Die Methode der nichtlokalen effektiven Wirkung in höherdimensionalen Raumzeitmodellen [Elektronische Ressource] / vorgelegt von Andreas Rathke
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Die Methode der nichtlokalen effektiven Wirkung in höherdimensionalen Raumzeitmodellen [Elektronische Ressource] / vorgelegt von Andreas Rathke

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Die Methode dernichtlokalen e ektiv en Wirkungin h oherdimensionalenRaumzeitmodellenINAUGURAL-DISSERTATIONzurErlangung des DoktorgradesderFakult at fur Mathematik und PhysikderAlbert-Ludwigs-Universit at Freiburg im Breisgauvorgelegt vonAndreas Rathkeaus HemerJuni 2003Dekan: Prof. Dr. Rolf SchneiderLeiter der Arbeit: Prof. Dr. Hartmann R omerReferent: Prof. Dr. R omerKorreferent: Prof. Dr. Jochum van der BijTag der mundlic hen Prufung: 23.10.2003The Method of the Nonlocal E ectiv e Actionin Higher-Dimensional Spacetime ModelsAndreas RathkeContents1 E ectiv e methods in braneworld dynamics 71.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.1.1 The role of the nonlocal e ectiv e action . . . . . . . . . . . . 101.1.2 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.2 The Randall-Sundrum models . . . . . . . . . . . . . . . . . . . . . 121.3 E ectiv e actions for warped braneworlds . . . . . . . . . . . . . . . . 141.3.1 Kaluza-Klein description . . . . . . . . . . . . . . . . . . . . . 151.3.2 Holographic . . . . . . . . . . . . . . . . . . . . . 181.3.2.1 One-brane e ectiv e gravity from AdS/CFT . . . . . 181.3.2.2 The AdS/CFT interpretation of the two-brane model 222 The nonlocal braneworld action 252.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.2 The e ectiv e action of brane-localized elds and the methods of itscalculation . . . . . . . . .

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Published 01 January 2004
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Die Methode der
nichtlokalen e ektiv en Wirkung
in h oherdimensionalen
Raumzeitmodellen
INAUGURAL-DISSERTATION
zur
Erlangung des Doktorgrades
der
Fakult at fur Mathematik und Physik
der
Albert-Ludwigs-Universit at Freiburg im Breisgau
vorgelegt von
Andreas Rathke
aus Hemer
Juni 2003Dekan: Prof. Dr. Rolf Schneider
Leiter der Arbeit: Prof. Dr. Hartmann R omer
Referent: Prof. Dr. R omer
Korreferent: Prof. Dr. Jochum van der Bij
Tag der mundlic hen Prufung: 23.10.2003The Method of the Nonlocal E ectiv e Action
in Higher-Dimensional Spacetime Models
Andreas RathkeContents
1 E ectiv e methods in braneworld dynamics 7
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1.1 The role of the nonlocal e ectiv e action . . . . . . . . . . . . 10
1.1.2 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2 The Randall-Sundrum models . . . . . . . . . . . . . . . . . . . . . 12
1.3 E ectiv e actions for warped braneworlds . . . . . . . . . . . . . . . . 14
1.3.1 Kaluza-Klein description . . . . . . . . . . . . . . . . . . . . . 15
1.3.2 Holographic . . . . . . . . . . . . . . . . . . . . . 18
1.3.2.1 One-brane e ectiv e gravity from AdS/CFT . . . . . 18
1.3.2.2 The AdS/CFT interpretation of the two-brane model 22
2 The nonlocal braneworld action 25
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2 The e ectiv e action of brane-localized elds and the methods of its
calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.2.1 The structure of the braneworld e ectiv e action . . . . . . . . 26
2.2.2 The role of radion elds . . . . . . . . . . . . . . . . . . . . . 29
2.3 Two-brane Randall-Sundrum model: the nal answer for the two- eld
braneworld action . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.4 The e ectiv e equations of motion . . . . . . . . . . . . . . . . . . . . 34
2.5 The recovery of the braneworld e ectiv e action . . . . . . . . . . . . 39
2.6 Green functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.6.1 The low-energy limit | recovery of Einstein theory . . . . . 43
2.6.2 Low-energy derivative expansion . . . . . . . . . . . . . . . . 45
2.7 The particle content of the two- eld braneworld action . . . . . . . . 47
2.7.1 The graviton sector . . . . . . . . . . . . . . . . . . . . . . . 48
2.7.2 Problems with the scalar sector of the theory . . . . . . . . . 49
2.7.3 Large interbrane distance . . . . . . . . . . . . . . . . . . . . 51
2.8 The reduced e ectiv e action . . . . . . . . . . . . . . . . . . . . . . . 53
2.8.1 Small interbrane distance . . . . . . . . . . . . . . . . . . . . 55
2.8.2 Large in and Hartle boundary conditions . 57
2.9 The nonlocal action for the RS one-brane model . . . . . . . . . . . 59
3 From nonlocal action to other methods 63
3.1 E ectiv e action of brane-localized elds vs. Kaluza-Klein reduction . 63
3.2 The recovery of the Kaluza-Klein tower . . . . . . . . . . . . . . . . 64
56
3.2.1 The particle interpretation of the transverse-traceless sector . 64
3.2.2 The eigenmode expansion of the Green function . . . . . . . 65
3.2.3 The spectrum and the eigenmodes of the e ectiv e action . . 67
3.2.4 The graviton e ectiv e action in the diagonalization approxi-
mation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.3 Phenomenological digression: radion-induced graviton oscillations . . 73
3.3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.3.2 Gravitational waves on the -brane . . . . . . . . . . . . . 75+
3.3.3 Quantum oscillations | an analogy . . . . . . . . . . . . . . 76
3.3.4 Gravitational-wave oscillations on the -brane . . . . . . . 78+
3.3.5 High-amplitude RIGO’s on the -brane from M-theory . . . 79+
3.3.6 Graviton oscillations on the -brane . . . . . . . . . . . . . 82
3.3.7 RIGO’s in bi-gravity models . . . . . . . . . . . . . . . . . . . 84
3.3.8 Summary on RIGO’s . . . . . . . . . . . . . . . . . . . . . . . 86
3.4 Holographic interpretation . . . . . . . . . . . . . . . . . . . . . . . . 87
3.4.1 The RS one-brane model . . . . . . . . . . . . . . . . . . . . 87
3.4.2 The RS two-brane model . . . . . . . . . . . . . . . . . . . . 88
4 Conclusions and Outlook 93
A Anti-deSitter space and the geometrical setting of the RS models
99
B Diagonalization of the kinetic and mass terms 101
Bibliography 1051
E ectiv e methods in braneworld
dynamics
1.1 Introduction
Among the most promising candidates for a uni ed description of quantum theory
and gravitation is superstring theory [1]. It predicts the existence of multidimensional
stable objects called \branes". On these branes open strings can end, which also
means that the gauge elds associated with the ends of a string reside on the branes.
Gauge elds, rather than spreading through the entire ten-dimensional spacetime
of superstring theory, are thus con ned to these lower-dimensional objects. This led
to the proposal that the universe we inhabit could be a three-dimensional brane
(\3-brane") on which all matter consisting of gauge elds is trapped [2, 3, 4]. In
string theory, gravity is mediated by the exchange of closed strings and therefore is
not restricted to the branes but propagates through the whole higher-dimensional
spacetime, commonly called bulk. The shape of the bulk can be constrained because
no deviations from four-dimensional gravity have yet been observed (see however
[5]).
Such a \braneworld" model can be constructed from compacti ed at higher di-
mensions, producing a Kaluza-Klein theory of gravity [6]. This results in one massless
graviton mode responsible for the observed behavior of gravity and a tower of mas-
sive graviton states, only observable at higher energies. It is imaginable that the
compacti cation is large enough in order to produce e ects at accelerators of the
next generation [4].
In a di eren t scenario [7, 8]. The higher-dimensional manifold does no longer
1have to \factorize" into our four-dimensional world and the additional dimensions.
In particular, one can assume that the bulk space has an anti-de Sitter (AdS) struc-
ture transverse to the branes. The brane showing the behavior of e ectiv e four-
2dimensional gravity (\our Universe") is put at an Z -orbifold point in the bulk2
1The term non-factorizable geometry is used in the context of higher-dimensional spacetime
models di eren tly from its actual geometrical meaning. A geometry is called
factorizable if its metric can be put in a block diagonal form | with one of these blocks being
the four-dimensional metric | in which the blocks do not depend on coordinates belonging to a
di eren t block of the full metric. cf. p. 16, Sec. 1.3.1.
2An orbifold is a coset space M=H, where H is a group of discrete symmetries of a manifold
78 1 Effective methods in braneworld dynamics
space. Consequently, the bulk metric has a re ection-symmetry with respect to the
position of the brane. The metric of AdS space yields an e ectiv e potential for the
linearized gravitational modes with high barriers on both sides of the brane. This
\volcano potential" leads to a zero-mass graviton state trapped at the position of
the brane and a continuous spectrum of massive modes, which are exponentially
suppressed on the brane [9] but can propagate through the whole bulk. The most
prominent models of this class are the Randall-Sundrum two-brane model [7] (com-
monly called RS1) and the Randall-Sundrum one-brane model [8] (called RS2).
There have been similar earlier proposals (e. g. [10]). The two Randall-Sundrum
(RS) models will form the framework for our present investigation.
The insight that even non-compact extra dimensions allow realistic e ectiv e four-
dimensional gravity has stired a lot of activity in the investigation of braneworlds.
The Realization of zero-mode localization in one-brane models has been explored in
[11]. In particular it was found that the appearance of a massless four-dimensional
graviton in the theory is not necessarily tied to an anti-de Sitter bulk but can also
be achieved in a deSitter bulk which seems more favorable from the point of view of
having a viable higher dimensional cosmic evolution [11, 12]. It has also been realized
that one can have a viable model of four-dimensional gravity even without an e ec-
tive massless graviton. In certain setups a tower of extremely light massive gravitons
may be indistinguishable from a massless graviton [13, 14]. Even gravity mediated by
a massless and an extremely light massive graviton at roughly equal strengths may
be viable and hard to distinguish from massless four-dimensional gravity [15, 16].
In these setups the Veltman-van-Dam-Zakharov discontinuity (VvDZ discontinuity)
[17] (for a recent analysis of VvDZ discontinuity at the classical level see [18]) is
either unobservable due to the setup [14] or cured by non-linear e ects [19], though
it is still under dispute if these models exhibit a realistic behavior in the limit of
strong gravitational elds [20].
The understanding of gravity in brane models can, however, only be considered
preliminary because the analysis is usually done studying only the zero-mode, ne-
glecting the continuous spectrum [21, 22]. A remarkable deviation from the usual
practice is the analysis of the RS one-brane model in [23]. There it was shown that
in the long-distance approximation the e ect of bulk gravity on the brane can be
replaced by a four-dimensional conformal eld theory with an ultraviolet cuto via
the anti-de Sitter/conformal- eld-theory (AdS/CFT) correspondence (cf. [24]). This
provides a direct technical connection between brane-world scenarios and string the-
ory. Only one-brane models have a zero-mode approximation that corresponds to
four-dimensional gravity without additional elds (such as Brans-Dicke scalars in
the RS two-brane model) [8, 21]. All such models predict a deviation from Einstein
gravity at short distances and may thereby be experimentally detected [25, 26].
The assumption of extra dimensions o ers a potential solution to the hierarchy
problem { the huge ratio between the magnitudes of the electroweak and gravitational
couplings. In models with compacti ed dimensions, this is due to the fact that gravity
propagates through the bulk and consequently is weakened by a factor of the inverse
1M. In the following we only consider the special case of a S =Z orbifold, i. e. re ection symmetry2
applied to a circle. We call this case aZ orbifold and the points of the circle with respect to which2
the re ection symmetry acts orbifold points.1.1 Introduction 9
size of the additional dimensions [4]. In models with extra dimensions having an
anti-de Sitter metric, an exponential weakening of the gravitational coupling can be
achieved by means of the exponential (\warp-") factor in the metric. Unfortuantely,
up to now all models with exponential bulk metrics generating this hierarchy and
leading to an e ectiv e four-dimensional behavior of gravity have been shown to
be unstable and to violate the null energy condition [27, 28] (but cf. [9]). In spite
of these problems, the study of anti-de Sitter brane-world models gained interest
because there is no mass gap compared with compacti cations on a circle. They
might therefore o er a new particle-physics phenomenology at much lower energies
(cf. [29] for an overview).
In addition to the hierarchy problem, the other major branch in brane-world
studies is cosmology, see for example [30, 31, 32, 33]. Emphasis is put on modi -
cations of the four-dimensional Friedmann equations due to the presence of higher
dimensions, and on scenarios for the very early universe. On the one hand, there are
more conservative models considering in ationary scenarios driven by scalar elds
on the brane [34, 35] or the trace anomaly of the matter elds on the brane [36, 37].
The latter analysis has heavily drawn on the technical bene ts of the AdS/CFT
correspondence. On the other hand, the in uence of bulk elds on the cosmologi-
cal evolution on the branes has been discussed. These bulk elds tend to drive the
branes closer to each other, nally resulting in a collision. Depending on the model,
this collision can be followed by a merge, a bounce, or a penetration of the colliding
branes [38, 39, 40, 41]. The last two types of collision, called ekpyrotic scenarios, have
been claimed to generate a perturbation spectrum consistent with the observed cos-
mic microwave background radiation. These models have, however, also been heavily
criticized [42, 43].
In models with compacti cation on a circle one has not yet succeeded to generate
viable in ationary scenarios in which the in ation on the brane is driven by the
modulus eld [44], i. e. the eld describing the size of the extra dimension. However,
a recent study has shown that it might be possible to generate in ation on the
brane without any additional scalar eld in the bulk or on the brane in a modi ed
RS two-brane model [45]. This is achieved using the scalar modulus of the extra
dimension, the \radion", as an in aton. The potential that provides the necessary
energy for the expansion is part of the tension of the brane on which in ation occurs.
During in ation on the brane, the branes move away from each other. The model
of [45] provides an in aton potential which tends to a xed value for in nite brane
separation and therefore might also be able to address the present-day accelerated
expansion of the universe [46, 47]. Modulus oscillations have been considered as a
driving force for reheating after in ation [48] and the stage of baryogenesis following
reheating [49]. Such a radion-induced reheating phase would naturally be the end
point of a radion-driven in ation.
There are also attempts to explain the creation of braneworlds by the methods
of quantum cosmology. In this approach the of two de-Sitter branes from
the euclidian regime has been analyzed in the tree-level approximation [50], see also
[51]. The details, however, remain far from being fully understood.10 1 Effective methods in braneworld dynamics
1.1.1 The role of the nonlocal e ectiv e action
In the context of this rapidly evolving eld of research we want to propose a new
method of deriving the four-dimensional e ectiv e action for the braneworld models
with a warped geometry. Our derivation will be performed for the Randall-Sundrum
models but will easily generalize to arbitrary one- and two-brane models with co-
dimension one. In order to relate the full higher-dimensional dynamics of the space-
time to the dynamics observed by an inhabitant of a brane, it is natural to reduce
the full dynamics to a description which only involves four-dimensional degrees of
freedom. This reduced also provides the framework for a convenient com-
parison between the cosmological observations in our Universe and the dynamics of
a braneworld model. There exist various prescriptions for obtaining the e ectiv e de-
grees of freedom on the brane from the bulk theory. The best known of these is the
so-called Kaluza-Klein (KK) reduction which will be described in some detail in Sec.
1.3.1. However, all of these prescriptions have drawbacks which make them of limited
use for cosmological considerations: in the KK description one has to undertake a re-
summation of the KK-tower in order to study the high-energy dynamics of the early
universe. The KK description will also fail if the extra dimension is in nite. Another
method [52] encodes the bulk dynamics in a so-called \dark-radiation" term, which
is a projection of the higher dimensional Weyl-tensor on the brane. Unfortunately,
the projective term prohibits the closure of the e ectiv e four-dimensional equations
of motion, i.e. the on-brane dynamics cannot entirely be expressed in terms of four-
dimensional quantities. Hence one has to make assumptions about the bulk dynamics
in addition to choosing initial conditions for the e ectiv e four-dimensional equations
of motion.
Another popular approach for the analysis of braneworld cosmologies is the use
of moduli e ectiv e actions [53, 54]. In this method the geometry of the branes is
chosen as xed and only the cosmological consequences of the motion of the brane
in the bulk are considered. This approach requires conformal correspondence of the
geometries of all branes because the only v e-dimensional degrees of freedom taken
into account are the moduli- elds of the higher dimensions.
In the particular case of an Anti-de Sitter bulk another possibility for obtaining
the on-brane e ectiv e dynamics opens up by making use of the AdS/CFT correspon-
dence [55]. It relates the gravitational dynamics on an Anti-de Sitter space to the
dynamics of a conformal eld theory on its boundary. Truncating the AdS bulk by
an end-of-the-world brane, as it is done in the RS one-brane model, can be accounted
for by breaking the conformal invariance of the CFT in the ultraviolet regime via
coupling it to four-dimensional gravity. Therefore, one obtains a description of the
on-brane dynamics of the RS one-brane model by four-dimensional gravity plus the
stress-energy tensor of a conformal eld theory. Unfortunately, this in principle clear
procedure remains technically involved. Not all parameters necessary to describe the
rst-order CFT contribution to the on-brane dynamics could yet be obtained in this
approach. Also the e ect of a second brane cutting o the other end of AdS space
has not yet been fully incorporated in the AdS/CFT description of braneworlds.
The holographic method to determine the e ectiv e four-dimensional action for the
RS model will be described in detail in Sec. 1.3.2.1.
The new method we propose provides us with a description of the dynamics of