolution of such a brane universe has been extensively investigated and effects such
as a quadratic density term in the Friedmann equations have been found [11]-[13]. This term arises
from the imposition of the Israel junction conditions which is a relationship between the extrinsic
curvature and energy-momentum tensor of the brane and results from the singular behavior in the
energy-momentum tensor. There has been concerns expressed over applying such junction conditions
in that they may not be unique. Indeed, other forms of junction conditions exist, so that different
conditions may lead to different physical results [14]. Furthermore, these conditions cannot be used
when more than one non-compact extra dimension is involved. To avoid such concerns, an interesting
higher-dimensional model was introduced in [15] where particles are trapped on a 4-dimensional
hypersurface by the action of a confining potential V. In [16], the dynamics of test particles confined
to a brane by the action of such potential at the classical and quantum levels were studied and the
effects of small perturbations along the extra dimensions investigated. Within the classical limits,
test particles remain stable under small perturbations and the effects of the extra dimensions are not
felt by them, making them undetectable in this way. The quantum fluctuations of the brane cause
the mass of a test particle to become quantized and, interestingly, the Yang-Mills fields appear as
quantum effects. Also, in [17], a braneworld model was studied in which the matter is confined to the
brane through the action of such a potential, rendering the use of any junction condition unnecessary
and predicting a geometrical explanation for the accelerated expansion of the universe.
In brane theories the covariant Einstein equations are derived by projecting the bulk equations
onto the brane. This was first done by Shiromizu, Maeda and Sasaki (SMS) [18] where the Gauss-
Codazzi equations together with Israel junction conditions were used to obtain the Einstein field
equations on the 3-brane. In a series of recent papers a number of authors [23, 24] have presented
detailed descriptions of the dynamic of homogeneous and anisotropic brane worlds in the SMS model.
The study of anisotropic homogeneous brane world cosmological models has shown an important
difference between these models and standard 4D general relativity, namely, that brane universes are
born in an isotropic state. For the anisotropic Bianchi type I and V geometries, with a conformally
flat bulk (vanishing Weyl tensor), this type of behavior has been found by exactly solving the field
equations [25].
In this paper, we follow [17] and consider an m-dimensional bulk space without imposing the
Z2 symmetry. As mentioned above, to localize the matter on the brane, a confining potential is
used rather than a delta-function in the energy-momentum tensor. The resulting equations on the
anisotropic brane are modified by an extra term that may be interpreted as the x-matter, providing
a possible phenomenological explanation for the accelerated expansion of the universe. The behavior
of the observationally important physical quantities such as anisotropy and deceleration parameter
is studied in this scenario. We should emphasize here that there is a difference between the model
presented in this work and models introduced in [19, 20] in that in the latter no mechanism is
introduced to account for the confinement of matter to the brane.
2 The model
In this section we present a brief review of the model proposed in [16]. Consider the background
manifold V 4 isometrically embedded in a pseudo-Riemannian manifold Vm by the map Y : V 4 → Vm
such that
,ν = ḡµν , GABY
a = 0, GABN
b = gab = ±1, (1)
where GAB (ḡµν) is the metric of the bulk (brane) space Vm(V 4) in arbitrary coordinates, {Y
A} ({xµ})
is the basis of the bulk (brane) and NAa are (m − 4) normal unit vectors, orthogonal to the brane.
Perturbation of V̄4 in a sufficiently small neighborhood of the brane along an arbitrary transverse
direction ξ is given by
ZA(xµ, ξa) = YA + (LξY)
A, (2)
where L represents the Lie derivative and ξa (a = 1, 2, ...,m− 4) is a small parameter along NAa that
parameterizes the extra noncompact dimensions. By choosing ξ orthogonal to the brane, we ensure
gauge independency [16] and have perturbations of the embedding along a single orthogonal extra
direction N̄a giving local coordinates of the perturbed brane as
ZA,µ(x
ν , ξa) = YA,µ + ξ
aN̄Aa,µ(x
ν). (3)
In a similar manner, one can find that since the vectors N̄A depend only on the local coordinates xµ,
they do not propagate along the extra dimensions. The above assumptions lead to the embedding
equations of the perturbed geometry
Gµν = GABZ
,ν , Gµa = GABZ
a, GABN
b = gab. (4)
If we set NAa = δ
a , the metric of the bulk space can be written in the following matrix form
GAB =
gµν +AµcA
ν Aµa
Aνb gab
, (5)
where
gµν = ḡµν − 2ξ
aK̄µνa + ξ
aξbḡαβK̄µαaK̄νβb, (6)
is the metric of the perturbed brane, so that
K̄µνa = −GABY
a;ν , (7)
represents the extrinsic curvature of the original brane (second fundamental form). We use the
notation Aµc = ξ
dAµcd, where
Aµcd = GABN
c = Āµcd, (8)
represents the twisting vector fields (the normal fundamental form). Any fixed ξa signifies a new
perturbed geometry, enabling us to define an extrinsic curvature similar to the original one by
K̃µνa = −GABZ
a;ν = K̄µνa − ξ
K̄µγaK̄
νb +AµcaA
. (9)
Note that definitions (5) and (9) require
K̃µνa = −
. (10)
In geometric language, the presence of gauge fields Aµa tilts the embedded family of sub-manifolds
with respect to the normal vector NA. According to our construction, the original brane is orthogonal
to the normal vector NA. However, equation (4) shows that this is not true for the deformed geometry.
Let us change the embedding coordinates and set
XA,µ = Z
,µ − g
abNAa Abµ. (11)
The coordinates XA describe a new family of embedded manifolds whose members are always or-
thogonal to NA. In this coordinates the embedding equations of the perturbed brane is similar to
the original one, described by equation (1), so that YA is replaced by XA. This new embedding of
the local coordinates is suitable for obtaining induced Einstein field equations on the brane. The
extrinsic curvature of a perturbed brane then becomes
Kµνa = −GABX
a;ν = K̄µνa − ξ
bK̄µγaK̄
νb = −
, (12)
which is the generalized York’s relation and shows how the extrinsic curvature propagates as a result
of the propagation of the metric in the direction of extra dimensions. The components of the Riemann
tensor of the bulk written in the embedding vielbein {XA,α,N
a }, lead to the Gauss-Codazzi equations
Rαβγδ = 2g
abKα[γaKδ]βb +RABCDX
,δ , (13)
2Kα[γc;δ] = 2g
abA[γacKδ]αb +RABCDX
,δ , (14)
where RABCD and Rαβγδ are the Riemann tensors for the bulk and the perturbed brane respectively.
Contracting the Gauss equation (13) on α and γ, we find
Rµν = (KµαcK
ν −KcK
µν ) +RABX
,ν − g
abRABCDN
b , (15)
and the Einstein tensor of the brane is given by
Gµν = GABX
,ν +Qµν + g
abRABN
b gµν − g
abRABCDN
b , (16)
where
Qµν = −g
KγµaKγνb −KaKµνb
KαβaK
αβa −KaK
gµν . (17)
As can be seen from the definition of Qµν , it is independently a conserved quantity, that is Q
;ν = 0
[19]. Using the decomposition of the Riemann tensor into the Weyl curvature, the Ricci tensor and
the scalar curvature
RABCD = CABCD −
(m− 2)
GB[DRC]A − GA[DRC]B
(m− 1)(m − 2)
R(GA[DGC]B), (18)
we obtain the 4D Einstein equations as
Gµν = GABX
,ν +Qµν − Eµν +
(m− 2)
gabRABN
b gµν
(m− 2)
(m− 1)(m − 2)
Rgµν , (19)
where
Eµν = g
abCABCDN
,ν , (20)
is the electric part of the Weyl tensor CABCD. Now, let us write the Einstein equation in the bulk
space as
AB + Λ
(b)GAB = α
∗SAB, (21)
where
SAB = TAB +
VGAB , (22)
here α∗ = 1
(M∗ is the fundamental scale of energy in the bulk space), Λ
(b) is the cosmological
constant of the bulk and TAB is the energy-momentum tensor of the matter confined to the brane
through the action of the confining potential V. We require V to satisfy three general conditions:
firstly, it has a deep minimum on the non-perturbed brane, secondly, depends only on extra coordi-
nates and thirdly, the gauge group representing the subgroup of the isometry group of the bulk space
is preserved by it [16]. The vielbein components of the energy-momentum tensor are given by
Sµν = SABX
,ν , Sµa = SABX
a , Sab = SABN
b . (23)
Use of equation (22) then gives
RAB = −
(m− 2)
GABS +
(m− 2)
Λ(b)GAB + α
∗SAB, (24)
R = −
(α∗S −mΛ(b)). (25)
Substituting RAB and R from the above into equation (19) and using equation (23), we obtain
Gµν = Qµν − Eµν +
(m− 3)
(m− 2)
α∗gabSabgµν +
(m− 2)
Sµν −
(m− 4)(m− 3)
(m− 1)(m− 2)
α∗Sgµν
(m− 7)
(m− 1)
Λ(b)gµν . (26)
On the other hand, again from equation (21), the trace of the Codazzi equation (14) gives the “gravi-
vector equation”
Kδaγ;δ −Ka,γ −AbaγK
b +AbaδK
bδ +Baγ =
3(m− 4)
α∗Saγ , (27)
where
Baγ = g
mnCABCDN
n . (28)
Finally, the “gravi-scalar equation” is obtained from the contraction of (15), (19) and using equation
S − gmnSmn
gab =
(Q+R+W ) gab −
Λ(b)gab, (29)
where
W = gabgmnCABCDN
a . (30)
Equations (26)-(30) represent the projections of the Einstein field equations on the brane-brane,
bulk-brane, and bulk-bulk directions.
As was mentioned in the introduction, localization of matter on the brane is realized in this model
by the action of a confining potential. Since the potential V is assumed to have a minimum on the
brane, which can be taken as zero, localization of matter may simply be realized by taking equation
(22) and consider its components on the brane, in which case we may write
ατµν =
(m− 2)
Tµν , Tµa = 0, Tab = 0, (31)
where α is the scale of energy on the brane. Now, the induced Einstein field equations on the original
brane can be written as
Gµν = ατµν −
(m− 4)(m− 3)
2(m− 1)
ατgµν − Λgµν +Qµν − Eµν , (32)
where Λ = −
(m−7)
(m−1)
Λ(b) and Qµν is a conserved quantity which according to [19] may be considered
as an energy-momentum tensor of a dark energy fluid representing the x-matter, the more common
phrase being ‘x-Cold-Dark Matter’ (xCDM). This matter has the most general form of the equation of
state which is characterized by the following conditions [21]: violation of the strong energy condition
at the present epoch for ωx < −1/3 where px = ωxρx, local stability i.e. c
s = δpx/δρx ≥ 0 and
preservation of causality i.e. cs ≤ 1. Ultimately, we have three different types of ‘matter’ on the
right hand side of equation (32), namely, ordinary confined conserved matter represented by τµν , the
matter represented by Qµν which will be discussed later and finally, the Weyl matter represented by
Eµν .
3 Field equations on the anisotropic brane
In the following we will investigate the influence of the extrinsic curvature terms on the anisotropic
universe described by Bianchi type I and V geometries. We restrict our analysis to a constant
curvature bulk, so that Eµν = 0. The constant curvature bulk is characterized by the Riemann tensor
RABCD = k∗(GACGBD − GADGBC), (33)
where GAB denotes the bulk metric components in arbitrary coordinates and k∗ is either zero for
the flat bulk, or proportional to a positive or negative bulk cosmological constant respectively, corre-
sponding to two possible signatures (4, 1) for the dS5 bulk and (3, 2) for the AdS5 bulk. We take, in
the embedding equations, g55 = ε = ±1. With this assumption the Gauss-Codazzi equations reduce
Rαβγδ =
(KαγKβδ −KαδKβγ) + k∗(gαγgβδ − gαδgβγ), (34)
Kα[β;γ] = 0. (35)
The effective equations derived in the previous section with constant curvature bulk can be written
Gµν = ατµν − λgµν +Qµν . (36)
Here, λ is the effective cosmological constant in four dimensions with Qµν being a completely geo-
metrical quantity given by
Qµν =
(KKµν −KραK
ν ) +
αβ −K2
, (37)
where K = gµνKµν . To proceed further, the confined source on the brane should now be specified.
The most common matter source used in cosmology is that of a perfect fluid which, in co-moving
coordinates, is given by
τµν = (ρ+ p)uµuν + pgµν , uµ = −δ
µ, p = (γ − 1)ρ, 1 ≤ γ ≤ 2, (38)
where γ = 2 represents the stiff cosmological fluid describing the high energy density regime of the
early universe.
From a formal point of view the Bianchi type I and V geometries are described by the line element
ds2 = −dt2 + a21(t)dx
2 + a22(t)e
−2βxdy2 + a23(t)e
−2βxdz2. (39)
The metric for the Bianchi type I geometry corresponds to the case β = 0, while for the Bianchi type
V case we have β = 1. Here ai(t), i = 1, 2, 3 are the expansion factors in different spatial directions.
For later convenience, we define the following variables [25]
ai, Hi =
, i = 1, 2, 3, 3H =
Hi, ∆Hi = Hi −H, i = 1, 2, 3. (40)
In equation (40), v is the volume scale factor, Hi, i = 1, 2, 3 are the directional Hubble parameters,
and H is the mean Hubble parameter. From equation (40) we also obtain H = v̇
. The physical
quantities of observational importance in cosmology are the expansion scalar Θ, the mean anisotropy
parameter A, and the deceleration parameter q, which are defined according to
Θ = 3H, 3A =
, q =
H−1 − 1 = −H−2
Ḣ +H2
. (41)
We note that A = 0 for an isotropic expansion. Moreover, the sign of the deceleration parameter
indicates how the universe expands. A positive sign for q corresponds to the standard decelerating
models whereas a negative sign indicates an accelerating expansion in late times.
Using the York’s relation
Kµνa = −
, (42)
we realize that in a diagonal metric, Kµνa is diagonal. After separating the spatial components, the
Codazzi equations reduce to (here α, β, γ, σ = 1, 2, 3)
Kαγa,σ +K
βσ = K
σa,γ +K
βγ , (43)
Kαγa,0 +
Kαγa =
K00a, i = 1, 2, 3. (44)
The first equation gives K11a,σ = 0 for σ 6= 1, since K
1a does not depend on the spatial coordinates.
Repeating the same procedure for α, γ = i, i = 2, 3, we obtain K22a,σ = 0 for σ 6= 2 and K
3a,σ = 0
for σ 6= 3. Assuming K11a = K
2a = K
3a = ba(t), where ba(t) are arbitrary functions of t, the second
equation gives
ḃa +
K00a, i = 1, 2, 3. (45)
Summing equations (45) we find
K00a = −
3ḃav
. (46)
For µ, ν = 1, 2, 3 we obtain
Kµνa = bagµν . (47)
Assuming further that the functions ba are equal and denoting ba = b, θ =
and Θ = v̇
, we find from
equation (37) that
αβ = b2
, K = b
, (48)
Qµν = −
gµν , µ, ν = 1, 2, 3, Q00 =
. (49)
Now, using these relations and equation (36), the modified Friedmann equations become
3Ḣ +
H2i = λ−
ρ(3γ − 2) +
, (50)
(vHi) = 2β
2v−2/3 + λ−
ρ(γ − 2) +
, i = 1, 2, 3. (51)
For β = 0 we obtain the field equations for Bianchi type I geometry, while β = 1 gives the Bianchi type
V equations on the brane world. These equations are modified with respect to the standard equations
by the components of the extrinsic curvature. Such term may be used to offer an explanation for
the x-matter. In the next section we discuss the ramifications of this term on the cosmology of our
model. We also note the implicit effects of the bulk signature ε on the expansion of the universe.
For the sake of completeness, let us compare the model presented in this work to the usual brane
worlds models where the Israel junction condition is used to calculate the extrinsic curvature in terms
of the energy-momentum tensor on the brane and its trace, that is
Kµν = −
α∗2(τµν −
τgµν), (52)
where α∗ is proportional to the gravitational constant in the bulk. If we did that, we would obtain
b(t) = −1
α∗2ρ, which, upon its substitution in equation (50), gives
3Ḣ +
H2i = λ+
ρ(2− 3γ) +
ρ2(1− 3γ), (53)
which is same as equation (16) in [25]. Therefore, the emergence of a ρ2 term in the Friedmann
equations is a consequence of the discontinuity in the bulk and the brane system. The existence of
this term either does not agree with observations or requires extra parameters and fine tuning.
4 Dark energy and role of extrinsic curvature
As we noted before, Qµν is an independently conserved quantity, suggesting that an analogy with the
energy momentum of an uncoupled non-conventional energy source would be in order. To evaluate
the compatibility of such geometrical model with the present experimental data, we identify Qµν with
x-matter [21] by defining Qµν as a perfect fluid and write
Qµν =
[(ρx + px)uµuν + pxgµν ] , px = (γx − 1)ρx. (54)
Comparing Qµν , µ, ν = 1, 2, 3 and Q00 from equation (54) with the components of Qµν and Q00 given
by equation (49), we obtain
px = −
, ρx =
. (55)
Equation (54) was chosen in accordance with the weak-energy condition corresponding to the positive
energy density and negative pressure with ε = +1. Use of the above equations leads to an equation
for b(t)
. (56)
If γx is taken as a constant, the solution for b(t) is
b(t) = b0v
−γx/2. (57)
Using equation (55) and this solution, the energy density of xCDM becomes
v−γx . (58)
A brief discussion on the energy-momentum conservation on the brane would be appropriate at
this point. The contracted Bianchi identities in the bulk space, GAB;A = 0, using equation (21),
imply
TAB +
= 0. (59)
Since the potential V has a minimum on the brane, the above conservation equation reduces to
τµν;µ = 0, (60)
and gives
γρ = 0. (61)
Thus, the time evolution of the energy density of the matter is given by
ρ = ρ0v
−γ . (62)
Using the geometrical energy density for Qµν and the evolution law of the matter energy density, the
field equations (50)-(51) become
3Ḣ +
H2i = λ+
ρ0(2− 3γ)v
b20(2− 3γx)v
−γx , (63)
(vHi) = 2β
2v−2/3 + λ+
ρ0(2− γ)v
b20(2− γx)v
−γx , i = 1, 2, 3. (64)
Summing equations (64) we find
(vH) = 2β2v−2/3 + λ+
ρ0(2− γ)v
b20(2− γx)v
−γx . (65)
Now, substituting back equation (65) into equations (64) we obtain
Hi = H +
, i = 1, 2, 3, (66)
with hi, i = 1, 2, 3 being constants of integration satisfying the consistency condition
i=1 hi = 0.
The basic equation describing the dynamics of the anisotropic brane world with a constant curvature
bulk can be written as
v̈ = 6β2v1/3 + 3λv +
ρ0(2− γ)v
1−γ +
b20(2− γx)v
1−γx . (67)
Here, we note that for a stiff fluid (γ = 2), the dynamics of the matter on the brane is solely
determined by the geometrical matter (xCDM). The general solution of equation (67) becomes
t− t0 =
9β2v4/3 + 3λv2 + 3αρ0v
2−γ + 9b20v
2−γx + C
)−1/2
dv, (68)
where C is a constant of integration. The time variation of the physically important parameters
described above in the exact parametric form, with v taken as a parameter, is given by
Θ = 3H =
9β2v4/3 + 3λv2 + 3αρ0v
2−γ + 9b20v
2−γx + C
, (69)
ai = a0iv
1/3 exp
9β2v4/3 + 3λv2 + 3αρ0v
2−γ + 9b20v
2−γx +C
)−1/2
, i = 1, 2, 3, (70)
A = 3h2
9β2v4/3 + 3λv2 + 3αρ0v
2−γ + 9b20v
2−γx + C
, (71)
9β2v4/3 +
2−γ +
2−γx + 3C
9β2v4/3 + 3λv2 + 3αρ0v2−γ + 9b
2−γx + C
) − 1, (72)
where h2 =
i=1 h
i . In addition, the integration constants hi and C must satisfy the consistency
condition h2 = 2
C. For β = 0 we obtain the general solutions for Bianchi type I geometry, while
β = 1 gives the Bianchi type V solutions on the anisotropic brane world.
In a matter dominated Bianchi type I universe, γ = 1 with γx = 0, equation (68) becomes
t− t0 =
3αρ0v + 9b
2 + C
, (73)
where for later convenience, we take C =
3α2̺2
. The time dependence of the volume scale factor of
the Bianchi type I universe is given by
v(t) = e3b0(t−t0) −
, (74)
which for t = t0+
becomes zero. By reparameterizing the initial value of the cosmological
time according to e−3b0t0 = αρ0
, the evolution of the anisotropic brane universe starts at t = 0 from
a singular state v(t = 0) = 0. Therefore the expansion scalar, scale factors, mean anisotropy, and
decelerating parameter are given by
Θ(t) =
3b0(t−t0)
e3b0(t−t0) − αρ0
, (75)
ai = a0i
e3b0(t−t0) −
]1/3 [
e−3b0(t−t0)
] 2b0hi
, i = 1, 2, 3, (76)
A(t) =
3b20e
6b0(t−t0)
, (77)
q(t) =
e−3b0(t−t0) − 1. (78)
We consider λ = 0 and show that, within the context of the present model, the extrinsic curvature
can be used to account for the accelerated expansion of the universe. In figure 1 we have plotted the
behavior of the deceleration parameter for different values of γ. The behavior of this parameter shows
that when the geometrical energy density is positive and the pressure is negative the AdS5 bulk is
not compatible with the expansion of the universe. Also, this behavior is much dependent on the
range of the values that γx can take. The use of the de Sitter bulk with ρx > 0 allows us to use the
wealth of available data from the recent measurements to determine limits on the values of γx in our
geometric model. For having an accelerating universe we distinguish γx <
. As mentioned before,
q(t) > 0 corresponds to the standard decelerating models whereas q(t) < 0 indicates an accelerating
expansion at late times. Therefore, the universe undergoes an accelerated expansion at late times in
0.5 1 1.5 2
0.5 1 1.5 2
Figure 1: Left, the deceleration parameter of the Bianchi type I universe and right, the same parameter in the Bianchi
type V universe for the de Sitter bulk with γ = 1 (solid line), γ = 4/3 (dashed line), γ = 2 (dot-dashed line), γx = 0.3
and λ = 0.
the absence of a positive cosmological constant. As has been noted in [20], it should be emphasised
here too that the geometrical approach considered here is based on three basic postulates, namely,
the confinement of the standard gauge interactions, the existence of quantum gravity in the bulk
and finally, the embedding of the brane world. All other model dependent properties such as warped
metric, mirror symmetries, radion or extra scalar fields, fine tuning parameters like the tension of the
brane and the choice of a junction condition are left out as much as possible in our calculations.
To understanding the behavior of the mean anisotropy parameter in our model, let us consider it
as a function of the volume scale factor
A(v) =
9β2v4/3 + 3λv2 + 3αρ0v
2−γ + 9b20v
2−γx +C
. (79)
The behavior of the anisotropy parameter at the initial state depends on the values of γ and γx. For
an accelerating universe we obtain γx <
. From equation (79), in the limit v → 0 (singular state)
and taking γx <
, we find
A(v) =
, 1 ≤ γ ≤ 2. (80)
Therefore for a brane world scenario with a confining potential, the initial state is always anisotropic.
In our model the behavior of the anisotropy parameter coincides with the standard 4D cosmology
and is different from the brane world models where a delta-function in the energy-momentum tensor
is used [26] to confine the mater on the brane. The behavior of the mean anisotropy parameter of
the Bianchi type I and V geometries are illustrated, for γx = 0.3 and different values of γ, in figure 2.
The behavior of this parameter shows that the universe starts from a singular state with maximum
anisotropy and ends up in an isotropic de Sitter inflationary phase at late times. In figure 3 we have
plotted the deceleration parameter and the anisotropy parameter of the Bianchi type I geometry for
γ = 1 and different values of γx = 0, 0.3, 0.5.
At this point it would be appropriate to compare our model with other forms of dark energy such
as the 4D quintessence. One may consider a 4D effective Lagrangian whose variation with respect
to gµν would result in the dynamical equations (36) compatible with the embedding and with the
confined matter hypotheses [20]. In contrast to the standard model, our model corresponds to an
Einstein-Hilbert Lagrangian which is modified by extrinsic curvature terms. The resulting Einstein
equations are thus modified by the term Qµν . Since, as was mentioned before, this quantity is
independently conserved, there is no exchange of energy between this geometrical correction and the
confined matter source. Such an aspect has one important consequence however; if Qµν is to be related
to dark energy, as we did in this paper, it does not exchange energy with ordinary matter, like the
coupled quintessence models [27]. The coupled scalar field models may avoid the cosmic coincidence
problem with the available data being used to fix the corresponding dynamics and, consequently, the
scalar field potential responsible for the present accelerating phase of the universe.
0.2 0.4 0.6 0.8 1 1.2 1.4
0.2 0.4 0.6 0.8 1 1.2 1.4
Figure 2: Left, the anisotropy parameter of the Bianchi type I universe and right, the same parameter in the Bianchi
type V universe for the de Sitter bulk with γ = 1 (solid line), γ = 4/3 (dashed line), γ = 2 (dot-dashed line), γx = 0.3
and λ = 0.
0.5 1 1.5 2 2.5 3
0.5 1 1.5 2
Figure 3: Left, the deceleration parameter of the Bianchi type I geometry and right, the anisotropy parameter in the
Bianchi type I geometry for the de Sitter bulk with γx = 0 (solid line), γx = 0.3 (dashed line), γx = 0.5 (dot-dashed
line), γ = 1 and λ = 0.
5 Conclusions
In this paper, we have studied an anisotropic brane world model in which the matter is confined to
the brane through the action of a confining potential, rendering the use of any junction condition
redundant. We have shown that in an anisotropic brane world embedded in a constant curvature dS5
bulk the accelerating expansion of the universe can be a consequence of the extrinsic curvature and
thus a purely geometrical effect. The study of the behavior of the anisotropy parameter shows that
in our model the universe starts as a singular state with maximum anisotropy and reaches, for both
Bianchi type I and V space-times, an isotropic state in the late time limit. The study of this scenario
in an inhomogeneous brane will be the subject of a future investigation.
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