dbi paperv rev

8/4/2019 DBI Paperv Rev

1/15

Dirac-Born-Infeld Field Trapped in the Braneworld

Ricardo Garca-Salcedo,1, a Dania Gonzalez,2, b Tame Gonzalez,3, c Claudia Moreno,4, d and Israel Quiros5, e

1Centro de Investigacion en Ciencia Aplicada y Tecnologia Avanzada - Legaria del IPN, Mexico D.F., Mexico.2Departamento de Matematica, Universidad Central de Las Villas, 54830 Santa Clara, Cuba.

3Departamento de Fsica, Centro de Investigacion y de Estudios Avanzados del IPN, A.P. 14-740, 07000 Mexico D.F., Mexico.4Departamento de Fsica y Matematicas, Centro Universitario de Ciencias Exactas e Ingenieras,

Corregidora 500 S.R., Universidad de Guadalajara, 44420 Guadalajara, Jalisco, Mexico.5Division de Ciencias e Ingeniera de la Universidad de Guanajuato, A.P. 150, 37150, Leon, Guanajuato, Mexico.

(Dated: July 24, 2011)

We apply the dynamical systems tools to study the (linear) cosmic dynamics of a Dirac-Born-Infeld-type field trapped in the braneworld. We focus, exclusively, in Randall-Sundrum and in Dvali-Gabadadze-Porrati brane models. We analyze the existence and stability of asymptotic solutionsfor the AdS throat and a particular relationship between the warp factor and the potential for

the DBI field (f() = 1/V()). It is demonstrated, in particular, that in the ultra-relativisticapproximation matter-scaling and scalar field-dominated solutions always arise. While the effect ofthe Randall-Sundrum brane is to re-introduce the initial curvature singularity that was removed bythe non-linear effect of the Dirac-Born-Infeld field, the extra-dimensional DGP brane effects modifythe stability of the critical points associated with the late-time cosmic dynamics. It is foud thatthe self-accelerating solution arising in the self-accelerating branch of the DGP scenario is a saddleequilibrium point, which means that this is not always the end point of the cosmic evolution in thatbranch of the model.

PACS numbers: 04.20.-q, 04.20.Cv, 04.20.Jb, 04.50.Kd, 11.25.-w, 11.25.Wx, 95.36.+x, 98.80.-k, 98.80.Bp,98.80.Cq, 98.80.Jk

I. INTRODUCTION

Recent observations from the Wilkinson MicrowaveAnisotropy Probe (WMAP) [1] offer strong supportingevidence in favor of the inflationary paradigm. In themost simple models of this kind, the energy density of theuniverse is dominated by the potential energy of a single(inflaton) scalar field that slowly rolls down in its self-interaction potential [2]. Restrictions imposed upon theclass of potentials which can lead to realistic inflationaryscenarios, are dictated by the slow-roll approximation,and hence, the result is that only sufficiently flat poten-tials can drive inflation. In order for the potential to besufficiently flat, these conventional inflationary modelsshould be fine-tuned. This simple picture of the early-time cosmic evolution can be drastically changed if oneconsiders models of inflation inspired in Unified Theo-ries like the Super String or M-theory. The most appea-ling models of this kind are the Randall-Sundrum brane-world model of type 2 (RS2) [3] and Dvali-Gabadadze-Porrati (DGP) brane worlds [4].

aElectronic address: [email protected] address: [email protected] address: [email protected]

Electronic address: [email protected] address: [email protected]

In the RS2 model a single co-dimension 1 brane withpositive tension is embedded in a five-dimensional anti-de

Sitter (AdS) space-time, which is infinite in the directionperpendicular to the brane. In general, the standard mo-del particles are confined to the brane, meanwhile gravi-tation can propagate in the bulk. In the low-energy limit,due to the curvature of the bulk, the graviton is confinedto the brane, and standard (four-dimensional) general re-lativity laws are recovered. RS2 braneworld models havean appreciable impact on early universe cosmology, inparticular, for the inflationary paradigm. In effect, a dis-tinctive feature of cosmology with a scalar field confinedto a RS2 brane is that the expansion rate of the universediffers at high energy from that predicted by standard ge-neral relativity. This is due to a term quadratic in the

energy density that produces enhancing of the frictionacting on the scalar field. This means that, in RS2 brane-world cosmology, inflation is possible for a wider class ofpotentials than in standard cosmology [5]. Even poten-tials that are not sufficiently flat from the point of viewof the conventional inflationary paradigm can producesuccessful inflation. At sufficiently low energies (muchless than the brane tension), the standard cosmic beha-vior is recovered prior to primordial nucleosynthesis scale(T 1 MeV) and a natural exit from inflation ensues as


2/15

2

the field accelerates down its potential [6].1

The DGP model describes a brane with 4D world-volume, that is embedded into a flat 5D bulk, and allowsfor infrared (IR)/large scale modifications of gravitatio-nal laws. A distinctive ingredient of the model is theinduced Einstein-Hilbert action on the brane, that is res-ponsible for the recovery of 4D Einstein gravity at mode-rate scales, even if the mechanism of this recovery is rat-her non-trivial [11]. Nevertheless, studying the dynamicsof DGP models continues being a very attractive subjectof research [23]. It is due, in part, to the very simple geo-metrical explanation to the dark energy problem andthe fact that it is one of a very few possible consistent IRmodifications of gravity that might be ever found. The

acceleration of the expansion at late times is explainedhere as a consequence of the leakage of gravity into thebulk at large (cosmological) scales, which is just a 5Dgeometrical effect.

Nonlinear scalar field theories of the Dirac-Born-Infeld(DBI) type have attracted much attention in recent yearsdue to their role in models of inflation based on stringtheory. DBI inflation [1317] is motivated by brane in-flationary models [18] in warped compactifications [19].These scenarios identify the inflaton with the positionof a mobile D-brane moving on a warped (compact) 6-dimensional submanifold of spacetime (for reviews andreferences see [20]), which means that the inflaton is in-

terpreted as an open string mode. In these models, thewarped space slows down the rolling of the inflaton oneven a steep potential, making easier inflation. This slo-wing down can also be understood as arising due to in-teractions between the inflaton and the strongly coupledlarge-N dual field theory [17]. This scenario can natura-lly arise in warped string compactifications [14]. Usuallyonly effective four-dimensional DBI cosmological modelsare studied.

It is our opinion that studying the impact higher-dimensional brane effects have on the cosmic dynamicsof DBI-type models, is a task worthy of attention. ADBI field trapped in a RS brane could be a nice scena-

rio to make early-time inflation easier, both, because ofthe interaction of the inflaton with the strongly coupled(large-N) dual field theory, and because of the UV braneeffects. Meanwhile a DBI scalar field confined to a (self-accelerating) DGP braneworld could be a useful arenawhere to address unified description of early-time infla-

1 In this scenario, reheating arises naturally even for potentials wit-hout a global minimum and radiation is created through gravita-tional particle production [7] and/or through curvaton reheating[8]. This last ingredient improves the brane steep inflationary

picture [9]. Other mechanisms such as preheating, for instance,have also been explored [10].

tion, fueled by the slowing down effect of the warped

space, and late-time speed-up, originated from UV lea-kage of gravity into the extra-space.Aim of this paper is, precisely, to investigate the dyna-

mics of a DBI-type field trapped in a Randall-Sundrumbrane and in a Dvali-Gabadadze-Porrati braneworld, res-pectively, by invoking the dynamical systems tools. Thestudy of the asymptotic properties of these models allowsto correlate such important dynamical systems conceptslike past and future attractors as well as saddle equi-librium points with generic cosmological evolution. Ina sense the present work might be considered as a na-tural completion of the one reported in Ref. [15]. Forthis reason, as in the above reference, here we concen-

trate in the case of an anti-de Sitter (AdS) throat andquadratic self-interaction potential for the inflaton and aparticular choice of the warp factor and of the potentialfor the DBI field (f() = 1/V()). In addition to thescalar field we also consider a background fluid trappedon the braneworld. Through the paper we use naturalunits (8G = 8/m2Pl = h = c = 1).

II. DBI ACTION

In the region where the back-reaction and stringy phy-sics can be ignored, the effective action for the DBI field

has the following form [17]:

SDBI =

d4x

|g|{f1()

1 + f()()2

f1() + V()}, (1)where is the inflaton, V() - its self-interaction poten-tial, and f() is the warping factor. For a spatially flat

FRW metric ()2 = 2, where the dot accounts forderivative with respect to the cosmic time. The equationof motion for the DBI inflaton can be written in thefollowing way:

+3f

2f2 f

f2+ 32L H

+3L

(V +

f

f2

)= 0, (2)

where the Lorentz boost L is defined as

L =1

1 f()2. (3)

Alternatively the equation of motion of the DBI-typefield can be written in the form of a continuity equation:


3/15

3

+ 3H( +p) = 0, (4)

where we have defined the following energy density andpressure of the DBI scalar field:

=L 1

f+ V(), p =

L 1Lf

V(), (5)

respectively.In this paper we concentrate just in two particularly

symple cases:

i) AdS throat which amounts to consider f() =/4, where in specific string constructions is a parameter which depends on the flux num-bers [17],2 and a quadratic self-interaction potentialV() = m22/2.

ii) Assumption of a particularly interesting rela-tionship between the warp factor and the potentialfor the DBI field: f() = 1/V().

III. AUTONOMOUS SYSTEM

The dynamical systems tools offer a very useful ap-proach to the study of the asymptotic properties of thecosmological models [22]. In order to be able to applythese tools one has to follow these steps: i) to identifythe phase space variables that allow writing the system ofcosmological equations in the form of an autonomous sys-tem of ordinary differential equations (ODE).3, ii) withthe help of the chosen phase space variables, to build anautonomous system of ODE out of the original system ofcosmological equations, and iii) (a some times forgottenor under-appreciated step) to identify the phase spacespanned by the chosen variables, that is relevant to thecosmological model under study. After this one is readyto apply the standard tools of the (linear) dynamical sys-tems analysis.

The goal of the dynamical systems study is to correlatesuch important concepts like past and future attractors(also, saddle critical points) with asymptotic cosmolo-gical solutions. If a given cosmological solution can be

2 In general, inflationary observables may depend on the detailsof the warp factor [21], however, if we assume that the last 60e-foldings of inflation occur far from the tip of the throat, theabove is a good approximation.

3 There can be several different possible choices, however, not all

of them allow for the minimum possible dimensionality of thephase space.

associated with a critical point in the phase space of the

model, this means that, independent of the initial data,the universes dynamics will evolve for a sufficiently longtime in the neighborhood of this solution, otherwise, itrepresents a quite generic phase of the cosmic dynamics.

In the following subsections we keep the expressions asgeneral as possible, and then, in section IV we substitutethe above mentioned expressions for f() and V().

A. DBI-RS Model

Here we will be concerned with the dynamics of a DBIinflaton that is trapped in a Randall-Sundrum brane of

type 2. The field equations in terms of the Friedmann-Robertson-Walker (FRW) metric are the following:

3H2 = T(1 +T2

), (6)

2H = (1 + T

)

L2 + (1 + m)m

, (7)

where m is the equation of state (EOS) parameter ofthe matter fluid, while T = + m the total energydensity on the brane. Additionally one has to considerthe continuity equations for the DBI-type field (equation(2) or, alternatively, (4)) and for the matter fluid:

m + 3(1 + m)Hm = 0. (8)

The model described by the above equations will becalled as DBI-RS model. Our aim will be to write thelatter system of second-order (partial) differential equa-tions, as an autonomous system of (first order) ordinarydifferential equations. For this purpose we introduce thefollowing phase space variables [15]:

x 1H

L3f

, y

L

H, z

V

3H, r T

3H2,

1 VV3/2f1/2

, 2 fV3/2f5/2

. (9)

It can be realized that, in terms of the variable r,

T

=2(1 r)

r, 0 < r 1. (10)

This means that the four-dimensional (low-energy/infra-red) limit of the Randall-Sundrum cosmological equa-tions corresponding to the formal limit is asso-ciated with the value r = 1, while the high-energy/ultra-violet limit

0, corresponds to r

0. We write the

Lorentz boost in terms of the variables of phase space as:


4/15

4

1L

=

1 y

2

3x2. (11)

Standard non-relativistic behavior corresponds to =1, while the ultra-relativistic (UR) regime is associatedwith = 0.

In terms of the variables that span the phase space(x,y,z,r,1, 2), the cosmological equations (4), (6), (7),and (8), can be written as an autonomous system of or-dinary differential equations (ODE):

x

= 1

2 (1 + 2)

yz3

x2 y2

2x xH

H , (12)

y = 32

[1(2 + 1) + 2( 1)2] z

3

x

32

(2 + 1)y y H

H, (13)

z =1

21

yz2

x z H

H, r =

2r(r 1)2 r

H

H, (14)

1 = 21

yz

x

V 3

2 ln f

ln V2

, (15)

2 =22

yz3

x3

f 5

2 ln V

3

ln f2

, (16)

where the prime denotes derivative with respect to thenumber of e-foldings ln a, while

V (V 2V)

(V)2, f

(f 2f)

(f)2,

and:

H

H= 2 r

2r

{y2 + 3(m + 1)

[r (1 )x2 z2]} .

(17)It will be helpful to have the parameters of observatio-

nal importance = /3H2 the scalar field dimen-sionless energy density parameter, and the equation ofstate (EOS) parameter = p/, written in terms ofthe variables of phase space:

= (1 )x2 + z2, = (1 )x2 z2

(1 )x2 + z2 . (18)

Recall, also, that the deceleration parameter q = (1 +H/H):

q = 1 +2

r

2r {y2

+ 3(m+1)[r (1)x2

z2

]}. (19)

B. DBI-DGP model

In this section we focus our attention in a braneworldmodel where the DBI inflaton is confined to a DGPbrane. In the (flat) FRW metric, the cosmological (field)equations are the following:

Q2 =1

3(m + ),

m = 3(1 + m)Hm, (20)

where, as before m is the EOS parameter of the matter

fluid, m is the energy density of the background baro-tropic fluid and is the energy density of DBI field.Also one has to consider the continuity equations for theDBI-type field (equation (2) or, alternatively, equation(4)). We have used the following definition:

Q2 H2 1

rcH, (21)

with rc being the so called crossover scale. In what fo-llows we will refer to this model as the DBI-DGP mo-

del. There are two possible branches of the DGP modelcorresponding to the two possible choices of the signs in(21): + is for the normal DGP models that are free ofghost, while - is for the self-accelerating solution.

As before, our goal will be to write the latter system ofsecond-order (partial) differential equations, as an auto-nomous system of (first order) ordinary differential equa-tions. For this purpose we introduce the following phasevariables:

x

1

QL

3f

, y

L

Q

, z

V

3Q, r

Q

H

, (22)

The expression determining the Lorentz boost coincideswith (11):

1L

=

1 y

2

3x2.

Hence, as before, = 1 is for the non-relativistic case,while = 0 is for the UR regime.

After the above choice of phase space variables thecosmological equations can be written as an autonomoussystem of ODE:


5/15

5

x = y2x2

[xy + z3r(1 + 2)

] x QQ

. (23)

y = 3z3r

2x

[1(

2 + 1) + 2( 1)2]

3y2

(2 + 1) y Q

Q, (24)

z =yz 2r

2x1 z Q

Q, r = r

(1 r21 + r2

)Q

Q. (25)

Recall that the prime denotes derivative with respect tothe number of e-foldings ln a, while

1 VV3/2f1/2

, 2 fV3/2f5/2

.

We have considered also the following relationship:

H

H=

2r2

1 + r2Q

Q,

where

Q

Q 1

2{3(1 + m)[1 x2(1 ) z2] + y2}. (26)

Equations (23)-(25) have to be complemented with theaddition of equations (15) and (16) above, which are theautonomous ordinary differential equations for 1 and 2,respectively. The equation (21) can be rewritten as:

r2 = 1 1rcH

. (27)

For the Minkowski phase, since 0 H (we con-sider just non-contracting universes), then 1 r .The case r 1 corresponds to the time reversalof the latter situation. For the self-accelerating phase,

r2 1, but since we want real valued r only,then 0 r2 1.4 As before, the case 1 r 0 re-presents time reversal of the case 0 r 1 that willbe investigated here.5 Both branches share the commonsubset (x,y,z,r = 1), which corresponds to the formal li-mit rc (see equation (21)), i. e., this represents just

4 In fact, fitting SN observations requires H r1c in order toachieve late-time acceleration (see, for instance, reference [25]and references therein). This means that r has to be real-valued.

5 Points with r = 0 and their neighborhood have to be carefully

analyzed due to the fact that at r = 0, other phase space variables(see Eq.(22)) and equations in (23-25) are undefined in general.

the standard behavior typical of Einstein-Hilbert theory

coupled to a self-interacting scalar field.To finalize this section we write useful magnitudes of

observational interest in terms of the dynamical variables(22). The effective dimensionless density parametersm = m/3Q

2 and = /3Q2:

m =mr2

= 1 x2(1 ) z2,

=r2

= x2(1 ) + z2. (28)

For the equation of state (EOS) = p/:

=x2(1 ) z2x2(1 ) + z2 . (29)

C. Brane Effects in the Phase Space

Higher-dimensional (brane) effects are encrypted in thevariable r for the DBI-RS model as well as for the DBI-

DGP model. In both cases the hyper-surfaces

inthe phase space, that are foliated by the value r = 1:=

(x,y,z,r = 1, 1, 2), represent the loci of theequilibrium configurations that can be associated withstandard general relativity behavior. For the DBI-DGPmodel the phase space hyper-plane

represents, addi-

tionally, the boundary in the phase space separating theMinkowski cosmological phase (r 1) from the the self-accelerating one (0 r 1). Trajectories in the phasespace that scape from

are associated with higher-

dimensional modifications of general relativity producedby the brane effects.

IV. EQUILIBRIUM POINTS IN THE PHASE

SPACE

In this section we will analyze in detail the existenceand stability of critical points of the autonomous sys-tems corresponding to both Randall-Sundrum and Dvali-Gabadadze-Porrari braneworld models. In both caseswe study i) an AdS throat often explored in the li-terature f() = /4, and a quadratic potential:V() = m22/2 [15], and ii) a particular relationshipbetween the warp factor and the potential for the DBIfield: f() = 1/V().


6/15

6

TABLE I: Properties of the critical points for the autonomous system (30).

Equilibrium point x y z r Existence m q

S 0 0 0 0 Always Undefined 0 0 Undefined 3m + 2

M 0 0 0 1 Undefined 1 0 Undefined (3m + 1)/2

U 1

3 0 1 0 0 1 0 1/2

TABLE II: Eigenvalues of the linearization matrices corresponding to the critical points in table I.

Equilibrium point 1 2 3 4

S 3m 3(m + 1) 3(m + 1) 3(m + 1)

M 3(1 + m) 3(m 1)/2 3(m + 1)/2 (3m + 1)/2

U 3m 3 3/2 3

A. AdS throat and a quadratic potential

Hence, the following relationship between f and V:

f =m4

4V2

,

is obtained. The above choice leads to 1 = being aconstant

=

2/(2/m),

while

2 = 2(x2/z2).Consequently, only two of the phase space variables x,z, and 2, are independent (say x and z). This factleads to considerable simplification of the problem sincethe dimension of the autonomous system reduces fromsix down to four. This is one of the reasons why thepresent particular case is generously considered in thebibliography.6

1. DBI-RS Model

As just noticed, after considering the specific form ofthe functions f() and V() above, the six-dimensional

6

The importance of the quadratic potential in the cosmologicalcontext is remarkable [26, 27].

autonomous dynamical system (12-16) can be reduceddown to a four-dimensional one:

x = yz2x2

(z2 2x2) y2

2x x H

H,

y = 3(1 )2xz 3(1 + 2)z32x

3(1 + 2)y

2 y H

H, (30)

z =yz2

2x z H

H, r =

2r(r 1)2 r

H

H,

where the ratio H/H = (q + 1) is given by Eq. (17).It arises the following constraint:

m =m

3H2= r (1 )x2 z2. (31)

Since m 0, then (1 )x2

+ z2

r. Besides, sincem 1 then (1 )x2 + z2 r 1. We will focus onexpanding FRW universes, so that x 0 and z 0. Theresulting four-dimensional phase space for the DBI-RSmodel is the following:

= {(x,y,z,r) : r 1 (1 )x2 + z2 r,x 0, y2 3x2, z 0, 0 r 1}. (32)

As properly noticed in the former section equilibriumpoints lying on the hyper-plane

= (x,y,z,r = 1)

are associated with standard general relativity dyna-mics. The remaining points in are related with higher-dimensional brane effects.


7/15

7

TABLE III: Properties of the equilibrium points of the autonomous system (33). Here m = m+1, while

(2 + 12).

Equilibrium point y z r Existence m q

S 0 0 0 Always 0 0 Undefined 3m + 2

M 0 0 1 1 0 Undefined 3m+12

K

3 0 1 0 1 0 12

SF 2 (23) 1 0 0 1 1 + 6 1 4M S 3m

mm

3m

1 1 < m < 0 1 + 3

2

m

2m

32m2

m

m

2m+1

3m+1

2

TABLE IV: Eigenvalues of the linearization matrices corresponding to the equilibrium points in table II I. The eigenvaluescorresponding to the fourth point (SF) in Tab.III, have not been included due to their overwhelming complexity. Here

2(92m + 6m + 1) + 24(3m + 32m + 3m + 1).

Equilibrium point 1 2 3

S 3(m + 1) 3(m + 1) 3(2m + 1)/2

M 3(m + 1) 3(m + 1)/2 3m/2

K 3m 3 3/2

M S 3m 34(1 m + /) 34(1 m /)

The relevant equilibrium points of the autonomous sys-tem of equations (30) are summarized in table I, while theeigenvalues of the corresponding linearization (Jacobian)matrices are shown in table II.

The source critical point S in table I is the past attrac-tor of the RS cosmological model. It is correlated with adecelerated solution whenever m > 2/3, otherwise it isan accelerated solution corresponding to a saddle criticalpoint. Existence of this solution is a distinctive featureof the higher-dimensional (brane) contributions. Actua-lly, according to equation (10), for finite non-vanishingT the case with r = 0 corresponds to the formal limit 0, i. e., the high-energy (ultra-violet) limit of theRS brane model. Since in this case x = y = z = r = 0,assuming finite non-vanishing f, L, V and total energydensity on the brane T, amounts to assuming that theHubble parameter has to be neccessarily infinitely largeH , a situation associated with the initial curva-ture singularity characterized by finite (non-null) energydensity on the braney. Notice that although in this limitboth and m vanish (in consequence T also vanis-hes), the usual definition for is irrelevant since the totalT is not constrained to be 1 as in usual cosmology, sothat there is no contradiction with the assumption that

T might be finite and non-vanishing in the 0-limit.7

0,62

0,42z

0,05

0,05

0,3

0,22

0,3

y

r

0,55

0,55

0,8

0,8

1,05

0,02

0,62

0,42 z5,0

0,05

2,5

0,3

0,22

0,0

y

t

0,55

2,5

0,8

5,0

7,5

1,05

0,02

10,0

7 This argument was suggested to us by one of the referees.


8/15

8

FIG. 1: Phase portrait generated by a given set of initial

data (m = 1/3, = 1) for the autonomous system of ODE(33), corresponding to the UR approximation of the DBI-RSmodel left-hand panel, and the corresponding flux in time right-hand panel. The values of the free parameter has beenchosen so that the scalar field-dominated solution (point SF(1.14, 0.75, 1)) is the late-time attractor.

The matter-dominated solution (equilibrium point M)and the ultra-relativistic (scalar field-dominated) phaseU, are always saddle critical points and are associa-ted with standard general relativity dynamics. The lat-ter is associated with decelerating dynamics while theformer M represents decelerating expansion whenever

m > 1/3 and inflationary solution otherwise. PointsM and U in Tab. I correspond to the equilibrium pointsA and B of Ref. [15], respectively.

More detailed information can be retrieved only af-ter further simplification of the case to study. One wayto achieve further simplification is to study the ultra-relativistic regime where = 0 L = . In the URregime, thanks to the relationship

= 0 x = y/

3,

the autonomous system of ODE (30) simplifies down toa set of three ordinary differential equations:

y = 3

3z3

2y 3y

2 y H

H,

z =

3z2

2 z H

H, r =

2r(r 1)2 r

H

H,

H

H= 2 r

2r{(m + 1)[3r y2 3z2] + y2}. (33)

The reduced (three-dimensional) phase space in thissimpler case is given by:

= {(y , z , r) : 3(r 1) y2 + 3z2 3r,y , z 0, 0 < r 1}. (34)

There are found five equilibrium points of the auto-nomous system of ODE (33) in . These points, to-gether with their properties, are listed in Tab.III, whilethe eigenvalues of the corresponding linearization matri-ces are shown in the table IV. By their overwhelmingcomplexity, the eigenvalues of the linearization matrixcorresponding to the fourth equilibrium point in tableIII (point SF) have not been included in Tab.IV. It isworth recalling that the standard general relativity beha-vior is associated with points in the phase space lying onthe phase plane = (y , z , r = 1).

The critical point S, which represents a singular cos-

mological solution, is associated with 5D brane effectssince, the only way the Hubble parameter H can be ar-bitrarily large while keeping T finite, is that the branetension 0, so that H2 2/ . Provided thatm > 1/2 this equilibrium point is always the past at-tractor in the phase space, i. e., it represents the sourcecritical point from which any phase path in origina-tes. It corresponds to decelerated expansion. In sectionV we will comment in more detail the difference betweenthis singular solution and the standard big-bang arisingin general relativity.

The matter-scaling solution (equilibrium point M S) isthe late-time attractor provided that (the definition of

the parameter can be found in the caption of the tableIV)

(1 m) < < (1 m).Otherwise, the scalar field-dominated solution SF is thelate-time attractor.

We have to point out that the matter-scaling solutionexists only if the equation of state (EOS) parameter of thebackground matter is a negative quantity: 1 < m < 0.This means that we can not have matter-scaling withbackground matter being dust (m = 0). Therefore,the usefulness of this equilibrium point to describe thecurrent phase of the cosmic evolution is unclear. Un-

like this, the scalar field-dominated solution SF is al-ways inflationary ( > 0 always) and could be associatedwith accelerated late-time cosmic dynamics. The matter-dominated solution M and the kinetic energy-dominatedphase (equilibrium point K in Tab. III), are alwayssaddle points in the phase space.

Worth noticing that the scalar field-dominated solution(point SF in Tab.III) and the matter-scaling phase M Scorrespond to the points C and D in reference [15]. Inthe standard 4D limit of the theory (r = 1) we obtainthe same behavior that was obtained in [15, 27].

In the Figure 1 the trajectories in the reducedphase space defined in (34) originated by a given set of

appropriated initial data, are drawn for the model of (30)in the ultra-relativistic approximation. The phase spacepictures in the figures 1 reveal the actual behavior of theRS dynamics: trajectories in phase space depart from theinitial curvature singularity, and, at late times, approachto the plane (y,z, 1), which is associated with standardfour-dimensional behavior, in particular the trajectoriesapproach to the SF point.

2. DBI-DGP Model

For an AdS throat and the quadratic self-interactionpotential, the autonomous system of ODE (23)-


9/15

9

TABLE V: Properties of the equilibrium points of the autonomous system (35).

Equilibrium point x y z r Existence m q

SA 0 0 0 0 Always Undefined 0 0 Undefined 1M 0 0 0 1 1 0 (1 + 3m)/2

U 1

3 0 1 0 0 1 0 1/2

TABLE VI: Eigenvalues of the linearization matrices corresponding to the critical points in table V.

Equilibrium point 1 2 3 4

SA 3(1 + m)/2 3(1 + m)/2 3(1 + m)/2 3(1 m)/2M 3(m 1)/2 3(1 + m)/2 3(1 + m)/2 3(1 + m)/2

U 3m 3 3/2 3/2

(25), (15), (16), reduces down to the following four-dimensional autonomous system:

x = y2

2x yzr

2x2(z2 2x2) x Q

Q,

y = 62xzr 3(2 + 1)

2x[(z2 2x2)zr +

+xy] y QQ

,

z =yz2r

2x z Q

Q, r = r

(1 r21 + r2

)Q

Q. (35)

The ratio Q/Q is given by Eq. (26).Recall that standard general relativity behavior is as-

sociated with points liying on the hyper-plane =(x,y,z,r = 1). The remaining points in the bulk of thephase space are associated with higher-dimensional(brane) effects.

The critical points of the autonomous system of ODE(35), together with their most important properties, aresummarized in the table V. The eigenvalues of the linea-rization matrices corresponding to the critical points inTab. V are shown in table VI.

Three critical points are found: the self-accelerating so-lution (point SA), the matter-dominated solution (M),and the UR phase (U) which is dominated by the DBIscalar field. The critical point SA the self-acceleratingsolution is always a saddle point and represents an infla-tionary solution. Actually, for r = 0 one gets a de Sittersolution H = 1/rc.8 This is the only equilibrium point

8 In fact, fitting SN observations requires H 1/rc in order to

that can be associated with extra-dimensional effects.The remaining points lie on the hyper-plane relatedwith standard general relativity behavior. The matter-dominated solution (critical point M in Tab. V) is thepast attractor if m > 1, otherwise it is also a saddlecritical point.9 The ultra-relativistic regime, which is do-minated by the scalar field, mimics the cosmic evolutionof a universe filled with dust. It can be the past attractor

only for negative m < 0. For > 0 it is a saddle equili-brium point in the phase space. The above results are tobe contracted with the results in the former subsection.

As before, a more detailed study of the asymptoticproperties of the model (35) requires additional simplifi-

cation. The ultra-relativistic approximation comes to ourrescue. As long as one considers just large Lorentz boosts(amounting to vanishing ) the relationship x = y/3is verified. This relationship allows for further simplifi-cation of the autonomus system of ODE (35). Actually,in the UR regime the above system of equations can besimplified to the following three-dimensional autonomoussystem of ODE:

achieve late time acceleration (see, for instance, reference [28]and references therein).

9

We do not consider unphysical situations where the EOS para-meter for the background fluid < 1.


10/15

10

TABLE VII: Properties of the equilibrium points of the autonomous system (36). Here

(2 + 12), while m

m+1.

Equilibrium point y z r Existence m q

M 0 0 1 1 0 Undefined 3m+12

K

3 0 1 0 1 0 12

SF 2

(23)

1 0 0 1 1 + 6

1 4

MS 3m

mm

3m

1 1 < m < 0 1 + 3

2

m

2m

32m2

m

m3m+1

2

TABLE VIII: Eigenvalues of the linearization matrices corresponding to the critical points in table VII. Here

12m(6m + ) + 2(6 ), while 24(1 + m)3 + 2(1 + 3m)2.Equilibrium point 1 2 3

M 3(m + 1)/2 3(m + 1)/2 3m/2

K 3m 3/2 3/2

SF /4 12(2+m)+38

+28

12(2+m)+38

28

MS 32

(1 + m)34

m 1 + 2

34

m 1 2

y = 32

y 3

3z3r

2y y Q

Q,

z =

3

2z2r z Q

Q,

r =r(1 r2)(1 + r2)

Q

Q. (36)

The phase space for the autonomous system (36) canbe defined in the following way. For the + - branch(the Minkowski cosmological phase):

+ = {(y , z , r) : 0 y2 + 3z2 3,y , z 0, r [1, )}, (37)

while, for the self-accelerating - - branch, it is given by:

= {(y , z , r) : 0 y2 + 3z2 3,y , z 0, r [0, 1]}. (38)

There are found four equilibrium points of the auto-nomous system of ODE (36). None of them is associa-ted with extra-dimensional effects since all of the criticalpoints lie on the phase hyper-plane with r = 1. This is

a expected result since in the UR limit the DBI effectsdominate over the brane effects at late times. These cri-tical points together with their most salient features are summarized in table VII. If, m > 0, the matter-dominated solution (point M) is the past attractor, whilefor negative m < 0, the kinetic energy-dominated solu-tion (point K) is the past attractor in the phase space(see in [15, 27]). The latter result is independent on whichbranch of the DGP is being considered since, for DGPmodels, at early times, the brane effects can be safelyignored which means that the standard GR cosmologicaldynamics is not modified.

The equilibrium points SF (scalar field-dominated so-

lution) and M S (scaling dominated-solution) are alwaysa saddle critical points. There is no future (late-time)attractor in the phase space of the model.

B. Special relationship f() = 1/V()

This case was studied in [23] where the authors usedan approach based on an existing (formal) mathematicalequivalence between the DBI model and standard tach-yon cosmology, under an appropriate transformation ofthe DBI field. The above relationship (f() = 1/V())leads to significant simplification of the field equations.The equation of motion of the DBI-type field and the


11/15

11

conservation equation are the following:

+ 32H = V(1 32/2V), (39)

where the modified Lorentz factor is defined as:

=1

1 2/V, (40)

and where we have defined the following energy densityand pressure of the DBI scalar field:

= V() , p = 1V().

From now on we shall call the model given by equations(39), (40), as modified tachyon cosmology (MTC). by as-suming f() = 1/V() the autonomous system of ordi-nary diffrential equations is a three-dimensional one sincewe can introduce a new dimensionless variable a y/x(1 = V /V = 2).

1. DBI-RS Model

With the aim to write an autonomous system out ofthe cosmological equations we introduce the following di-mensionless phase space variables:

a =V

, z =

V

3H, r =

T3H2

. (41)

As before

T

=2(1 r)

r, 0 < r 1.

Then we can write the above system as:

a = (a2 1)[3a +

3 (ln V)z]

z =

3

4az2 (ln V)

z(2 r)2r

3z2(m a2)

1 a2 3mr

r = (r 1)

3z2(m a2)1 a2 3mr

(42)

where m = m+1 is the barotropic index for the matter(0

m

2), and the tilde accounts for derivative in

respect to the cosmological time .

In terms of the above phase space variables the cons-

traint (Friedmann) equation can be written as:

m = r z2

1 a2 .

Other cosmological parameters such as the DBI-fieldenergy density parameter , the equation of state pa-rameter , and the deceleration parameter q look like:

=z2

1 a2 , = a2 1

q = 1 + (2 r)2r [

3mr + 3z 2(m a2)]. (43)

The phase space spanned by the above variables is defi-ned as follows:

= {(a,z,r) : |a| 1, 0 z4 1 a2, 0 r 1}.For an exponential self-interaction potential of the

form:

V() = V0 exp(),since ln V = = const, then the system (42) is aclosed autonomous system:

a = (a2

1)[3a

3z]

z =

3

4az2 z(2 r)

2r

3z2(m a2)

1 a2 3mr

r = (r 1)

3z2(m a2)1 a2 3mr

. (44)

The critical points corresponding to the exponentialpotential are summarized in Table IX, while their respec-tive eigenvalues are displayed in Table X. These pointscoincide with the ones reported in [23] but for the ad-ditional high-energy/ultra-violet critical point S relatedwith the brane effects. While the points M, T and M Sshow similar stability properties than reported in [23],

the stability of the critical point U differs in that it is asaddle critical point unlike in standard cosmology whereit was a inflationary past attractor.

In Figure 2 we show the trajectories in phase spacefor different sets of initial conditions. As clearly seenfrom this figure, the trajectories in phase space emergefrom the source critical point S = (a,z,r) = (1, 0, 0) the kinetic/potential energy-scaling solution meaningthat this is the past attractor of the Randall-Sundrumcosmological model. This equilibrium point is associatedwith a initial curvature singularity of higher-dimensionalorigin since, assuming finite (non-null) V, and T,

a = 1, z = r = 0 H .


12/15

12

TABLE IX: Properties of the critical points for the autonomous system (42). Where z =

36+4

2

6 .

Equilibrium Point a z r Existence q

S 1 0 0 Always undefined 0 undefined

M 0 0 1 0 1 1 + 3m2U 1 0 1 0 0 1 + 3m2M S

m

3m

1 3m12m

m 1 + 3m2T z

3z 1 1 1

2z2

3 1 + 3m/2

TABLE X: Eigenvalues of the linearization matrices corresponding to last four equilibrium points in Tab.IX.

Here

482m

1

m

2+ 4 + m(17m 20).

Equilibrium Point 1 2 3

M 3 3m 3m/2

U 3m 3m/2 6

M S 3m 34 [(2 m) + ] 34 [(2 m)]

T 2(

4+362)

123 + 2

12

(4 + 36 2) 3m + 212

(4 + 36 2)

0,84

0,64

0,44 z

0,05

0,05

0,15

0,3

0,25

0,24

0,35

a

r

0,45

0,55

0,55

0,65

0,8

0,75

0,85

0,04

0,95

0,84

0,64

0,44 z5,0

0,08

2,5

0,28

0,24

0,0

a

0,48

t 2,5

0,68

5,0

0,88

7,5

0,04

10,0

FIG. 2: Phase portrait generated by a given set of initialdata (m = 1, = 1) for the autonomous system of ODE(44), corresponding to the UR approximation of the DBI-RSmodel left-hand panel, and the corresponding flux in time right-hand panel. The trajectories in phase space emerge fromthe point S = (a ,z,r) = (1, 0, 0) the kinetic/potentialenergy-scaling solution.

2. DBI-DGP Model

Here we use the same variables a and z, and redefiner = Q/H. In terms of these variables we can write thefollowing system of ODE:

a = (a2 1)[3a +

3 (ln V)z]

z = z

3

4az (ln V) 2r

2

r2

1

Q

Q

r =r(1 r2)

r2 + 1

Q

Q(45)

where

Q

Q= 3

2r2

mr

2 +z2(a2 m)

1 a2

.

The DBI-field energy density parameter , and theequation of state parameter are the same than in theformer subsection. The deceleration parameter q is:

q = 1 2r2

r2 + 1

Q

Q .


13/15

13

TABLE XI: Eigenvalues of the linearization matrice corresponding to the equilibrium points U, T and M S to the system 45.

Here

482m1m2

+ 4 + m(17m 20).

Equilibrium Point 1 2 3

U 3m/2 3m/2 6

M S 3m2

34

[(2 m) + ] 34 [(2 m)]

T2(

4+362)

12 3 + 2

12

(4 + 36 2

)3m + 212

(4 + 36 2

)

The phase space for the autonomous system (45) can

be defined in the following way. For the + - branch(the Minkowski cosmological phase):

+ = {(a,z,r) : |a| 1, 0 z4 + a2 1,z 0, r [1, )}, (46)

while, for the self-accelerating - - branch, it is given by:

= {(a,z,r) : |a| 1, 0 z4 + a2 1,y , z 0, r [0, 1]}. (47)

The system (45) is not a closed autonomous systemand, for this reason, we focus in the exponential poten-tial, V = V0 exp(), for which V /V = , and thesystem of ODE is closed.

We obtain the same critical points M, U, MS and T,that were obtained in the former subsection within theRandall-Sundrum Model. However in this case the sta-bility of the critical points U, T and MS differ from theones in the former subsection as shown in the table XI.The equilibrium point U = (1, 0, 1) represents an in-flationary solution: a past attractor in the phase space.This point represents a scaling of the potential and ofthe kinetic energy of the scalar field. These results coin-

cide with the ones reported in [23]. On the contrary, thematter-scaling solution is always a saddle point unlikein the case explored in [23] were this point (whenever itexists) was a stable equilibrium point the future attrac-tor. The equilibrium point T (the tachyon-dominatedsolution) is the late-time attractor in the present case.

V. RESULTS AND DISCUSSION

The importance of the brane effects for the cosmic dy-namics is well known. These effects can modify the gene-ral relativity laws at early times (UV modifications), aswell as at late times (IR modifications). Actually, while

the Randall-Sundrum brane model produces UV modi-

fications of general relativity, in the Dvali-Gabadadze-Porrati braneworld IR modifications of the laws of gra-vity arise instead. In a similar way, the introduction of anon-linear Dirac-Born-Infeld type of field might modifythe cosmic dynamics at early as well as at late times. Thismakes even more interesting the study of the combinedeffect of a DBI-type field trapped in the braneworld. Aimof the present paper has been, precisely, the study of theasymptotic properties of the latter kind of cosmologicalmodels.

A. RS Model

The main result of Sec.IV A 1 and Sec.IV B 1 can besummarized as follows: The initial curvature singula-rity (critical point S in Tab.I) is the past attractor inthe phase space, it represents the source critical pointfrom which any phase path in originates. For a widerange of parameters it is a decelerated solution. We haveto differentiate this cosmological singularity from stan-dard general relativity big-bang singularity. Actually, ina standard 4D model (no brane effects), the big-bangsingularity which is characterized by infinitely large Hand total energy density T , is removed by the effect of

the DBI field. This can be corroborated by noting thaton the hyper-surface where GR solutions lie, there areno critical points that can be associated with a curvaturesingularity.10 What the results of our study show is thatthe effect of the RS brane is to restore the occurrenceof a curvature singularity at very high energy/curvature.The fact that the latter (initial, cosmological) singularityis correlated with a past attractor means that the big-bang is a generic property of DBI-RS models. For the

10 Absence of critical points that can be correlated with a curvature

singularity does not mean that particular solutions with curva-ture singularities might arise.


14/15

14

particularly simple relationship between the warp factor

and the potential of the DBI field f() = 1/V() (theexponential potential was considered), the past attrac-tor is the kinetic/potential energy-scaling solution withthe occurrence of a curvature singularity due to the UVbrane effects.

In the Sec.IV A 1 the matter dominated solution (cri-tical point M in Tab.I) and the ultra-relativistic (sca-lar field-dominated) phase (critical point U) are al-ways saddle critical points and are associated with stan-dard general relativity dynamics. The late-time behaviorin Sec.IV A 1 is correlated with the critical points MS(matter-scaling solution) and SF (scalar field solution).The point MS is the late-time attractor provided that

(1 m) < < (1 m).11

Otherwise, the scalarfield-dominated solution SF is the late-time attractor.This result coincides with the one obtained within thestandard 4D limit of the theory and reported in [15, 27].The critical points M, T and M S in Sec.IV B 1 have simi-lar stability properties than in standard cosmology [23].On the contrary, the stability of the point U differs fromthe one in standard cosmology where it was a inflationarypast attractor. In the case where the RS brane effect isconsidered this is a saddle critical point instead.

In general, the dynamical behavior of the Randall-Sundrum model differs from the standard behavior wit-hin four-dimensional Einstein-Hilbert gravity only at

early times (high-energy regime). The existence of theinitial curvature singularity (the past attractor S) is tobe contrasted with the standard four-dimensional resultwithin general relativity fuelled by a DBI scalar field (anda background fluid), where either the fluid-dominated orthe kinetic-dominated, singularity-free solutions can bethe past attractor [15, 23, 27]. The late-time cosmologi-cal dynamics, on the contrary, is not affected by the RSbrane effects in any essential way.

B. DGP Model

The following important results can be summarized inthis case: The new ingredient added by the DGP braneeffect is the inflationary self-accelerating solution (criticalpoint SA). It is always a saddle point in the phase space.This means that the self-accelerating solution is not al-ways the end point of the cosmic evolution even for theself-accelerating branch of the DGP model. The matter-dominated, decelerated solution (critical point M) is al-ways a saddle critical point. The early time behavior in

11

The definition of the parameter can be found in the caption ofthe table IV.

the DGP model can be correlated either with the matter-

dominated solution or with the kinetic energy-dominatedsolution as in [15, 27]. This result is independent ofwhich branch of the DGP is being considered since atearly times the DGP brane effects can be safely ignored,i. e., the standard cosmological dynamics is not modi-fied. There is no future (late-time) attractor in the phasespace of the model. The scalar field-dominated solution(equilibrium point SF) and the matter-scaling solution(equilibrium point MS) are always saddle equilibriumpoints. In Sec.IV B 2 we obtained the same four criticalpoints correlated with GR solutions, that were obtainedin the Randall-Sundrum model (Sec.IV B 1). However,in this case the stability properties of the critical points

U, T and MS have been modified by the DGP braneeffects. The latter result is independent of which branchof the DGP is being considered. The critical point U inSec.IV B 2 a past attractor in the phase space repre-sents an inflationary solution. This point represents ascaling of the potential and of the kinetic energy of theDBI scalar field. The stability properties are the samethat were found and reported in [23].

Summing up: the effect of the DGP brane is to mo-dify the late-time cosmological dynamics through chan-ging the stability of the corresponding (late-time) criti-cal points. Actually, in the present case the scalar field-dominated solution, as well as the scaling dominated-

solution, are always saddle critical points. This resulthas to be confronted with the classical general relativityresult where the above-mentioned solutions can be late-time attractors.12

VI. CONCLUSION

In the present paper we aimed at studying the asym-ptotic properties of a DBI-type field trapped in the bra-neworld for two particularly interesting and somewhatsimple cases: i) an AdS throat (f() = /4) and a qua-

dratic potential (V() = m2

2

/2) [15], and ii) assum-ption of a special relationship between the warp factorand the potential for the DBI field: f() = 1/V(). Thecombined effect of the non-linear nature of the DBI fieldand of the higher-dimensional brane effects seem to pro-duce a rich dynamics. Both brane contributions and non-linear DBI effects can modify the general relativity lawsof gravity at late times as well as at early times. Herewe focused in Randall-Sundrum and in Dvali-Gabadadze-

12 In the DBI-DGP case, at early times, the dynamics is general re-

lativistic so that the stability properties of the matter-dominatedphase in tables III and IV just coincide with the results of [15, 27].


15/15

15

Porrati braneworlds exclusively. In a sense this work can

be considered as a natural completion of the Ref. [15, 27]to consider the combined effect of the DBI-type field andof the braneworld.

We performed a thorough study of the phase space co-rresponding to the above two scenarios. It is revealedthat RS brane effects conspire to re-introduce the initialcurvature singularity this time of higher-dimesnionalorigin that was removed by the non-linear effects ofthe DBI field at very high energy. At the oposite end ofthe energy/curvature spectrum, the DGP brane effectsmodify the late-time cosmic dynamics: while in the self-accelerating branch the self-accelerating solution is corre-

lated with a saddle critical point (point SA), in general

the stability of the standard general relativity solutionsis modified by the extra-dimensional effects.

The present study can be considered as a natural com-plection of the work of Ref. [15].

This work was partly supported by CONACyTMexico, under grants 49924-J, 52327, 105079, InstitutoAvanzado de Cosmologia (IAC) collaboration. R G-Sacknowledges partial support from COFAA-IPN, EDI-IPN and IPN grant SIP-20100610. D G, and T G ak-nowledge also the MES of Cuba for partial support ofthe research.

[1] C. L. Bennett et al., Astrophys. J. Suppl. Ser. 148 (2003)1; G. Hinshaw et al., Astrophys. J. Suppl. Ser. 148 (2003)135; D. N. Spergel et al., Astrophys. J. Suppl. Ser. 148(2003) 175; H. V. Peiris et al., Astrophys. J. Suppl. Ser.148 (2003) 213; A. Kogut et al., Astrophys. J. Suppl.Ser. 148 (2003) 161; E. Komatsu et al., Astrophys. J.Suppl. Ser. 148 (2003) 119

[2] A. A. Starobinsky, Phys. Lett. B 91 (1980) 99; A. H.Guth, Phys. Rev. D 23 (1981) 347; A. Albrecht, P. J.Steinhardt, Phys. Rev. Lett. 48 (1982) 1220; A. D. Linde,Phys. Lett. B 108 (1982) 389; Phys. Lett. B 129 (1983)

177.[3] L. Randall, R. Sundrum, Phys. Rev. Lett. 83 (1999)3370.

[4] G. R. Dvali, G. Gabadadze, M. Porrati, Phys. Lett. B485208-214 (2000), [hep-th/0005016].

[5] R. M. Hawkins, J. E. Lidsey, Phys. Rev. D 63 (2001)041301.

[6] G. Huey, J. E. Lidsey, Phys. Lett. B514 (2001) 217.[7] L. H. Ford, Phys. Rev. D35 (1987) 2955.[8] B. Feng, M. Li, Phys. Lett. B564 (2003) 169.[9] A. R. Liddle, L. A. Urena-Lopez, Phys. Rev. D68 (2003)

043517.[10] M. Sami, V. Sahni, Phys. Rev. D70 (2004) 083513.[11] C. Deffayet, G. R. Dvali, G. Gabadadze, A. I. Vainsh-

tein, Phys. Rev. D65 (2002) 044026 [hep-th/0106001];A. Nicolis, R. Rattazzi, JHEP 0406 (2004) 059, [hep-th/0404159].

[12] I. Quiros, R. Garca-Salcedo, T. Matos, C. Moreno, Phys.Lett. B 670 (2009) 259265.

[13] E. Silverstein, D. Tong, Phys. Rev. D 70 (2004) 103505;M. Alishahiha, E. Silverstein, D. Tong, Phys. Rev. D 70(2004) 123505.

[14] X. Chen, Phys. Rev. D 71 (2005) 063506 [hep-th/0408084]; JHEP 08 (2005) 045 [hep-th/0501184].

[15] Z-K. Guo, N. Ohta,JCAP 04 (2008) 035.[16] M. C. Bento, O. Bertolami, A. A. Sen, Phys. Rev. D 67

(2003) 063511; X. Chen, Phys. Rev. D 71 (2005) 063506;S. E. Shandera, S-H. H. Tye, JCAP 05 (2006) 007.

[17] X. Chen, M. Huang, S. Kachru, G. Shiu, JCAP 01 (2007)002.

[18] G. R. Dvali, S. H. H. Tye, Phys. Lett. B 450 (1999) 72[hep-ph/9812483]. S. Kachru, R. Kallosh, A. Linde, J.M. Maldacena, L. McAllister, S. P. Trivedi, JCAP 0310(2003) 013 [hep-th/0308055].

[19] H. Verlinde, Nucl. Phys. B 580 (2000) 264 [hep-th/9906182]; S. Gukov, C. Vafa, E. Witten, Nucl. Phys.B 584 (2000) 69 [hep-th/9906070]; S. Gukov S, C. Vafa,E. Witten E, Nucl. Phys. B 608 (2001) 477 (erratum); K.Dasgupta, G. Rajesh, S. Sethi, JHEP 08(1999) 023 [hep-

th/9908088]; B. R. Greene, K. Schalm, G. Shiu, Nucl.Phys. B 584 (2000) 480 [hep-th/0004103]; S. B. Gid-dings, S. Kachru, J. Polchinski, Phys. Rev. D 66 (2002)106006 [hep-th/0105097].

[20] J. M. Cline, hep-th/0612129; S. H. Henry Tye, Lect. No-tes Phys. 737 (2008) 949-974 [hep-th/0610221]. L. McA-llister, E. Silverstein, Gen. Rel. Grav. 40 (2008) 565-605(2008) [arXiv:0710.2951].

[21] S. Kecskemeti, J. Maiden, G. Shiu, B. Underwood, JHEP0609, (2006), 076, [arXiv:hep-th/0605189].

[22] A. A. Coley, Dynamical systems and cosmology,Dordrecht-Kluwer, Netherlands (2003).

[23] I. Quiros, T. Gonzalez, D. Gonzalez, Y. Napoles, R.Garcia-Salcedo, C. Moreno, Class. Quantum Grav. 27(2010) 215021, [arXiv:0906.2617].

[24] W. Fang, Y. Li, K. Zhang, H.-Q. Lu, Class. Quant. Grav.26 (2009), 155005, [arXiv:0810.4193].

[25] K. Koyama, Class. Quantum Grav. 24 (2007) R231 [ar-Xiv:0709.2399].

[26] C. Ahn, C. Kim, E. V. Linder, Phys. Rev. D80, (2009),123016, [arXiv:0909.2637].

[27] E. J. Copeland, S. Mizuno, M. Shaeri, [arXiv:1003.2881][28] K. Koyama, Class. Quantum Grav. 24 (2007) R231, [ar-

Xiv:0709.2399v2][29] T. Gonzalez, T. Matos, I. Quiros, A. Vazquez-Gonzalez,

Phys. Lett. B 676 (2009) 161167.

dbi paperv rev

Documents