Page 1

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

1

Page 2

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

2

Page 3

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

3

Page 4

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

4

Page 5

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

5

Page 6

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)

6

Page 7

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)

7

Page 8

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

8

Page 9

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),

9

Page 10

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)

10

Page 11

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

11

Page 12

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.

12

Page 13

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