Non-linear Gravity on Branes and Effective Action

5 Pages·0.1 MB·English

by  Jiro Soda

Non-linear Gravity on Branes and Effective Action
Jiro Soda,1 Sugumi Kanno 2
1Department of Fundamental Sciences, FIHS, Kyoto University, Kyoto 606-8501, Japan
2 Graduate School of Human and Environmental Studies, Kyoto University, Kyoto 606-8501, Japan
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Abstract
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Wedevelopthegeneralformalism tostudythelowenergyregimeofthebraneworld.
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We apply our formalism to the single brane model where the AdS/CFT correspon-
n
dence will take an important role. We also consider the two-brane system and show
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thesystemisdescribedbythequasi-scalartensorgravity. Ourresultprovidesabasis
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for predicting CMB fluctuationsin thebraneworld models.
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1 Introduction

1
v Theexistenceoftheinitialsingularityinthestandardcosmologyisanotoriousproblemwhichisexpected
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to be solved by taking into account the quantum effects of the gravity. It is widely accepted that the
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superstring theory is the most promising candidate for the quantum theory of gravity. One prominent
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1 feature of the superstring theory is the existence of the extra dimensions. To fill the gap between this
0 theoreticalpredictionandour4-dimensionaluniverse,weneedtohidetheextradimensionssomeway. The
3 conventionalideaistheKaluza-Kleincompactificationscenariowheretheinternaldimensionsareassumed
0
to be compactified to the Planck scale. Recently, however, a new picture, the so-called braneworld, has
/
c emerged thanks to the developments of the non-perturbative aspects of the superstring theory [1, 2]. In
q
the braneworld picture, the ordinary matter exists on the brane while the gravity can propagate in the
-
r bulk. Inparticular,weareonthebrane! Hence,whatwewouldliketoknowishowthenon-lineargravity
g
appearsonthebrane. Ingeneral,itwouldbedifficulttogetsuchadescriptionduetothestrongcoupling
:
v ofthe bulk degreesoffreedomwith those of the brane. However,in the lowenergy regime ρ ℓ2R 1,
Xi itispossibleto obtainthe 4-dimensionaleffectivetheoryapproximately[3,4,5]. Indeed,thσis∼is suffic≪ient
except for the extreme situation for which we need more profound understanding of the string theory.
r
a For example, if we take the curvature scale ℓ as 10−16cm, our approximation is valid at the energy scale
below 1011GeV or the gravitational radius greater than 10−21km. In fact, this is the interesting regime
for most astrophysical and cosmological phenomena. In this paper, we review our recent results on this
issue [4, 5].
In the next section, we will develop the general formalism. In sec.3, we apply our formalism to the
singlebranemodelwheretheAdS/CFTcorrespondencewillplayanimportantrole. Insec.4,weconsider
thetwo-branesystemandshowthesystemisdescribedbythequasi-scalartensorgravity. Sec.5isdevoted
to the conclusion.
2 General Formalism
If there exists no matter, the ground state is the Minkowski spacetime in the 4-dimensional theory.
Correspondingly, if there exists no matter on the brane, we would expect the induced metric on the
brane is Minkowskiandthe bulk geometry is the Anti-deSitter spacetime. Indeed, we have suchsolution
ds2 =dy2+Ω2(y)η dxµdxν ,
µν
where Ω2 = exp[ 2y/ℓ] is the warp factor. The brane is located at y = 0 in this coordinate system. To
−
obtain the above solution we have imposed the relation κσ =6/ℓ.
1E-mail:[email protected]
2E-mail:[email protected]
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Let us put the small amount of matter on the brane. Then, the brane will be curved and the bulk
geometry will be deformed as
ds2 =dy2+ Ω2(y)h (x)+δg (y,xµ) dxµdxν , (1)
µν µν
wheretheboundaryconditionδg (y(cid:0)=0,xµ)=0isimposedsot(cid:1)hath becomestheinducedmetricon
µν µν
the brane. How the geometry will be deformed is determined by the 5-dimensional Einstein equations:
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G(5) = g +δ(y)8πG ℓ( σg +T )δµδν , A=(y,µ) , (2)
AB ℓ2 AB N − µν µν A B
where T is the energy-momentum tensor of the matter. As we are considering the deviation from the
µν
Anti-deSitter spacetime, it is convenient to define the variables
1 1
δKµ = δ[gµαg ] δΣµ + δµ δK , (3)
ν −2 αν,y ≡ ν 4 ν
where δΣµ =0. In terms of these variables, the Hamiltonian constraint equation becomes
µ
ℓ 3 (4)
δK = δK2 δΣµ δΣν R (4)
−6 4 − ν µ−
(cid:20) (cid:21)
and the momentum constraint equation reads
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δΣλ δK =0 . (5)
∇λ µ− 4∇µ
The evolution equation in the direction to y is
1 (4)
Ω4δΣµ =δKδΣµ Rµ . (6)
Ω4 ν ,y ν −" ν#
traceless
(cid:2) (cid:3)
As we have the singular source at the brane position, we must take into account the junction condition,
2 3
δΣµ δµδK =8πG Tµ , (7)
ℓ ν − 4 ν y=0 N ν
(cid:20) (cid:21)(cid:12)
(cid:12)
where we have imposed the Z symmetry on the sp(cid:12)acetime.
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After solving the bulk equations of motion, the junction condition gives the effective equations of
motion for the induced metric. By integrating the evolution equation, we have
2 3 ℓ2χµ 2 y (4) 1 (4)
δΣµ δµδK = ν dyΩ4 Rµ δµ R δKδΣµ
ℓ (cid:20) ν − 4 ν (cid:21) − Ω4 − ℓΩ4 Z " ν − 4 ν − ν#
1 (4) 1 3
δµ R + δµ δK2 δΣµ δΣν
−4 ν 4 ν 4 − ν µ
(cid:20) (cid:21)
ℓ2χµ (x) (4)
= ν +Gµ (Ω2(y)h (x))+ (ℓ4R2) ,
− Ω4(y) ν µν O
whereweintroducedtheconstantsofintegrationχ . Thisintegrationcanbe performediterativelywith
µν
the expansion parameter ℓ2R. The resulting equations take the form
(4) ℓ2
Gµ (Ω2 h )=8πG Tµ + χµ +tµ , (8)
ν |y=0 µν N ν Ω4 ν ν
y=0
|
where we have included the trivial factor Ω =1 for memorizing how the warpfactor comes in to the
y=0
|
effective theory. Here we have decomposed the corrections to the conventional Einstein theory into the
nonlocal part χ and the local part t .
µν µν
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We can expand these corrections in the order of the ℓ2R:
Nonlocal χ = χ(1) + χ(2) + (9)
µν µν µν ···
(ℓ2R) (ℓ4R2)
O O
Local t = |{z} t|({2z)} + t(3) + (10)
µν µν µν ···
(ℓ4R2) (ℓ6R3)
O O
|{z} |{z}
wheretheexpansionofthelocaltensorstartsfromthesecondorderbecausethefirstorderpartisalready
included as the Einstein tensor. Notice that we have χ(1)µ =0 because of Σµ =0.
µ µ
3 Single Brane Model (RS2): AdS/CFT correspondence
Now we shall apply the general formalism to the single brane model [4]. A natural boundary condition
for the single brane model is to impose the regularity at the Cauchy horizon, namely asymptotically
AdS boundary condition. Taking this boundary condition, we obtain χ(1) =0. Thus, we have recovered
µν
Einstein theory at the leading order.
At the next order (l4R2), the traceless part of t(2) is proportional to
µν
O
1 1 1
µ = Rµ Rα RRµ δµ(Rα Rβ R2)
Sν α ν − 3 ν − 4 ν β α− 3
1 2 1
Rαµ +Rα |µ R|µ 2Rµ + δµ2R (11)
−2 |να ν |α− 3 |ν − ν 6 ν
(cid:18) (cid:19)
where µ istransverseandtraceless: µ =0, µ =0.Thenonlocalpartχ˜(2) =χ(2)+1/4h t(2)µ
S ν|µ S ν|µ S µ µν µν µν µ
can not be determined by the Einstein equations, but constrained as
1 1
χ˜(2)µ = Rα Rβ R2 .
µ −8 β α− 3
(cid:18) (cid:19)
Here, the AdS/CFT correspondence comes in. As the trace anomaly for some supersymmetric theories
proportional to the above result, we make the following identification
κ2
χ˜(2) = TCFT . (12)
µν ℓ3 µν
Note that the effective number of the CFT fields ℓ3/κ2 ℓ2/G 1066 is so huge. This is the regime
∼ ∼
where AdS/CFT correspondence holds.
Thus, the effective equations of motion become
G(4) =8πG T +8πG TCFT+α µ . (13)
µν N µν N µν S ν
Seeing Eq.(11), one can read off the effective action
1 αℓ2 1
S = d4x√ hR+S +S + d4x√ h RµνR R2 . (14)
eff matter CFT µν
16πG − 16πG − − 3
N Z N Z (cid:20) (cid:21)
Nowonecanconsiderthecosmologyusingtheaboveeffectivetheory. Onecanobtaintherenormalized
action for the CFT, SCFT, then we can deduce the one point function from the formula
2 δSCFT
<TCFT >= .
µν −√ g δgµν
−
The two point correlation function can be also calculated as
2 δ <TCFT(x)>
<TCFT(x)TCFT(y)>= µν .
µν λρ −√ g δgλρ(y)
−
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Thus, we obtain the perturbed effective Einstein Equations
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δG =8πG δT d4y g(y)<TCFT(x)TCFTλρ(y)>δg +αδt(2) . (15)
µν N µν − 2 − µν λρ µν
Z
p
This is nothing but the integro-differential equation. It is possible in principle to solve numerically the
linearized equations of motion in the cosmologicalsituation. We leave this for the future work.
4 Two Brane Model (RS1): Radion
In this section, we will consider the two-branesystem which is more realistic from the M-theory point of
view [3, 5](see also [6]). In the two-brane system, the radion field plays an important role. The radion
is defined as the distance between the positive tension brane and the negative tension brane, d(x). Now
the warp factor Ω2 =exp[ 2d(x)/ℓ] becomes dynamical variable.
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The general formula gives the equation on the positive tension brane:
κ2
G(4)µ (h )= T⊕µ +ℓ2χ(1)µ (16)
ν µν ℓ ν ν
where χ represents the effect of the bulk geometry on the brane. Importantly, this equation holds
µν
irrespectiveofthe existenceofthe otherbrane. Similarly,the equationofmotiononthe negativetension
brane is given by
κ2 ℓ2
G(4)µ (f =Ω2h )= T⊖µ + χ(1)µ (17)
ν µν µν − ℓ ν Ω4 ν
where f is the induced metric on the negative tension brane. Here, the effect of the bulk geometry
µν
enhanced by the factor 1/Ω4 since the bulk geometry shrinks towards to the negative tension brane.
(1)
Although Eqs. (16) and (17) are non-local individually, with undetermined χ , one can combine both
µν
(1)
equationstoreducethemtolocalequationsforeachbrane. Thishappenstobepossiblesinceχ appears
µν
onlyalgebraically;onecaneasilyeliminateχ(1) fromEqs.(16)and(17). DefininganewfieldΨ=1 Ω2,
µν
−
we find
ℓ3 κ2(1 Ψ)
χ(1)µ = − T⊕µ +(1 Ψ)T⊖µ
2 ν − 2Ψ ν − ν
l (cid:0) 3 (cid:1) 1
Ψ|µ δµΨ|α + Ψ|µΨ δµΨ|αΨ .
−2Ψ |ν − ν |α 2(1 Ψ) |ν − 2 ν |α
(cid:20)(cid:16) (cid:17) − (cid:18) (cid:19)(cid:21)
The condition χ(1)µ =0 gives the equations of motion for the radion field:
µ
κ2 1
2Ψ= (1 Ψ) T⊕+(1 Ψ)T⊖ Ψ|µΨ . (18)
3ℓ − − − 2(1 Ψ) |µ
−
(cid:8) (cid:9)
Interestingly, we can rearrange the above equations as
κ2 κ2(1 Ψ)2 1 ω(Ψ) 1
Gµ (h)= T⊕µ + − T⊖µ + Ψ|µ δµΨ|α + Ψ|µΨ δµΨ|αΨ (19)
ν ℓΨ ν lΨ ν Ψ |ν − ν |α Ψ2 |ν − 2 ν |α
(cid:16) (cid:17) (cid:18) (cid:19)
and
κ2T⊕+(1 Ψ)T⊖ 1 dω(Ψ)
2Ψ= − Ψ|µΨ , (20)
ℓ 2ω(Ψ)+3 − 2ω(Ψ)+3 dΨ |µ
wherethecouplingfunctionω(Ψ)takesthefollowingform: ω(Ψ)=3Ψ/2(1 Ψ).Wenamedthissystem
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as the quasi-scalar-tensortheory. Eqs.(19) and (20) can be derived from
l 3
S = d4x√ h ΨR(h) Ψ|αΨ + d4x√ h ⊕+ d4x√ h(1 Ψ)2 ⊖ . (21)
A 2κ2 − − 2(1 Ψ) |α − L − − L
Z (cid:20) − (cid:21) Z Z
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This is the effective action on the positive tension brane.
The effective action on the negative tension brane can be also derived in the similar way as
l 3
S = d4x f ΦR(f)+ Φ;αΦ + d4x f ⊖+ d4x f(1+Φ)2 ⊕ , (22)
B 2κ2 − 2(1+Φ) ;α − L − L
Z (cid:20) (cid:21) Z Z
p p p
where Φ=1/Ω2 1.
−
Thus,wehavederivedaclosedsetofequations(19)and(20). Bysolvingtheseequations,wecanknow
the anisotropic stress χ(1) explicitly. Now, we can make a precise predictions on the CMB fluctuations!
µν
This will be reported somewhere else.
5 Conclusion
We have developed the general formalism to obtain the effective action in the low energy regime.
In the case of the single brane model, by imposing asymptotically AdS boundary condition, we have
obtained the Einstein theory with corrections represented by CFT and higher curvature polynomial.
It is suggested that the cosmological perturbation theory in the brane world can be formulated as the
integro-differential equations.
In the case of the two-brane model, we have shown that the system is described by the quasi-scalar-
tensor theory. Equivalently, it can be regarded as the Einstein theory with the extra energy source,
χ(1) corresponding to the dark radiation in the homogeneous cosmological case. This turns out to be
µν
determinedbytheradionfieldandtheenergymomentumtensorsonpositiveandnegativetensionbranes.
ThenextordercorrectionsduetoKalza-Kleinmassivemodescanbe representedbythehighercurvature
terms. Itisinterestingtostudythecosmologicalscenariobasedontheeffectiveactionwehavederived[7].
Acknowledgements
ThisworkwassupportedinpartbytheMonbukagakushoGrant-in-AidforScientificResearch,Nos.14540258.
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