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arXiv:1005.3101v1 [gr-qc] 18 May 2010
Inflation and late time acceleration in braneworld cosmological
models with varying brane tension
K. C. Wong∗
Department of Physics, University of Hong Kong,
Pok Fu Lam Road, Hong Kong, P. R. China
K. S. Cheng†and T. Harko‡
Department of Physics and Center for Theoretical and Computational Physics,
University of Hong Kong, Pok Fu Lam Road, Hong Kong, P. R. China
(Dated: May 19, 2010)
Abstract
Braneworld models with variable brane tension λ introduce a new degree of freedom that allows
for evolving gravitational and cosmological constants, the latter being a natural candidate for dark
energy. We consider a thermodynamic interpretation of the varying brane tension models, by
showing that the field equations with variable λ can be interpreted as describing matter creation
in a cosmological framework. The particle creation rate is determined by the variation rate of the
brane tension, as well as by the brane-bulk energy-matter transfer rate. We investigate the effect
of a variable brane tension on the cosmological evolution of the Universe, in the framework of a
particular model in which the brane tension is an exponentially dependent function of the scale
factor. The resulting cosmology shows the presence of an initial inflationary expansion, followed
by a decelerating phase, and by a smooth transition towards a late accelerated de Sitter type
expansion. The varying brane tension is also responsible for the generation of the matter in the
Universe (reheating period). The physical constraints on the model parameters, resulted from the
observational cosmological data, are also investigated.
PACS numbers: 04.50.-h, 04.20.Jb, 04.20.Cv, 95.35.+d
∗Electronic address: fankywong@gmail.com
†Electronic address: hrspksc@hkucc.hku.hk
‡Electronic address: harko@hkucc.hku.hk
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I.INTRODUCTION
The idea of embedding our Universe in a higher dimensional space has attracted a con-
siderable interest recently, due to the proposal by Randall and Sundrum that our four-
dimensional (4D) spacetime is a three-brane, embedded in a 5D spacetime (the bulk) [1, 2].
This proposal is based on early studies on superstring theory and M-theory, which have sug-
gested that our four dimensional world is embedded into a higher dimensional spacetime.
Particularly, the 10 dimensional E8⊗ E8heterotic superstring theory is a low-energy limit
of the 11 dimensional supergravity, under the compactification scheme M10× S1/Z2[3, 4].
Thus, the 10 dimensional spacetime is compactified as M4× CY6× S1/Z2, implying that
our Universe (a brane) is embedded into a higher dimensional bulk. In this paradigm, the
standard model particles are open strings, confined on the braneworld, whilst the gravitons
and the closed strings can freely propagate into the bulk [5].
The Randall-Sundrum Type II model has the virtue of providing a new type of compacti-
fication of gravity [1, 2]. Standard 4D gravity can be recovered in the low-energy limit of the
model, with a 3-brane of positive tension embedded in 5D anti-de Sitter bulk. The covariant
formulation of the braneworld models has been formulated in [6], leading to the modification
of the standard Friedmann equations on the brane. It turns out that the dynamics of the
early Universe is altered by the quadratic terms in the energy density and by the contri-
bution of the components of the bulk Weyl tensor, which both give a contribution in the
energy momentum tensor. This implies a modification of the basic equations describing the
cosmological and astrophysical dynamics, which has been extensively considered recently
[7].
The recent observations of the CMB anisotropy by WMAP [10] have provided convincing
evidence for the inflationary paradigm [11], according to which in its very early stages the
Universe experienced an accelerated (de Sitter) expansionary phase (for recent reviews on
inflation see [12]).
At the end of inflation, the Universe is in a cold and low-entropy phase, which is utterly
different from the present hot high-entropy Universe. Therefore the Universe should be
reheated, or defrosted, to a high enough temperature, in order to recover the standard Hot
Big Bang [13]. The reheating process may be envisioned as follows: the energy density in
zero-momentum mode of the scalar field decays into normal particles with decay rate Γ. The
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decay products then scattered and thermalize to form a plasma [12].
Apart from the behavior of the inflaton field, the evolutions of dark energy and dark
matter in reheating stage were also considered. In [14], dark energy and dark matter were
originated from a scalar field in different stages of the inflation, according to a special form
of potential. Meanwhile, the conditions for unifying the description of inflation, dark matter
and dark energy were considered in [15]. A specific model was later proposed in [16], by
using a modified quadratic scalar potential. The candidates of dark matter in [15] and [16]
were oscillations of a scalar field. However, it may be possible that dark matter existed on
its own without originating from the scalar field. This may pose less stringent constraint on
the scalar field, so that dark matter can be included in inflation paradigm in a easier way.
On the other hand, it was proposed that the decay products of scalar field acquired thermal
mass [17].
The reheating in the braneworld models has also been considered recently. In the context
of the braneworld inflation driven by a bulk scalar field, the energy dissipation from the
bulk scalar field into the matter on the brane was studied in [18]. The obtained results
supports the idea that the brane inflation model, caused by a bulk scalar field, may be a
viable alternative scenario of the early Universe. The inflation and reheating in a braneworld
model derived from Type IIA string theory was studied in [19]. In this model the inflaton
can decay into scalar and spinor particles, thus reheating the Universe. A model in which
high energy brane corrections allow a single scalar field to describe inflation at early epochs
and quintessence at late times was discussed in [20]. The reheating mechanism in the model
originates from Born-Infeld matter, whose energy density mimics cosmological constant at
very early times and manifests itself as radiation subsequently. The particle production at
the collision of two domain walls in a 5-dimensional Minkowski spacetime was studied in
[21]. This may provide the reheating mechanism of an ekpyrotic (or cyclic) brane Universe,
in which two BPS branes collide and evolve into a hot big bang Universe. The reheating
temperature TRHin models in which the Universe exits reheating at temperatures in the
MeV regime was studied in [22], and a minimum bound on TRH was obtained. The de-
rived lower bound on the reheating temperature also leads to very stringent bounds on the
compactification scale in models with n large extra dimensions. The dark matter problem
in the Randall-Sundrum type II braneworld scenario was discussed in [23], by assuming
that the lightest supersymmetric particle is the axino. The axinos can play the role of cold
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dark matter in the Universe, due to the higher reheating temperatures in the braneworld
model, as compared to the conventional four-dimensional cosmology. The impact of the
non-conventional brane cosmology on the relic abundance of non-relativistic stable particles
in high and low reheating scenarios was investigated in [24]. In the case of high reheating
temperatures, the brane cosmology may enhance the dark matter relic density by many order
of magnitudes, and a stringent lower bound on the five dimensional scale may be obtained.
In the non-equilibrium case, the resulting relic density is very small. The curvaton dynamics
in brane-world cosmologies was studied in [25].
Brane-worlds with non-constant tension, based on the analogy with fluid membranes,
which exhibit a temperature-dependence according to the empirical law established by
E¨ otv¨ os, were introduced in [26]. This new degree of freedom allows for evolving gravi-
tational and cosmological constants, the latter being a natural candidate for dark energy.
The covariant dynamics on a brane with variable tension was studied in its full general-
ity, by considering asymmetrically embedded branes, and allowing for non-standard model
fields in the 5-dimensional space-time. This formalism was applied for a perfect fluid on
a Friedmann brane, which is embedded in a 5-dimensional charged Vaidya-Anti de Sitter
space-time. For cosmological branes a variable brane tension leads to several important
consequences. A variable brane tension may remove the initial singularity of the Universe,
since the brane Universe was created at a finite temperature Tcand scale factor amin[27].
Both the brane tension and the 4-dimensional gravitational coupling ’constant’ increase with
the scale factor from zero to asymptotic values. The 4-dimensional cosmological constant is
dynamical, evolving with a, starting with a huge negative value, passing through zero, and
finally reaching a small positive value. Such a scale–factor dependent cosmological constant
has the potential to generate additional attraction at small a (as dark matter does) and late-
time repulsion at large a (dark energy). The evolution of the brane tension is compensated
by energy interchange between the brane and the fifth dimension, such that the continu-
ity equation holds for the cosmological fluid [27]. The resulting cosmology closely mimics
the standard model at late times, a decelerated phase being followed by an accelerated ex-
pansion. The energy absorption of the brane drives the 5D space-time towards maximal
symmetry, thus becoming Anti de Sitter. Other physical and cosmological implications of a
varying brane tension have been considered in [28].
It is the purpose of the present paper to further investigate the cosmological implications
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of a varying brane tension. As a first step in our study, we consider a thermodynamic
interpretation of the varying brane tension models, by showing that the field equations with
variable λ can be interpreted as describing matter creation in a cosmological framework.
The particle creation rate is determined by the variation rate of the brane tension, as well
as by the brane-bulk energy-matter transfer rate. In particular, by adopting a theoretical
model in which the brane tension is a simple function of the scale factor of the Universe, we
consider the possibility that the early inflationary era in the evolution of the brane Universe
was driven by a varying brane tension. A varying brane tension may also be responsible for
the generation of the matter after reheating, as well as for the late time acceleration of the
Universe.
The present paper is organized as follows. In Section II we present the field equations of
the brane world models with varying brane tension and we write down the basic equations
describing the cosmological dynamics of a flat Friedmann-Robertson-Walker Universe. The
thermodynamic interpretation of the brane-world models with varying brane tension and
brane-bulk matter-energy exchange is considered in Section III. A power-law inflationary
brane-world model with varying brane tension and non-zero bulk pressure is obtained in
Section IV. The analytical behavior of the cosmological model with varying brane tension
is considered in Section V, by using the small and large time approximations for the brane
tension. The numerical analysis of the model is performed in Section VII. We discuss and
conclude our results in Section VIII.
II. GEOMETRY AND FIELD EQUATIONS IN THE VARIABLE BRANE TEN-
SION MODELS
In the present Section we present the field equations for brane world models with varying
brane tension, and the corresponding cosmological field equations for a flat Robertson-
Walker space-time.
A. Gravitational field equations
We start by considering a five dimensional (5D) spacetime (the bulk), with a large neg-
ative 5D cosmological constant(5)Λ and a single four-dimensional (4D) brane, on which
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usual (baryonic) matter and physical fields are confined. The 4D braneworld ((4)M,(4)gµν)
is located at a hypersurface
?B?XA?= 0?
in the 5D bulk spacetime ((5)M,(5)gAB) with
mirror symmetry, and with coordinates XA,A = 0,1,...,4. The induced 4D coordinates on
the brane are xµ,µ = 0,1,2,3. We choose normal Gaussian coordinates, and therefore the
5D metric is related to the 4D metric by the relation(5)gMN=(4)gMN+ nMnN, where nM
is the normal vector.
The induced 4D metric is gIJ=(5)gIJ−nInJ, where nIis the space-like unit vector field
normal to the brane hypersurface(4)M. The basic equations on the brane are obtained by
projections onto the brane world with Gauss equation, Codazzi equation and Israel junction
condition, the projected Einstein equation are given by
Gµν= −Λgµν+ k2Tµν+¯k4Sµν− ¯ ǫµν+¯LTF
µν+¯Pµν+ Fµν, (1)
where
Sµν=1
2TTµν−1
εµν= CABCDnCnDgA
4TµαTα
ν+3TαβTαβ− T2
24
gµν, (2)
µgB
ν, (3)
and
Fµν=(5)TABgA
µgB
ν+
?
(5)TABnAnB−1
4
(5)
T
?
gµν, (4)
respectively.
Apart from the terms quadratic in the brane energy-momentum tensor, in the field equa-
tions on the brane there are two supplementary terms, corresponding to the projection of
the 5D Weyl tensor εµν and of the projected tensor Fµν, which contains the bulk matter
contribution. Both terms induce bulk effects on the brane.
Also, the possible asymmetric embedding is characterized by the tensor
¯Lµν=¯Kµν¯K −¯Kµσ¯Kσ
ν−gµν
2
?¯K2−¯Kαβ¯Kαβ?,
µν=¯Kµν¯K −¯Kµσ¯Kσ
(5)
with trace¯L =¯Kαβ¯Kαβ−¯K2, and trace-free part¯LTF
tively.
For a Z2symmetric embedding¯Kµν= 0, and thus¯Lµν= 0.¯Pµνis given by the pull-back
ν+¯Lgµν/4, respec-
to the brane of the energy-momentum tensor characterizing possible non-standard model
fields (e. g. scalar, dilaton, moduli, radiation of quantum origin) living in 5D,
¯Pµν=2˜k2
3
?
gα
µgβ
ν(5)Tαβ
?TF, (6)
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which is traceless by definition. Another projection of the 5D sources appears in the brane
cosmological constant Λ. which is defined as
Λ = Λ0−
¯L
4−2˜k2
3(nαnβ(5)Tαβ), (7)
where 2Λ0= k2
5λ + k2
5Λ5.
In the case of a variable brane tension, the projected gravitational field equations on the
brane have a similar form to the general case,
Gµν= −Λgµν+ k2Tµν+¯k4Sµν− ¯ ǫµν+¯LTF
µν+¯Pµν+ Fµν. (8)
However, the evolution of the brane tension appears in the Codazzi equation, and in the
differential Bianchi identity. The Codazzi equation is
∇µ¯
Kµ
ν − ∇ν¯K = k2
5(gρ
νnσ(5)Tρσ), (9)
and it gives the conservation equation of the matter on the brane as
∇µTµ
ν= ∇νλ − ∆(gρ
νnσ(5)Tρσ). (10)
The differential Bianchi identity, written as ∇µRρµ=1
2∇ρR, gives
∇µ(¯ ǫµν− L
TF
µν−¯Pµν)=∇ν¯L
4
+
k2
2∇ν(nρnσ(5)Tρσ) −
ν−T
3(Tµν∇µT − T∇νT)] −
5
k4
5λ
6∆(gσ
νnρTσρ)
+k4
5
4(Tµ
3gµ
ν)∆(gσ
µnρ(5)Tσρ) +
k4
4[2Tµσ∇[νTµ]σ
12(Tµ
5
+1
k4
5
ν− Tgµ
ν)∇µλ).(11)
From Eq. (3), one can introduce an effective non-local energy density U, which can be
obtained by assuming that εµν in the projected Einstein equation behaves as an effective
radiation fluid,
− εµν=k4
5
6λU(uµuν+a2
3hµν),(12)
where uµis the matter four-velocity, and hµν= gµν+ uµν, respectively.
B.Cosmological models with dynamic brane tension
We assume that the metric on the brane is given by the flat Robertson-Walker-Friedmann
metric, with
(4)gµνdxµdxν= −dt2+ a2(t)(dx2+ dy2+ dz2),(13)
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where a is the scale factor. The matter on the brane is assumed to consist of a perfect
fluid, with energy density ρ, and pressure p, respectively. The gravitational field equations,
governing the evolution of the brane Universe with variable brane tension, in the presence
of brane-bulk energy transfer, and with a non-zero bulk pressure, are then given by [26, 27]
?˙ a
a
?2
¨ a
a
=
(4)Λ
3
+k4
5λ
18
?
?
ρ +ρ2
2λ+ U
1 +2ρ
?
, (14)
=
(4)Λ
3
−k4
5λ
36
ρ
?
λ
?
+ 3p
?
1 +ρ
λ
?
+ 2U
?
, (15)
˙ ρ + 3H (ρ + p) = −˙λ − 2P5,
˙U + 4U˙ a
a+ U
λ
(16)
k4
5λ
6
?
˙λ
?
=
k2
2
5
˙¯PB+k4
5λ
3
?
1 +ρ
λ
?
P5, (17)
(4)Λ =
k2
2Λ5+k4
5
5
12λ2−k2
5
2
¯PB,(18)
where P5describes the bulk-brane matter-energy transfer, while PBis the bulk pressure.
An important observational parameter, which is an indicator of the rate of expansion of
the Universe, is the deceleration parameter q, defined as
q =d
dt
?1
H
?
− 1 = −a¨ a
˙ a2= −
¨ a/a
(˙ a/a)2. (19)
If q < 0, the expansion of the Universe is accelerating, while q > 0 indicates a decelerating
phase.
III.THERMODYNAMIC INTERPRETATION OF THE VARYING TENSION IN
BRANE-WORLD MODELS
For the sake of generality we also assume that there is an effective energy-matter trans-
fer between the brane and the bulk, and the brane-bulk matter-energy exchange can be
described as
P5= −αbb
2ρcr
?a0
a
?3w
H, (20)
where αbbis a constant, ρcris the present day critical density of the Universe, and a0is the
present day value of the scale factor.
In the presence of a varying brane tension and of the bulk-brane matter and energy
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exchange, the energy conservation equation on the brane can be written as
˙ ρ + 3(ρ + p)H = −ρ
?˙λ
ρ− αbbH
?
, (21)
where we have used Eq. (20) for the description of brane bulk energy transfer, by taking into
account that ρ = ρcr(a0/a)3w. We suppose that the matter content of the early Universe
is formed from m non-interacting comoving relativistic fluids with energy densities and
thermodynamic pressures ρi(t) and pi(t), i = 1,2,...,m, respectively, with each fluid formed
from particles having a particle number density ni(t), i = 1,2,...,m, and obeying equations
of state of the form ρi(t) = kinγi
i, pi(t) = (γi− 1)ρi, i = 1,2,...,m, where ki= ρ0i/nγi
0i≥
0, i = 1,2,...,m, are constants, and 1 ≤ γi ≤ 2, i = 1,2,...,m. For example, we can
consider that the particle content of the early Universe is determined by pure radiation (i.
e., different types of massless particles, or massive matter (baryonic and dark) in equilibrium
with electromagnetic radiation and decoupled massive particles. The total energy density
and pressure of the cosmological fluid results from summing the contribution of the l simple
fluid components, and are given by ρ(t) =?m
For a multicomponent comoving cosmological fluid and in the presence of variable brane
i=1ρi(t) and p(t) =?m
i=1pi(t), respectively.
tension and bulk-brane energy exchange, Eq. (21) becomes
l?
i=1
[ ˙ ρi+ 3(ρi+ pi)H] = −
m
?
i=1
ρi(t)
?
˙λ
?m
i=1ρi(t)− αbbH
?
. (22)
Eq. (22) can be recast into the form of m particle balance equations,
˙ ni(t) + 3ni(t)H = Γi(t)ni(t),i = 1,2,...,m,(23)
where Γi(t), i = 1,2,...,m, are the particle production rates, given by
Γi(t) = −1
γi
?
˙λ
mρi(t)− αbbH
?
,i = 1,2,...,m.(24)
In order for Eq. (23) to describe particle production the condition Γi(t) ≥ 0, i = 1,2,...,m,
is required to be satisfied, leading to the following restriction imposed to the time variation
rate of the brane tension
˙λ ≤ αbblρi(t)H,i = 1,2,...,m.(25)
Note that if Γi(t) = 0, i = 1,2,...,m, we obtain the usual particle conservation law of the
standard cosmology. Of course, the casting of Eq. (22) is not unique. In Eqs. (23) and (24),
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we consider the simultaneous creation of a multicomponent comoving cosmological fluid,
but other possibilities can be formulated in the same way (for example, creation of a single
component in a mixture of fluids).
The entropy Si, generated during particle creation at temperatures Ti, i = 1,2,...,m, can
be obtained from Eq. (23), and for each species of particles has the expression
TidSi
dt
= −1
γi
?
˙λ
mρi(t)− αbbH
?
ρi(t)V,i = 1,2,...,m,(26)
where V is the volume of the Universe, or, equivalently,
dSi
dt
=γiρi(t)V
Ti
Γi(t),i = 1,2,...,m.(27)
In a cosmological fluid where the density and pressure are functions of the temperature
only, ρ = ρ(T), p = p(T), the entropy of the fluid is given by S = (ρ + p)V/T = γρ(t)V/T.
Therefore we can express the total entropy S(t) of the multicomponent cosmological fluid
filled brane Universe as a function of the particle production rate only,
S(t) =
m
?
i=1
S0iexp
??t
t0
Γi(t′)dt′
?
, (28)
where S0i≥ 0, i = 1,2,...,m, are constants of integration. In the case of a general perfect
comoving multicomponent cosmological fluid with two essential thermodynamical variables,
the particle number densities ni, i = 1,2,...,m, and the temperatures Ti, i = 1,2,...,m, it is
conventional to express ρiand piin terms of niand Tiby means of the equilibrium equations
of state ρi= ρi(ni,Ti), pi= pi(ni,Ti), i = 1,2,...,m. By using the general thermodynamic
relation
∂ρi
∂ni
=ρi+ pi
ni
−Ti
ni
∂pi
∂Ti,i = 1,2,...,m, (29)
in the case of a general comoving multicomponent cosmological fluid Eq. (21) can also be
rewritten in the form of m particle balance equations,
˙ ni(t) + 3ni(t)H = Γi(t)ni(t),i = 1,2,...,m, (30)
with the particle production rates Γi(t) given by some complicated functions of the thermo-
dynamical parameters, brane tension and brane-bulk energy exchange rate,
Γi(t) = −
ρi
ρi+ pi
?
˙λ
lρi(t)− αbbH + Ti∂ lnρi
∂Ti
?˙Ti
Ti
− C2
i
˙ ni
ni
??
,i = 1,2,...,m,(31)
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where C2
i = (∂pi/∂Ti)/(∂ρi/∂Ti). The requirement that the particle balance equation
Eq. (30) describes particle production, Γi(t) ≥ 0, i = 1,2,...,m, imposes in this case the
following constraint on the time variation of the brane tension,
˙λ < αbblρi(t)H − lρi(t)Ti∂ lnρi
∂Ti
?˙Ti
Ti
− C2
i
˙ ni
ni
?
,i = 1,2,...,m. (32)
In the general case the entropy generated during the reheating period due to the variation
of the brane tension and the bulk-brane energy exchange can be obtained for each component
of the cosmological fluid from the equations
dSi
dt
=(ρi+ pi)V
Ti
?
˙λ
lρi(t)− αbbH + Ti∂ lnρi
∂Ti
?˙Ti
Ti
− C2
i
˙ ni
ni
??
,i = 1,2,...,m, (33)
while the total entropy of the Universe is given by S(t) =?m
The entropy flux vector of the kth component of the cosmological fluid is given by
i=1Si(t).
S(k)α= nkσkuα,k = 1,2,..,m, (34)
where σk, k = 1,2,...,m, is the specific entropy (per particle) of the corresponding cosmo-
logical fluid component and uαis the four-velocity of the fluid. By using the Gibbs equation
nTdσ = dρ−[(ρ + p)/n]dn for each component of the fluid, and assuming that the entropy
density σ does not depend on the brane tension, we obtain
S(k)α
;α
= −1
Tk
?˙λ − αbbHρ
?
−µkΓknk
Tk
,k = 1,2,..,m, (35)
where µkis the chemical potential defined by µk = [(ρk+ pk)/nk] − Tkσk. The chemical
potential is zero for radiation. For each component of the cosmological fluid the second law
of thermodynamics requires that the condition
S(k)α
;α
≥ 0,k = 1,2,..,m,(36)
has to be satisfied.
IV.POWER LAW INFLATION IN BRANE WORLD MODELS WITH VARYING
BRANE TENSION AND BULK PRESSURE
For a vacuum Universe with ρ = p = 0, in the presence of a non-zero bulk pressure and
matter- energy exchange between the brane and the bulk, the field equations Eqs. (14) take
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the form
3
?1
31
a
da
dτ
?2
=l2
2− pB+ lu, (37)
a
d2a
dτ2=l2
2− pB− lu,
?2
(38)
dl
dτ= −2
3p5, (39)
and
d
dτ
?lua4?= −a4d
dτ
?l2
2− pB
?
, (40)
where we have introduced a set of dimensionless variables (τ,l,pB,u,p5) defined as
τ =
?3
2t,λ =
3
k2
5
l,PB=
3
k2
5
pB,U =3
k2
5
u,P5=
3
k2
5
p5. (41)
Moreover, we consider that the five-dimensional cosmological constant Λ5= 0. We assume
that the inflationary evolution is of the power law type, and therefore a = τα, where α is a
constant. Then Eqs. (37) and (38) give
2
?l2
2− pB
?
=3α(2α − 1)
τ2
,(42)
and
2lu =3α
τ2. (43)
Eq. (40) is identically satisfied. In order to completely solve the problem, we need to
specify the form of the energy matter-transfer from the bulk to the brane. By assuming a
functional form given by p5= p05τ−β, where β > 0 and p05> 0 are constants, we obtain
immediately
l(τ) =
?8
?27
3
1
β − 1τ−β+1,pB(τ) =4
32α(β − 1)τβ−3.
3
1
(β − 1)2τ−2(β−1)−3α(2α − 1)
2τ2
,
u(τ) =
(44)
The Hubble parameter of the Universe during the inflationary phase is given by H = α/t.
The deceleration parameter is obtained as q = d(1/H)/dt − 1 = (1 − α)/α. Therefore, if
α > 1, q < 0, and the brane world Universe experiences an inflationary expansion.
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V. SCALE FACTOR DEPENDENT BRANE TENSION MODELS
In the following we assume that there is no matter-energy exchange between the bulk
and the brane, P5= 0, and that the bulk pressure is also zero, PB= 0. For the matter on
the brane we adopt as equation of state a linear barotropic relation between density and
pressure, given by
p = (w − 1)ρ, (45)
where w = constant and w ∈ [1,2]. Therefore Eq. (16) gives
˙ ρ + 3Hwρ = −˙λ,(46)
while Eq. (17) gives immediately
λU =U0
a4, (47)
where U0is an arbitrary constant of integration. In the following, in order to simplify the
analysis, we assume that U0= 0.
In order to explain the main observational features of modern cosmology (inflation, re-
heating, deceleration period and late time acceleration, respectively), we assume that the
brane tension varies as a function of the scale factor a according to the equation
λ2= λ2
0e−2βa2−6(5)Λ
k2
5
+ λ2
1, (48)
where β, λ0and λ1are constants.
Suppose tin,ainand ρinare the values of the time, of the scale factor, and of the energy
density before inflation. Generally, in the present paper we use the subscript “in” to denote
the values of the cosmological parameters before the inflation, and the subscript “en” to
denote values after inflation. Thus, for example, N = ln(aen/ain) is the e-folding number.
The basic physical parameters of our model are tin, ain, ρin, N,k5,
(5)Λ, λ0, λ1, and β,
respectively. The coupling constant k5and the five-dimensional cosmological constant(5)Λ
are constrained by the present value of the gravitational constant,
k4
6
5
?
−6(5)Λ
k2
5
+ λ2
1≈ k3
5
?
−
(5)Λ
6
= 8πG ≈ 1.68 × 10−55eV−2, (49)
and by the constraints on the 5D cosmological constant [9]
k2
2
5
(5)Λ ≈ −
6
(0.1mm)2≈ −2.3 × 10−5eV2, (50)
13
Page 14
where we have used the natural system of units with ? = c = 1. ¿From these two conditions,
we obtain k4
5≈ 3.6 × 10−105eV−6and(5)Λ ≈ −7.7 × 1046eV5. Besides, the value of λ1can
be obtained from the value of the present day dark energy ρdark≈ 10−12eV4[8],
k4
5
12λ2
1= 8πGρdark≈ 1.6 × 10−67eV2, (51)
which gives λ1≈ 1.6×1019eV4. We can have a backward checking on Eq. (49), from which
it follows that the condition λ2
1≪ −6(5)Λ/k2
5is indeed satisfied. We also choose λ0to be of
the same order of magnitude as the vacuum energy ρvac∽ 10100eV4at GUT scale [8], [33],
k4
12λ2
5
0= 8πGρvac∽ 1045eV2, (52)
which gives λ0∽ 1.3 × 1075eV. The scale difference between λ0and
λ0/?−6(5)Λ/k2
For the remaining model parameters tin,ain,ρin,N,β, we constraint them in the next section.
?−6(5)Λ/k2
5+ λ2
1is
5∽ 1025. The differences in the scales of λ0and λ1are of the order of ∽ 56.
When a is very small, the brane tension λ ≈ λ0dominates the early Universe at the time
of inflation. Due to the exponential expansion of the Universe, the brane tension quickly
decays to a constant just after the inflation. The decay of the brane tension will generate
the matter content of the Universe, according to Eq. (16). This happens also during the
accelerated expansion period of the Universe. Matter is created during all periods of the
expansion of the Universe, but the most important epoch for matter creation is near the
end of inflation. In the evolution of the Universe there is one moment when ¨ a = 0, which
corresponds to the moment when the Universe switches from the accelerating expansion to
a decelerating phase. After the matter (which is mainly in the form of radiation) energy
density reaches its maximum, the Universe enters into a radiation dominated phase, and the
quadratic term in Eq. (14) will become dominant first. The matter energy density continue
to decrease due to the expansion. When the linear term in matter equals the quadratic
term, the Universe switches back to the ΛCDM model. Therefore, the Universe enters in
the matter dominated epoch at about 4.7 × 104yr [34]. Then the matter term equals the
residue term in Eq. (8) at about 10Gyr [34]. This is the second moment in the evolution
of the Universe when ¨ a = 0. After this moment, the Universe enters in an accelerating
expansionary phase again, and its dynamics is controlled by the term λ1.
14
Page 15
VI.QUALITATIVE ANALYSIS OF THE MODEL
In the present Section we consider the approximate behavior of the cosmological model
with varying brane tension in the different cosmological epochs.
A.Early inflationary phase: 2βa2≪ 1
When the scale factor a is very small, the exponential factor e−2βa2in Eq. (48) can
be approximated by 1. Therefore, the brane tension is given by λ2≈ λ2
corresponds to the vacuum energy necessary to give an exponential inflation. Since k2
0, and physically it
5λ2
0/6 ≫
(5)Λ, from Eq. (14) the scale factor evolves in time as an exponential function of time, given
by
a = aine(k2
5λ0/6)t, (53)
where ainis the value of the scale factor prior to inflation. The e-folding number is given
by N = ln(a/ain), which should be roughly of the order of N ? 60 in order to solve the
flatness, Horizon problem, etc. In the present paper we adopt for N the value N = 70. Since
2βa2
en∽ 1 at the end of inflation, we can roughly estimate the value of aento be
aen∽
1
√2β.
(54)
From the value of N we adopted, we obtain the value of ainas
ain= aene−N. (55)
According to Eq. (53), the end time of the inflation tencan be estimated as
k2
5λ0(ten− tin)/6 ≈ k2
5λ0ten/6 ≈ N, (56)
which implies that ten≈ 10−36s. The value of tinis insensitive to the variation of the initial
conditions, provided that tinis at least one order smaller than ten. With the adopted value
of the e-folding N and β, we can fix the values of ainand of tin, respectively. Since at the
beginning of the inflationary stage the matter is not yet generated, we have ρin= 0. For the
deceleration parameter, from Eq. (19) we find
q ≈−λ2
0e−2βa2− λ2
0e−2βa2+ λ2
1
λ2
1
= −1. (57)
15
Page 16
By substituting the tension we can rewrite Eq. (46) as
dρ
dt+ 3w1
a
da
dtρ = −dλ
dt=
2βλ2
0a˙ ae−2βa2
?
λ2
0e−2βa2−6(5)Λ
k2
5
+ λ2
1
. (58)
For 2βa2≪ 1, the exponential factor does not change much as a increase. Therefore,
dρ
dt+ 3w1
a
?
where we have also used Eq. (53). Conversion of the matter from the brane tension energy
da
dtρ =
2βλ2
0a˙ a
λ2
0−6(5)Λ
k2
5
+ λ2
1
≈2βλ2
0a˙ a
?λ2
0
= βk2
5λ2
0a2/3. (59)
begins already during the inflationary stage, and the rate of the conversion is proportional to
a2during inflation. Therefore, the matter density also rises exponentially in the late stages
of the inflationary phase. The matter generation rate becomes most important at the end
of inflation.
B.Reheating period: 2βa2≈ 1
During the reheating period the evolution of the matter gradually changes from an ex-
ponential increase to a power law decrease, ρ ∝ a−3w. In our model the matter density
is a smooth function, which is strictly increasing from the beginning of inflation, and then
strictly decreases after the end of the reheating phase. Therefore there must be a maximum
value of the density ρmaxat a time tmax. After tmax, the Universe was dominated by matter
which is in the form of radiation, and almost all the energy of the brane tension converted
into matter. The temperature of matter, corresponding to a radiation dominated Universe
at tmax, is denoted TRH, and is given by
ρmax=π2
15(kTRH)4,(60)
where k is Boltzmann’s constant. Current theory on gravitinos production constraints TRH
to be TRH< 109− 1010GeV [30–32]. Therefore the maximum density of the Universe must
satisfy the condition ρmax< 1036−1040GeV4. The maximum density can be obtained from
the condition ˙ ρ|t=tmax= 0, and, with the use of Eq. (14) it is given as a solution of the
equation
3w1
a
da
dtρmax= −dλ
dt=
2βλ2
0a˙ ae−2βa2
?
λ2
0e−2βa2−6(5)Λ
k2
5
+ λ2
1
??????
t=tmax
.(61)
16
Page 17
With the rough approximation βa2|t=tmax≈ βa2
en∽ 1, Eq. (61) can be written as
3wρmax=
2βλ2
0a2e−2βa2
?
λ2
0e−2βa2−6(5)Λ
k2
5
+ λ2
1
??????
t=tmax
≈
2λ2
0e−2βa2
?
λ2
0e−2βa2−6(5)Λ
k2
5
+ λ2
1
??????
t=tmax
≈ 0.74λ0. (62)
This relation gives the maximum matter density of the Universe. And the value of λ0∽
1039GeV4that we have chosen is consistent with the maximum value of the matter energy
density.
C.Matter Domination period: 2βa2≫ 1 and 2λρ ≫ λ2
1
At this stage, the Universe is dominated by matter and the brane tension is roughly a
constant. During this period, the key difference with the conventional cosmological models
is the presence of the quadratic term in matter density, which will dominate the dynamics
of the Universe at the beginning of this period. The evolution equation of the scale factor is
da
dt= ak2
5
6
??
2λρ + ρ2?
≈ ak2
5
6ρ, (63)
and the evolution of the matter density is given by
dρ
dt+ 41
a
da
dtρ = 0, (64)
where we have assumed that immediately after the matter energy reaches its maximum the
matter is in the form of radiation. The solution is ρ = constant/a4≡ ρmaxa4(tmax)/a4≈
λ0/β2a4. Therefore, the scale factor and deceleration parameter evolve as
a(t) =
?2k2
5
3
ρmaxa4(tmax)t
?1
4
≈k2
5λ0
6β2t, (65)
q = −a¨ a
˙ a2=
k4
36
5λ
?ρ(1 +2ρ
λ) + 3(w − 1)ρ(1 +ρ
k4
5λ
18(ρ +ρ2
λ)?
λ)
≈ 1 + 3w = 5. (66)
After the quadratic density and linear density equality 2λρ = ρ2, the linear term in matter
will take over, i.e. 2λρ ≫ ρ2. The Universe enters in the ΛCDM model at this radiation
dominated phase, and its dynamics is described by the equations
da
dt= a
??
k4
18λρ
5
?
, (67)
17
Page 18
and
dρ
dt+ 41
a
da
dtρ = 0, (68)
respectively. The solution for the scale factor is
a2(t)
2
≈
?
k3
5
√−6(5)Λλ0
18β2
?1/2
t,(69)
with ρ = ρmaxa4(tmax)/a4. During this period the deceleration parameter evolves as
q ≈
k4
36[ρ + 3(w − 1)ρ]
k4
5
18λρ
5λ
=3w − 2
2
= 1. (70)
After this stage, the Universe switches from radiation dominated to non-relativistic matter
dominated at trm= 1.5×1012s [34]. To simulate the transition from the radiation dominated
period to the baryonic matter dominated period one can introduce a time varying w given
by [29]
w =4trm/3 + t
trm+ t
. (71)
In the matter dominated era (w = 1), the deceleration parameter shifts to q = 0.5,
according to Eq. (70). Recall that β is still free, and we can use it to match the scale factor
at the radiation-matter equality. With the use of Eq. (69), and by taking arm= 2.8 × 10−4
[34], we obtain
?
which gives λ0≈ 10−15× β2. This gives the estimate of β ∽ 1045.
2.6 × 10−20s−1∽
k3
5
√−6(5)Λλ0
18β2
?1/2
, (72)
D.Dark Energy Domination era (2λρ < λ2
1)
With the increase of the cosmological time, the constant term λ1in Eq. (14) will dominate
over matter. Thus this term plays the role of the dark energy of the standard ΛCDM models.
In its late stages of evolution the Universe becomes “dark energy” dominated. The Universe
turn to an exponential acceleration again, with
a ∝ e(k2
5/6)λ1t,(73)
but with a time scale much longer than the inflationary time scale. The deceleration pa-
rameter will converge to
q →−k4
5λ2
5λ2
1
k4
1
= −1, (74)
18
Page 19
showing that the expansion of the universe is accelerating.
VII. NUMERICAL ANALYSIS OF THE MODEL
The field equation can be rewritten in a simple form by introducing as set of dimensionless
variables τ, l, and r, defined as
τ = k5
?
(−Λ5)
2
t,ρ = k−1
5
?
3 × (−Λ5)r,λ = k−1
5
?
3 × (−Λ5)l, (75)
respectively. Then the field equations Eqs. (14)-(18) can be written in a dimensionless form
as
3
?1
6−1
a
da
dτ
?2
= −1 +l2
2+ lr +r2
2, (76)
1
a
d2a
dτ2= −1
3+l2
6l
?
r
?
1 +2r
l
?
+ 3(w − 1)r
?
1 +r
l
??
, (77)
dr
dτ+ 3w1
a
da
dτr = −dl
dτ.
(78)
By rescaling the four-dimensional cosmological constant so that(4)Λ = [k2
5(−Λ5)/2]leff, it
follows that leff= −1 + l2/2. In the dimensionless variables, the deceleration parameter is
given by q = −(ad2a/dτ2)/(da/dτ)2, and can be explicitly expressed as a function of the
physical parameters of the model as
q =1 − l2/2 + l[r (1 + 2r/l) + 3(w − 1)r(1 + r/l)]/2
−1 + l2/2 + lr + r2/2
To obtain the numerical solution, one should also consider the rescaled form of λ,
. (79)
l2= 2(l2
0e−2βa2+ 1 + l2
1), (80)
where l0and l1are constants, with l0≈ 1025≫ l1≈ 10−31. Eq. (71) becomes
w =4τrm/3 + τ
τrm+ τ
, (81)
with τrm= (4.8 × 10−3eV)trm= 1025. Note that the conversion in Eq. (75) can be written
out numerically
t = (208 eV−1)τ = (1.4 × 10−13s)τ,ρ = (2.0 × 1050eV)r,λ = (2.0 × 1050eV)l.(82)
The time variations of the scale factor, of the energy density, of the brane tension, and
of the deceleration parameter of the Universe are presented, for different scales of vacuum
19
Page 20
energy l0, in Figs. 1 respectively.From the plot of a and q, we find that there are 5
stages in the evolution of the Universe. Namely, the inflation and reheating stage, the
quadratic density stage, the radiation domination stage, the non-relativistic matter stage,
and the late time acceleration stage. Comparing plots with different l0’s, we see the effects
of different vacuum energy scales on the evolution of the Universe. According to the plot
with different l0’s, we find that a larger vacuum energy would provide a faster inflation, and
it also generates more matter. The matter energy density reaches its maximum at earlier
times. Although there are many changes in the evolution of the Universe caused by changing
l0, these characteristics are hard to be constraint by observations. On the other hand, l0
cannot be arbitrary due to the theoretical constraint, e.g. vacuum energy density, gravitinos
production, etc.
FIG. 1: Time variations of the scale factor a, deceleration parameter q, and energy density r of
the Universe in braneworld models with varying scale factor dependent brane tension. Different l0
are plotted: l0= 1024(dotted curve), l0= 1025(solid curve), and l0= 1026(dashed curve).
The time variations of the scale factor, and of the deceleration parameter of the Universe
are presented, for different values of β, in Figs. 2 respectively. If we assume λ0 is fixed
20
Page 21
by the vacuum energy scale, the value of β is well confined. Varying β would affect the
observational constraint of a. From the graph of r, we can cross check that the value of
ρmaxindeed fulfills the requirement of Eq. (60). Besides β, we could examine the effect of
adopting a different e-folding N. Since aenis determined by β according to the condition
Eq. (54), it follows that it is determined by observational constraints. Different e-foldings
could be a result of differences in ain. However, we find from Fig. 3 that aindoes not affect
the post-inflationary epochs. Therefore ainis not a robust parameters, and we cannot fully
determine it.
FIG. 2: Time variations of the scale factor a, deceleration parameter q, and energy density r of
the Universe in braneworld models with varying scale factor dependent brane tension. Different β
are plotted: β = 1043(dotted curve), β = 1045(solid curve), and β = 1047(dashed curve).
VIII.DISCUSSIONS AND FINAL REMARKS
In the present paper we have considered some cosmological implications of the braneworld
models with variable tension. We have considered a thermodynamic interpretation of the
21
Page 22
FIG. 3: Time variations of the scale factor a of the Universe in braneworld models with varying scale
factor dependent brane tension. Different ainare plotted: ain= 10−51(dotted curve), ain= 10−53
(solid curve), and ain= 10−55(dashed curve).
model, and we have shown that from a thermodynamic point of view a variable brane
tension can describe particle production processes in the early Universe, as well as the
entropy production in the early stages of the cosmological evolution. A simple power-law
inflationary cosmological model has also been obtained. By adopting a simple analytical
form for λ, we have obtained a complete description of the dynamics and evolution of the
Universe from the stage of the inflation to the phase of the late acceleration. Moreover, the
differences between the energy scales of the theoretical vacuum energy at inflationary epoch
and during late time acceleration are studied.
A variable brane tension can drive the inflationary evolution of the Universe, and also be
responsible for matter creation in the post-inflationary phase (the reheating of the Universe).
In the model we have adopted the brane tension converges to a constant that gives the
gravitational constant, and the residue after cancelation with the Λ5term would give the
dark energy. It is interesting to note that, as opposed to the standard cosmological scenarios,
in the present model matter creation takes place during the entire inflationary phase, but
reaches a maximum after the exponential expansion of the Universe ends. Therefore in the
braneworld models with varying brane tension there can be no clear distinction between
inflation and reheating.By using numerical analysis as well as approximate analytical
22
Page 23
methods, we have obtained the result that in this variable tension model there are 5 phases
in the cosmological evolution of the Universe. At the beginning, there are the inflationary
and the reheating phases. During these phases, the brane tension is the dominant energy
in the Universe, and its magnitude is of the order of the vacuum energy at GUT scale. As
the brane tension decays, the matter is created, and the Universe enters in the hot Universe
phase of the Big Bang picture. The third phase is the quadratic density domination phase.
This is a unique characteristic of the brane world scenario, during which the Universe is
dominated by a quadratic term in the energy density of the radiation. The deceleration
parameter increases from −1 during inflation to 5 at this phase. To study the cosmological
dynamics we have introduced a set of dimensionless quantities, which describe the evolution
of the scaled densities in terms of a dimensionless time parameter τ. By using the results
of the numerical simulations for the rescaled variables, one can obtain some constraints on
the physical parameters of the model.
Acknowledgments
This work is supported by the RGC grant HKU 701808P of the Government of the Hong
Kong SAR.
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