Bouncing Model in Brane World Theory

0.32 MB·English

by  Rodrigo Maier

Bouncing Model in Brane World Theory
Rodrigo Maier1,2, Nelson Pinto-Neto2 and Ivano Damia˜o Soares3
1Institute of Cosmology and Gravitation, University of Portsmouth,
Dennis Sciama Building, Portsmouth, PO1 3FX, United Kingdom
2ICRA - Centro Brasileiro de Pesquisas F´ısicas – CBPF,
Rua Dr. Xavier Sigaud, 150, Urca, CEP22290-180, Rio de Janeiro, Brazil and
3Centro Brasileiro de Pesquisas F´ısicas – CBPF,
Rua Dr. Xavier Sigaud, 150, Urca, CEP22290-180, Rio de Janeiro, Brazil
(Dated: January 23, 2013)
We examine the nonlinear dynamics of a closed Friedmann-Robertson-Walker universe in the
framework of Brane World formalism with a timelike extra dimension. In this scenario, the Fried-
mann equations contain additional terms arising from the bulk-brane interaction which provide a
concrete model for nonsingular bounces in theearly phase of theUniverse. We construct a nonsin-
3
gularcosmologicalscenariosourcedwithdust,radiationandacosmologicalconstant. Thestructure
1
of the phase space shows a nonsingular orbit with two accelerated phases, separated by a smooth
0
transition corresponding to a decelerated expansion. Given observational parameters we connect
2
such phases to a primordial accelerated phase, a soft transition to Friedmann (where the classical

n regime is valid),and a graceful exit to a de Sitteraccelerated phase.
a PACS numbers: 98.80.Cq, 04.60.Ds
J

2
2 I. INTRODUCTION our universe has evolved should depend crucially on the
adopted version of the theory to describe the dynamics

] around the singularity.
c AlthoughGeneralRelativityisthemostsuccessfulthe-
q One of the most important characteristics of our uni-
orythatpresentlydescribesgravitation,itpresentssome
- verse, supported by observational data, is its large scale
r intrinsiccrucialproblemswhenwetrytoconstructacos-
g of homogeneity and isotropy. In fact, the scale of ho-
mological model in accordance with observational data.
[ mogeneity and isotropy is empirically well accepted for
In cosmology, the ΛCDM model gives us important pre-
distances above 100 Mpc. Indeed this is the main rea-
1 dictions about the evolution of the universe and about
v its current state[1]. However, let us assume that the ini- son that makes the geometry of Friedmann-Lematre-
0 Robertson-Walker (FLRW) a powerful theoretical tool
tial conditions of our universe were fixed when the early
5 for the construction of a cosmological scenario[1]. How-
universe emerged from the semi-Planckian regime and
2 ever, when we consider a homogeneous and isotropic
started its classical expansion. Evolving back such ini-
5
model filled with baryonic matter, we find several dif-
. tial conditions using the Einstein field equations, we see
1 ficulties when we take into account the primordial state
that our universe is driven towards an initial singularity
0 ofouruniverse. Amongsuchdifficulties, we canmention
where the classical regime is no longer valid[2].
3
the horizon and flatness problems[1].
1 Notwithstanding the cosmic censorship conjecture[3],
As a possible solution to these problems emerged the
: there is no doubt that General Relativity must be prop-
v socalledInflationaryParadigm[1,10]. Althoughthisfun-
i erly correctedor evenreplacedby a completely new the-
X damental paradigm allows to solve the horizon and flat-
ory, let us say a quantum theory of gravity. This de-
ness problems, inflationary cosmologydoes not solve the
r mand is in order to solve the issue of the presence of
a problemofthe initialsingularity. Therefore,nonsingular
the initial singulariy predicted by classical General Rel-
models from a new theory which provide alternative so-
ativity, either in the formation of a black hole or in the
lutions to these problems should be strongly considered.
beginning of the universe. While a full quantum gravity
theory remains presently an elusive theoretical problem, In this paper we adhere to the brane world scenario,
quantumgravitycorrectionsnearsingularitiesformedby where a timelike noncompact extra dimension is intro-
gravitational collapse have been the object of much re- duced, constituting the bulk space, and all the mat-
cent research, from quantum cosmology[4, 5] to D-brane ter content of our universe would be trapped on a 4-
theory[6–9]. In the latter scenario, extra dimensions are dimensional spacetime embedded in the bulk. At low
introduced constituting the bulk space. In the case of energies General Relativity is recovered [8], but at high
spatiallyhomogeneousandisotropiccosmologies,theba- energy scales significant changes are introduced into the
sic resulting distinction between the two approaches lies gravitational dynamics and the singularities can be re-
in the corrections introduced in the Friedmann Hamil- moved [6].
tonian constraint, leading either to modifications in the While spacelike extra dimensions theories have re-
kinetic energy terms or to extra potential energy terms. ceived more attention in the last decades[9], stud-
In both cases we may have bounces in the scale factor ies involving extra timelike dimensions have been
corresponding to the avoidance of a singularity in the considered[11] despite from the fact that propagating
models. Inthiscontext,theinitialconditionsfromwhich tachyonic modes or negative norm states may arise
2
due to timelike extra dimensions. These modes have form
been regarded as problematic once they might violate
1 (5)
cpaaurtsiacllietsy.[1I2s]subeys lcikoenstihdeereinxgceeindtinergalyctisomnasllalmowoenrgbuosuunadl S = 2κ25nZM1p−ǫ (5)gh R−2Λ5+2κ25L5id5x
on the size of timelike extra dimensions[13], the imagi- + ǫ (5)g (5)R 2Λ +2κ2L d5x
naryself-energyofchargedfermionsinducedbytachyonic ZM2p− h − 5 5 5i
modes – whichseemsto causedisappearanceoffermions
icnletsoinnodtuhcinedgb–yantadcthhyeonspsownittahnneoeguastdiveecaeyneorfgsytaabrleempaarjtoir- +2ǫZΣp−(4)gK2d4x−2ǫZΣp−(4)gK1d4xo
1 1 (4)
difficulties[12]. Nevertheless,in orderto addressthe cos- + (4)g R 2σ d4x
mological constant problem in Kaluza-Klein theories[14] 2ZΣp− (cid:16)2κ24 − (cid:17)
or reconcile a solutionof the hierarchyproblem with the
+ (4)gL (g ,ρ)d4x. (1)
cosmological expansion of the universe[15], timelike ex- ZΣp− 4 αβ
tradimensionshavebeenconsidered. Ontheotherhand,
it has been shown in [16] that the appearance of mass- In the above (5)R is the Ricci scalar of the Lorentzian
less ghosts in an effective four-dimensional theory can 5-dim metric gAB on , and (4)R is the scalar curva-
M
be avoidedby considering topologicalcriteria in Kaluza- ture of the induced metric gab on Σ. The parameter σ is
Klein theories with extra compactified time-like dimen- denoted the brane tension. The unit vector nA normal
sions. Moreover, avoidance of propagating tachyonic or to the boundary Σ has norm ǫ. If ǫ = 1 the signature
−
negativenormstatescanalsobe achievedbyconsidering of the bulk space is ( , ,+,+,+), so that the extra
a noncompact timelike extra dimension[7], which is the dimension is timelike.−T−he quantity K = Kab gab is
case in the model of this paper. the trace of the symmetric tensor of extrinsic curvature
K =Y C Y D n , where YA(xa) are the embed-
ab ,a ,b C D
∇
We organize the paper as follows. In section II we ding functions of Σ in [17]. While L4(gab,ρ) is the
M
presentabriefreviewofthemodifiedEinsteinfieldequa- Lagrangean density of the perfect fluid[18](with equa-
tionsintheBraneWorldscenario. InsectionIII,wecon- tion of state p = αρ), whose dynamics is restricted to
struct a nonsingular cosmological scenario sourced with the brane Σ, L5 denotes the lagrangian of matter in the
dust, radiation and a cosmological constant. In section bulk. All integrations over the bulk and the brane are
IV,we showthatgiventhe observationalparameters,we taken with the natural volume elements ǫ (5)g d5x
−
can connect such phases to a primordial accelerated ex- and (4)g d4x respectively. κ and κpare Einstein
5 4
pansion, a soft transition to Friedmann (where the clas- constpan−tsinfiveandfour-dimensions. We useunits such
sical regime is valid), and a graceful exit to a de Sitter that c=1.
acceleratedphase. Asourfinalremarks,wediscusssome Variations that leave the induced metric on Σ intact
ofitspossibleimprintsinthephysicsofcosmologicalper- result in the equations
turbations.
(5)G +Λ (5)g =κ2(5)T , (2)
AB 5 AB 5 AB
while considering arbitrary variations of g and taking
AB
into account (2) we obtain
κ2
(4)G +ǫ 4 S(1)+S(2) =κ2 τ σg , (3)
ab κ2(cid:16) ab ab (cid:17) 4(cid:16) ab− ab(cid:17)
5
where S K Kg , and τ is the energy mo-
ab ab ab ab
II. FIELD EQUATIONS mentum ten≡sor ass−ociated to L . In the limit κ
4 4
→
equation (3) reduces to the Israel-Darmois junction
∞
condition[19]
For the sake of completeness let us give a brief intro-
duction to braneworld theory, making explicit the spe- (cid:16)Sa(1b)+Sa(2b)(cid:17)=ǫ κ25(cid:16)τab−σgab(cid:17) (4)
cific assumptions used in obtaining the dynamics of the
WeimposetheZ -symmetry[9]andusethejunctioncon-
model. We relyonreferences[6,9], andournotationba- 2
ditions (4) to determine the extrinsic curvature on the
sically follows [2]. Let us start with a 4-dim Lorentzian
brane,
brane Σ with metric g , embedded in a 5-dim confor-
ab
mallyflatbulkMwithmetricgAB. CapitalLatinindices K = ǫκ2 (τ 1τg )+ σg . (5)
range from 0 to 4, small Latin indices range from 0 to 3. ab −2 5h ab− 3 ab 3 abi
We regard Σ as a common boundary of two pieces
M1 Now using Gauss equation
and of and the metric g induced on the brane
2 ab
M M
by the metric of the two pieces should coincide although (4)R = (5)R YMYNYRYS
abcd MNRS ,a ,b ,c ,d
the extrinsic curvatures of Σ in and are allowed
1 2
tobe different. The actionforthMetheoryMhasthe general +ǫ KacKbd KadKbc (6)
(cid:16) − (cid:17)
3
together with equations (2) and (5) we arrive at the in-
duced field equations on the brane
(4)G = Λ (4)g +8πG τ +ǫκ4Π
ab − 4 ab N ab 5 ab
ǫE +ǫF (7)
ab ab
−
where we define
1 Λ 1
Λ := κ2 5 + ǫκ2σ2 , (8)
4 2 5(cid:16)κ2 6 5 (cid:17)
5
κ4σ
G :=ǫ 5 , (9)
N
48π
1 1 1
Π := τcτ + ττ + (4)g τcdτ
ab −4 a bc 12 ab 8 ab cd
1
τ2(4)g , (10)
−24 ab
2
F := κ2 ǫ (5)T YBYD
ab 3 5n BD ,a ,b
1
+ (5)T nBnD ǫ (5)T (4)g ,(11) FIG. 1: Illustration of the behavior of V(a) for parameters
h BD − 4 i abo k = 0.8, Λ4 = 1.5 Erad = 0.1, and for Edust = 0.001 (con-
tinuous line) E =0.09 (dashed line). The increase of the
E is the electric part of the Weyl tensor in the bulk dust
ab dustcontentforafixedkexcludesthepresenceofperpetually
induced in the brane, T is the energy-momentum in
AB bouncingsolutions.
the bulk, and G defines the Newton’s constant on the
N
brane. For a timelike extra dimension we have that
ǫ = 1 in our conventions implying that σ < 0 in ac-
−
cordance with observations. We also remark that the ef-
The matter content of the model, restricted to the
fective 4-dimcosmologicalconstantmightbe setzero,or brane, is given by noninteracting perfect fluids, namely,
made conveniently small, in the present case of an extra dust and radiation, with respective equations of state
timelikedimensionbyproperlyfixingthe bulkcosmolog- p = 0, p = ρ /3, and energy momentum tensor
dust rad rad
ical constant as Λ 1κ4 σ2. It is important to note τab:=τab +τab satisfying τab =0= τab .
5 ≃ 6 5 dust rad ∇b dust ∇b rad
that for a 4-dim brane embedded in a conformally flat In this situation we have that
empty bulk we have the absence of the Weyl conformal
tensor projection Eab and of Fab in Eq. (8). Π00 = 1 (ρdust+ρrad)2,
12
Accordingly, Codazzi’s equations imply that
1 2
1 Πij = h12(ρdust+ρrad)
∇aK−∇bKab =−2ǫκ25∇bτab, (12) + 61(ρdust+ρrad)(pdust+prad) igij, (15)
resulting in
and Codazzi’s equations (12) imply that Πa = 0,
∇aEab =∇bτab+κ45∇aΠab+∇aFab, (13) consistent with the contracted Bianchi’s iden∇tiatiesb in (7)
and Codazzi’s equation (13). The modified Friedmann
where is the covariant derivative with respect to the
∇a equations have the first integral
induced metric g . Equations (7) and (13) are the dy-
ab
namicalequationsofthe gravitationalfieldonthebrane.
k Λ 8πG
H2 + 4 = N(ρ +ρ )
a2 − 3 3 dust rad
4πG
III. THE MODEL N(ρ +ρ )2 , (16)
dust rad
− 3σ
| |
Let us consider a FLRW geometry on the four-
where H := a˙/a with a˙ da/dt. It is worth noting
dimensional brane embedded in a five-dimensional de- ≡
that the bounce is solely engendereddue to the presence
Sitter bulk with a timelike extra dimension (ǫ = 1)[6].
− of a timelike extra dimension which induces the minus
Considering comoving coordinateson the brane, the line
sign in the last term of (16). By assuming a spacelike
element is given by
extradimension,wewouldobtainaplussigninsteadthat
1 provides a singular model.
ds2 = dt2+a2(t) dr2+r2dΩ2 , (14)
− h1 kr2 i Expressing
−
where a(t) is the scale factor, k is the spatial curvature E E
dust rad
and dΩ2 is the solid angle. ρdust = a3 , ρrad = a4 , (17)
4
4 largeErad thepotentialV(a)presentsnolocalmaximum
Escape to deSitter or minimum[20].
The critical points in the phase space are stationary
S
solutions of (20), namely, the points of the phase space
2
II S (a=a ,p =0)correspondingtothezerosoftheright-
crit a
hand-side of (20). Here, a stands for the real positive
crit
I III rootsofdV/da. Byconsideringthe caseofclosedgeome-
P1 P2
Pa 0 tries (k > 0), it is not difficult to verify that, depend-
ingonthe valuesoftheparameters(Λ , σ ,E ,E ),
4 rad dust
| |
there are at most two critical points associated with one
minimum and one local maximum of V(a). In this case,
II S
–2
the minimum of the potential corresponds to a center
S while the maximum corresponds to a saddle. This con-
figurationallowsustoobtaindifferenttypesoforbitsthat
–4 describestheevolutionofuniversesinthismodel. InFig.
0.5 1 1.5 2 we illustrate the phase space portrait of the model for
a Λ = 1.5, σ = 6000, E = 0.15 and E = 0.05, and
4 rad dust
for varying k. The value of E is sufficiently bounded
dust
so that V(a) has a well. The critical points P (center)
FIG.2: Phaseportraitofthedynamicswiththecriticalpoint 1
P1 (center)and P2 (saddle). Orbitsin region IIare solutions and P2 (saddle) correspond to stable and unstable Ein-
ofone-bounceuniverseswithagracefulexittoanaccelerated stein universes. Typically the model allows for the pres-
(inflationary) phasealong theseparatrix S. ence of perpetually bouncing universes (periodic orbits
in region I), and one-bounce universes (region II). Re-
gionI is bounded by the separatrix emergingfrom the
S
saddleP . AseparatrixalsoemergefromP towardsthe
2 2
where Edust and Erad are constants of motion, the first deSitter attractor at infinity, defining a graceful exit of
integralof motion(16) can be expressedas the Hamilto- orbits in regionII to an (inflationary)acceleratedphase.
nian constraint From now on we will restrict ourselves to the case of
closedgeometries. Inthenextsectionwewillexamwhat
p2
= a +V(a)=0, (18) kind oforbit wouldbe generatedwhen one considers the
H 2
observational values of (Λ ,E ,E ).
4 rad dust
where
k Λ4a2 8πGN Edust Erad IV. OBSERVATIONAL COSMOLOGY
V(a)= +
2 − 6 − 6 (cid:16) a a2 (cid:17)
8πGN Edust Erad 2 As observationalcosmologyasserts,the domain of ho-
+ + . (19)
12σ (cid:16) a2 a3 (cid:17) mogeneity and isotropy of our present Universe is well
| |
accepted for scales around the present horizon, which is
From (18) we derive the dynamical system given by a 1028 cm (here, the subscript 0 denotes
0
∼
the present epoch). In this case, we obtain the following
dV
a˙ =p , p˙ = . (20) observational parameters
a a
−da
Λ 1.34 10−56cm−2 , (21)
Itisworthnotingthatthelastterminthepotential(19) 4 ≃ ×
acts asaninfinite potentialbarrierandis responsiblefor Edust 2.6 1054 g , (22)
≃ ×
the avoidance of the singularity a = 0. These potential E 4 1078 g cm, (23)
rad
≃ ×
corrections are equivalent to fluids with negative energy
densities. This is in accordance with the fact that quan- where the Hubble radius is fixed to H 0.77
0
tum effects can violate the classical energy conditions, 10−28 cm−1. ∼ ×
andmayavoidcurvaturesingularitieswhereclassicalgen- From Ref. [22], the brane tension has a lower bound
eral relativity breaks down[21]. Such violations tend to which corresponds to σ 1022 g cm−3. That is,
min
| | ∼
occuronshortscalesand/orathighcurvatures,whichis a star with the Chandrasekhar mass will not form an
the case of the present models. eventhorizonif the branetension is smallerthan σ .
min
| |
ThebehaviorofthepotentialV(a)isillustratedinFig. Itturns outthatthis value furnishesus with acurvature
1fork =0.8,Λ =1.5,E =0.1,andforE =0.001 scale l 1/√R = ( a/a¨) 1034 l at the bounce
4 rad dust c b b P
≡ ∼
(continuous line) E =0.09(dashed line). We can see (where l is the Planckplenghtand R is the Ricci scalar
dust P b
thattheincreaseofthedustcontentforafixedkexcludes at the bounce). In order to guarantee that l at the
c
thepresenceofperpetuallybouncingsolutionsbydriving bounceisnotsmallerthan103l ,thebranetensionmust
P
themaximumoutofthephysicalspace. Forasufficiently be less than1085 g cm−3. Therefore,we havethe follow-
5
1´1022 -0.2
VHaL VHaL
-0.4
0
-0.6
0 a21´~110.51e4+14cm 4´1014 6´1014 8´1014 1´1015 -0.8
a
-1.0
-2´1022
-1.2
-1.4
-3´1022 a2~5.6e+27cm
5.0´1027 1.0´1028 1.5´1028 2.0´1028 2.5´1028
(a) (a)
4´1011 1.5
Pa Pa
a1~1.5e+14cm
1.0 a2~5.6e+27cm
2´1011
0.5
1.0´1014 1.5´1014 2.0´1014 2.5´1014 3.0´1014 3.5´1014 4.0´a1014 5.0´1027 1.0´1028 1.5´1028 2.0´a1028
-0.5
-2´1011
-1.0
-4´1011 -1.5
(b) (b)
4.´10-28
0.002 HHzL
HHzL
0.001 2.´10-28
4´1013 5´1013 6´1013 7´1013 8´1013 z 9´1013 -1 1 2 3
z
-0.001 -2.´10-28
-0.002
-4.´10-28
(c) (c)
FIG. 3: (a) The potential V(a) and the phase space (b) in FIG. 4: (a)The potential V(a) and the phase space (b) in
theregionthatencompassesthecriticalpointa1,considering the region that encompasses the critical point a2 and com-
the observational values (21) and (24). For |σ| ∼ |σ|min, we pletes Figs. 3. This critical point is of the order of 1028 cm,
obtain a1 ∼ 1014 cm. Although this primordial accelerated coinciding with the domain of homogeneity and isotropy of
phase does not correspond to usual inflation (0.27 being the ourpresent Universe,regardless ofthevalueof |σ|. In (c) we
numberofe-folds),itisimportanttoremarkthatasouruni- showthebehavioroftheHubblefactorasfunctionofredshift
versehasnobeginningoftimeandthecosmological constant in a neighborhood of the saddle a2 given the normalization
is small, the particle horizon before the bounce was already a0 =1 and the parameters (30). The domain 0≥z >−1 of
biggerthanthescalesofcosmologicalinterest. In(c)weshow theH >0branchcorrespondstothefinalacceleration phase
thebehavior of the Hubblefactor as function of redshift in a approachingdeSitterasz→−1,withH =const.atz =−1.
neighborhood of the bounce given the normalization a0 = 1
and the parameters (30).
6
ingphysicaldomain(notspoilingthenucleosynthesis)for starting from (a = a ,p < 0) is an initial acceleration
1 a1
the brane tension phase that leads the universe through the bounce and
ends in (a = a ,p > 0), when the universe enters in a
1022g/cm3 . σ . 1085 g/cm3, (24) long and smoot1h dae1celerated expansion phase. This pri-
| |
mordialbouncing acceleratedphase does not correspond
where we have set c = 1. Feeding the Hamiltonian con-
tousualinflation,thenumberofe-foldsbeing0.27. Note,
strain (18) with σ and the parameters (21), we ob-
| |min however, that there is no horizon problem in the model.
tain that the spatial curvature is k 0.002 for σ
≃ | | ≥ In fact, before the bounce, due to its cosmological con-
σ .
| |min stant dominated contraction from the infinity past until
Considering the lower bound limit for σ , numerical
| | a2, the particle horizon dp is given by
calculations show that the potential V(a) has always a
local minimum at a1 (corresponding to a center) and a d = a¯ a¯ 1 da 1028 cm, (31)
local maximum at a2 (corresponding to a saddle). – cf. p (cid:12) Z∞ aa˙ (cid:12)≃
Figs. 3(a) and 4(a). If we increase σ by four orders of (cid:12) (cid:12)
mofamgnaigtnuidtue,dwe.eOobntathineaovthaelurehoafnad1,dthe|cer|leoacsaeldmbayxoinmeuomrdaer oifra¯de≥r oaf2.ΛT−1h/e2r,ewfo(cid:12)hreicthhiespcaorntsi(cid:12)tcrleaihnoerdizboynpisreaslerneatdoybsoefrtvhae-
2
is of the order of 1028 cm (cf. Fig. 4(a)) for σ σ tionstobeoftheorderoftheHubbleradiustoday. Hence
min
| |≥| |
(regardless of the value of σ ). We exhibit the behavior there is no horizon problem for the scales of cosmologi-
of V(a) and the phase spa|ce|(a,p ) trajectory – for the calinterest. ThedeceleratedexpansionFriedmannphase
a
parameters(21)-(24)and σ σ min –inFigs. 3(a)and ends in the neighborhood of a2 with a graceful exit to a
4(a), and Figs. 3(b) and|4(|b∼),|re|spectively. We should late de Sitter accelerated phase.
note that Figs. 3 and Figs. 4 display the same potential From (8) and (21), we see that the parameter Λ5 of
V(a) and the same universe phase space trajectory in the model must be adjusted in a very precise way. In
distinct ranges of a, complementing each other. fact, Λ5 must be very close to 8πGN σ , which has the
Given the redshift relation minimum value 105cm−2 (see Eq. (24)|)|and increases as
σ increases, in order to yield the observed value of Λ
4
a(0) | |
=1+z, (25) given in Eq. (21). This is the usual problematic fine-
a(z) tuning of the cosmologicalconstant,ofatleast60 orders
of magnitude as we have seen above, which the present
equation(16) can be rewritten as
model, at least in this first approach, does not solve. It
turns outthat this is similarto anissue containedin the
H2 =H2 Ω (1+z)3+Ω (1+z)4
0 0dust 0rad Randall and Sundrum model [24]. In this scenario the
n
+Ω Ω (1+z)2 brane is embedded in an anti-de Sitter (4+1) spacetime
0Λ− 0k and the fine-tuning relation Λ = κ4σ2/6 has to be
3H2 5 − 5
0 (1+z)6[Ω +Ω (1+z)]2 (26) satisfied. Itwasshownin[25]thattheRandall-Sundrum
0dust 0rad
−16πGσ o model is unstable under small deviations from this fine-
| |
tuning. As a future investigation we will examine if the
where
same happens in our model.
ρ ρ
0dust 0rad
Ω , Ω , (27)
0dust ≡ ρcr. 0rad ≡ ρcr.
0 0
ρ Λ 1 V. CONCLUSIONS
Λ
Ω , (28)
0Λ ≡ ρcr. ≡ 8πGρcr.
0 0
In the framework of a Brane World formalism with a
k
Ω , (29) timelike extra dimension, we have obtained a homoge-
0k ≡ a2H2
0 0 neous and isotropic bouncing model compatible with all
observations at the background level. It starts with a
and ρcr. 3H2/8πG. By fixing the normalization a =
0 ≡ 0 0 de Sitter contraction from the infinity past, experiences
1, we obtain the following parameters according to the
a bounce at very small scales, turning to the usual stan-
WMAP 7 year results[23]:
dardexpandingdeceleratingphasesofradiationandmat-
Ω 0.26 , Ω 10−5 , Ω 0.73. (30) terdomination,andarecenttransitiontoanaccelerating
0dust 0rad 0Λ
≃ ≃ ≃ expansion. Thebounceitselfiscausedbytheappearance
Substituting these parameters in (26), we obtain Ω ofnewtermscomingfromtheextratimelikedimensionof
0k
≃
0.004. In figures 3(c) and 4(c) we show the behavior of thebulkinthe4-dimensionalFriedmannequation,which
the Hubble scale factor H with respect to the redshift z. becomeimportantathighcuravturescalesandavoidthe
It is remarkable that considering the interval of 62 or- cosmological singularity, inducing a gravitational repul-
ders of magnitude of σ (cf. (24)), the trajectory in the sion due to the timelike nature of the extra dimension.
phase space of the ab|ov|e observable universe belongs to We have two free parameters: the brane tension σ and
region II of the phase space (cf. Fig. 2) corresponding thefivedimensionalcosmologicalconstantΛ . Thebrane
5
to a one-single-bounce orbit. The part of the trajectory tensioncanassumeawidevarietyofvalues,seeEq.(24),
7
but Λ mustbe highly fine-tuned to the value of σ in or- tensorprojectedonthe branewhichwillmodify the per-
5
der to yield an effective 4-dimensional cosmologicalcon- turbed field equations. This is a technicaland conceptu-
stant compatible with observations. Hence the model allyinvolvedproblemwhichwillbeinvestigatedinfuture
solvesthesingularityproblemofthestandardcosmologi- publications.
calmodel,togetherwiththehorizonandflatnesspuzzles,
but it does not solve the cosmologicalconstant problem.
Ournextstepwillbetoperturbthemodelandinvesti-
Acknowledgements
gate the evolution of cosmological perturbations in such
cosmological background. Indeed, our work in progress
shows that, if one imposes an unperturbed de Sitter The authors acknowledgethe partialfinancialsupport
bulk, a numerical treatment of linear hydrodynamical from CNPq/MCTI-Brasil, through a post-doctoral re-
perturbation in the universe indicates that the bounce searchgrantno. 201907/2011-9(RM)andresearchgrant
hastheeffectofsubstantiallyenhancetheperturbations, no. 306527/2009-0(IDS). NPN also would like to thank
nonetheless these perturbations remain bounded with CNPq of Brazil for financial support. RM acknowledges
δρ/ρ 1 and δp/p 1. However, a general analysis the Institute of Cosmology and Gravitation, University
≪ ≪
of cosmological perturbations in this scenario demands of Portsmouth, for their hospitality. Figures were gen-
also a perturbed de Sitter bulk. In this case the 5-D erated using the Wolfram Mathematica 7 and MAPLE
scalar perturbations will induce fluctuations of the Weyl 13.
[1] V.Mukhanov,PhysicalFoundationsofCosmology(Cam- Phys. A9, 1457 (1994).
bridge UniversityPress, 2005). [12] G.R.Dvali,G.Gabadadze,G.Senjanovic,inManyFaces
[2] R. M. Wald, General Relativity (University of Chicago of the Superworld:Yuri Golfand Memorial Volume 525-
Press, Chicago, 1984). 532,Eds.Y.Golfand,M.Shifman,M.A.Shifman(World
[3] R.Penrose, Phys.Rev. Lett.14, 57 (1965). Scientific, 1999)
[4] J. Acacio de Barros, N. Pinto-Neto, and M. A. Sagioro- [13] F. J. Yndurain,Phys.Lett. B256, 15 (1991)
Leal, Phys. Lett. A 241, 229 (1998); R. Colistete Jr., [14] Ya. Aref’eva, B.G. Dragovi´c and I.V. Volovich, Phys.
J. C. Fabris, N. Pinto-Neto, Phys. Rev. D62, 083507 Lett. B177, 357 (1986).
(2000); F.G. Alvarenga, J.C. Fabris, N.A. Lemos and [15] M. Chaichian and A. B. Kobakhidze,Phys. Lett.B488,
G.A. Monerat, Gen.Rel.Grav. 34, 651 (2002); N. Pinto- 117 (2000).
Neto,F.T.Falciano,RobertoPereiraandE.SergioSan- [16] Ya. Aref’eva and I.V. Volovich, Phys. Lett. B164, 287
tiniPhys. Rev.D86, 063504 (2012). (1985).
[5] M. Bojowald (Loop Quantum Cosmology Collab- [17] L. P. Eisenhart, Riemannian Geometry (Princeton Uni-
oration), Living Rev. Relativity 8, 11 (2002), versity Press, New Jersey, 1997).
http://relativity.livingreviews.org/Articles/lrr-2005/11; [18] A. H.Taub, Phys. Rev. 94, 6 (1954).
M. Bojowald and R. Tavakol, arXiv:gr-qc/08024274. [19] W. Israel, Il Nuovo Cimento 44B, 1 (1966).
[6] Y. V. Shtanov, hep-th/0005193 (2000); Y. V. Shtanov. [20] R.Maier,I.Damia˜oSoaresandE.V.Tonini,Phys.Rev.
Phys.Lett.B541,177(2002);Yu.ShtanovandV.Sahni, D79, 023522 (2009).
Phys.Lett. B557, 1 (2003). [21] N. Pinto-Neto and P. Peter, Phys. Rev. D78, 063506
[7] A. Iglesias, Z. Kakushadze, Phys. Lett. B515, 477 (2008) ; L. H. ford, arXiv:gr-qc/0514096.
(2001). [22] R.Maier andI.Damia˜o Soares, InternationalJournal of
[8] T.Shiromizu,K.MaedaandM.Sasaki,Phys.Rev.D62, Modern Physics D Vol. 21, No. 5 (2012) 1250050.
024012 (2000). [23] E. Komatsu et al., The Astrophysical Journal Supple-
[9] R. Maartens, Phys. Rev. D62, 084023 (2000); Living ment Series 192 18 (2011).
Rev.Rel. 7:7,2004 (gr-qc/0312059). [24] L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 4690
[10] L. F. Abbott and So-Young Pi, Inflationary Cosmology (1999).
(World ScientificPublishing, 1986). [25] Timon Boehm, Ruth Durrer and Carsten van de Bruck,
[11] A.D.Sakharov,Zh.Eksp.Teor.Fiz.87,375(1984)[Sov. Phys. Rev.D 64, 063504 (2001).
Phys.JETP60,214(1984)]; J.Barrett,G.W.Gibbons,
M. J. Perry, C. N. Pope and P. J. Ruback, Int. J. Mod.