Complete cosmic scenario in the Randall-Sundrum braneworld from the dynamical systems perspective

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by  Jibitesh Dutta

Complete cosmic scenario in the Randall-Sundrum braneworld from
the dynamical systems perspective
Jibitesh Dutta 1,2 and H. Zonunmawia 2 ,
∗ †
1Mathematics Division, Department of Basic Sciences and Social Sciences,
North Eastern Hill University, NEHU Campus, Shillong - 793022 ( INDIA ) and
2Department of Mathematics, North Eastern Hill University,
NEHU Campus, Shillong - 793022 ( INDIA )
(Dated: January 5, 2016)
The paper deals with dynamical system analysis of a coupled scalar field in the Randall-
Sundrum(RS)2braneworld. Thelatetimeattractordescribesthefinalstateofthecosmicevolution.
6
In RS2 based phantom model there is no late-time attractor and consequently there is uncertainty
1
incosmicevolution. Inthispaper,wehaveshownthatitispossibletogetlate-timeattractorwhen
0
gravity is coupled to scalar field. Finally, in order to predict the final evolution of the universe,
2
we have also studied classical stability of the model. It is found that there are late time attractors

n whicharebothlocally aswellasclassically stableandsoourmodelcanrealisethelatetimecosmic
a acceleration.
J

3 PACSnumbers: 98.80.Cq,98.80.-k

]
c Keywords : RS braneworld; phantom scalar field ; dynamical system; attractor; phase space.
q
-
r
g I. INTRODUCTION
[

1 The modified theories of gravity which can explain the present observed accelerated expansion of the uni-
v verse [1–3] without any need of dark energy(DE) [4, 5] have received lot of success in recent years. Brane
3 gravity(BG)/Braneworldscenario inspired by advances of String/M-Theory is a prominent theory of modified
8
gravity. In this scenario our observable universe is assumed to be a sub-manifold embedded in compactified
2
higherdimensionalspacetimeknownasbulk. Hereexceptgravity,allthestandardmodelparticlesareconfined
0
0 to sub- manifold called brane. Braneworldmodels provides a novel way to correct General Relativity(GR) and
. gives answer to several unsolved problems of cosmology [6].
1
Among different braneworld models, the Randall-Sundrum type II model(RS2) is very popular among cos-
0
mologists due to its rich conceptual base [7]. RS2 model gave a framework often referred as braneworlds with
6
1 wrapped extra-dimension in contrast to Kaluza-Klein mechanism of compactification where extra-dimension is
: very small [8]. In this set up big-rip singularity of phantom model can be avoided and accelerationis transient
v
in nature [9]. In contrast to GR based models here inflation is possible for a large class of potentials due to
i
X modified Friedmann equation[10].
r Another braneworld model that received considerable attention is the Dvali-Gabadadze-Porrati(DGP)
a
braneworld model [11–13]. This model can explain the present acceleration without using mysterious DE.
Usually RS2 model modifies GR at small cosmologicalscales and DGP model modifies GR at large cosmologi-
cal distance. Consequently RS2 models have more impact on cosmology of early universe. The interesting and
recentfeature is thatRS2modelcanalsomodify late time cosmicexpansionifthe energydensity ofthe matter
contentincreasesatlate time e.g., phantomscalarfield [9, 14]. Thus for scalarfields likephantom, RS2 models
can have impact on late universe and it can give a complete cosmic scenario.
Dynamicalsystemstoolshaveacentralroleforqualitativeanalysisofhomogeneouscosmologicalmodelswhose
evolution is governed by finite dimensional autonomous system of ordinary differential equations (ASODE)
[15, 16]. The goal of the dynamical systems approach is to correlate asymptotic cosmological solutions with
important concepts like past and future attractors along with saddle points. In the phase space of models, if a
criticalpointcanbeassociatedwithasolution,itmeansthatintheneighbourhoodofthissolutiontheuniverse
dynamicswillevolveforasufficientlylongtime irrespectiveofinitialdata. Todescribeacompletecosmological
scenario,acosmologicalmodeldescribedbyanautonomoussystemmusthaveapasttimeattractor(torepresent
∗ [email protected],[email protected]
† [email protected]
2
inflationary epoch), one or two saddle points (to represent radiation and matter dominated universe) and an
late time attractor (to represent accelerated expansion universe) [17].
In cosmologyscalar fields play an active role to model DE, inflation and other cosmologicalfeatures [18, 19].
ThedynamicalstudyofascalarfieldcoupledwithperfectfluidhasfoundmanyapplicationsinGRcontext. For
a nice introduction [17, 20] and application of this technique in GR based cosmologyone can see recent papers
[21–23]. In RS2 model, using dynamical system approachRudra et al investigated role of modified Chaplyging
gas in explaining accelerated expansion of the universe [24]. They showed that around fixed points universe
expandswithapower-law. AdynamicalstudyofDiracBornInfield(DBI)trappedinthebraneworldisexplored
in [25] and it is found that in ultra relativistic regime scalarfield dominated and matter scaling solutions exist.
In braneworld context dynamics of self-interacting scalar field trapped in a RS2 model for a wide variety of
potentialshavebeenstudiedin[26–28]. Escobaretal didarefinedstudyofascalarfieldtrappedinRS2model
using Centre Manifold Theory with an arbitrary potential. They proved that there are no late time attractors
with 5Dmodifications [28]. Furthermore,in a recentpaper Biswaset al studied interacting darkenergyin RS2
braneworld using dynamical systems tools [29]. All the above studies were done for a quintessence scalar field.
It is found that only early times cosmology is modified with respect to GR based models. On the contrary
for usual phantom scalar field, the higher dimensional Brane effects of RS2 usual phantom scalar field model
modifies past and future asymptotic properties so that we get past time attractor and saddle point only [14].
In GR basedphantomscalarfield modelwe get saddle point andlate time attractorbut it does notadmit past
time attractor [14, 30].
In this paper we shall show that in RS2 model when gravity is coupled to scalar field with scalar coupling
functionandapotential,thelateuniverseisalsomodifiedandwegetacompletecosmicscenario. Thecoupling
function additionally helps to control the behaviour of scalar field and have been used extensively in unifying
phantom inflation with late time acceleration [5, 31–33]. Recently Mahata et al [34] studied the dynamics of
phantom cosmological model where gravity is coupled to scalar field having scalar coupling function in GR
contextusing dynamicalsystemtools. The aimofthis paperis tostudy the dynamics ofcoupledscalarfieldon
RS2 braneworld by using dynamical system tools. Furthermore, we have also examined the classical stability
of the modelin addition to calculating variousobservablequantities inthe asymptotic solutions. In contrastto
Ref [34], we shallshow that coupled scalarfield give richdynamics andlate time accelerationsolutions are also
classically stable.
The paper is organised as follows: The basic and the evolution equations of the autonomous system are
presented in section (II). Sections (III) and (IV) deals with the analysis of the critical points of the system in
4D and 3D respectively. The classical stability of the model has been studied in section (V). Final section is
devoted to discussion and conclusion.
II. BASIC EQUATIONS
The total action of the field equations in RS2 model is written as
S =S +S +S (1)
RS φ m
where
S = d5x g(5) (2R(5)+Λ )+ d4x √ g λ (2)
RS Z p− 5 Z −
is the action of RS2 model, g(5) and g are the bulk metric and the brane metric respectively, R(5) is the
µν µν
Ricci scalar in the bulk and Λ is the bulk cosmologicalconstant, S is the action of the scalar field coupled to
5 φ
gravity given by
1
S = d4x√ g µ(φ)gµν( φ)( φ) V(φ) (3)
φ Z − h − 2 ∇µ ∇ν − i
µ(φ) is the coupling parameter, V(φ) is the potential of the scalar field and S = d4x √ g L is the action
m m
−
of the matter field which is chosen as the cold dark matter (CDM). R
Observations support homogeneous and isotropic model of the late universe, given by the line-element [36]
ds2 =dt2 a2(t)[dx2+dy2+dz2] (4)
−
3
wherea(t)isthescalefactor. Inthisspace-time,fromaction(1)wegetthefollowingRS2modelbasedmodified
Friedmann field equations [6]
ρ
3H2 =ρ 1+ T (5)
T
(cid:16) 2λ(cid:17)
Here λ is the brane tension and ρ = ρ +ρ , ρ and ρ are being the energy density of CDM and the
T φ m φ m
scalar field respectively. It should be noted that we have neglected the cosmological constant in the 4D brane
by choosing RS fine tuning condition and for simplicity of calculation we have chosen AdS bulk where dark
radiation term vanishes.
When gravity is coupled to the scalar field with scalar coupling function, the energy density and thermody-
namic pressure are given by
1 1
ρ = µ(φ)φ˙2+V(φ) and p = µ(φ)φ˙2 V(φ) (6)
φ φ
2 2 −
where µ(φ) is the coupling parameter chosen for arbitrary function of φ and V(φ) is the potential of the scalar
field.
Here we assumethat the scalarfield does notinteractwithCDM andthe energyconservationrelationstakes
the form.
ρ˙ +3H(ρ +p )=0 (7)
φ φ φ
ρ˙ +3Hρ =0 (8)
m m
Using Eqs.(5) , (6) , (7) and (8), we get
ρ
2H˙ = 1+ T (µ(φ)φ˙2+ρ ) (9)
m
−(cid:16) λ (cid:17)
This implies
2H˙
µ(φ)φ˙2 = +ρ (10)
−(cid:16)1+ ρT m(cid:17)
λ
AsclearfromEq.(10),whenH˙ ispositive,wehavephantomphasewhereµ(φ)isnegativeandusingconservation
Eqs.(7) and (8), evolution equation of the scalar field is given by
1dµ(φ) dV
µ(φ)φ¨+ φ˙2+3Hµ(φ)φ˙ + =0 (11)
2 dφ dφ
Following [27], we introduce the following dimensionless phase space variables in order to build ASODE of
the above cosmologicalequation.
µ(φ) φ˙ V(φ) ρ √6
x , y , z T and s (12)
≡ p√6 H ≡ p√3 H ≡ 3H2 ≡ φ
After these choice of phase space variables, we can write following ASODE for above evolution equations
3 2 z
x′ =−3x−β1(s) y2+ 2x(cid:16) −z (cid:17)(z+x2−y2)
3 2 z
y′ =β1(s) xy+ 2y(cid:16) −z (cid:17)(z+x2−y2)
4
z =3(1 z)+(z+x2 y2)
′
− −
xs2
s = (13)
′
− µ (s)
1
p
where β (s)= β(φ) = 3 dlnV , µ (s)= µ(φ) and the primes denotes differentiation with respect to new
1 2µ(φ) dφ 1
q
time variable U =lna. After the above choice of variables one can realise that
ρ 2(1 z)
T = − , 0<z 1 (14)
λ z ⇒ ≤
Eq.(14) tells us that at z = 1, brane effects vanish and GR is recovered. It may be noted that at the point
z = 0 the ratio ρT is undefined but as we are interested in asymptotic behaviour, we can study the limiting
λ
case z 0 separately.
→
The effective equation of state for the scalar field (i.e. p =(ν 1)ρ ) is given by
φ φ φ
−
2x2
ν = (15)
φ x2+y2
Fromthe abovedimensionlessphasespacevariablestheequationofstate(EoS)parameterhastheexpression
p x2 y2
ω = φ = − (16)
φ ρ x2+y2
φ
The density parameter for the scalar field has the same expression as that of GR [34]
Ω =x2+y2 (17)
φ
Also we have
H˙
Ω =z x2 y2 and q = 1 (18)
m − − − − H2
where H˙ = 3 2 z (z+x2 y2) so that the deceleration parameter q has the expression
H2 −2(cid:16) −z (cid:17) −
3 2 z
q = 1+ − (z+x2 y2) (19)
− 2(cid:16) z (cid:17) −
Fornumericalcalculationswehaveestimatedinitialconditionsinsuchawaythatitisconsistentwithpresent
observations ( Ω =0.27 and the deceleration parameter q = 0.53 [35] ), so
m
−
Ω =x2+y2 =0.73 (20)
φ 0 0
Using these values in the constraint Eq.(18), we get z = 1, where x , y and z are the present values of x, y
0 0 0 0
and z respectively. Eq.(19) and (20) yields a lower bound set of present work x =0.489 and y =0.7.
0 0
HereweseethattheexpressionsofCDMenergydensity parameteranddecelerationparameterinvolvebrane
effects and it differs from that of GR [14].
The phase space for the autonomous dynamical system driven by the evolution Eq.(13) can be defined as:
Ψ= (x,y,z):0 x2+y2 z, 1 x 1,0 y,0<z 1 s R (21)
{ ≤ ≤ − ≤ ≤ ≤ ≤ }×{ ∈ }
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III. CRITICAL POINTS OF 4D AUTONOMOUS SYSTEM
In this section we will analyse in detail the critical points of the autonomous system corresponding to RS2
model and their stability properties. Let y = g(y) be a given non-linear autonomous system of differential
′
equations with a critical point y Rn, where y = dy, y (y ,y ,.......,y ) Rn and g : Rn Rn. The
0 ∈ ′ dt ≡ 1 2 n ∈ →
system can be linearised around the critical point as u = Au = Df(y )u, where u = y y , A = Df(y ) and
′ 0 0 0
−
Df(y)= ∂gi, i,j =1,2,3,.......n and A is the Jacobian matrix of the system. A critical point of the system is
∂yj
said to be hyperbolic if none of the eigenvalues of the Jacobian matrix A have zero real part. If the Jacobian
matrix have at least one zero eigenvalue, then the critical point is non-hyperbolic criticalpoint. We summarise
the stability criteria of the hyperbolic critical point which can be determined from the sign of eigenvalues of A
as follows:
If all the eigenvalues have negative real part then the critical point is stable (i.e., late time attractor).
•
If all the eigenvalues have positive real part then the critical point is unstable (i.e., past time attractor).
•
If some eigenvalues of the Jacobian matrix have positive real part and other have negative real part then
•
the critical point is saddle.
Theabovelinearstabilityanalysiscannotbeusedfornon-hyperbolicpoints,sosomeothertechniquessuchas
CentreManifoldTheory,Lyapunov’sfunctions[17,37]andnumericallyperturbationofsolutionsaroundcritical
points [38] may be employed to study the stability of such points. The method of perturbations of solutions
around critical points is quite standard and has been extensively used in recent cosmological studies [21–23]
and so this technique will be used in our present work.
InordertosolvetheaboveASODE(13)weneedtoconsiderspecificpotentialV(φ)andcouplingfunctionµ(φ).
Regarding V(φ), the general popular choice in literature is to assume an exponential potential. This potential
has the advantage that the phase space remains four dimensional and field equations can be studied easily
qualitatively using dynamical systems tools. In order to consider V(φ) beyond the constant and exponential
potential, one new variable v 1 dV has to be introduced [39–41]. Further, exponential potential in case of
≡ V dφ
phantom field in standard cosmology admits late time attractor but in the RS2 model yields saddle point [14].
In this paper we shall show that it is possible to get late time attractor under this potential in RS2 model
when gravity is coupled to scalar field. As exponential potentials are known to be significant in dynamical
investigations [43–45], we assume V(φ) = V e αφ. Concerning coupling function µ(φ), the popular choice in
0 −
literature are exponential and power-law forms [34, 42]. In what follows as in [34], in order to remain general,
we choose two coupling functions (exponential and power-law forms) corresponding to same the exponential
potential and the corresponding analysis will be done separately in following two subsections.
A. Scenario 1 : Exponential potential and exponential coupling function
Here we choose
V(φ)=V e αφ and µ(φ)=µ eνφ (22)
0 − 0
In this scenario, the autonomous system (13) simplifies to
3 2 z
x′ =−3x+α0 y2e−sb + 2x(cid:16) −z (cid:17)(z+x2−y2)
3 2 z
y′ =−α0 xye−sb + 2y(cid:16) −z (cid:17)(z+x2−y2)
z′ =3(1 z)(z+x2 y2)
− −
s = xs2 e−sb (23)
′
− √µ0
6
where α =α 3 and b= 3 ν
0 q2µ0 q2
The critical points of the present autonomous system are:
C1(0,0,0,0) , C2(0,0,1,0) , C3±(±1,0,1,0), C4(0,1,1,s) , C5(0,√z,z,0) , C6(0,0,0,s) , C7(0,0,1,s)
Table 1. Scenario 1 : Exponential potential and exponential coupling function. The critical points for ASODE
(23) and values of the relevant parameters
Critical point x y z s Existence Ω Ω ω
φ m φ
C 0 0 0 0 Always 0 0 Undefined
1
C 0 0 1 0 ,, 0 1 Undefined
2
C3± ±1 0 1 0 ,, 1 0 1
C 0 1 1 0 ,, 1 0 -1
4
C 0 √z z ]0,1] 0 ,, z 0 -1
5
∈
C 0 0 0 s ,, 0 0 Undefined
6
C 0 0 1 s ,, 0 1 Undefined
7
Table 2. Scenario 1 : Exponential potential and exponential coupling function. Eigenvalues of the linearised
matrix for the critical points ASODE (23) and corresponding values of dimension of Stable manifold and
deceleration parameters.
Critical point λ λ λ λ Stable manifold q
1 2 3 4
C 0 3 3 0 No stable manifold 2
1
C 3 3 3 0 2D 1
2 −2 2 − 2
C3± 3 3 −6 0 1D 2
C 3 3 0 0 2D -1
4
− −
C 3 3 0 0 2D -1
5
− −
C 0 3 3 0 No stable manifold 2
6
C 3 3 3 0 2D 1
7 −2 2 − 2
1. Critical points and their properties for Scenario 1:
The critical points of the above ASODE (23) and the eigenvalues of the corresponding Jacobian matrix are
listed in Table 1 and Table 2 respectively. Here we see that all the critical points are non hyperbolic in nature.
7
Solinearstabilitytheorycannotbeusedtostudythestabilityofthesystem. Butusingcentremanifoldtheorem
we can find dimension of stable manifold in this case [37]. It may be noted that the five dimensionaleffects are
presented when z =1 so C , C and C only reflects the brane effects.
1 5 6
6
We note that the point C =(0,0,0,0)is a critical point of the ASODE (23). But at this point the first two
1
equations of ASODE (23) are undefined. To study the nature of the critical point C , we can apply a secure
1
approach as reported in [14] and expand ASODE (23) in the neighbourhood of C = (0,0,0,0) by putting
1
x 0+ǫ , y 0+ǫ , z 0+ǫ and s 0+ǫ in the above ASODE (23). If we neglect higher
x y z s
→ → → →
order terms, then the ASODE (23) reduces to following system ǫ = 0 , ǫ = 3ǫ , ǫ = 3ǫ and ǫ = 0.
′x ′y y ′z z ′s
So, the point C is non hyperbolic critical point. We can only say in this case is that the point (0,0,0,0) is
1
unstable as perturbations will grow exponentially. So this critical point will behave as past time attractor and
ischaracterisedbyvanishingmattercontentandwehaveempty(Misner-RS)universe. Inbraneworld,existence
of empty universe is a distinctive feature. Furthermore, we have a decelerating solution with (q = 2) (limiting
case) at this point. The behaviour of C is same as C and C is not interesting from physical point of view.
6 1 3
In what follows we summarise the remaining critical points of scenario 1 and their properties.
As seen from Tables 1 and 2, the saddle critical points C and C indicate that universe is completely
2 7
•
dominated by dark matter (Ω = 1) and expansion of the universe around these points is decelerated
m
(q = 1) having 2D stable manifolds. These points are saddle in nature.
2
Atthe criticalpointC ,weseethatthe universeiscompletelydominatedbythe scalarfield, atthis point
4
•
expansionoftheuniverseisaccelerated(q = 1)havingtwonegativeeigenvaluesandtwozeroeigenvalues.
−
Sincethispointisnon-hyperbolic,wenumericallyperturbedthesolutionsnearthiscriticalpointandcheck
thestability. Tothisendweplottheprojectionsofperturbationsonx,y,zandsaxesseparatelyinFIG.(1)
toFIG.(4). FromFIG.(1)andFIG.(4)weseethattheprojectionofperturbationssolutionsasymptotically
approach x=0 and s=0 respectively as U approaches to infinity. Similarly, from FIG.(2) and FIG.(3),
y and z approachto 1 as U approachesto infinity, so that the whole system asymptotically approachthe
critical point C = (0,1,1,0) as U approaches to infinity. From this perturbation plots we can conclude
4
that the point C =(0,1,1,0) is a stable critical point of the ASODE (23).
4
For the critical point C , we have (x,y,z,s)= (0,√z,z,0). This implies z = y2 (x = 0 , s = 0), so that
5
•
this point is a curves of critical points. If we compare C and C , we see that C is a particular case of
4 5 4
C . At this point ρ =0 and ρ =V, so that the above Eq.(5) reduces to
5 m T
3H2 =V 1+ V . Itrepresentsthe slow-rollFriedmannequationwheretheHubble parameterisrelated
(cid:16) 2λ(cid:17)
to the potential of the inflation field having 2D stable manifolds. When potential is much larger than
the brane tension V >> λ H = V , so that at early time expansion rate H get enhanced with
⇒ RS √6λ RS
respect to the H ( GR rate of expansion) in the RS2 model and we have HRS = V . Thus in this
GR HGR q2λ
way brane effects fuel early inflation in RS2 model. FIG.(5) shows that the nature of curves of critical
points of C
5
B. Scenario 2 : Exponential potential and power law coupling parameter
Here we choose
V(φ)=V0e−αφ and µ(φ)=µ0φn (24)
where V , µ and n are constants.
0 0
Then the autonomous system (13) becomes
3 2 z
x′ =−3x+β2 y2sn2 + 2x(cid:16) −z (cid:17)(z+x2−y2)
3 2 z
y′ =−β2 xysn2 + 2y(cid:16) −z (cid:17)(z+x2−y2)
z′ =3(1 z)(z+x2 y2)
− −
8
FIG. 1: The plot of X against U for Scenario 1 nearthe critical point C4
.
FIG. 2: The plot of Y against U for Scenario 1 near thecritical point C4
.
s′ = µ2 xs2+n2 (25)
−
where β2 = α06−n4 and µ2 = 6√−µn40. The critical points of the present autonomous system are
D1(0,0,0,0) , D2(0,0,1,0) , D3±(±1,0,1,0) , D4(0,1,1,s) , D5(0,√z,z,0) , D6(0,0,0,s) , D7(0,0,1,s).
Table 3. Scenario 2 : Exponential potential and power law coupling function. The critical points for ASODE
9
FIG. 3: The plot of Z against U for Scenario 1 near thecritical point C4
.
FIG. 4: The plot of S against U for Scenario 1 nearthe critical point C4
.
(25) and values of the relevant parameters
10
FIG. 5: Phase portrait of Scenario 1 near thecritical point C5
.
Critical point x y z s Existence Ω Ω ω
φ m φ
D 0 0 0 0 Always 0 0 Undefined
1
D 0 0 1 0 ,, 0 1 Undefined
2
D3± ±1 0 1 0 ,, 1 0 1
D 0 1 1 0 ,, 1 0 -1
4
D 0 √z z ]0,1] 0 ,, z 0 -1
5
∈
D 0 0 0 s ,, 0 0 Undefined
6
D 0 0 1 s ,, 0 1 Undefined
7
Table 4. Scenario 2 : Exponential potential and power law coupling function. Eigenvalues of the linearised
matrix for the critical points ASODE (25) and corresponding values of dimension of Stable manifold and
deceleration parameters.