Asymmetric brane-worlds with induced gravity

0.16 MB·English

by  Laszlo Gergely

Asymmetric brane-worlds with induced gravity
La´szl´o A´. Gergely1,2 and Roy Maartens1
1Institute of Cosmology and Gravitation, University of Portsmouth, Portsmouth PO1 2EG, UK and
2Departments of Theoretical and Experimental Physics,
University of Szeged, Szeged 6720, D´om t´er 9, Hungary
(Dated: February 7, 2008)
TheRandall-Sundrumscenario,witha1+3-dimensionalbraneina5-dimensionalbulkspacetime,
can be generalized in various ways. We consider the case where the Z2-symmetry at the brane is
relaxed, and in addition the gravitational action is generalized to include an induced gravity term
on the brane. We derive the complete set of equations governing the gravitational dynamics for a
general brane and bulk, and identify how the asymmetry and the induced gravity act as effective
source terms in the projected field equations on the brane. For a Friedmann brane in an anti de
Sitter bulk, the solution of the Friedmann equation is given by the solution of a quartic equation.
Wefindthe perturbativesolutions for small asymmetry,which has an effect at late times.
5
0
0
I. INTRODUCTION contain two branches, labelled ǫ = 1, with very differ-
2 ±
ent properties. The ǫ = 1 branch has an RS limit, in-
n cludingthecasewhereind−ucedgravityisasub-dominant
Developments in particle physics suggest that our ob-
a
correction to the RS model [8, 9, 10].
J servable world may be a 1+3-dimensional “brane” sur-
faceembeddedinahigher-dimensional“bulk”spacetime,
1 In the present paper we lift the Z -symmetry of the
2
2 with matter fields confined on the brane while gravity embedding,andderivethecompletesystemofequations,
propagates in the bulk. An important family of such including the effective Einstein equations on the brane.
2 brane-worldmodelsisprovidedbytheRandall-Sundrum
Some earlier results for this scenario in the particular
v (RS) scenario, with a single warped and non-compact
case of Friedmann branes embedded in a vacuum bulk
7
extra dimension [1]. The effective Einstein equations on
9 were presented in [9, 10].
the braneinthe generalcasewerederivedviathecovari-
0
ant Shiromizu-Maeda-Sasakiapproach[2], together with InSection2,byuseofthe Lanczos-Sen-Darmois-Israel
1
1 equations governingthe bulk degrees of freedom. A gen- junctionconditions[11,12,13,14],wederivethegeneral-
4 eralization of these equations for the case when the Z izedeffectiveEinsteinequationsinaformconsistentwith
2
0 -symmetry of the embedding is lifted and the bulk con- previous works. We show that, remarkably, the asym-
/ tains matter, was recently presented in [3]. Due to the metric contributions add linearly to the induced gravity
c
q asymmetry new terms appear in the effective Einstein corrections.
- equations, including a varying cosmological“constant”.
r In Section 3 we apply the formalism to the case of
g A correction arising from the quantum interaction of
Friedmann branes. We derive the generalized Fried-
: bulk gravitons and brane matter can be introduced into
v mann and Raychaudhuri equations together with the
RS models by adding the so-called induced gravity con-
i constraints emerging from the Codazzi equation and
X tribution to the gravitationalaction:
other equations of the asymmetric case. Among the
r
a 1 asymmetric features, we find a variable 4-dimensional
S = d5x g R 2Λ cosmological “constant” Λ and contributions of the em-
2κ2 − −
Z p h i γ beddingtothedarkradiationU. AsintheZ2-symmetric
+ d4x√e eg λe+ R . (1) induced gravity models, the non-linearity of the Fried-
e brane − − 2κ2 mann equation in H2 leads to two branches, ǫ = 1.
Z h i ±
TheasymmetricRSmodelcanberecoveredintheǫ= 1
Tildes indicate the 5-dimensional counterparts of stan- −
case, as the γ 0 limit.
dard4-dimensionalquantities,λisthebranetensionand →
γ is the dimensionless induced-gravity parameter, with Following the method of [3], in Section 4 we present
γ =0 givingthe standardRSmodel. The brane gravita- an independent derivation of the Friedmann equation in
tional constant is given by [2] the simplest case when the bulk is 5-dimensional anti
de Sitter space-time (AdS), with different cosmological
κ4λ constants on the two sides of the brane. The compari-
κ2 = . (2)
6 son of the two forms of the Friedmann equation yields a
quarticequationinH2,asin[9,10]. Wefindaperturba-
e
The effective Einstein equations (with Z2-symmetry) tivesolutionofthequarticequationforsmallasymmetry.
were derived for the general case in [4, 5], generalizing Then, by use of the twice contractedBianchiidentity we
the earlier special cases, such as the Dvali-Gabadadze- identifybothunknownfunctionsΛandU. Finallywean-
Porrati models [6, 7], in which γ = 1 and Eq. (2) does alyze the high- and low-energy limits of the Friedmann
not hold, since λ = 0 = Λ. The induced gravity models equation.
e
2
II. EFFECTIVE EINSTEIN EQUATIONS The sums of the effective Einstein equation, Codacci
equation and twice contracted Gauss equation, taken on
A. The general projection formalism the two sides of the brane, lead to
1 F
In this subsection we summarize the relevant results G = Λ˜g +κ2 F g , (16)
ab ab ab ab ab
−2 − 2 −E
of [3]. (cid:18) (cid:19)
The first and second fundamental forms of the brane b
are ∇bKa = ∇aK, e (17)
10
R = Λ˜ +F 2J. (18)
gab = gab nanb, (3) 3 −
−
1
Kab = anb = ngab, (4) The differences of the same equations give:
∇e 2L
where na is the unit normal to the brane, with acceler- ∆ = ∆F , (19)
ab {ab}
E
ation αa = nb∇bna = gbaαb. Following [3] we introduce κ2Kabτab = ∆Λ, (20)
the tensors −
τb = 0, (21)
FJaabb ==eKKaKcKabbc−−KLancKKabcb.+∇bαa−αbαa, ((65)) κe2Ka∇bτbaab = ∆Je− 35∆Λ. (22)
As shown in [3], the 5-dimensional Einstein equations We have furtheer decomposed the differeence of the effec-
Gab = Λgab are equivalent to the system comprised of tive Einstein equations into tracefree and trace parts, in
−
the effective 4-dimensional Einstein equations, the Co- Eqs. (19) and (20).
deazziequeaetionand the twice contractedGauss equation:
2 1
Gab = 3G{ab}+ 2ncndGcdgab B. Induced gravity equations
1
+eFab Fgab e ab, (7) Up to this point, all results are unchanged compared
− 2 −E
to the asymmetric RS model, presented in [3] (here we
Kb K = gbncG , (8)
∇b a−∇a a bc have omitted the arbitrary bulk energy-momentum ten-
R = R+F 2J. (9) sor in [3]). This is simply because induced gravity does
e−
not modify bulk physics, as seen in Eq. (1). However,
Here
e Eq.(1)alsoshowsthattheinducedgravityscenarioadds
1
Q gcgd g gcd Q , (10) a new term to the brane total energy-momentum [5]:
{ab} ≡ a b − 4 ab cd
(cid:18) (cid:19) γ
denotes the projected trace-free part, and the “electric” τab =−λgab+Tab− κ2Gab, (23)
part of the bulk Weyl tensor is
whereT representsordinarymatteronthebrane. Then
1 ab
Eab ≡Cacbdncnd =J{ab}− 3G{ab}. (11) it follows that
The brane divides the bulk into two distinct regions. δF λ
e e δF g =κ2 g +T γG
The jump andaverageofaquantityacrossthe braneare ab− 2 ab −2 ab ab − ab
∆diQtio=nsQim+p−oQse−caonndtinQui=ty21o(fQt+h+e fiQr−st).fuTnhdeajmuenncttaiolnfocromn-, + κ(cid:18)62λS[κ2T −γG(cid:19)]ab, (24)
∆g =0, and the Lanczos equation [11]
ab
τ where the tensor functional [Q] is defined by [5]
∆K = κ2 τ g , (12) S
ab ab ab
− − 3
(cid:16) (cid:17) 1 Q
which relates the jump of the second fundamental form [Q] = Q Qc+ Q
to the total brane energye-momentum. (These conditions S ab 4"− ac b 3 ab
were first formulated in a coordinate-independent man-
1 Q2
ner in [14].) Then [3]: g Q Qcd+ . (25)
ab cd
τ − 2 (cid:18)− 3 (cid:19)#
∆F = κ2 K τ g
ab ab ab
− − 3
The tensor [T] that is quadratic in T , was intro-
+e τ3hKa(cid:16)b−2Kc(aτbc(cid:17)) , (13) duced for thSe Z2-asbymmetric case in [2]. ab
We define the brane cosmological “constant” [3]:
c i
F = K K K K +δF , (14)
ab ab − ac b ab
κ4 τ L
δF τ τc τ . (15) Λ=Λ , (26)
ab ≡ − 4 ac b − 3 ab 0− 4
(cid:16) (cid:17)
e
3
where Λ is a constant, III. FRIEDMANN BRANE
0
κ2λ Λ˜ For a Friedmann brane,
Λ = + , (27)
0
2 2 g = u u +a2h , (35)
ab a b ab
−
T = ρu u +pa2h , (36)
while L is a variable determined by the asymmetric em- ab a b ab
bedding, being the trace of the following tensor: 3(a˙2+k)
G = u u 2aa¨+a˙2+k h , (37)
ab a2 a b− ab
Lab ≡KKab−KacKcb− g2ab K2−KcdKcd . (28) where ua is the geodesic 4-vel(cid:0)ocity, a is the(cid:1) scale fac-
(cid:16) (cid:17) tor and hab is the 3-metric with constantcurvature (and
Then the effective Einstein equations are curvature index k = 0, 1) of the maximally symmetric
±
spacialslices,onwhichtheenergydensityρandpressure
(1+γ)G = Λg +κ2T p are constant. Then Eq. (25) implies
ab ab ab ab
− −E
+ κ62λS[κ2T −γG]ab+L{ab}. (29) S[κ2T −γG]ab =Vuaub+Wa2hab, (38)
where
The asymmetric contributions [3] in Λ and in the source 2
κ2ρ √3γ(a˙2+k)
termL{ab} adduplinearly withtheinducedgravitycon- V = , (39)
tributions,derivedfirstin[5]. Notethat,asdefinedhere, "√12 − 2a2 #
the cosmological “constant” is a function of the embed- κ4ρ(ρ+2p) γκ2[(2ρ+3p)(a˙2+k) 2ρaa¨]
ding of the brane. W = −
12 − 6a2
Both the Codazzi equation (17) and Eq. (18), giving
γ2(a˙2+k)(a˙2+k 4aa¨)
J, are insensitive to brane physics, such as the choice + − . (40)
4a4
ofthematterenergy-momentumtensorandthepresence
of the induced gravitycorrection. However the following We can introduce an effective “non-local” energy den-
equations do depend on these features. Using Eq. (23) sity U, which imprints bulk effects on the brane (gener-
inEq.(19) andcombiningEqs. (20) and(22), we obtain alizing the Z -symmetric procedure [15]), via
2
∆ and ∆J:
Eab a2
+L =κ2U u u + h . (41)
κ2 −Eab {ab} a b 3 ab
∆ = K κ2T γG (cid:18) (cid:19)
κ2 Eab − {ab}− {ab} In the Z -symmetric case, U arises from the Coulomb
2
1(cid:2)K κ2(2λ+T)(cid:3) γG field of a bulk black hole, and is called “dark radiation”;
e − 3 {ab} − for an AdS bulk, U = 0. Asymmetry allows for a non-
+2K κ(cid:2)2Tc γGc , (cid:3) (30) zeroU inanAdSbulk. ThentheeffectiveEinsteinequa-
c{a b}− b} tion (29) gives the generalized Friedmann equation,
2 h i
∆J = 3∆Λ. (31) ρ (a˙2+k) 3γ2 a˙2+k 2
1+γ 1+ =
λ a2 2λκ2 a2
Theseequations,togetheerwithEq.(18),characterizethe h (cid:16) (cid:17)i (cid:18) (cid:19)
1 ρ
off-braneevolutionofK ,andwewillnotdealwiththem + Λ+κ2ρ 1+ +κ2U , (42)
ab
3 2λ
further here.
h (cid:16) (cid:17) i
In the asymmetric case there are also the additional and the generalized Raychaudhuri equation,
constraints, Eqs. (20) and (21), to be imposed on the
ρ a¨ ρ+p (a˙2+k)
matter fields and geometrical properties of the brane: 1+γ 1+ 3γ =
λ a − 2λ a2
h (cid:16) (cid:17)i (cid:18) (cid:19)
∇bTab = 0, (32) 1 2Λ κ2 3p 1+ ρ +ρ 1+2ρ 2κ2U
λK+Kab T γ G = 1 ∆Λ. (33) 6( − λ λ − )
− ab− κ2 ab −κ2 h (cid:16) (cid:17) (cid:16) (cid:17)i
3γ2 a˙2+k a¨ a˙2+k
(cid:16) (cid:17)
+ 2 . (43)
From the twice-contracted Bianchi identity ine4 dimen- 2λκ2 a2 a − a2
e (cid:18) (cid:19)(cid:18) (cid:19)
sions, using Eq. (32), we find
These have the same form as Eqs. (3.20,21) of [5]. All
asymmetric features are absorbed in Λ and U, which
6
a L = a [κ2T γG] . (34) are different from those of [5]. The conservation equa-
∇ Eab− {ab} −κ2λ∇ S − ab
tion (32) has the usual form,
(cid:0) (cid:1)
The cosmological consequences of this equation will be a˙
exploited in the next section. ρ˙+3 (ρ+p)=0, (44)
a
4
and the other constraint (33) with no symmetric coun- It is not immediate to obtain the RS limit from the
terpart gives above equations, since we need to go to second order in
γ. In the limit of small induced gravity,
1 γ a¨ a˙2+k
∆Λ= λ p 2 + K ρ
κ2 − − κ2 a a2 (1+γ) 1+γ 1+
(cid:20) (cid:18) (cid:19)(cid:21) A → λ
ρe+p+2 γ a¨ a˙2+k u u Kab. (45) γ2(cid:16) (cid:17) ρ Λ
e− κ2 a − a2 a b ρ 1+ + +U , (51)
(cid:20) (cid:18) (cid:19)(cid:21) − λ 2λ κ2
(cid:20) (cid:16) (cid:17) (cid:21)
In the RS model (γ 0 limit) this is just a constraint
→ andto leadingorderwerecoverthe FriedmannandRay-
on the embedding and the perfect fluid. Induced gravity
chaudhuri equations of the asymmetric RS model in the
introduces a˙ and a¨ contributions, transforming Eq. (45)
ǫ = 1 branch. The other branch does not allow this
into a dynamical equation, to be added to the system −
limit.
of the energy-balance equation (44) and the generalized
We can now eliminate the derivatives of a from
Friedmann and Raychaudhuri equations (42) and (43).
Eq. (45), to obtain an algebraic relation among U, Λ,
We will discuss the significance of this equation below.
ρ, a and the embedding:
The “non-local”conservationequation(34) reduces to
K+u u Kab κ2 (ρ+3p)+2 Λ κ2U
U˙ +4a˙ U = 1 L˙ , (46) a b " γ −
a 4κ2 (cid:16) (cid:17) (cid:0) (cid:1)
(1+γ)2 (1+γ) ∆Λ λ
which shows how asymmetry can be a source for U even κ2 λ 3ǫκ2 + K
in the absence of a bulk black hole. For a Z -symmetric − γ2 #− γ " κ2 γ #A
2 e
demenbtewddaiyntgo, dweerihvaevEeqL.(=46)0iasntdotUak∝etah−e4t.imAendienrdiveapteivne- +κ2(1+γ2γ)2λ −2K+uaubKab eA2 =0. (52)
of the generalized Friedmann equation, then eliminate ρ˙ h i
by Eq. (44) and eliminate the derivatives of a from the To leading order, this constraint agrees with the corre-
terms not containing γ by Eqs. (42) and (43). sponding constraint of the asymmetric RS model [3].
The Lanczos equation becomes The Lanczos equation (47) becomes
κ2 3γ a¨ a˙2+k 3ǫ (1+γ)2
(1+γ) ∆K = ρ+3p λ
∆Kab =− 3 ((cid:20)2ρ+3p−λ+ κ2 (cid:18)2a − a2 (cid:19)(cid:21)uaub κ2 A ab h − γ
2γ (1+γ)
e+ ρ+λ 3γ a˙2+k a2h . (47) e+ κ2 Λ−κ2U −ǫ γ λA uaub
− κ2 a2 ab) (1+(cid:0)γ) (cid:1) i
(cid:20) (cid:18) (cid:19)(cid:21) + λ [ǫ+(1+γ) ]a2h . (53)
ab
γ A A
Uptothis pointthediscussionhasfollowed[3],there-
sults ofwhich canbe recoveredinthe γ 0limit. Since
→
theFriedmannequation(42)isquadraticin a˙2+k ,we IV. ANTI DE SITTER BULK
can take its square root, following [5], and obtain
(cid:0) (cid:1)
k κ2 (1+γ) Since the bulk equations are unchanged by the intro-
H2+ = ρ+ λ(1+ǫ ) , (48) ductionoftheinducedgravitycontributiononthebrane,
a2 3γ γ A
(cid:20) (cid:21) the static bulk solution admitting spatial sections with
where cosmological symmetry will be AdS. Then the off-brane
evolution equations (30), (31) and (18) are trivially sat-
2γ γ
2 =1+ ρ Λ+κ2U , (49) isfied and ab =0.
A (1+γ)2λ − κ2 E
h (cid:0) (cid:1)i
and ǫ = 1 distinguishes the two branches. Although
± A. Determining the functions U and Λ
the above equations have the same form as those in [5],
the function = (ρ,U,Λ) depends on the asymmetric
character of Athe eAmbedding. Employing Eq. (48), the From the extrinsic curvature of a generic Friedmann
Raychaudhuri equation (43) can also be simplified, brane [3] we find
a¨ κ2 hab∆Kab =6aB, (54)
6ǫ(1+γ) = (ρ+3p)+2(Λ κ2U)
− Aa γ −
where
(1+γ)2 (1+γ)
κ2 λ(1+ǫ )2+ǫκ2 (ρ+3p) . 1/2
− γ2 A γ A Λ±
B± = a˙2+k a2 . (55)
(50) −" − 6 #
e
5
(We have chosen the normal to the brane pointing to- with 2 as given by Eq. (49). With this procedure, the
A
wards the + region.) Then explicit dependence on ǫ is lost, and the two branches
will re-emerge as roots of Eq. (59).
12B∆B+a2∆Λ=0, (56) The functions U and Λ Λ obey Eq. (46), which be-
0
−
comes
and
e
3(ρ+p)x˙
a˙2+k Λ0 κ2λ B 2 ∆B 2 κ2U = 4ρ˙ . (63)
= + + , (57)
a2 3 − 6 a 2a
(cid:18) (cid:19) (cid:18) (cid:19)
Once the solutions of Eq. (59) are found, U is given by
which is the Friedmann equation, written in terms of B Eq. (63). This is an alternative route to the lengthy
and ∆B. These can be determined from Eq. (56) and procedure, described in [3], of comparing the two forms
the hab -projection of the Lanczos equation (53): of the Raychaudhuri equation,.
In the Z -symmetric limit, with ∆Λ = 0 , the coeffi-
2
k Λ κ2λ 1+ǫ(1+γ) 2 cients of the polynomial (59) reduce to
H2+ = 0 + A 1
a2 3 6 ((cid:20) γ (cid:21) − ) c = c =0, c = 1024λ2β2e,
0 1 2
+ γ2(∆Λ)2 . (58) c3 = 2048γκ−2λ2β,−c4 = 1024γ2κ−4λ2. (64)
96κ2λ[1+ǫ(1+γ) ]2 −
A
e
Theonlysolutionisthedoublerootx=0. ThenEq.(63)
In the γ 0 limit, using Eq. (51), we recover the Fried-
→ issolvedforU =0andwerecovertheresultof[5],inthe
mann equation of the asymmetric RS model with AdS
AdS case =0.
0
bulk[3]. Wehavecheckedthattheremainingconstraints, E
Eqs. (52) and the Codazzi equation, are satisfied.
A comparison of the generic form of the Friedmann
B. Perturbative solutions: small asymmetry
equation, Eq. (48), with the specific form for an AdS
bulk,Eq.(58),givesanalgebraicrelationbetweenΛand
U. We obtain a quartic polynomial AsmalldeviationfromZ2-symmetrycanbe described
via the dimensionless parameter
4
cixi =0, (59) (∆Λ)2
α= 1. (65)
Xi=0 6κ4λ2 ≪
in the variable e
To leading order, the solution of Eq. (59) is
x=Λ Λ +κ2U, (60)
− 0 3κ2λ2
x= λ+γβ ǫ λ(λ+2γβ) α. (66)
with coefficients 32β2 −
h p i
γ2λ2(∆Λ)4 From Eq. (63) we obtain
c = ,
0 − 4κ4
c = 32λ2(λ+eγβ)(∆Λ)2 , κ2U = 9κ2(ρ+p)λ2 (λ+2γβ)(2λ+γβ)
1 κ2 −128β3(λ+2γβ)
h
γ2(∆Λe)2 ǫ(2λ+3γβ) λ(λ+2γβ) α. (67)
c = 1024λ2 +β2 , −
2 − " 32κ4 # p i
e Switching off the induced gravity in this perturbative
2048γλ2β
c = , solution gives
3 κ2
1024γ2λ2 3(1 ǫ)κ2λ3
c = . (61)
4 − κ4 x = 32(−ρ+λ)2 α, (68)
Here we have introduced the notation 9(1 ǫ)κ2(ρ+p)λ3
κ2U = − α. (69)
γ γΛ − 64(ρ+λ)3
0
β =ρ+λ 1+ . (62)
2 − κ2
(cid:16) (cid:17) The perturbative solution is in fact exact in the limit
Equation(59)wasobtainedbyeliminatinga˙2 fromthe without induced gravity, as can be verified by direct in-
twoformsoftheFriedmannequation,Eqs.(48)and(58), spectionofEqs.(59)and(63)intheγ 0limit. Forthe
then inserting 2 from Eq. (49), expressing from the ǫ = 1 branch the results of the asym→metric RS model,
A A −
resultingequation,takingitssquareandcomparingagain presented in [3], are recovered.
6
Having obtained x and U, the explicit forms of the Tolowestorder,thefirstcorrectionisfromasymmetry,in
Friedmann and Raychaudhuri equations, Eqs. (48) and contrast to the high-energy case. The Friedmann equa-
(50), can be written. For this, we remark that tion (48) becomes
2γ
2 =1+ κ2ρ γx γΛ . (70) 2κ2(1+γ)λ 3γ4
A (1+γ)2κ2λ − − 0 H2 = 1 α , (78)
(cid:0) (cid:1) 3γ2 " −32(1+γ)2(2+γ)2 #
In the Z -symmetric limit this agrees with Eq. (3.26)
2
of [5], for vanishing dark radiation term. for ǫ=1, and
κ2λ
H2 = α, (79)
C. High energy limit 16(1+γ)
We now consider the perturbative small-asymmetry for ǫ= 1.
−
case at high energies (ρ λ) in the early universe. We It follows that for ǫ = 1, the late-time acceleration
assume that there is no≫bare cosmological constant on from induced gravity is slightly reduced by asymmetry,
the brane, i.e., Λ = 0, and that the spatial sections are whereas for the ǫ = 1 branch (with an RS limit), the
0 −
flat, k =0. By Eqs. (62) and (66), asymmetry introduces a small acceleration.
3 λ
β ρ, x γκ2λ α. (71)
≈ ≈ 32 ρ V. CONCLUDING REMARKS
(cid:18) (cid:19)
Then from Eq. (70),
Combining the method developed in [3] with that
2γ ρ λ λ2 of [5], we have developed a general formalism describing
2 = 1+O ,α , (72)
A (1+γ)2 λ ρ ρ2 gravitationaldynamics onthe asymmetricallyembedded
(cid:20) (cid:18) (cid:19)(cid:21) brane with induced gravity. The decomposition of the
so that to lowest order, there is no effect from asymme- bulk Einstein equations yielded the tensorial, vectorial
try. The Friedmann equation (48) is the same as in the and scalar projections. They are equivalent to the pro-
symmetric case: jected Einstein, the Codazzi and the twice contracted
Gauss equations. In the derivation of the final set of
H2 = κ2ρ 1+ǫ 2λ 1+O λ,αλ2 . (73) equations the generalized junction conditions were ap-
3γ " sγρ( sρ ρ2!)# plied. The resulting effective Einstein equation contains
all terms from the asymmetric RS model, to which the
induced gravity contributions add linearly.
D. Low energy limit ForFriedmannbranes,the generalizedFriedmannand
Raychaudhuri equations were derived, together with the
In the low-energy, late-universe regime, induced grav- relevantconstraintequations. Theycontaintwoundeter-
ity models with ǫ = 1 are interesting because they can mined functions functions U and Λ Λ0. These depend
−
produce acceleration even without dark energy [7, 8]. onthebulkgeometry. Asanapplicationoftheformalism,
Asymmetric RS models can also produce acceleration wediscussedindetailtheAdSbulkwithdifferentcosmo-
(see [3] and references therein). We consider ǫ = 1 logical constants on the two sides of the brane. Deter-
models in the perturbative small-asymmetry case, w±ith miningthefunctionsU andΛ Λ0reducesinthiscaseto
−
Λ =0=k. We assume that solving a quartic polynomial. This feature of the model
0
was already remarked on and exploited in [9] , where a
ρ
α 1, (74) perturbativesolutioninterms ofasmallinducedgravity
λ ≪ ≪
contribution was analyzed. Similarly, in [10] a quartic
which is readily achieved by choosing the brane tension equationinH2+k/a2 waspresented. Ourmethodinde-
high enough relative to the very low energy scale of the termining the Friedmann equation also yielded a quartic
late universe. equation. We have found and discussed those solutions
By Eqs. (62) and (66), of the quartic equation for which the deviation from Z
2
-symmetry is small. In the high-energy, early-universe
γ
β λ 1+ , (75) regime, the asymmetry does not affect the Friedmann
≈ 2
equationtolowestorder. Inthelow-energy,late-universe
(cid:16) (cid:17) 2
3 1 ǫ+γ
x κ2λ − α. (76) regime,theasymmetrydoeshaveaneffect. Intheǫ= 1
≈ 16 2+γ branch, it contributes a late-time acceleration, while−in
(cid:18) (cid:19)
the ǫ = 1 branch is slightly reduces the late-time accel-
Then from Eqs. (70) and (74),
eration arising from induced gravity.
3 γ(1 ǫ+γ) 2
1 − α. (77)
A≈ − 16 (1+γ)(2+γ)
(cid:20) (cid:21)
7
Acknowledgments OTKA grants Nos. T046939 and TS044665 and a
We thank Cedric Deffayet for useful comments. LA´G PPARC visitor grant at Portsmouth. RM is supported
was supported by the Hungarian Scholarship Board, by PPARC.
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